Research Article Axially Symmetric-dS Solution in...
9
Research Article Axially Symmetric-dS Solution in Teleparallel () Gravity Theories Gamal G. L. Nashed 1,2 1 Centre for eoretical Physics, e British University in Egypt, P.O. Box 43, Shorouk City 11837, Egypt 2 Mathematics Department, Faculty of Science, Ain Shams University, Cairo 11566, Egypt Correspondence should be addressed to Gamal G. L. Nashed; [email protected] Received 2 February 2015; Revised 11 March 2015; Accepted 11 March 2015 Academic Editor: Elias C. Vagenas Copyright © 2015 Gamal G. L. Nashed. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP 3 . We apply a tetrad field with six unknown functions to Einstein field equations. Exact vacuum solution, which represents axially symmetric-dS spacetime, is derived. We multiply the tetrad field of the derived solution by a local Lorentz transformation which involves a generalization of the angle and get a new tetrad field. Using this tetrad, we get a differential equation from the scalar torsion = ] ] . Solving this differential equation we obtain a solution to the () gravity theories under certain conditions on the form of () and its first derivatives. Finally, we calculate the scalars of Riemann Christoffel tensor, Ricci tensor, Ricci scalar, torsion tensor, and its contraction to explain the singularities associated with this solution. 1. Introduction e discovery of the acceleration of the universe through the SNeIa Hubble diagram has been later confirmed by wide range of data, from more recent SNeIa data to BAOs and CMBR anisotropies [1–12]. Such overwhelming abundance of observational evidences in favor of the cosmic speed up does not fit in the framework of GR making clear that our theoretical background is seriously flawed [13]. e idea of unifying the gravitation and electromag- netism was made by Einstein [14] in 1928. is attempt was based on the mathematical structure of teleparallelism, also referred to as distant or absolute parallelism. In other words, the idea was the introduction of a tetrad field, a field of orthonormal bases on the tangent spaces at each point of the four-dimensional spacetime. e tetrad has sixteen components whereas the gravitational field, represented by the spacetime metric, has only ten. e six additional degrees of freedom of the tetrad were then supposed by Einstein to be related to the six components of the electromagnetic field [15–20]. is attempt of unification did not succeed because the additional six degrees of freedom of the tetrad are actu- ally eliminated by the 6-parameter local Lorentz invariance of the theory. However, Einstein introduced concepts that remain important to the present day. Teleparallelism could be considered by using the Weitzenb¨ ock connection that is curvatureless but has torsion, rather than the curvature defined by the Levi-Civita connection [21]. Similar to the exotic dark energy and other modified gravity models, it is found that the cosmic acceleration can be obtained successfully from another gravitational scenario, () theory [22]. It is based on the teleparallel equivalent of general relativity (TEGR) which is known as teleparallel gravity (TG). A scalar torsion is the Lagrangian of teleparallel gravity. e teleparallel gravity is not a new theory of gravity but an alternative geometric formulation of general relativity (GR). In teleparallel gravity, the Levi- Civita connection used in Einstein’s GR is replaced by the Weitzenb¨ ock connection with torsion. However, the torsion vanishes in the dark energy and modified gravity models. Moreover, () theories have several interesting features. ey not only can explain the late accelerating expansion, but also always have second order differential equations, which is simpler than the () gravity. In addition, when certain conditions are satisfied, the behavior of () will be similar to quintessence [23]. Although () gravity has attracted Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 915928, 8 pages http://dx.doi.org/10.1155/2015/915928
Transcript of Research Article Axially Symmetric-dS Solution in...
Research ArticleAxially Symmetric-dS Solution in Teleparallel 119891(119879)Gravity Theories
Gamal G L Nashed12
1Centre for Theoretical Physics The British University in Egypt PO Box 43 Shorouk City 11837 Egypt2Mathematics Department Faculty of Science Ain Shams University Cairo 11566 Egypt
Correspondence should be addressed to Gamal G L Nashed nashedbueedueg
Received 2 February 2015 Revised 11 March 2015 Accepted 11 March 2015
Academic Editor Elias C Vagenas
Copyright copy 2015 Gamal G L NashedThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3
We apply a tetrad field with six unknown functions to Einstein field equations Exact vacuum solution which represents axiallysymmetric-dS spacetime is derived We multiply the tetrad field of the derived solution by a local Lorentz transformation whichinvolves a generalization of the angle 120601 and get a new tetrad field Using this tetrad we get a differential equation from the scalartorsion 119879 = 119879120572
120583]119878120572120583] Solving this differential equation we obtain a solution to the 119891(119879) gravity theories under certain conditions
on the form of 119891(119879) and its first derivatives Finally we calculate the scalars of Riemann Christoffel tensor Ricci tensor Ricci scalartorsion tensor and its contraction to explain the singularities associated with this solution
1 Introduction
The discovery of the acceleration of the universe throughthe SNeIa Hubble diagram has been later confirmed by widerange of data from more recent SNeIa data to BAOs andCMBR anisotropies [1ndash12] Such overwhelming abundanceof observational evidences in favor of the cosmic speed updoes not fit in the framework of GR making clear that ourtheoretical background is seriously flawed [13]
The idea of unifying the gravitation and electromag-netism was made by Einstein [14] in 1928 This attemptwas based on the mathematical structure of teleparallelismalso referred to as distant or absolute parallelism In otherwords the idea was the introduction of a tetrad field a fieldof orthonormal bases on the tangent spaces at each pointof the four-dimensional spacetime The tetrad has sixteencomponents whereas the gravitational field represented bythe spacetimemetric has only tenThe six additional degreesof freedom of the tetrad were then supposed by Einstein tobe related to the six components of the electromagnetic field[15ndash20] This attempt of unification did not succeed becausethe additional six degrees of freedom of the tetrad are actu-ally eliminated by the 6-parameter local Lorentz invariance
of the theory However Einstein introduced concepts thatremain important to the present day Teleparallelism couldbe considered by using the Weitzenbock connection thatis curvatureless but has torsion rather than the curvaturedefined by the Levi-Civita connection [21]
Similar to the exotic dark energy and other modifiedgravity models it is found that the cosmic accelerationcan be obtained successfully from another gravitationalscenario 119891(119879) theory [22] It is based on the teleparallelequivalent of general relativity (TEGR) which is known asteleparallel gravity (TG) A scalar torsion 119879 is the Lagrangianof teleparallel gravity The teleparallel gravity is not a newtheory of gravity but an alternative geometric formulationof general relativity (GR) In teleparallel gravity the Levi-Civita connection used in Einsteinrsquos GR is replaced by theWeitzenbock connection with torsion However the torsionvanishes in the dark energy and modified gravity modelsMoreover 119891(119879) theories have several interesting featuresThey not only can explain the late accelerating expansion butalso always have second order differential equations whichis simpler than the 119891(119877) gravity In addition when certainconditions are satisfied the behavior of 119891(119879) will be similarto quintessence [23] Although 119891(119879) gravity has attracted
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 915928 8 pageshttpdxdoiorg1011552015915928
2 Advances in High Energy Physics
wide attention a disadvantage which has been pointed out in[24 25] is that the action and its field equations do not satisfylocal Lorentz symmetry However the investigation about119891(119879) theories ismeaningful because it can provide some hintsabout Lorentz violation
Up till now a number of 119891(119879) theories have beenproposed [22 26ndash56] Under these cases it is shown that119891(119879) theories are not dynamically equivalent to teleparallelaction plus a scalar field [50] Like other gravity theories andmodels the 119891(119879) theories also have been investigated usingthe popular observational data Investigations show that the119891(119879) theories are compatible with observations (see eg [5758] and references therein) So we note that the new type of119891(119879) theory in [54] was proposed to explain the accelerationIt behaves like a cosmological constant but is free from thecoincidence problem In order to have a better study of 119891(119879)gravitational theories we try to search for exact solutions forthe field equations of 119891(119879) Specifically we are interested inaxially symmetric solutions which asymptotically behave asdS
Within 119891(119879) gravitational theory there are many solu-tions spherically symmetric [59 60] spherically symmetriccharged [61] homogenous anisotropic [62] and stability ofthe Einstein static closed and open universe [63] Some cos-mological features of the ΛCDM model in the framework ofthe119891(119879) are investigated [64]However up till now no axiallysymmetric-dS solution derived in this theory It is the aimof the present study to find an analytic axially symmetric-dS solution in 119891(119879) gravitational theories In Section 2preliminaries of 119891(119879) gravitational theory are presented InSection 3 a tetrad field with six unknown functions is appliedto the field equation of 119891(119879) New analytic solution vacuumone with two constants of integration for Einstein fieldequation is derived In Section 4 we multiply this solutionby a local Lorentz transformation with a generalization ofthe angle 120601 that is 119873(120601) Using the calculations carried outin Section 3 we succeed to derive special solution for 119891(119879)gravitational theories under certain conditions on 119891(119879) andits first derivative Final section is devoted to sum up theresults obtained
2 Preliminaries of 119891(119879)
In a spacetime with absolute parallelism the parallel vectorfield 119890
119894
120583 [21] defines the nonsymmetric affine connection
Γ120583
]120588def = 119890119894
120583
119890119894
]120588 (1)
where 119890119894120583] = 120597]119890119894120583 ldquoWe use the Greek indices 120583 ] for
local holonomic spacetime coordinates and the Latin indices119894 119895 label (co)-frame componentsrdquo
The curvature tensor defined by Γ120583]120588 given by (1) isidentically vanishing The metric tensor 119892
120583] is defined by
119892120583]
def = 120578119894119895119890119894
120583119890119895
] (2)
with 120578119894119895
= (minus1 +1 +1 +1) being the metric of Minkowskispacetime Defining the torsion and the contortion compo-nents as
119879120583
]120588def = Γ120583
120588] minus Γ120583
]120588 = 119890119894
120583
(120597]119890119894
120588minus 120597120588119890119894
])
119870120583]120588
def = minus
1
2(119879120583]120588minus 119879
]120583120588minus 119879120588
120583])
(3)
where the contortion equals the difference between Weitz-enbock and Levi-Civita connection that is 119870120583]120588 = Γ120583]120588 minus
120583]120588The skew symmetric tensor 119878
120583
]120588 is defined as
119878120583
]120588 def =
1
2(119870
]120588120583+ 120575
]120583119879120582120588
120582minus 120575120588
120583119879120582]120582) (4)
The torsion scalar is defined as
119879def = 119879120583
]120588119878120583]120588 (5)
The action of 119891(119879) theory is given by
L (119890119886
120583 Φ119860) = int119889
4
119909119890 [1
16120587(119891 (119879) minus 2Λ) +LMatter (Φ119860)]
(6)
where 119890 = radicminus119892 = det(119890119886120583) LMatter is the Lagrangian
of matter field Λ is the cosmological constant and Φ119860are
matter fieldsSimilar to the 119891(119877) theory one can define the action of
119891(119879) theory as a function of the fields 119890119886120583and by putting the
variation of the function with respect to the field 119890119886120583to be
vanishing one can obtain the following equations of motion
119878120583
]120588119879120588119891 (119879)119879119879
+ [119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582] 119891 (119879)
119879
minus1
4120575]120583(119891 (119879) minus 2Λ) = 8120587T
120583
]
(7)
where
119879120588
=120597119879
120597119909120588 119891 (119879)
119879=
120597119891 (119879)
120597119879 119891 (119879)
119879119879=
1205972119891 (119879)
1205971198792
(8)
andT120583
] is the energy momentum tensorNow we are going to rewrite (7) in another form the
field equations (7) are written in terms of the tetrad and itspartial derivatives These equations appear to be differentfrom Einsteinrsquos field equations Following [24 25] one canobtain an equation relating 119879 with the Ricci scalar of themetric 119877 These will make the equivalence between TG andGR clear On the other hand the tetrad cannot be eliminatedcompletely in favor of the metric in (7) because of the lackof local Lorentz symmetry but the latter can be brought ina form that closely resembles Einsteinrsquos equation This formis more suitable for constructing analytic solutions in the119891(119879) theory To start writing the field equations in a covariantversion one must replace partial derivatives in the tensors by
Advances in High Energy Physics 3
covariant derivatives compatible with the metric 119892120583] that is
nabla120590where nabla
120590119892120583] = 0 Thus (3) can be written as
119879120583
]120588 = 119890119886
120583
(nabla]119890119886
120588minus nabla120588119890119886
]) (9)
Using (9) in (3) and (4) one can get
119870120583
]120588 = 119890119886
120583
nabla120588119890119886
]
119878120583]120588= 120578119894119895
119890119894
120583
nabla120588119890119895
]+ 120575
]120588120578119894119895
119890119894
120590
nabla120590119890119895
120583
minus 120575120583
120588120578119894119895
119890119894
120590
nabla120590ℎ119895
]
(10)
On the other hand from the relation betweenWeitzenbock connection and the Levi-Civita connection onecan write the Riemann tensor for the Levi-Civita connection120588
120583] in terms of the nonsymmetric connection Γ120588120583] in the
form
119877120588
120583120582] = 120597120582Γ120588
120583] minus 120597]Γ120588
120583120582+ Γ120588
120590120582Γ120590
120583] minus Γ120588
120590]Γ120590
120583120582
= nabla120582119870120588
120583] minus nabla]119870120588
120583120582+ 119870120588
120590120582119870120590
120583] minus 119870120588
120590]119870120590
120583120582
(11)
The associated Ricci tensor can then be written as
119877120583] = nabla]119870
120588
120583120588minus nabla120588119870120588
120583] + 119870120588
120590]119870120590
120583120588minus 119870120588
120590120588119870120590
120583] (12)
Now by using 119870120590120583] given by (3) along with the relations
119870(120583])120590 = 119879120583(]120590) = 119878120583(]120590) = 0 and considering that 119878120583120588120583
=
2119870120583120588120583
= minus119879120583120588120583
one has [24ndash53]
119877120583] = minusnabla
120582
119878]120582120583 minus 119892120583]nabla120582
119879120588
120582120588minus 119878120588120590
120583119870120590120588]
119877 = minus119879 minus 2nabla120583
119879120588
120583120588= minus119879 minus
2
119890120597120583
(119890119879120588
120583120588)
(13)
Equation (13) implies that the 119879 and 119877 differ only by a covari-ant divergence of a spacetime vector Therefore the Einstein-Hilbert action and the teleparallel action (ie LTEGR =
int1198894119909|119890|119879) will both lead to the same field equations and aredynamically equivalent theories In [24 25] the authors haveshown that this equivalence is directly at the level of the fieldequations By using the equations listed above and after somealgebraic manipulations one can get
119866120583] minus
119892120583]
2119879 = minusnabla
120588
119878]120588120583 minus 119878120590120588
120583119870120588120590] (14)
where 119866120583] = 119877
120583] minus (12)119892120583]119877 is the Einstein tensor
With the aid of the equations listed above it can be shownafter some algebraic manipulations that [24 25]
119866120583] = 119890
minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572]120588
) minus 119879120572
120583120582119878120572]120582
+1
2119892120583]119879 (15)
Finally by using (14) and (15) the field equations for 119891(119879)gravity (7) can be rewritten in the form
119891 (119879)119879119866120583] minus
1
2(119891 (119879) minus 2Λ minus 119879119891 (119879)
119879) 119892120583]
+ 119878]120588
120583119879120588119891 (119879)119879119879
= 8120587T120583]
(16)
Equation (16) can be taken as the starting point of the119891(119879) modified gravity model and it has a structure similar
to the field equation of 119891(119877) gravity Note that in the moregeneral case with 119891(119879) = 119879 the field equations are covariantform Nevertheless the theory is not local Lorentz invariantIn case of 119891(119879) = 119879 and constant scalar torsion 119891(119879
0) GR is
recovered and field equations are covariant and the theory isLorentz invariant
3 Axially Symmetric Solution in119891(119879) Gravity Theory
The tetrad field that is stationary and has axial symmetrytakes the form
(ℎ119894
120583) =
[[[[[
[
1198781(119903 120579) 0 0 119878
2(119903 120579)
0 1198783(119903 120579) 0 0
0 0 1198784(119903 120579) 0
1198785(119903 120579) 0 0 119878
6(119903 120579)
]]]]]
]
(17)
where 119878119894(119903 120579) 119894 = 1 sdot sdot sdot 6 are unknown functions of the radial
coordinate 119903 and the azimuthal angle 120579 Applying tetrad field(17) to the field equations of GR119877
120583]minus(12)119892120583](119877minus2Λ) we get
a system of lengthy partial nonlinear differential equationsThe solution of these system has the form
1198781(119903 120579) = radic
120575 (119903)
Σ (119903 120579) 119878
2(119903 120579) = 119878
7(119903 120579) 119878
1(119903 120579)
1198783(119903 120579) =
1
1198781(119903 120579)
1198784(119903 120579) =
radicΣ (119903 120579)
119891 (120579)
1198785(119903 120579) =
120573 sin 120579
radicΣ (119903 120579) 119878
6(119903 120579) =
1198788(119903 120579)
radicΣ (119903 120579)
where 120575 (119903) = (1199032
+ 1198882
2
)(1 minusΛ1199032
3) minus 2119888
1119903
Σ (119903 120579) = 1199032
+ 1198882
2cos2120579 119891 (120579) = 1 +1198882
2Λcos21205793
1198787= minusΩ119888
2sin2120579 Ω = 1 +
1198882
2Λ
3
120573 = 1198882radic119891 (120579) 119878
8(119903 120579) = minusΩ sin 120579radic(1199032 + 119888
2
2) 119891 (120579)
(18)
Themetric associated with tetrad field (17) after using (18) hasthe following form ldquoFor brevity we write 119878
119894(119903 120579) equiv 119878
119894 119894 =
1 sdot sdot sdot 6rdquo
1198891199042
= [1198781
2
minus 1198785
2
] 1198891199052
minus 1198783
2
1198891199032
minus 1198784
2
1198891205792
minus [1198786
2
minus 1198782
2
] 1198891206012
minus [11987851198786minus 11987811198782] 119889119905119889120601
(19)
which is the Kerr-dS spacetime provided that 1198881
= 119872 and1198882= 119886where119872 and 119886 are themass of the gravitational system
and the rotation parameter [65] Solution (18) is a solution toEinstein field equations however themain task of the presentstudy is to find an axially symmetric-dS solution for 119891(119879) To
4 Advances in High Energy Physics
do such aim we multiply tetrad field (17) with the following119904119900(3)matrix
(Λ119894
119895) = (
1 0 0 0
0 sin 120579 cosΦ cos 120579 sinΦ minus sinΦ
0 sin 120579 sinΦ cos 120579 sinΦ cosΦ0 cos 120579 minus sin 120579 0
) (20)
where Φ is a function of azimuthal angle 120601 that is Φ =119873(120601) ldquoHere we consider Φ to be function of 120601 only whichis sufficient to carry out our aim However if Φ becomesa general function that is depends on 119903 120579 and 120601 thecalculations will be very complicatedrdquo The new tetrad field
(119890119894
120583)1
= Λ119894
119895119890119895
120583 (21)
takes the form
(119890119894
120583)1
=
((((((((
(
radic120575(119903)
radicΣ (119903 120579)0 0
119886sin2120579radic120575 (119903)
ΩradicΣ (119903 120579)
minus119886 sin 120579 sinΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 cosΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 cosΦradicΣ (119903 120579)
radic119891 (120579)minus(1199032 + 1198862) sin 120579 sinΦradic119891 (120579)
ΩradicΣ (119903 120579)
119886 sin 120579 cosΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 sinΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 sinΦradicΣ (119903 120579)
radic119891 (120579)
(1199032 + 1198862) sin 120579 cosΦradic119891 (120579)
ΩradicΣ (119903 120579)
0cos 120579radicΣ (119903 120579)
radic120575 (119903)
sin 120579radicΣ (119903 120579)
radic119891 (120579)0
))))))))
)
(22)
We rewrite (16) as
119866120583] +
1
119891 (119879)119879
Λ119892120583]
=1
119891 (119879)119879
(1
2(119891 (119879) minus 119879119891 (119879)
119879) 119892120583]
minus 119878]120588
120583119879120588119891 (119879)119879119879
+ 8120587T120583])
997904rArr 119866120583] +
1
119891 (119879)119879
Λ119892120583] =
1
119891 (119879)119879
(4120587T120583] + 119879
eff120583])
(23)
where 119879eff120583] = (12)[119891(119879) minus 119879119891(119879)
119879]119892120583] minus 119878]
120588
120583119879120588119891(119879)119879119879
isthe effective energy momentum tensor
To find a vacuum solution within 119891(119879) theories it issufficient to find the condition that makes the effectiveenergy momentum tensor 119879eff
120583] vanishing The simplestcondition that satisfies this aim is the vanishing of the scalartorsion 119879 The imposition of 119879 = 0 is just one (simplest)solution and in principle there can exist many more forexample when the torsion scalar is constant the effectiveenergy momentum tensor 119879eff
120583] will also be vanishing undersome constraints on the form of 119891 (constant) and its firstderivative Using (6) and tetrad field (22) one can obtainℎ = det(ℎ119886
120583) = (1199032 + 1198862cos2120579) sin 120579(1 + Λ1198862) With the use
of (3) (4) and (5) we obtain the torsion scalar as
119879 = radic119891 (120579) 120575 (119903) (2ΩΣ2
1198731015840
minus 21198866
Λcos6120579
minus 2 [1 +Λ
3(51199032
minus 1198862
)] 1198864cos4120579
minus 2119903 (5Λ
31199033
+ 4119872)1198862cos2120579
+ 21199034
[1 minusΛ
3(31199032
+ 1198862
)])
+ 2Σ2
(3 + 1198862
Λcos2120579 + 3Ω1198731015840
)
sdot [119903Λ
3(21199032
+ 1198862
) minus 119903 + 119872]
(24)
where 1198731015840 = 119889119873(120601)119889120601 The exact solution of (24) has theform
119873(120601) =120601radic119891 (120579)
ΩΣ2 (2Λ1199033 minus 3119903 + 1199031198862Λ + 3119872 + 3radic119891 (120579) 120575 (119903))
sdot (1198862
119891 (120579) (21198862cos4120579 + cos2120579 [3119903
2
+ 1198862cos2120579] minus 119903
2
)
+ radic120575 (119903)Σ [21199033
Λ + 1198862
Λ minus 3 119903 + 119872] 119903
minus 1198864sin2120579cos2120579Λ
minus 9radic1205753 (119903) [1199032
minus 1198862cos2120579]
minus 3Σ2radic119891 (120579) (3119872 + 2119903
3
Λ minus 3119903 + 1199031198862
Λ))
(25)
Using (25) and (22) in (23) we get a solution to 119891(119879) gravita-tional theory provided that 119891(0) = 0
4 Singularities
Now we are going to study the singularities of the derivedsolution For this purpose we search the value at which119903 makes 119892
00and 119892
11tend to zero or infin This procedure
may reproduce singularities corresponding to coordinatesingularities Therefore to procedure correct singularitieswe are going to demonstrate the invariants In orthodox
Advances in High Energy Physics 5
general relativity or its modifications the invariants are theRicci scalar the Kretschmann scalar or other invariantsconstructed from Riemann tensor and its contractions Inthe teleparallel geometry of gravity we have two approachesof finding invariants In the first approach one uses thesolution of the vierbein and the Weitzenbock connectionto calculate torsion invariants such as the torsion scalar 119879In the second approach one uses the solution to constructmetric and then construct the Levi-Civita connection andfinally calculate curvature invariants such as the Ricci andKretschmann scalars The comparison of the two approachesis able to show differences between curvature and torsiongravity In teleparallel theories we mean by singularity ofspacetime the singularity of the scalar concomitants of thetorsion and curvature tensors
The torsion invariant 119879 that is the torsion scalar thatarises from the vierbein solution (22) with the use ofWeitzenbockrsquos connection (25) is vanishing The curvatureinvariants that arise from themetric solution (19) through thecalculation of the Levi-Civita connection are [61]
119877120583]120582120590
119877120583]120582120590
=1
3Σ6[811988612
Λ2cos12120579 + 48119886
10
1199032
Λ2cos10120579
+ 1201198868
1199034
Λ2cos8120579 + 16119886
6cos6120579 [101199036
Λ2
minus 91198722
]
+ 1201198864
1199032cos4120579 [119903
6
Λ2
+ 181198722
]
+ 481198862
1199034cos2120579 [119903
6
Λ2
minus 451198722
]
+ 1441199036
1198722
+ 811990312
Λ2
]
119877120583]119877120583] = 4Λ
2
119877 = minus4Λ2
= minus2nabla120572119879120572
119879120583]120582
119879120583]120582 =
minus119891 (119903 120579)
27sin2120579120575 (119903) Σ5119891 (120579)32
119879120583
119879120583=
minus1198911(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119905120583]120582
119905120583]120582 =
minus1198912(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119886120583
119886120583=
21198862 [1199032120575 (119903) sin2120579 + cos2120579119891 (120579)]
9120575 (119903) Σ119891 (120579)
(26)
where 119891(119903 120579) 1198911(119903 120579) and 119891
2(119903 120579) are functions of 119903 and 120579
Here 119905120583]120582 119879120583 and 119886
120583are the irreducible representation of the
torsion tensor
119905120582120583] =
1
2(119879120582120583] + 119879
120583120582]) +1
6(119892]120582119881120583 + 119892
120583]119881120582) minus1
3119892120582120583
119881]
119879120583= 119879120582
120582]
119886120583=
1
6120598120583]120588120590119879
]120588120590
(27)
where 120598120583]120588120590 is defined by
120598120583]120588120590
def = radicminus119892120575
120583]120588120590 (28)
with 120575120583]120588120590 being completely antisymmetric and normalized
as 1205750123
= minus1Observing the forms of Ricci and Kretschmann torsion
and its irreducible representation scalars in (26) we deducethat in the Kerr-dS case there are no divergence points ateither 119903 = 0 or 119903 = infin
5 Main Results and Discussion
119891(119879) gravitational theories are modifications of the TEGRthat try to deal with the recent problems appearing incosmology In these theories it is not easy to find exactsolutions We have rewritten the field equations of thesetheories in a simple form This form enables us to show theextra terms that are responsible for the deviation from GRtheory In the nonvacuum case those terms can be regardedas the effective energy momentum tensor and generally theydepend on the scalar torsion and its derivatives For vacuumsolution one must show that the effective energy momentumtensors are vanishing So if the torsion scalar is vanishing thenit turns out that the extra terms are vanishing provided someconstraints on the form of the zero function that is 119891(0) andits first derivative
The equivalence between GR and TEGR emerges fromthe property that Einstein-Hilbert Lagrangian differs fromTEGR Lagrangian by a four-divergence term By using theLevi-Civita scalar curvature 119877 for the metric (2) one gets theresult of (13) Therefore TEGR and Einstein-Hilbert actionsare equivalent up to the equations of motion This meansthat GR and TEGR possess the same number of degrees offreedom In spite of the fact that the tetrad field contains 16components (6more than themetric field) TEGR is invariantunder local Lorentz transformations of the tetrad due to theexistence of the divergence term In the TEGR the tetrad fieldchanges under local Lorentz transformation as
119890119894
120583= Λ119894
119895119890119895
120583 119890
119894
120583
= Λ119894
119895
119890119895
120583
(29)
This transformation adds a four-divergence to the TEGRLagrangian 119890119879 This behavior is evident in (13) because 119890119877 isinvariant under local Lorentz transformations Instead 119891(119879)gravity like other theories of amended gravity possessesextra degrees of freedom In fact the dynamical equations(7) are sensitive to local Lorentz transformations of thetetrad except for the case 119891(119879) = 119879 (ie TEGR) Thisimplies that the dynamical equations of 119891(119879) gravitationaltheories contain information not only about the evolutionof the metric but also about some extra degrees of freedomexclusively associated with the tetrad that are not present inthe undeformed theory [24 25 33ndash51] For 119891(119879) theories theLagrangian changes under a local Lorentz transformation as
119890119891 (119879) = 119890119891 (119879 + divergence term) (30)
In this case the divergence term remains free inside thefunction 119891 damaging the principle of invariance under
6 Advances in High Energy Physics
local Lorentz transformation The loss of the local Lorentzinvariance leads to a preferred global reference frame definedby the autoparallel curves of the manifold that consistentlysolve the dynamical equations This means that (7) not onlydetermines themetric but also chooses some other propertiesof the tetrad field The tetrads connected by local Lorentztransformations lead to the same metric however they aredifferent with respect to the parallel framework Becauseof this fundamental property of 119891(119879) theories when oneis searching for solutions of a given symmetry it is quitedifficult to do an ansatz for the tetrad field For themetric casesymmetry helps us to choose suitable coordinates to write themetric However this does not say much about the ansatzfor the tetrad due to local Lorentz transformation [66 67]Certainly in the context of 119891(119879) theories the proper framewhich parallelizes the spacetime for a given symmetry ofthe geometry must be independent of the function 119891 [68]This work is focused on how to find the parallelization foraxially symmetric solutions in 119891(119879) theories In particularwe want to know whether Kerr-dS geometry survives or notin 119891(119879) gravity To answer this question we should find thecorrect ansatz to solve (7) This search is greatly facilitated byinvoking the following argument concerning the survival ofcertain TEGR solutions [67] if a vacuum solution of 119891(119879)gravity has 119879 = 0 then it will be a solution of TEGR aswell (a cosmological constantmight be necessary) In fact thereplacement of119879 = 0 in (7) after neglecting the cosmologicalconstant leads to
119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582minus
1
4120575]120583
119891 (0)
1198911015840 (0)= 0 (31)
which is a TEGR vacuum equation with cosmological con-stant 2Λ = 119891(0)1198911015840(0) We can neglect the cosmologicalconstant term by restricting the family of functions 119891 to have119891(0) = 0 and 1198911015840(0) = 0 In other words we can utilizethe freedom to do local Lorentz transformations in TEGRto search a tetrad having 119879 = 0 if we succeed then wewill state that such solution survives in 119891(119879) gravity Noticethat TEGR vacuum solutions do not force 119879 to vanish 119877must vanish Thus (13) says that 119879 is a four-divergence Sothe former argument is based on the sensitivity of 119879 to localLorentz transformations The above argument means thatTEGR vacuum solutions having 119879 = 0 (or 119879 = constant)cannot be deformed by119891(119879) gravityWe have shown that thisis the case for Kerr-dS geometry what is meant that 119891(119879)gravity is unable to smooth the singularity of a black hole[67 69 70]
In this study we have used a diagonal tetrad field withsix unknown functions of redial coordinate 119903 and azimuthalangle 120579 This type of tetrad does not depend on local Lorentztransformation This tetrad field has been applied to thefield equations of GR with a cosmological constant and gota system of nonlinear partial differential equations Thissystem have been solved and a solution with two constants ofintegration is derived The associated metric of this solutionis similar to Kerr-dS solution provided that the two constantsof integration are the gravitational mass of the system and therotating parameter [65] The tetrad field of this solution hasbeen multiplied by 119904119900(3) which contains a generalization of
the angle120601 that is119873(120601)Thenew tetrad field has beenused tocalculate the scalar torsion and differential equation with oneunknown 119873(120601) has been derived This differential equationhas been solved assuming the vanishing of the scalar torsionand the form of the unknown function has been derivedThisform of the unknown function has been used in the newtetrad field which has been shown that the new tetrad is aspecial solution to 119891(119879) provided that 119891(0) = 0
The singularities of the derived solution have been dis-cussed and has been shown that the derived solution has nosingularities at either 119903 = 0 or 119903 = infin This is done throughthe calculations of the scalars of Ricci Kretschman torsiontensor and its irreducible representationThe situation of thesingularities is expected to change for higher order of 119891(119879)
This is a first step in deriving a special solution within119891(119879) gravitational theories which proves that any GR solu-tion can be regarded as a solution in 119891(119879) under someconditions [58] This procedure can be generalized by tryingto derive solutions for quadratic 119891(119879) by trying to solve theeffective energy momentum tensor which will be responsiblefor such quadratic theory This will be done elsewhere
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by the Egyptian Ministry ofScientific Research under Project no 24-2-12
References
[1] D N Spergel L Verde H V Peiris et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003
[2] H V Peiris E Komatsu L Verde et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations implica-tions for inflationrdquoTheAstrophysical Journal Supplement Seriesvol 148 no 1 pp 213ndash232 2003
[3] D N Spergel R Bean O Dore et al ldquoThree-year Wilkinsonmicrowave anisotropy probe (WMAP) Observations impli-cations for cosmologyrdquo The Astrophysical Journal SupplementSeries vol 170 no 2 pp 377ndash408 2007
[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical interpretationrdquo Astrophysics Journal vol 180 pp 330ndash3762009
[5] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 no 6662 pp 51ndash54 1998
[6] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999
[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Superconductivity
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GravityJournal of
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Physics Research International
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Solid State PhysicsJournal of
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Soft MatterJournal of
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Journal of
Biophysics
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ThermodynamicsJournal of
2 Advances in High Energy Physics
wide attention a disadvantage which has been pointed out in[24 25] is that the action and its field equations do not satisfylocal Lorentz symmetry However the investigation about119891(119879) theories ismeaningful because it can provide some hintsabout Lorentz violation
Up till now a number of 119891(119879) theories have beenproposed [22 26ndash56] Under these cases it is shown that119891(119879) theories are not dynamically equivalent to teleparallelaction plus a scalar field [50] Like other gravity theories andmodels the 119891(119879) theories also have been investigated usingthe popular observational data Investigations show that the119891(119879) theories are compatible with observations (see eg [5758] and references therein) So we note that the new type of119891(119879) theory in [54] was proposed to explain the accelerationIt behaves like a cosmological constant but is free from thecoincidence problem In order to have a better study of 119891(119879)gravitational theories we try to search for exact solutions forthe field equations of 119891(119879) Specifically we are interested inaxially symmetric solutions which asymptotically behave asdS
Within 119891(119879) gravitational theory there are many solu-tions spherically symmetric [59 60] spherically symmetriccharged [61] homogenous anisotropic [62] and stability ofthe Einstein static closed and open universe [63] Some cos-mological features of the ΛCDM model in the framework ofthe119891(119879) are investigated [64]However up till now no axiallysymmetric-dS solution derived in this theory It is the aimof the present study to find an analytic axially symmetric-dS solution in 119891(119879) gravitational theories In Section 2preliminaries of 119891(119879) gravitational theory are presented InSection 3 a tetrad field with six unknown functions is appliedto the field equation of 119891(119879) New analytic solution vacuumone with two constants of integration for Einstein fieldequation is derived In Section 4 we multiply this solutionby a local Lorentz transformation with a generalization ofthe angle 120601 that is 119873(120601) Using the calculations carried outin Section 3 we succeed to derive special solution for 119891(119879)gravitational theories under certain conditions on 119891(119879) andits first derivative Final section is devoted to sum up theresults obtained
2 Preliminaries of 119891(119879)
In a spacetime with absolute parallelism the parallel vectorfield 119890
119894
120583 [21] defines the nonsymmetric affine connection
Γ120583
]120588def = 119890119894
120583
119890119894
]120588 (1)
where 119890119894120583] = 120597]119890119894120583 ldquoWe use the Greek indices 120583 ] for
local holonomic spacetime coordinates and the Latin indices119894 119895 label (co)-frame componentsrdquo
The curvature tensor defined by Γ120583]120588 given by (1) isidentically vanishing The metric tensor 119892
120583] is defined by
119892120583]
def = 120578119894119895119890119894
120583119890119895
] (2)
with 120578119894119895
= (minus1 +1 +1 +1) being the metric of Minkowskispacetime Defining the torsion and the contortion compo-nents as
119879120583
]120588def = Γ120583
120588] minus Γ120583
]120588 = 119890119894
120583
(120597]119890119894
120588minus 120597120588119890119894
])
119870120583]120588
def = minus
1
2(119879120583]120588minus 119879
]120583120588minus 119879120588
120583])
(3)
where the contortion equals the difference between Weitz-enbock and Levi-Civita connection that is 119870120583]120588 = Γ120583]120588 minus
120583]120588The skew symmetric tensor 119878
120583
]120588 is defined as
119878120583
]120588 def =
1
2(119870
]120588120583+ 120575
]120583119879120582120588
120582minus 120575120588
120583119879120582]120582) (4)
The torsion scalar is defined as
119879def = 119879120583
]120588119878120583]120588 (5)
The action of 119891(119879) theory is given by
L (119890119886
120583 Φ119860) = int119889
4
119909119890 [1
16120587(119891 (119879) minus 2Λ) +LMatter (Φ119860)]
(6)
where 119890 = radicminus119892 = det(119890119886120583) LMatter is the Lagrangian
of matter field Λ is the cosmological constant and Φ119860are
matter fieldsSimilar to the 119891(119877) theory one can define the action of
119891(119879) theory as a function of the fields 119890119886120583and by putting the
variation of the function with respect to the field 119890119886120583to be
vanishing one can obtain the following equations of motion
119878120583
]120588119879120588119891 (119879)119879119879
+ [119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582] 119891 (119879)
119879
minus1
4120575]120583(119891 (119879) minus 2Λ) = 8120587T
120583
]
(7)
where
119879120588
=120597119879
120597119909120588 119891 (119879)
119879=
120597119891 (119879)
120597119879 119891 (119879)
119879119879=
1205972119891 (119879)
1205971198792
(8)
andT120583
] is the energy momentum tensorNow we are going to rewrite (7) in another form the
field equations (7) are written in terms of the tetrad and itspartial derivatives These equations appear to be differentfrom Einsteinrsquos field equations Following [24 25] one canobtain an equation relating 119879 with the Ricci scalar of themetric 119877 These will make the equivalence between TG andGR clear On the other hand the tetrad cannot be eliminatedcompletely in favor of the metric in (7) because of the lackof local Lorentz symmetry but the latter can be brought ina form that closely resembles Einsteinrsquos equation This formis more suitable for constructing analytic solutions in the119891(119879) theory To start writing the field equations in a covariantversion one must replace partial derivatives in the tensors by
Advances in High Energy Physics 3
covariant derivatives compatible with the metric 119892120583] that is
nabla120590where nabla
120590119892120583] = 0 Thus (3) can be written as
119879120583
]120588 = 119890119886
120583
(nabla]119890119886
120588minus nabla120588119890119886
]) (9)
Using (9) in (3) and (4) one can get
119870120583
]120588 = 119890119886
120583
nabla120588119890119886
]
119878120583]120588= 120578119894119895
119890119894
120583
nabla120588119890119895
]+ 120575
]120588120578119894119895
119890119894
120590
nabla120590119890119895
120583
minus 120575120583
120588120578119894119895
119890119894
120590
nabla120590ℎ119895
]
(10)
On the other hand from the relation betweenWeitzenbock connection and the Levi-Civita connection onecan write the Riemann tensor for the Levi-Civita connection120588
120583] in terms of the nonsymmetric connection Γ120588120583] in the
form
119877120588
120583120582] = 120597120582Γ120588
120583] minus 120597]Γ120588
120583120582+ Γ120588
120590120582Γ120590
120583] minus Γ120588
120590]Γ120590
120583120582
= nabla120582119870120588
120583] minus nabla]119870120588
120583120582+ 119870120588
120590120582119870120590
120583] minus 119870120588
120590]119870120590
120583120582
(11)
The associated Ricci tensor can then be written as
119877120583] = nabla]119870
120588
120583120588minus nabla120588119870120588
120583] + 119870120588
120590]119870120590
120583120588minus 119870120588
120590120588119870120590
120583] (12)
Now by using 119870120590120583] given by (3) along with the relations
119870(120583])120590 = 119879120583(]120590) = 119878120583(]120590) = 0 and considering that 119878120583120588120583
=
2119870120583120588120583
= minus119879120583120588120583
one has [24ndash53]
119877120583] = minusnabla
120582
119878]120582120583 minus 119892120583]nabla120582
119879120588
120582120588minus 119878120588120590
120583119870120590120588]
119877 = minus119879 minus 2nabla120583
119879120588
120583120588= minus119879 minus
2
119890120597120583
(119890119879120588
120583120588)
(13)
Equation (13) implies that the 119879 and 119877 differ only by a covari-ant divergence of a spacetime vector Therefore the Einstein-Hilbert action and the teleparallel action (ie LTEGR =
int1198894119909|119890|119879) will both lead to the same field equations and aredynamically equivalent theories In [24 25] the authors haveshown that this equivalence is directly at the level of the fieldequations By using the equations listed above and after somealgebraic manipulations one can get
119866120583] minus
119892120583]
2119879 = minusnabla
120588
119878]120588120583 minus 119878120590120588
120583119870120588120590] (14)
where 119866120583] = 119877
120583] minus (12)119892120583]119877 is the Einstein tensor
With the aid of the equations listed above it can be shownafter some algebraic manipulations that [24 25]
119866120583] = 119890
minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572]120588
) minus 119879120572
120583120582119878120572]120582
+1
2119892120583]119879 (15)
Finally by using (14) and (15) the field equations for 119891(119879)gravity (7) can be rewritten in the form
119891 (119879)119879119866120583] minus
1
2(119891 (119879) minus 2Λ minus 119879119891 (119879)
119879) 119892120583]
+ 119878]120588
120583119879120588119891 (119879)119879119879
= 8120587T120583]
(16)
Equation (16) can be taken as the starting point of the119891(119879) modified gravity model and it has a structure similar
to the field equation of 119891(119877) gravity Note that in the moregeneral case with 119891(119879) = 119879 the field equations are covariantform Nevertheless the theory is not local Lorentz invariantIn case of 119891(119879) = 119879 and constant scalar torsion 119891(119879
0) GR is
recovered and field equations are covariant and the theory isLorentz invariant
3 Axially Symmetric Solution in119891(119879) Gravity Theory
The tetrad field that is stationary and has axial symmetrytakes the form
(ℎ119894
120583) =
[[[[[
[
1198781(119903 120579) 0 0 119878
2(119903 120579)
0 1198783(119903 120579) 0 0
0 0 1198784(119903 120579) 0
1198785(119903 120579) 0 0 119878
6(119903 120579)
]]]]]
]
(17)
where 119878119894(119903 120579) 119894 = 1 sdot sdot sdot 6 are unknown functions of the radial
coordinate 119903 and the azimuthal angle 120579 Applying tetrad field(17) to the field equations of GR119877
120583]minus(12)119892120583](119877minus2Λ) we get
a system of lengthy partial nonlinear differential equationsThe solution of these system has the form
1198781(119903 120579) = radic
120575 (119903)
Σ (119903 120579) 119878
2(119903 120579) = 119878
7(119903 120579) 119878
1(119903 120579)
1198783(119903 120579) =
1
1198781(119903 120579)
1198784(119903 120579) =
radicΣ (119903 120579)
119891 (120579)
1198785(119903 120579) =
120573 sin 120579
radicΣ (119903 120579) 119878
6(119903 120579) =
1198788(119903 120579)
radicΣ (119903 120579)
where 120575 (119903) = (1199032
+ 1198882
2
)(1 minusΛ1199032
3) minus 2119888
1119903
Σ (119903 120579) = 1199032
+ 1198882
2cos2120579 119891 (120579) = 1 +1198882
2Λcos21205793
1198787= minusΩ119888
2sin2120579 Ω = 1 +
1198882
2Λ
3
120573 = 1198882radic119891 (120579) 119878
8(119903 120579) = minusΩ sin 120579radic(1199032 + 119888
2
2) 119891 (120579)
(18)
Themetric associated with tetrad field (17) after using (18) hasthe following form ldquoFor brevity we write 119878
119894(119903 120579) equiv 119878
119894 119894 =
1 sdot sdot sdot 6rdquo
1198891199042
= [1198781
2
minus 1198785
2
] 1198891199052
minus 1198783
2
1198891199032
minus 1198784
2
1198891205792
minus [1198786
2
minus 1198782
2
] 1198891206012
minus [11987851198786minus 11987811198782] 119889119905119889120601
(19)
which is the Kerr-dS spacetime provided that 1198881
= 119872 and1198882= 119886where119872 and 119886 are themass of the gravitational system
and the rotation parameter [65] Solution (18) is a solution toEinstein field equations however themain task of the presentstudy is to find an axially symmetric-dS solution for 119891(119879) To
4 Advances in High Energy Physics
do such aim we multiply tetrad field (17) with the following119904119900(3)matrix
(Λ119894
119895) = (
1 0 0 0
0 sin 120579 cosΦ cos 120579 sinΦ minus sinΦ
0 sin 120579 sinΦ cos 120579 sinΦ cosΦ0 cos 120579 minus sin 120579 0
) (20)
where Φ is a function of azimuthal angle 120601 that is Φ =119873(120601) ldquoHere we consider Φ to be function of 120601 only whichis sufficient to carry out our aim However if Φ becomesa general function that is depends on 119903 120579 and 120601 thecalculations will be very complicatedrdquo The new tetrad field
(119890119894
120583)1
= Λ119894
119895119890119895
120583 (21)
takes the form
(119890119894
120583)1
=
((((((((
(
radic120575(119903)
radicΣ (119903 120579)0 0
119886sin2120579radic120575 (119903)
ΩradicΣ (119903 120579)
minus119886 sin 120579 sinΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 cosΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 cosΦradicΣ (119903 120579)
radic119891 (120579)minus(1199032 + 1198862) sin 120579 sinΦradic119891 (120579)
ΩradicΣ (119903 120579)
119886 sin 120579 cosΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 sinΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 sinΦradicΣ (119903 120579)
radic119891 (120579)
(1199032 + 1198862) sin 120579 cosΦradic119891 (120579)
ΩradicΣ (119903 120579)
0cos 120579radicΣ (119903 120579)
radic120575 (119903)
sin 120579radicΣ (119903 120579)
radic119891 (120579)0
))))))))
)
(22)
We rewrite (16) as
119866120583] +
1
119891 (119879)119879
Λ119892120583]
=1
119891 (119879)119879
(1
2(119891 (119879) minus 119879119891 (119879)
119879) 119892120583]
minus 119878]120588
120583119879120588119891 (119879)119879119879
+ 8120587T120583])
997904rArr 119866120583] +
1
119891 (119879)119879
Λ119892120583] =
1
119891 (119879)119879
(4120587T120583] + 119879
eff120583])
(23)
where 119879eff120583] = (12)[119891(119879) minus 119879119891(119879)
119879]119892120583] minus 119878]
120588
120583119879120588119891(119879)119879119879
isthe effective energy momentum tensor
To find a vacuum solution within 119891(119879) theories it issufficient to find the condition that makes the effectiveenergy momentum tensor 119879eff
120583] vanishing The simplestcondition that satisfies this aim is the vanishing of the scalartorsion 119879 The imposition of 119879 = 0 is just one (simplest)solution and in principle there can exist many more forexample when the torsion scalar is constant the effectiveenergy momentum tensor 119879eff
120583] will also be vanishing undersome constraints on the form of 119891 (constant) and its firstderivative Using (6) and tetrad field (22) one can obtainℎ = det(ℎ119886
120583) = (1199032 + 1198862cos2120579) sin 120579(1 + Λ1198862) With the use
of (3) (4) and (5) we obtain the torsion scalar as
119879 = radic119891 (120579) 120575 (119903) (2ΩΣ2
1198731015840
minus 21198866
Λcos6120579
minus 2 [1 +Λ
3(51199032
minus 1198862
)] 1198864cos4120579
minus 2119903 (5Λ
31199033
+ 4119872)1198862cos2120579
+ 21199034
[1 minusΛ
3(31199032
+ 1198862
)])
+ 2Σ2
(3 + 1198862
Λcos2120579 + 3Ω1198731015840
)
sdot [119903Λ
3(21199032
+ 1198862
) minus 119903 + 119872]
(24)
where 1198731015840 = 119889119873(120601)119889120601 The exact solution of (24) has theform
119873(120601) =120601radic119891 (120579)
ΩΣ2 (2Λ1199033 minus 3119903 + 1199031198862Λ + 3119872 + 3radic119891 (120579) 120575 (119903))
sdot (1198862
119891 (120579) (21198862cos4120579 + cos2120579 [3119903
2
+ 1198862cos2120579] minus 119903
2
)
+ radic120575 (119903)Σ [21199033
Λ + 1198862
Λ minus 3 119903 + 119872] 119903
minus 1198864sin2120579cos2120579Λ
minus 9radic1205753 (119903) [1199032
minus 1198862cos2120579]
minus 3Σ2radic119891 (120579) (3119872 + 2119903
3
Λ minus 3119903 + 1199031198862
Λ))
(25)
Using (25) and (22) in (23) we get a solution to 119891(119879) gravita-tional theory provided that 119891(0) = 0
4 Singularities
Now we are going to study the singularities of the derivedsolution For this purpose we search the value at which119903 makes 119892
00and 119892
11tend to zero or infin This procedure
may reproduce singularities corresponding to coordinatesingularities Therefore to procedure correct singularitieswe are going to demonstrate the invariants In orthodox
Advances in High Energy Physics 5
general relativity or its modifications the invariants are theRicci scalar the Kretschmann scalar or other invariantsconstructed from Riemann tensor and its contractions Inthe teleparallel geometry of gravity we have two approachesof finding invariants In the first approach one uses thesolution of the vierbein and the Weitzenbock connectionto calculate torsion invariants such as the torsion scalar 119879In the second approach one uses the solution to constructmetric and then construct the Levi-Civita connection andfinally calculate curvature invariants such as the Ricci andKretschmann scalars The comparison of the two approachesis able to show differences between curvature and torsiongravity In teleparallel theories we mean by singularity ofspacetime the singularity of the scalar concomitants of thetorsion and curvature tensors
The torsion invariant 119879 that is the torsion scalar thatarises from the vierbein solution (22) with the use ofWeitzenbockrsquos connection (25) is vanishing The curvatureinvariants that arise from themetric solution (19) through thecalculation of the Levi-Civita connection are [61]
119877120583]120582120590
119877120583]120582120590
=1
3Σ6[811988612
Λ2cos12120579 + 48119886
10
1199032
Λ2cos10120579
+ 1201198868
1199034
Λ2cos8120579 + 16119886
6cos6120579 [101199036
Λ2
minus 91198722
]
+ 1201198864
1199032cos4120579 [119903
6
Λ2
+ 181198722
]
+ 481198862
1199034cos2120579 [119903
6
Λ2
minus 451198722
]
+ 1441199036
1198722
+ 811990312
Λ2
]
119877120583]119877120583] = 4Λ
2
119877 = minus4Λ2
= minus2nabla120572119879120572
119879120583]120582
119879120583]120582 =
minus119891 (119903 120579)
27sin2120579120575 (119903) Σ5119891 (120579)32
119879120583
119879120583=
minus1198911(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119905120583]120582
119905120583]120582 =
minus1198912(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119886120583
119886120583=
21198862 [1199032120575 (119903) sin2120579 + cos2120579119891 (120579)]
9120575 (119903) Σ119891 (120579)
(26)
where 119891(119903 120579) 1198911(119903 120579) and 119891
2(119903 120579) are functions of 119903 and 120579
Here 119905120583]120582 119879120583 and 119886
120583are the irreducible representation of the
torsion tensor
119905120582120583] =
1
2(119879120582120583] + 119879
120583120582]) +1
6(119892]120582119881120583 + 119892
120583]119881120582) minus1
3119892120582120583
119881]
119879120583= 119879120582
120582]
119886120583=
1
6120598120583]120588120590119879
]120588120590
(27)
where 120598120583]120588120590 is defined by
120598120583]120588120590
def = radicminus119892120575
120583]120588120590 (28)
with 120575120583]120588120590 being completely antisymmetric and normalized
as 1205750123
= minus1Observing the forms of Ricci and Kretschmann torsion
and its irreducible representation scalars in (26) we deducethat in the Kerr-dS case there are no divergence points ateither 119903 = 0 or 119903 = infin
5 Main Results and Discussion
119891(119879) gravitational theories are modifications of the TEGRthat try to deal with the recent problems appearing incosmology In these theories it is not easy to find exactsolutions We have rewritten the field equations of thesetheories in a simple form This form enables us to show theextra terms that are responsible for the deviation from GRtheory In the nonvacuum case those terms can be regardedas the effective energy momentum tensor and generally theydepend on the scalar torsion and its derivatives For vacuumsolution one must show that the effective energy momentumtensors are vanishing So if the torsion scalar is vanishing thenit turns out that the extra terms are vanishing provided someconstraints on the form of the zero function that is 119891(0) andits first derivative
The equivalence between GR and TEGR emerges fromthe property that Einstein-Hilbert Lagrangian differs fromTEGR Lagrangian by a four-divergence term By using theLevi-Civita scalar curvature 119877 for the metric (2) one gets theresult of (13) Therefore TEGR and Einstein-Hilbert actionsare equivalent up to the equations of motion This meansthat GR and TEGR possess the same number of degrees offreedom In spite of the fact that the tetrad field contains 16components (6more than themetric field) TEGR is invariantunder local Lorentz transformations of the tetrad due to theexistence of the divergence term In the TEGR the tetrad fieldchanges under local Lorentz transformation as
119890119894
120583= Λ119894
119895119890119895
120583 119890
119894
120583
= Λ119894
119895
119890119895
120583
(29)
This transformation adds a four-divergence to the TEGRLagrangian 119890119879 This behavior is evident in (13) because 119890119877 isinvariant under local Lorentz transformations Instead 119891(119879)gravity like other theories of amended gravity possessesextra degrees of freedom In fact the dynamical equations(7) are sensitive to local Lorentz transformations of thetetrad except for the case 119891(119879) = 119879 (ie TEGR) Thisimplies that the dynamical equations of 119891(119879) gravitationaltheories contain information not only about the evolutionof the metric but also about some extra degrees of freedomexclusively associated with the tetrad that are not present inthe undeformed theory [24 25 33ndash51] For 119891(119879) theories theLagrangian changes under a local Lorentz transformation as
119890119891 (119879) = 119890119891 (119879 + divergence term) (30)
In this case the divergence term remains free inside thefunction 119891 damaging the principle of invariance under
6 Advances in High Energy Physics
local Lorentz transformation The loss of the local Lorentzinvariance leads to a preferred global reference frame definedby the autoparallel curves of the manifold that consistentlysolve the dynamical equations This means that (7) not onlydetermines themetric but also chooses some other propertiesof the tetrad field The tetrads connected by local Lorentztransformations lead to the same metric however they aredifferent with respect to the parallel framework Becauseof this fundamental property of 119891(119879) theories when oneis searching for solutions of a given symmetry it is quitedifficult to do an ansatz for the tetrad field For themetric casesymmetry helps us to choose suitable coordinates to write themetric However this does not say much about the ansatzfor the tetrad due to local Lorentz transformation [66 67]Certainly in the context of 119891(119879) theories the proper framewhich parallelizes the spacetime for a given symmetry ofthe geometry must be independent of the function 119891 [68]This work is focused on how to find the parallelization foraxially symmetric solutions in 119891(119879) theories In particularwe want to know whether Kerr-dS geometry survives or notin 119891(119879) gravity To answer this question we should find thecorrect ansatz to solve (7) This search is greatly facilitated byinvoking the following argument concerning the survival ofcertain TEGR solutions [67] if a vacuum solution of 119891(119879)gravity has 119879 = 0 then it will be a solution of TEGR aswell (a cosmological constantmight be necessary) In fact thereplacement of119879 = 0 in (7) after neglecting the cosmologicalconstant leads to
119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582minus
1
4120575]120583
119891 (0)
1198911015840 (0)= 0 (31)
which is a TEGR vacuum equation with cosmological con-stant 2Λ = 119891(0)1198911015840(0) We can neglect the cosmologicalconstant term by restricting the family of functions 119891 to have119891(0) = 0 and 1198911015840(0) = 0 In other words we can utilizethe freedom to do local Lorentz transformations in TEGRto search a tetrad having 119879 = 0 if we succeed then wewill state that such solution survives in 119891(119879) gravity Noticethat TEGR vacuum solutions do not force 119879 to vanish 119877must vanish Thus (13) says that 119879 is a four-divergence Sothe former argument is based on the sensitivity of 119879 to localLorentz transformations The above argument means thatTEGR vacuum solutions having 119879 = 0 (or 119879 = constant)cannot be deformed by119891(119879) gravityWe have shown that thisis the case for Kerr-dS geometry what is meant that 119891(119879)gravity is unable to smooth the singularity of a black hole[67 69 70]
In this study we have used a diagonal tetrad field withsix unknown functions of redial coordinate 119903 and azimuthalangle 120579 This type of tetrad does not depend on local Lorentztransformation This tetrad field has been applied to thefield equations of GR with a cosmological constant and gota system of nonlinear partial differential equations Thissystem have been solved and a solution with two constants ofintegration is derived The associated metric of this solutionis similar to Kerr-dS solution provided that the two constantsof integration are the gravitational mass of the system and therotating parameter [65] The tetrad field of this solution hasbeen multiplied by 119904119900(3) which contains a generalization of
the angle120601 that is119873(120601)Thenew tetrad field has beenused tocalculate the scalar torsion and differential equation with oneunknown 119873(120601) has been derived This differential equationhas been solved assuming the vanishing of the scalar torsionand the form of the unknown function has been derivedThisform of the unknown function has been used in the newtetrad field which has been shown that the new tetrad is aspecial solution to 119891(119879) provided that 119891(0) = 0
The singularities of the derived solution have been dis-cussed and has been shown that the derived solution has nosingularities at either 119903 = 0 or 119903 = infin This is done throughthe calculations of the scalars of Ricci Kretschman torsiontensor and its irreducible representationThe situation of thesingularities is expected to change for higher order of 119891(119879)
This is a first step in deriving a special solution within119891(119879) gravitational theories which proves that any GR solu-tion can be regarded as a solution in 119891(119879) under someconditions [58] This procedure can be generalized by tryingto derive solutions for quadratic 119891(119879) by trying to solve theeffective energy momentum tensor which will be responsiblefor such quadratic theory This will be done elsewhere
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by the Egyptian Ministry ofScientific Research under Project no 24-2-12
References
[1] D N Spergel L Verde H V Peiris et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003
[2] H V Peiris E Komatsu L Verde et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations implica-tions for inflationrdquoTheAstrophysical Journal Supplement Seriesvol 148 no 1 pp 213ndash232 2003
[3] D N Spergel R Bean O Dore et al ldquoThree-year Wilkinsonmicrowave anisotropy probe (WMAP) Observations impli-cations for cosmologyrdquo The Astrophysical Journal SupplementSeries vol 170 no 2 pp 377ndash408 2007
[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical interpretationrdquo Astrophysics Journal vol 180 pp 330ndash3762009
[5] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 no 6662 pp 51ndash54 1998
[6] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999
[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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FluidsJournal of
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Advances in Condensed Matter Physics
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
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GravityJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Biophysics
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ThermodynamicsJournal of
Advances in High Energy Physics 3
covariant derivatives compatible with the metric 119892120583] that is
nabla120590where nabla
120590119892120583] = 0 Thus (3) can be written as
119879120583
]120588 = 119890119886
120583
(nabla]119890119886
120588minus nabla120588119890119886
]) (9)
Using (9) in (3) and (4) one can get
119870120583
]120588 = 119890119886
120583
nabla120588119890119886
]
119878120583]120588= 120578119894119895
119890119894
120583
nabla120588119890119895
]+ 120575
]120588120578119894119895
119890119894
120590
nabla120590119890119895
120583
minus 120575120583
120588120578119894119895
119890119894
120590
nabla120590ℎ119895
]
(10)
On the other hand from the relation betweenWeitzenbock connection and the Levi-Civita connection onecan write the Riemann tensor for the Levi-Civita connection120588
120583] in terms of the nonsymmetric connection Γ120588120583] in the
form
119877120588
120583120582] = 120597120582Γ120588
120583] minus 120597]Γ120588
120583120582+ Γ120588
120590120582Γ120590
120583] minus Γ120588
120590]Γ120590
120583120582
= nabla120582119870120588
120583] minus nabla]119870120588
120583120582+ 119870120588
120590120582119870120590
120583] minus 119870120588
120590]119870120590
120583120582
(11)
The associated Ricci tensor can then be written as
119877120583] = nabla]119870
120588
120583120588minus nabla120588119870120588
120583] + 119870120588
120590]119870120590
120583120588minus 119870120588
120590120588119870120590
120583] (12)
Now by using 119870120590120583] given by (3) along with the relations
119870(120583])120590 = 119879120583(]120590) = 119878120583(]120590) = 0 and considering that 119878120583120588120583
=
2119870120583120588120583
= minus119879120583120588120583
one has [24ndash53]
119877120583] = minusnabla
120582
119878]120582120583 minus 119892120583]nabla120582
119879120588
120582120588minus 119878120588120590
120583119870120590120588]
119877 = minus119879 minus 2nabla120583
119879120588
120583120588= minus119879 minus
2
119890120597120583
(119890119879120588
120583120588)
(13)
Equation (13) implies that the 119879 and 119877 differ only by a covari-ant divergence of a spacetime vector Therefore the Einstein-Hilbert action and the teleparallel action (ie LTEGR =
int1198894119909|119890|119879) will both lead to the same field equations and aredynamically equivalent theories In [24 25] the authors haveshown that this equivalence is directly at the level of the fieldequations By using the equations listed above and after somealgebraic manipulations one can get
119866120583] minus
119892120583]
2119879 = minusnabla
120588
119878]120588120583 minus 119878120590120588
120583119870120588120590] (14)
where 119866120583] = 119877
120583] minus (12)119892120583]119877 is the Einstein tensor
With the aid of the equations listed above it can be shownafter some algebraic manipulations that [24 25]
119866120583] = 119890
minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572]120588
) minus 119879120572
120583120582119878120572]120582
+1
2119892120583]119879 (15)
Finally by using (14) and (15) the field equations for 119891(119879)gravity (7) can be rewritten in the form
119891 (119879)119879119866120583] minus
1
2(119891 (119879) minus 2Λ minus 119879119891 (119879)
119879) 119892120583]
+ 119878]120588
120583119879120588119891 (119879)119879119879
= 8120587T120583]
(16)
Equation (16) can be taken as the starting point of the119891(119879) modified gravity model and it has a structure similar
to the field equation of 119891(119877) gravity Note that in the moregeneral case with 119891(119879) = 119879 the field equations are covariantform Nevertheless the theory is not local Lorentz invariantIn case of 119891(119879) = 119879 and constant scalar torsion 119891(119879
0) GR is
recovered and field equations are covariant and the theory isLorentz invariant
3 Axially Symmetric Solution in119891(119879) Gravity Theory
The tetrad field that is stationary and has axial symmetrytakes the form
(ℎ119894
120583) =
[[[[[
[
1198781(119903 120579) 0 0 119878
2(119903 120579)
0 1198783(119903 120579) 0 0
0 0 1198784(119903 120579) 0
1198785(119903 120579) 0 0 119878
6(119903 120579)
]]]]]
]
(17)
where 119878119894(119903 120579) 119894 = 1 sdot sdot sdot 6 are unknown functions of the radial
coordinate 119903 and the azimuthal angle 120579 Applying tetrad field(17) to the field equations of GR119877
120583]minus(12)119892120583](119877minus2Λ) we get
a system of lengthy partial nonlinear differential equationsThe solution of these system has the form
1198781(119903 120579) = radic
120575 (119903)
Σ (119903 120579) 119878
2(119903 120579) = 119878
7(119903 120579) 119878
1(119903 120579)
1198783(119903 120579) =
1
1198781(119903 120579)
1198784(119903 120579) =
radicΣ (119903 120579)
119891 (120579)
1198785(119903 120579) =
120573 sin 120579
radicΣ (119903 120579) 119878
6(119903 120579) =
1198788(119903 120579)
radicΣ (119903 120579)
where 120575 (119903) = (1199032
+ 1198882
2
)(1 minusΛ1199032
3) minus 2119888
1119903
Σ (119903 120579) = 1199032
+ 1198882
2cos2120579 119891 (120579) = 1 +1198882
2Λcos21205793
1198787= minusΩ119888
2sin2120579 Ω = 1 +
1198882
2Λ
3
120573 = 1198882radic119891 (120579) 119878
8(119903 120579) = minusΩ sin 120579radic(1199032 + 119888
2
2) 119891 (120579)
(18)
Themetric associated with tetrad field (17) after using (18) hasthe following form ldquoFor brevity we write 119878
119894(119903 120579) equiv 119878
119894 119894 =
1 sdot sdot sdot 6rdquo
1198891199042
= [1198781
2
minus 1198785
2
] 1198891199052
minus 1198783
2
1198891199032
minus 1198784
2
1198891205792
minus [1198786
2
minus 1198782
2
] 1198891206012
minus [11987851198786minus 11987811198782] 119889119905119889120601
(19)
which is the Kerr-dS spacetime provided that 1198881
= 119872 and1198882= 119886where119872 and 119886 are themass of the gravitational system
and the rotation parameter [65] Solution (18) is a solution toEinstein field equations however themain task of the presentstudy is to find an axially symmetric-dS solution for 119891(119879) To
4 Advances in High Energy Physics
do such aim we multiply tetrad field (17) with the following119904119900(3)matrix
(Λ119894
119895) = (
1 0 0 0
0 sin 120579 cosΦ cos 120579 sinΦ minus sinΦ
0 sin 120579 sinΦ cos 120579 sinΦ cosΦ0 cos 120579 minus sin 120579 0
) (20)
where Φ is a function of azimuthal angle 120601 that is Φ =119873(120601) ldquoHere we consider Φ to be function of 120601 only whichis sufficient to carry out our aim However if Φ becomesa general function that is depends on 119903 120579 and 120601 thecalculations will be very complicatedrdquo The new tetrad field
(119890119894
120583)1
= Λ119894
119895119890119895
120583 (21)
takes the form
(119890119894
120583)1
=
((((((((
(
radic120575(119903)
radicΣ (119903 120579)0 0
119886sin2120579radic120575 (119903)
ΩradicΣ (119903 120579)
minus119886 sin 120579 sinΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 cosΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 cosΦradicΣ (119903 120579)
radic119891 (120579)minus(1199032 + 1198862) sin 120579 sinΦradic119891 (120579)
ΩradicΣ (119903 120579)
119886 sin 120579 cosΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 sinΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 sinΦradicΣ (119903 120579)
radic119891 (120579)
(1199032 + 1198862) sin 120579 cosΦradic119891 (120579)
ΩradicΣ (119903 120579)
0cos 120579radicΣ (119903 120579)
radic120575 (119903)
sin 120579radicΣ (119903 120579)
radic119891 (120579)0
))))))))
)
(22)
We rewrite (16) as
119866120583] +
1
119891 (119879)119879
Λ119892120583]
=1
119891 (119879)119879
(1
2(119891 (119879) minus 119879119891 (119879)
119879) 119892120583]
minus 119878]120588
120583119879120588119891 (119879)119879119879
+ 8120587T120583])
997904rArr 119866120583] +
1
119891 (119879)119879
Λ119892120583] =
1
119891 (119879)119879
(4120587T120583] + 119879
eff120583])
(23)
where 119879eff120583] = (12)[119891(119879) minus 119879119891(119879)
119879]119892120583] minus 119878]
120588
120583119879120588119891(119879)119879119879
isthe effective energy momentum tensor
To find a vacuum solution within 119891(119879) theories it issufficient to find the condition that makes the effectiveenergy momentum tensor 119879eff
120583] vanishing The simplestcondition that satisfies this aim is the vanishing of the scalartorsion 119879 The imposition of 119879 = 0 is just one (simplest)solution and in principle there can exist many more forexample when the torsion scalar is constant the effectiveenergy momentum tensor 119879eff
120583] will also be vanishing undersome constraints on the form of 119891 (constant) and its firstderivative Using (6) and tetrad field (22) one can obtainℎ = det(ℎ119886
120583) = (1199032 + 1198862cos2120579) sin 120579(1 + Λ1198862) With the use
of (3) (4) and (5) we obtain the torsion scalar as
119879 = radic119891 (120579) 120575 (119903) (2ΩΣ2
1198731015840
minus 21198866
Λcos6120579
minus 2 [1 +Λ
3(51199032
minus 1198862
)] 1198864cos4120579
minus 2119903 (5Λ
31199033
+ 4119872)1198862cos2120579
+ 21199034
[1 minusΛ
3(31199032
+ 1198862
)])
+ 2Σ2
(3 + 1198862
Λcos2120579 + 3Ω1198731015840
)
sdot [119903Λ
3(21199032
+ 1198862
) minus 119903 + 119872]
(24)
where 1198731015840 = 119889119873(120601)119889120601 The exact solution of (24) has theform
119873(120601) =120601radic119891 (120579)
ΩΣ2 (2Λ1199033 minus 3119903 + 1199031198862Λ + 3119872 + 3radic119891 (120579) 120575 (119903))
sdot (1198862
119891 (120579) (21198862cos4120579 + cos2120579 [3119903
2
+ 1198862cos2120579] minus 119903
2
)
+ radic120575 (119903)Σ [21199033
Λ + 1198862
Λ minus 3 119903 + 119872] 119903
minus 1198864sin2120579cos2120579Λ
minus 9radic1205753 (119903) [1199032
minus 1198862cos2120579]
minus 3Σ2radic119891 (120579) (3119872 + 2119903
3
Λ minus 3119903 + 1199031198862
Λ))
(25)
Using (25) and (22) in (23) we get a solution to 119891(119879) gravita-tional theory provided that 119891(0) = 0
4 Singularities
Now we are going to study the singularities of the derivedsolution For this purpose we search the value at which119903 makes 119892
00and 119892
11tend to zero or infin This procedure
may reproduce singularities corresponding to coordinatesingularities Therefore to procedure correct singularitieswe are going to demonstrate the invariants In orthodox
Advances in High Energy Physics 5
general relativity or its modifications the invariants are theRicci scalar the Kretschmann scalar or other invariantsconstructed from Riemann tensor and its contractions Inthe teleparallel geometry of gravity we have two approachesof finding invariants In the first approach one uses thesolution of the vierbein and the Weitzenbock connectionto calculate torsion invariants such as the torsion scalar 119879In the second approach one uses the solution to constructmetric and then construct the Levi-Civita connection andfinally calculate curvature invariants such as the Ricci andKretschmann scalars The comparison of the two approachesis able to show differences between curvature and torsiongravity In teleparallel theories we mean by singularity ofspacetime the singularity of the scalar concomitants of thetorsion and curvature tensors
The torsion invariant 119879 that is the torsion scalar thatarises from the vierbein solution (22) with the use ofWeitzenbockrsquos connection (25) is vanishing The curvatureinvariants that arise from themetric solution (19) through thecalculation of the Levi-Civita connection are [61]
119877120583]120582120590
119877120583]120582120590
=1
3Σ6[811988612
Λ2cos12120579 + 48119886
10
1199032
Λ2cos10120579
+ 1201198868
1199034
Λ2cos8120579 + 16119886
6cos6120579 [101199036
Λ2
minus 91198722
]
+ 1201198864
1199032cos4120579 [119903
6
Λ2
+ 181198722
]
+ 481198862
1199034cos2120579 [119903
6
Λ2
minus 451198722
]
+ 1441199036
1198722
+ 811990312
Λ2
]
119877120583]119877120583] = 4Λ
2
119877 = minus4Λ2
= minus2nabla120572119879120572
119879120583]120582
119879120583]120582 =
minus119891 (119903 120579)
27sin2120579120575 (119903) Σ5119891 (120579)32
119879120583
119879120583=
minus1198911(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119905120583]120582
119905120583]120582 =
minus1198912(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119886120583
119886120583=
21198862 [1199032120575 (119903) sin2120579 + cos2120579119891 (120579)]
9120575 (119903) Σ119891 (120579)
(26)
where 119891(119903 120579) 1198911(119903 120579) and 119891
2(119903 120579) are functions of 119903 and 120579
Here 119905120583]120582 119879120583 and 119886
120583are the irreducible representation of the
torsion tensor
119905120582120583] =
1
2(119879120582120583] + 119879
120583120582]) +1
6(119892]120582119881120583 + 119892
120583]119881120582) minus1
3119892120582120583
119881]
119879120583= 119879120582
120582]
119886120583=
1
6120598120583]120588120590119879
]120588120590
(27)
where 120598120583]120588120590 is defined by
120598120583]120588120590
def = radicminus119892120575
120583]120588120590 (28)
with 120575120583]120588120590 being completely antisymmetric and normalized
as 1205750123
= minus1Observing the forms of Ricci and Kretschmann torsion
and its irreducible representation scalars in (26) we deducethat in the Kerr-dS case there are no divergence points ateither 119903 = 0 or 119903 = infin
5 Main Results and Discussion
119891(119879) gravitational theories are modifications of the TEGRthat try to deal with the recent problems appearing incosmology In these theories it is not easy to find exactsolutions We have rewritten the field equations of thesetheories in a simple form This form enables us to show theextra terms that are responsible for the deviation from GRtheory In the nonvacuum case those terms can be regardedas the effective energy momentum tensor and generally theydepend on the scalar torsion and its derivatives For vacuumsolution one must show that the effective energy momentumtensors are vanishing So if the torsion scalar is vanishing thenit turns out that the extra terms are vanishing provided someconstraints on the form of the zero function that is 119891(0) andits first derivative
The equivalence between GR and TEGR emerges fromthe property that Einstein-Hilbert Lagrangian differs fromTEGR Lagrangian by a four-divergence term By using theLevi-Civita scalar curvature 119877 for the metric (2) one gets theresult of (13) Therefore TEGR and Einstein-Hilbert actionsare equivalent up to the equations of motion This meansthat GR and TEGR possess the same number of degrees offreedom In spite of the fact that the tetrad field contains 16components (6more than themetric field) TEGR is invariantunder local Lorentz transformations of the tetrad due to theexistence of the divergence term In the TEGR the tetrad fieldchanges under local Lorentz transformation as
119890119894
120583= Λ119894
119895119890119895
120583 119890
119894
120583
= Λ119894
119895
119890119895
120583
(29)
This transformation adds a four-divergence to the TEGRLagrangian 119890119879 This behavior is evident in (13) because 119890119877 isinvariant under local Lorentz transformations Instead 119891(119879)gravity like other theories of amended gravity possessesextra degrees of freedom In fact the dynamical equations(7) are sensitive to local Lorentz transformations of thetetrad except for the case 119891(119879) = 119879 (ie TEGR) Thisimplies that the dynamical equations of 119891(119879) gravitationaltheories contain information not only about the evolutionof the metric but also about some extra degrees of freedomexclusively associated with the tetrad that are not present inthe undeformed theory [24 25 33ndash51] For 119891(119879) theories theLagrangian changes under a local Lorentz transformation as
119890119891 (119879) = 119890119891 (119879 + divergence term) (30)
In this case the divergence term remains free inside thefunction 119891 damaging the principle of invariance under
6 Advances in High Energy Physics
local Lorentz transformation The loss of the local Lorentzinvariance leads to a preferred global reference frame definedby the autoparallel curves of the manifold that consistentlysolve the dynamical equations This means that (7) not onlydetermines themetric but also chooses some other propertiesof the tetrad field The tetrads connected by local Lorentztransformations lead to the same metric however they aredifferent with respect to the parallel framework Becauseof this fundamental property of 119891(119879) theories when oneis searching for solutions of a given symmetry it is quitedifficult to do an ansatz for the tetrad field For themetric casesymmetry helps us to choose suitable coordinates to write themetric However this does not say much about the ansatzfor the tetrad due to local Lorentz transformation [66 67]Certainly in the context of 119891(119879) theories the proper framewhich parallelizes the spacetime for a given symmetry ofthe geometry must be independent of the function 119891 [68]This work is focused on how to find the parallelization foraxially symmetric solutions in 119891(119879) theories In particularwe want to know whether Kerr-dS geometry survives or notin 119891(119879) gravity To answer this question we should find thecorrect ansatz to solve (7) This search is greatly facilitated byinvoking the following argument concerning the survival ofcertain TEGR solutions [67] if a vacuum solution of 119891(119879)gravity has 119879 = 0 then it will be a solution of TEGR aswell (a cosmological constantmight be necessary) In fact thereplacement of119879 = 0 in (7) after neglecting the cosmologicalconstant leads to
119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582minus
1
4120575]120583
119891 (0)
1198911015840 (0)= 0 (31)
which is a TEGR vacuum equation with cosmological con-stant 2Λ = 119891(0)1198911015840(0) We can neglect the cosmologicalconstant term by restricting the family of functions 119891 to have119891(0) = 0 and 1198911015840(0) = 0 In other words we can utilizethe freedom to do local Lorentz transformations in TEGRto search a tetrad having 119879 = 0 if we succeed then wewill state that such solution survives in 119891(119879) gravity Noticethat TEGR vacuum solutions do not force 119879 to vanish 119877must vanish Thus (13) says that 119879 is a four-divergence Sothe former argument is based on the sensitivity of 119879 to localLorentz transformations The above argument means thatTEGR vacuum solutions having 119879 = 0 (or 119879 = constant)cannot be deformed by119891(119879) gravityWe have shown that thisis the case for Kerr-dS geometry what is meant that 119891(119879)gravity is unable to smooth the singularity of a black hole[67 69 70]
In this study we have used a diagonal tetrad field withsix unknown functions of redial coordinate 119903 and azimuthalangle 120579 This type of tetrad does not depend on local Lorentztransformation This tetrad field has been applied to thefield equations of GR with a cosmological constant and gota system of nonlinear partial differential equations Thissystem have been solved and a solution with two constants ofintegration is derived The associated metric of this solutionis similar to Kerr-dS solution provided that the two constantsof integration are the gravitational mass of the system and therotating parameter [65] The tetrad field of this solution hasbeen multiplied by 119904119900(3) which contains a generalization of
the angle120601 that is119873(120601)Thenew tetrad field has beenused tocalculate the scalar torsion and differential equation with oneunknown 119873(120601) has been derived This differential equationhas been solved assuming the vanishing of the scalar torsionand the form of the unknown function has been derivedThisform of the unknown function has been used in the newtetrad field which has been shown that the new tetrad is aspecial solution to 119891(119879) provided that 119891(0) = 0
The singularities of the derived solution have been dis-cussed and has been shown that the derived solution has nosingularities at either 119903 = 0 or 119903 = infin This is done throughthe calculations of the scalars of Ricci Kretschman torsiontensor and its irreducible representationThe situation of thesingularities is expected to change for higher order of 119891(119879)
This is a first step in deriving a special solution within119891(119879) gravitational theories which proves that any GR solu-tion can be regarded as a solution in 119891(119879) under someconditions [58] This procedure can be generalized by tryingto derive solutions for quadratic 119891(119879) by trying to solve theeffective energy momentum tensor which will be responsiblefor such quadratic theory This will be done elsewhere
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by the Egyptian Ministry ofScientific Research under Project no 24-2-12
References
[1] D N Spergel L Verde H V Peiris et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003
[2] H V Peiris E Komatsu L Verde et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations implica-tions for inflationrdquoTheAstrophysical Journal Supplement Seriesvol 148 no 1 pp 213ndash232 2003
[3] D N Spergel R Bean O Dore et al ldquoThree-year Wilkinsonmicrowave anisotropy probe (WMAP) Observations impli-cations for cosmologyrdquo The Astrophysical Journal SupplementSeries vol 170 no 2 pp 377ndash408 2007
[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical interpretationrdquo Astrophysics Journal vol 180 pp 330ndash3762009
[5] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 no 6662 pp 51ndash54 1998
[6] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999
[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
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FluidsJournal of
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AstronomyAdvances in
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Superconductivity
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Statistical MechanicsInternational Journal of
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GravityJournal of
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Physics Research International
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Soft MatterJournal of
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PhotonicsJournal of
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Journal of
Biophysics
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ThermodynamicsJournal of
4 Advances in High Energy Physics
do such aim we multiply tetrad field (17) with the following119904119900(3)matrix
(Λ119894
119895) = (
1 0 0 0
0 sin 120579 cosΦ cos 120579 sinΦ minus sinΦ
0 sin 120579 sinΦ cos 120579 sinΦ cosΦ0 cos 120579 minus sin 120579 0
) (20)
where Φ is a function of azimuthal angle 120601 that is Φ =119873(120601) ldquoHere we consider Φ to be function of 120601 only whichis sufficient to carry out our aim However if Φ becomesa general function that is depends on 119903 120579 and 120601 thecalculations will be very complicatedrdquo The new tetrad field
(119890119894
120583)1
= Λ119894
119895119890119895
120583 (21)
takes the form
(119890119894
120583)1
=
((((((((
(
radic120575(119903)
radicΣ (119903 120579)0 0
119886sin2120579radic120575 (119903)
ΩradicΣ (119903 120579)
minus119886 sin 120579 sinΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 cosΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 cosΦradicΣ (119903 120579)
radic119891 (120579)minus(1199032 + 1198862) sin 120579 sinΦradic119891 (120579)
ΩradicΣ (119903 120579)
119886 sin 120579 cosΦradic119891 (120579)
radicΣ (119903 120579)
sin 120579 sinΦradicΣ (119903 120579)
radic120575 (119903)
cos 120579 sinΦradicΣ (119903 120579)
radic119891 (120579)
(1199032 + 1198862) sin 120579 cosΦradic119891 (120579)
ΩradicΣ (119903 120579)
0cos 120579radicΣ (119903 120579)
radic120575 (119903)
sin 120579radicΣ (119903 120579)
radic119891 (120579)0
))))))))
)
(22)
We rewrite (16) as
119866120583] +
1
119891 (119879)119879
Λ119892120583]
=1
119891 (119879)119879
(1
2(119891 (119879) minus 119879119891 (119879)
119879) 119892120583]
minus 119878]120588
120583119879120588119891 (119879)119879119879
+ 8120587T120583])
997904rArr 119866120583] +
1
119891 (119879)119879
Λ119892120583] =
1
119891 (119879)119879
(4120587T120583] + 119879
eff120583])
(23)
where 119879eff120583] = (12)[119891(119879) minus 119879119891(119879)
119879]119892120583] minus 119878]
120588
120583119879120588119891(119879)119879119879
isthe effective energy momentum tensor
To find a vacuum solution within 119891(119879) theories it issufficient to find the condition that makes the effectiveenergy momentum tensor 119879eff
120583] vanishing The simplestcondition that satisfies this aim is the vanishing of the scalartorsion 119879 The imposition of 119879 = 0 is just one (simplest)solution and in principle there can exist many more forexample when the torsion scalar is constant the effectiveenergy momentum tensor 119879eff
120583] will also be vanishing undersome constraints on the form of 119891 (constant) and its firstderivative Using (6) and tetrad field (22) one can obtainℎ = det(ℎ119886
120583) = (1199032 + 1198862cos2120579) sin 120579(1 + Λ1198862) With the use
of (3) (4) and (5) we obtain the torsion scalar as
119879 = radic119891 (120579) 120575 (119903) (2ΩΣ2
1198731015840
minus 21198866
Λcos6120579
minus 2 [1 +Λ
3(51199032
minus 1198862
)] 1198864cos4120579
minus 2119903 (5Λ
31199033
+ 4119872)1198862cos2120579
+ 21199034
[1 minusΛ
3(31199032
+ 1198862
)])
+ 2Σ2
(3 + 1198862
Λcos2120579 + 3Ω1198731015840
)
sdot [119903Λ
3(21199032
+ 1198862
) minus 119903 + 119872]
(24)
where 1198731015840 = 119889119873(120601)119889120601 The exact solution of (24) has theform
119873(120601) =120601radic119891 (120579)
ΩΣ2 (2Λ1199033 minus 3119903 + 1199031198862Λ + 3119872 + 3radic119891 (120579) 120575 (119903))
sdot (1198862
119891 (120579) (21198862cos4120579 + cos2120579 [3119903
2
+ 1198862cos2120579] minus 119903
2
)
+ radic120575 (119903)Σ [21199033
Λ + 1198862
Λ minus 3 119903 + 119872] 119903
minus 1198864sin2120579cos2120579Λ
minus 9radic1205753 (119903) [1199032
minus 1198862cos2120579]
minus 3Σ2radic119891 (120579) (3119872 + 2119903
3
Λ minus 3119903 + 1199031198862
Λ))
(25)
Using (25) and (22) in (23) we get a solution to 119891(119879) gravita-tional theory provided that 119891(0) = 0
4 Singularities
Now we are going to study the singularities of the derivedsolution For this purpose we search the value at which119903 makes 119892
00and 119892
11tend to zero or infin This procedure
may reproduce singularities corresponding to coordinatesingularities Therefore to procedure correct singularitieswe are going to demonstrate the invariants In orthodox
Advances in High Energy Physics 5
general relativity or its modifications the invariants are theRicci scalar the Kretschmann scalar or other invariantsconstructed from Riemann tensor and its contractions Inthe teleparallel geometry of gravity we have two approachesof finding invariants In the first approach one uses thesolution of the vierbein and the Weitzenbock connectionto calculate torsion invariants such as the torsion scalar 119879In the second approach one uses the solution to constructmetric and then construct the Levi-Civita connection andfinally calculate curvature invariants such as the Ricci andKretschmann scalars The comparison of the two approachesis able to show differences between curvature and torsiongravity In teleparallel theories we mean by singularity ofspacetime the singularity of the scalar concomitants of thetorsion and curvature tensors
The torsion invariant 119879 that is the torsion scalar thatarises from the vierbein solution (22) with the use ofWeitzenbockrsquos connection (25) is vanishing The curvatureinvariants that arise from themetric solution (19) through thecalculation of the Levi-Civita connection are [61]
119877120583]120582120590
119877120583]120582120590
=1
3Σ6[811988612
Λ2cos12120579 + 48119886
10
1199032
Λ2cos10120579
+ 1201198868
1199034
Λ2cos8120579 + 16119886
6cos6120579 [101199036
Λ2
minus 91198722
]
+ 1201198864
1199032cos4120579 [119903
6
Λ2
+ 181198722
]
+ 481198862
1199034cos2120579 [119903
6
Λ2
minus 451198722
]
+ 1441199036
1198722
+ 811990312
Λ2
]
119877120583]119877120583] = 4Λ
2
119877 = minus4Λ2
= minus2nabla120572119879120572
119879120583]120582
119879120583]120582 =
minus119891 (119903 120579)
27sin2120579120575 (119903) Σ5119891 (120579)32
119879120583
119879120583=
minus1198911(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119905120583]120582
119905120583]120582 =
minus1198912(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119886120583
119886120583=
21198862 [1199032120575 (119903) sin2120579 + cos2120579119891 (120579)]
9120575 (119903) Σ119891 (120579)
(26)
where 119891(119903 120579) 1198911(119903 120579) and 119891
2(119903 120579) are functions of 119903 and 120579
Here 119905120583]120582 119879120583 and 119886
120583are the irreducible representation of the
torsion tensor
119905120582120583] =
1
2(119879120582120583] + 119879
120583120582]) +1
6(119892]120582119881120583 + 119892
120583]119881120582) minus1
3119892120582120583
119881]
119879120583= 119879120582
120582]
119886120583=
1
6120598120583]120588120590119879
]120588120590
(27)
where 120598120583]120588120590 is defined by
120598120583]120588120590
def = radicminus119892120575
120583]120588120590 (28)
with 120575120583]120588120590 being completely antisymmetric and normalized
as 1205750123
= minus1Observing the forms of Ricci and Kretschmann torsion
and its irreducible representation scalars in (26) we deducethat in the Kerr-dS case there are no divergence points ateither 119903 = 0 or 119903 = infin
5 Main Results and Discussion
119891(119879) gravitational theories are modifications of the TEGRthat try to deal with the recent problems appearing incosmology In these theories it is not easy to find exactsolutions We have rewritten the field equations of thesetheories in a simple form This form enables us to show theextra terms that are responsible for the deviation from GRtheory In the nonvacuum case those terms can be regardedas the effective energy momentum tensor and generally theydepend on the scalar torsion and its derivatives For vacuumsolution one must show that the effective energy momentumtensors are vanishing So if the torsion scalar is vanishing thenit turns out that the extra terms are vanishing provided someconstraints on the form of the zero function that is 119891(0) andits first derivative
The equivalence between GR and TEGR emerges fromthe property that Einstein-Hilbert Lagrangian differs fromTEGR Lagrangian by a four-divergence term By using theLevi-Civita scalar curvature 119877 for the metric (2) one gets theresult of (13) Therefore TEGR and Einstein-Hilbert actionsare equivalent up to the equations of motion This meansthat GR and TEGR possess the same number of degrees offreedom In spite of the fact that the tetrad field contains 16components (6more than themetric field) TEGR is invariantunder local Lorentz transformations of the tetrad due to theexistence of the divergence term In the TEGR the tetrad fieldchanges under local Lorentz transformation as
119890119894
120583= Λ119894
119895119890119895
120583 119890
119894
120583
= Λ119894
119895
119890119895
120583
(29)
This transformation adds a four-divergence to the TEGRLagrangian 119890119879 This behavior is evident in (13) because 119890119877 isinvariant under local Lorentz transformations Instead 119891(119879)gravity like other theories of amended gravity possessesextra degrees of freedom In fact the dynamical equations(7) are sensitive to local Lorentz transformations of thetetrad except for the case 119891(119879) = 119879 (ie TEGR) Thisimplies that the dynamical equations of 119891(119879) gravitationaltheories contain information not only about the evolutionof the metric but also about some extra degrees of freedomexclusively associated with the tetrad that are not present inthe undeformed theory [24 25 33ndash51] For 119891(119879) theories theLagrangian changes under a local Lorentz transformation as
119890119891 (119879) = 119890119891 (119879 + divergence term) (30)
In this case the divergence term remains free inside thefunction 119891 damaging the principle of invariance under
6 Advances in High Energy Physics
local Lorentz transformation The loss of the local Lorentzinvariance leads to a preferred global reference frame definedby the autoparallel curves of the manifold that consistentlysolve the dynamical equations This means that (7) not onlydetermines themetric but also chooses some other propertiesof the tetrad field The tetrads connected by local Lorentztransformations lead to the same metric however they aredifferent with respect to the parallel framework Becauseof this fundamental property of 119891(119879) theories when oneis searching for solutions of a given symmetry it is quitedifficult to do an ansatz for the tetrad field For themetric casesymmetry helps us to choose suitable coordinates to write themetric However this does not say much about the ansatzfor the tetrad due to local Lorentz transformation [66 67]Certainly in the context of 119891(119879) theories the proper framewhich parallelizes the spacetime for a given symmetry ofthe geometry must be independent of the function 119891 [68]This work is focused on how to find the parallelization foraxially symmetric solutions in 119891(119879) theories In particularwe want to know whether Kerr-dS geometry survives or notin 119891(119879) gravity To answer this question we should find thecorrect ansatz to solve (7) This search is greatly facilitated byinvoking the following argument concerning the survival ofcertain TEGR solutions [67] if a vacuum solution of 119891(119879)gravity has 119879 = 0 then it will be a solution of TEGR aswell (a cosmological constantmight be necessary) In fact thereplacement of119879 = 0 in (7) after neglecting the cosmologicalconstant leads to
119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582minus
1
4120575]120583
119891 (0)
1198911015840 (0)= 0 (31)
which is a TEGR vacuum equation with cosmological con-stant 2Λ = 119891(0)1198911015840(0) We can neglect the cosmologicalconstant term by restricting the family of functions 119891 to have119891(0) = 0 and 1198911015840(0) = 0 In other words we can utilizethe freedom to do local Lorentz transformations in TEGRto search a tetrad having 119879 = 0 if we succeed then wewill state that such solution survives in 119891(119879) gravity Noticethat TEGR vacuum solutions do not force 119879 to vanish 119877must vanish Thus (13) says that 119879 is a four-divergence Sothe former argument is based on the sensitivity of 119879 to localLorentz transformations The above argument means thatTEGR vacuum solutions having 119879 = 0 (or 119879 = constant)cannot be deformed by119891(119879) gravityWe have shown that thisis the case for Kerr-dS geometry what is meant that 119891(119879)gravity is unable to smooth the singularity of a black hole[67 69 70]
In this study we have used a diagonal tetrad field withsix unknown functions of redial coordinate 119903 and azimuthalangle 120579 This type of tetrad does not depend on local Lorentztransformation This tetrad field has been applied to thefield equations of GR with a cosmological constant and gota system of nonlinear partial differential equations Thissystem have been solved and a solution with two constants ofintegration is derived The associated metric of this solutionis similar to Kerr-dS solution provided that the two constantsof integration are the gravitational mass of the system and therotating parameter [65] The tetrad field of this solution hasbeen multiplied by 119904119900(3) which contains a generalization of
the angle120601 that is119873(120601)Thenew tetrad field has beenused tocalculate the scalar torsion and differential equation with oneunknown 119873(120601) has been derived This differential equationhas been solved assuming the vanishing of the scalar torsionand the form of the unknown function has been derivedThisform of the unknown function has been used in the newtetrad field which has been shown that the new tetrad is aspecial solution to 119891(119879) provided that 119891(0) = 0
The singularities of the derived solution have been dis-cussed and has been shown that the derived solution has nosingularities at either 119903 = 0 or 119903 = infin This is done throughthe calculations of the scalars of Ricci Kretschman torsiontensor and its irreducible representationThe situation of thesingularities is expected to change for higher order of 119891(119879)
This is a first step in deriving a special solution within119891(119879) gravitational theories which proves that any GR solu-tion can be regarded as a solution in 119891(119879) under someconditions [58] This procedure can be generalized by tryingto derive solutions for quadratic 119891(119879) by trying to solve theeffective energy momentum tensor which will be responsiblefor such quadratic theory This will be done elsewhere
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by the Egyptian Ministry ofScientific Research under Project no 24-2-12
References
[1] D N Spergel L Verde H V Peiris et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003
[2] H V Peiris E Komatsu L Verde et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations implica-tions for inflationrdquoTheAstrophysical Journal Supplement Seriesvol 148 no 1 pp 213ndash232 2003
[3] D N Spergel R Bean O Dore et al ldquoThree-year Wilkinsonmicrowave anisotropy probe (WMAP) Observations impli-cations for cosmologyrdquo The Astrophysical Journal SupplementSeries vol 170 no 2 pp 377ndash408 2007
[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical interpretationrdquo Astrophysics Journal vol 180 pp 330ndash3762009
[5] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 no 6662 pp 51ndash54 1998
[6] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999
[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
general relativity or its modifications the invariants are theRicci scalar the Kretschmann scalar or other invariantsconstructed from Riemann tensor and its contractions Inthe teleparallel geometry of gravity we have two approachesof finding invariants In the first approach one uses thesolution of the vierbein and the Weitzenbock connectionto calculate torsion invariants such as the torsion scalar 119879In the second approach one uses the solution to constructmetric and then construct the Levi-Civita connection andfinally calculate curvature invariants such as the Ricci andKretschmann scalars The comparison of the two approachesis able to show differences between curvature and torsiongravity In teleparallel theories we mean by singularity ofspacetime the singularity of the scalar concomitants of thetorsion and curvature tensors
The torsion invariant 119879 that is the torsion scalar thatarises from the vierbein solution (22) with the use ofWeitzenbockrsquos connection (25) is vanishing The curvatureinvariants that arise from themetric solution (19) through thecalculation of the Levi-Civita connection are [61]
119877120583]120582120590
119877120583]120582120590
=1
3Σ6[811988612
Λ2cos12120579 + 48119886
10
1199032
Λ2cos10120579
+ 1201198868
1199034
Λ2cos8120579 + 16119886
6cos6120579 [101199036
Λ2
minus 91198722
]
+ 1201198864
1199032cos4120579 [119903
6
Λ2
+ 181198722
]
+ 481198862
1199034cos2120579 [119903
6
Λ2
minus 451198722
]
+ 1441199036
1198722
+ 811990312
Λ2
]
119877120583]119877120583] = 4Λ
2
119877 = minus4Λ2
= minus2nabla120572119879120572
119879120583]120582
119879120583]120582 =
minus119891 (119903 120579)
27sin2120579120575 (119903) Σ5119891 (120579)32
119879120583
119879120583=
minus1198911(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119905120583]120582
119905120583]120582 =
minus1198912(119903 120579)
27sin2120579120575 (119903) Σ3119891 (120579)
119886120583
119886120583=
21198862 [1199032120575 (119903) sin2120579 + cos2120579119891 (120579)]
9120575 (119903) Σ119891 (120579)
(26)
where 119891(119903 120579) 1198911(119903 120579) and 119891
2(119903 120579) are functions of 119903 and 120579
Here 119905120583]120582 119879120583 and 119886
120583are the irreducible representation of the
torsion tensor
119905120582120583] =
1
2(119879120582120583] + 119879
120583120582]) +1
6(119892]120582119881120583 + 119892
120583]119881120582) minus1
3119892120582120583
119881]
119879120583= 119879120582
120582]
119886120583=
1
6120598120583]120588120590119879
]120588120590
(27)
where 120598120583]120588120590 is defined by
120598120583]120588120590
def = radicminus119892120575
120583]120588120590 (28)
with 120575120583]120588120590 being completely antisymmetric and normalized
as 1205750123
= minus1Observing the forms of Ricci and Kretschmann torsion
and its irreducible representation scalars in (26) we deducethat in the Kerr-dS case there are no divergence points ateither 119903 = 0 or 119903 = infin
5 Main Results and Discussion
119891(119879) gravitational theories are modifications of the TEGRthat try to deal with the recent problems appearing incosmology In these theories it is not easy to find exactsolutions We have rewritten the field equations of thesetheories in a simple form This form enables us to show theextra terms that are responsible for the deviation from GRtheory In the nonvacuum case those terms can be regardedas the effective energy momentum tensor and generally theydepend on the scalar torsion and its derivatives For vacuumsolution one must show that the effective energy momentumtensors are vanishing So if the torsion scalar is vanishing thenit turns out that the extra terms are vanishing provided someconstraints on the form of the zero function that is 119891(0) andits first derivative
The equivalence between GR and TEGR emerges fromthe property that Einstein-Hilbert Lagrangian differs fromTEGR Lagrangian by a four-divergence term By using theLevi-Civita scalar curvature 119877 for the metric (2) one gets theresult of (13) Therefore TEGR and Einstein-Hilbert actionsare equivalent up to the equations of motion This meansthat GR and TEGR possess the same number of degrees offreedom In spite of the fact that the tetrad field contains 16components (6more than themetric field) TEGR is invariantunder local Lorentz transformations of the tetrad due to theexistence of the divergence term In the TEGR the tetrad fieldchanges under local Lorentz transformation as
119890119894
120583= Λ119894
119895119890119895
120583 119890
119894
120583
= Λ119894
119895
119890119895
120583
(29)
This transformation adds a four-divergence to the TEGRLagrangian 119890119879 This behavior is evident in (13) because 119890119877 isinvariant under local Lorentz transformations Instead 119891(119879)gravity like other theories of amended gravity possessesextra degrees of freedom In fact the dynamical equations(7) are sensitive to local Lorentz transformations of thetetrad except for the case 119891(119879) = 119879 (ie TEGR) Thisimplies that the dynamical equations of 119891(119879) gravitationaltheories contain information not only about the evolutionof the metric but also about some extra degrees of freedomexclusively associated with the tetrad that are not present inthe undeformed theory [24 25 33ndash51] For 119891(119879) theories theLagrangian changes under a local Lorentz transformation as
119890119891 (119879) = 119890119891 (119879 + divergence term) (30)
In this case the divergence term remains free inside thefunction 119891 damaging the principle of invariance under
6 Advances in High Energy Physics
local Lorentz transformation The loss of the local Lorentzinvariance leads to a preferred global reference frame definedby the autoparallel curves of the manifold that consistentlysolve the dynamical equations This means that (7) not onlydetermines themetric but also chooses some other propertiesof the tetrad field The tetrads connected by local Lorentztransformations lead to the same metric however they aredifferent with respect to the parallel framework Becauseof this fundamental property of 119891(119879) theories when oneis searching for solutions of a given symmetry it is quitedifficult to do an ansatz for the tetrad field For themetric casesymmetry helps us to choose suitable coordinates to write themetric However this does not say much about the ansatzfor the tetrad due to local Lorentz transformation [66 67]Certainly in the context of 119891(119879) theories the proper framewhich parallelizes the spacetime for a given symmetry ofthe geometry must be independent of the function 119891 [68]This work is focused on how to find the parallelization foraxially symmetric solutions in 119891(119879) theories In particularwe want to know whether Kerr-dS geometry survives or notin 119891(119879) gravity To answer this question we should find thecorrect ansatz to solve (7) This search is greatly facilitated byinvoking the following argument concerning the survival ofcertain TEGR solutions [67] if a vacuum solution of 119891(119879)gravity has 119879 = 0 then it will be a solution of TEGR aswell (a cosmological constantmight be necessary) In fact thereplacement of119879 = 0 in (7) after neglecting the cosmologicalconstant leads to
119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582minus
1
4120575]120583
119891 (0)
1198911015840 (0)= 0 (31)
which is a TEGR vacuum equation with cosmological con-stant 2Λ = 119891(0)1198911015840(0) We can neglect the cosmologicalconstant term by restricting the family of functions 119891 to have119891(0) = 0 and 1198911015840(0) = 0 In other words we can utilizethe freedom to do local Lorentz transformations in TEGRto search a tetrad having 119879 = 0 if we succeed then wewill state that such solution survives in 119891(119879) gravity Noticethat TEGR vacuum solutions do not force 119879 to vanish 119877must vanish Thus (13) says that 119879 is a four-divergence Sothe former argument is based on the sensitivity of 119879 to localLorentz transformations The above argument means thatTEGR vacuum solutions having 119879 = 0 (or 119879 = constant)cannot be deformed by119891(119879) gravityWe have shown that thisis the case for Kerr-dS geometry what is meant that 119891(119879)gravity is unable to smooth the singularity of a black hole[67 69 70]
In this study we have used a diagonal tetrad field withsix unknown functions of redial coordinate 119903 and azimuthalangle 120579 This type of tetrad does not depend on local Lorentztransformation This tetrad field has been applied to thefield equations of GR with a cosmological constant and gota system of nonlinear partial differential equations Thissystem have been solved and a solution with two constants ofintegration is derived The associated metric of this solutionis similar to Kerr-dS solution provided that the two constantsof integration are the gravitational mass of the system and therotating parameter [65] The tetrad field of this solution hasbeen multiplied by 119904119900(3) which contains a generalization of
the angle120601 that is119873(120601)Thenew tetrad field has beenused tocalculate the scalar torsion and differential equation with oneunknown 119873(120601) has been derived This differential equationhas been solved assuming the vanishing of the scalar torsionand the form of the unknown function has been derivedThisform of the unknown function has been used in the newtetrad field which has been shown that the new tetrad is aspecial solution to 119891(119879) provided that 119891(0) = 0
The singularities of the derived solution have been dis-cussed and has been shown that the derived solution has nosingularities at either 119903 = 0 or 119903 = infin This is done throughthe calculations of the scalars of Ricci Kretschman torsiontensor and its irreducible representationThe situation of thesingularities is expected to change for higher order of 119891(119879)
This is a first step in deriving a special solution within119891(119879) gravitational theories which proves that any GR solu-tion can be regarded as a solution in 119891(119879) under someconditions [58] This procedure can be generalized by tryingto derive solutions for quadratic 119891(119879) by trying to solve theeffective energy momentum tensor which will be responsiblefor such quadratic theory This will be done elsewhere
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by the Egyptian Ministry ofScientific Research under Project no 24-2-12
References
[1] D N Spergel L Verde H V Peiris et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003
[2] H V Peiris E Komatsu L Verde et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations implica-tions for inflationrdquoTheAstrophysical Journal Supplement Seriesvol 148 no 1 pp 213ndash232 2003
[3] D N Spergel R Bean O Dore et al ldquoThree-year Wilkinsonmicrowave anisotropy probe (WMAP) Observations impli-cations for cosmologyrdquo The Astrophysical Journal SupplementSeries vol 170 no 2 pp 377ndash408 2007
[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical interpretationrdquo Astrophysics Journal vol 180 pp 330ndash3762009
[5] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 no 6662 pp 51ndash54 1998
[6] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999
[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
local Lorentz transformation The loss of the local Lorentzinvariance leads to a preferred global reference frame definedby the autoparallel curves of the manifold that consistentlysolve the dynamical equations This means that (7) not onlydetermines themetric but also chooses some other propertiesof the tetrad field The tetrads connected by local Lorentztransformations lead to the same metric however they aredifferent with respect to the parallel framework Becauseof this fundamental property of 119891(119879) theories when oneis searching for solutions of a given symmetry it is quitedifficult to do an ansatz for the tetrad field For themetric casesymmetry helps us to choose suitable coordinates to write themetric However this does not say much about the ansatzfor the tetrad due to local Lorentz transformation [66 67]Certainly in the context of 119891(119879) theories the proper framewhich parallelizes the spacetime for a given symmetry ofthe geometry must be independent of the function 119891 [68]This work is focused on how to find the parallelization foraxially symmetric solutions in 119891(119879) theories In particularwe want to know whether Kerr-dS geometry survives or notin 119891(119879) gravity To answer this question we should find thecorrect ansatz to solve (7) This search is greatly facilitated byinvoking the following argument concerning the survival ofcertain TEGR solutions [67] if a vacuum solution of 119891(119879)gravity has 119879 = 0 then it will be a solution of TEGR aswell (a cosmological constantmight be necessary) In fact thereplacement of119879 = 0 in (7) after neglecting the cosmologicalconstant leads to
119890minus1
119890119886
120583120597120588(119890119890119886
120572
119878120572
]120588) minus 119879120572
120583120582119878120572
]120582minus
1
4120575]120583
119891 (0)
1198911015840 (0)= 0 (31)
which is a TEGR vacuum equation with cosmological con-stant 2Λ = 119891(0)1198911015840(0) We can neglect the cosmologicalconstant term by restricting the family of functions 119891 to have119891(0) = 0 and 1198911015840(0) = 0 In other words we can utilizethe freedom to do local Lorentz transformations in TEGRto search a tetrad having 119879 = 0 if we succeed then wewill state that such solution survives in 119891(119879) gravity Noticethat TEGR vacuum solutions do not force 119879 to vanish 119877must vanish Thus (13) says that 119879 is a four-divergence Sothe former argument is based on the sensitivity of 119879 to localLorentz transformations The above argument means thatTEGR vacuum solutions having 119879 = 0 (or 119879 = constant)cannot be deformed by119891(119879) gravityWe have shown that thisis the case for Kerr-dS geometry what is meant that 119891(119879)gravity is unable to smooth the singularity of a black hole[67 69 70]
In this study we have used a diagonal tetrad field withsix unknown functions of redial coordinate 119903 and azimuthalangle 120579 This type of tetrad does not depend on local Lorentztransformation This tetrad field has been applied to thefield equations of GR with a cosmological constant and gota system of nonlinear partial differential equations Thissystem have been solved and a solution with two constants ofintegration is derived The associated metric of this solutionis similar to Kerr-dS solution provided that the two constantsof integration are the gravitational mass of the system and therotating parameter [65] The tetrad field of this solution hasbeen multiplied by 119904119900(3) which contains a generalization of
the angle120601 that is119873(120601)Thenew tetrad field has beenused tocalculate the scalar torsion and differential equation with oneunknown 119873(120601) has been derived This differential equationhas been solved assuming the vanishing of the scalar torsionand the form of the unknown function has been derivedThisform of the unknown function has been used in the newtetrad field which has been shown that the new tetrad is aspecial solution to 119891(119879) provided that 119891(0) = 0
The singularities of the derived solution have been dis-cussed and has been shown that the derived solution has nosingularities at either 119903 = 0 or 119903 = infin This is done throughthe calculations of the scalars of Ricci Kretschman torsiontensor and its irreducible representationThe situation of thesingularities is expected to change for higher order of 119891(119879)
This is a first step in deriving a special solution within119891(119879) gravitational theories which proves that any GR solu-tion can be regarded as a solution in 119891(119879) under someconditions [58] This procedure can be generalized by tryingto derive solutions for quadratic 119891(119879) by trying to solve theeffective energy momentum tensor which will be responsiblefor such quadratic theory This will be done elsewhere
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work is partially supported by the Egyptian Ministry ofScientific Research under Project no 24-2-12
References
[1] D N Spergel L Verde H V Peiris et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations determi-nation of cosmological parametersrdquo The Astrophysical JournalSupplement Series vol 148 no 1 pp 175ndash194 2003
[2] H V Peiris E Komatsu L Verde et al ldquoFirst-year Wilkinsonmicrowave anisotropy probe (WMAP)lowast observations implica-tions for inflationrdquoTheAstrophysical Journal Supplement Seriesvol 148 no 1 pp 213ndash232 2003
[3] D N Spergel R Bean O Dore et al ldquoThree-year Wilkinsonmicrowave anisotropy probe (WMAP) Observations impli-cations for cosmologyrdquo The Astrophysical Journal SupplementSeries vol 170 no 2 pp 377ndash408 2007
[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe (WMAP) observations cosmolog-ical interpretationrdquo Astrophysics Journal vol 180 pp 330ndash3762009
[5] S Perlmutter G Aldering M Della Valle et al ldquoDiscovery ofa supernova explosion at half the age of the Universerdquo Naturevol 391 no 6662 pp 51ndash54 1998
[6] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 p 565 1999
[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 7
[8] P Astier J Guy N Regnault et al ldquoThe Supernova LegacySurvey measurement of Ω
119872ΩΛand w from the first year data
setrdquo Astronomy amp Astrophysics vol 447 no 1 pp 31ndash48 2006[9] A G Riess L-G Strolger S Casertano et al ldquoNew hubble
space telescope discoveries of type Ia supernovae at 119911 ge 1narrowing constraints on the early behavior of dark energyrdquoTheAstrophysical Journal vol 659 no 1 p 98 2007
[10] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 2004
[11] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe second data release of the Sloan digital sky surveyrdquo TheAstronomical Journal vol 128 p 502 2004
[12] K Abazajian J K Adelman-McCarthy M A Agueros et alldquoThe third data release of the sloan digital sky surveyrdquo TheAstronomical Journal vol 129 no 3 p 1755 2005
[13] N Radicella V F Cardone and S Camera ldquoAccelerating f(T)gravity models constrained by recent cosmological datardquo inProgress inMathematical Relativity Gravitation and Cosmologyvol 60 pp 367ndash370 Springer Berlin Germany 2014
[14] A Einstein ldquoRiemannian geometry with preservation of theconcept of distant parallelismrdquo in Sitzungsberichte der Deut-schen Akademie der Wissenschaften zu Berlin Mathematisch-Naturwissenschaftliche Klasse vol 17 p 217 1928
[15] A Einstein ldquoZurTheorie der Raume mit Riemann-Metrik undFernparallelis-musrdquo Sitzungsberichte der Preussischen Akadem-ie der Wissenschaften Physikalisch-Mathematische Klasse pp401ndash402 1930
[16] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[17] C Pellegrini and J Plebanski ldquoTetrad fields and gravitationalfieldsrdquo Matematisk-fysiske Skrifter udgivet af Det KongeligeDanske Videnskabernes Selskab vol 2 no 4 1963
[18] K Hayashi and T Nakano ldquoExtended translation invarianceand associated gauge fieldsrdquo Progress of Theoretical Physics vol38 no 2 pp 491ndash507 1967
[19] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 no 12 pp 3524ndash3553 1979
[20] M I Mikhail and M I Wanas ldquoA generalized field theory IField equationsrdquo Proceedings of the Royal Society of London Avol 471 no 2176 1977
[21] R Weitzenbock Invariance Theorie Nordhoff Groningen TheNetherlands 1923
[22] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D vol 79 no 12 Article ID 1240195 pages 2009
[23] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 23 pages 2012
[24] B Li T P Sotirious and J D Barrow ldquo119891(119879) gravity andlocal Lorentz invariancerdquo Physical Review D vol 83 Article ID064035 2011
[25] T P Sotirious B Li and J D Barrow ldquoGeneralizationsof teleparallel gravity and local Lorentz symmetryrdquo PhysicalReview D vol 83 Article ID 104030 2011
[26] E V Linder ldquoErratum Einsteinrsquos other gravity and the acceler-ation of the universe [Phys Rev D 81 127301 (2010)]rdquo PhysicalReview D vol 82 no 10 Article ID 109902 2010
[27] P Wu and H Yu ldquo119891(119879) models with phantom divide linecrossingrdquoThe European Physical Journal C vol 71 article 15522011
[28] K Bamba C Q Geng C C Lee and L W Luo ldquoEquation ofstate for dark energy in f(T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 article 021 2011
[29] Y-F Cai S-H Chen J B Dent S Dutta and E N SaridakisldquoMatter bounce cosmology with the f (T) gravityrdquo Classical andQuantum Gravity vol 28 no 21 Article ID 215011 2011
[30] R Aldrovandi and J G Pereira An Introduction to TeleparallelGravity Instituto de Fisica Teorica UNSEP Sao Paulo Prazil2014 httpwwwiftunespbrgcgtelepdf
[31] M H Daouda M E Rodrigues and M J S Houndjo ldquoStaticanisotropic solutions in f(T) theoryrdquo European Physical JournalC vol 72 no 2 pp 1ndash12 2012
[32] M Hamani Daouda M E Rodrigues and M J S HoundjoldquoAnisotropic fluid for a set of non-diagonal tetrads in f(T)gravityrdquo Physics Letters Section B Nuclear Elementary Particleand High-Energy Physics vol 715 no 1ndash3 pp 241ndash245 2012
[33] J Yang Y-L Li Y Zhong and Y Li ldquoThick brane split causedby spacetime torsionrdquo Physical Review D vol 85 Article ID084033 2012
[34] K Karami and A Abdolmaleki ldquoGeneralized second law ofthermodynamics in 119891(119879) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 4 article 007 2012
[35] K Atazadeh and F Darabi ldquo119891(119879) cosmology via Noethersymmetryrdquo The European Physical Journal C vol 72 no 5 pp1ndash7 2012
[36] M I Wanas ldquoThe other side of gravity and geometry anti-gravity and anticurvaturerdquoAdvances inHigh Energy Physics vol2012 Article ID 752613 10 pages 2012
[37] M I Wanas and H A Hassan ldquoTorsion and particle horizonsrdquoInternational Journal of Theoretical Physics vol 53 no 11 pp3901ndash3909 2014
[38] M I Wanas N L Youssef and A M Sid-Ahmed ldquoTeleparallelLagrange geometry and a unified field theoryrdquo Classical andQuantum Gravity vol 27 no 4 Article ID 045005 2010
[39] N L Youssef and A M Sid-Ahmed ldquoExtended absolute paral-lelism geometryrdquo International Journal of Geometric Methods inModern Physics vol 5 no 7 pp 1109ndash1135 2008
[40] N L Youssef and A M Sid-Ahmed ldquoLinear connections andcurvature tensors in the geometry of parallelizable manifoldsrdquoReports on Mathematical Physics vol 60 no 1 pp 39ndash53 2007
[41] H Wei X-J Guo and L-F Wang ldquoNoether symmetry in f(T)theoryrdquo Physics Letters B vol 707 no 2 pp 298ndash304 2012
[42] K Karami S Asadzadeh A Abdolmaleki and Z Safari ldquoHolo-graphic f(T)-gravity model with power-law entropy correctionrdquoPhysical Review D vol 88 Article ID 084034 2013
[43] P A Gonzalez E N Saridakis and Y Vasquez ldquoCircularlysymmetric solutions in three-dimensional teleparallel f(T) andMaxwell-f(T) gravityrdquo Journal of High Energy Physics vol 2012article 53 2012
[44] S Capozziello V F Cardone H Farajollahi and A RavanpakldquoCosmography in f(T) gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 84 no 4 Article ID043527 2011
[45] R-X Miao M Li and Y-G Miao ldquoViolation of the firstlaw of black hole thermodynamics in 119891(119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 11 article 033 2011
[46] X-H Meng and Y-B Wang ldquoBirkhoff rsquos theorem in f(T)gravityrdquo The European Physical Journal C vol 71 article 17552011
[47] H Wei X-P Ma and H-Y Qi ldquof(T) theories and varying finestructure constantrdquo Physics Letters B vol 703 no 1 pp 74ndash802011
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
8 Advances in High Energy Physics
[48] M Li R-X Miao and Y-G Miao ldquoDegrees of freedom of f(T)gravityrdquo Journal of High Energy Physics vol 2011 article 1082011
[49] S Chattopadhyay and U Debnath ldquoEmergent universe inthe chameleon 119891(119877) and 119891(119879) gravity theoriesrdquo InternationalJournal of Modern Physics D vol 20 no 6 pp 1135ndash1152 2011
[50] R-J Yang ldquoConformal transformation in f(T) theoriesrdquo Euro-physics Letters vol 93 no 6 Article ID 60001 2011
[51] C G Bohmer A Mussa and N Tamanini ldquoExistence ofrelativistic stars in f(T) gravityrdquoClassical and QuantumGravityvol 28 no 24 Article ID 245020 2011
[52] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 2012
[53] K Atazadeh and M Mousavi ldquoVacuum spherically symmetricsolutions in 119891(119879) gravityrdquoThe European Physical Journal C vol73 no 1 article 2272 2013
[54] R-J Yang ldquoNew types of 119891(119879) gravityrdquo The European PhysicalJournal C vol 71 article 1797 2011
[55] K Bamba C-Q Geng C-C Lee and L-W Luo ldquoEquation ofstate for dark energy in f (T) gravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 1 article 021 2011
[56] S Nesseris S Basilakos E Saridakis and L PerivolaropoulosldquoViable 119891(119879) models are practically indistinguishable fromΛ119862119863119872rdquo Physical Review D vol 88 Article ID 103010 2013
[57] R Zheng and Q-G Huang ldquoGrowth factor in f(T) gravityrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 3article 2 2011
[58] R Ferraro and F Fiorini ldquoCosmological frames for theorieswith absolute parallelismrdquo International Journal of ModernPhysics Conference Series vol 3 p 227 2011
[59] G G L Nashed ldquoA special exact spherically symmetric solutionin f(T) gravity theoriesrdquoGeneral Relativity and Gravitation vol45 no 10 pp 1887ndash1899 2013
[60] G G L Nashed ldquoStationary axisymmetric solutions and theirenergy contents in teleparallel equivalent of Einstein theoryrdquoAstrophysics and Space Science vol 330 no 1-2 pp 173ndash1812010
[61] S Capozziello P A Gonzalez E N Saridakis and Y VasquezldquoExact charged black-hole solutions in D-dimensional 119891(119879)gravity torsion vs curvature analysisrdquo Journal of High EnergyPhysics vol 2013 article 39 2013
[62] M E Rodrigues M J S Houndjo D Saez-Gomez andF Rahaman ldquoAnisotropic universe models in 119891(119879) gravityrdquoPhysical Review D vol 86 no 10 Article ID 104059 2012
[63] J-T Li C-C Lee and C-Q Geng ldquoEinstein static universe inexponential f (T) gravityrdquoThe European Physical Journal C vol73 article 2315 2013
[64] I G Salako M E Rodrigues A V Kpadonou M J SHoundjo and J Tossa ldquoΛCDM model in f(T) gravity recon-struction thermodynamics and stabilityrdquo Journal of Cosmologyand Astroparticle Physics vol 2013 no 11 article 60 2013
[65] H Suzuki E Takasugi and H Umetsu ldquoAnalytic solutions ofthe Teukolsky equation in KERr-de SITter and KERr-Newman-de SITter geometriesrdquo Progress of Theoretical Physics vol 102no 2 pp 253ndash272 1999
[66] R Ferraro and F Fiorini ldquoNon-trivial frames for 119891(119879) theoriesof gravity and beyondrdquo Physics Letters B vol 702 no 1 pp 75ndash80 2011
[67] R Ferraro and F Fiorini ldquoSpherically symmetric static space-times in vacuum f(T) gravityrdquo Physical ReviewD vol 84 ArticleID 083518 2011
[68] N Tamanini and C G Boehmer ldquoGood and bad tetrads in f(T)gravityrdquo Physical Review D vol 86 Article ID 044009 2012
[69] R Ferraro and F Fiorini ldquoRemnant group of local Lorentztransformations in f (T) theoriesrdquo Physics Letters D vol 91 no6 Article ID 064019 10 pages 2015
[70] R Ferraro and F Fiorini ldquoKerr geometry in f (T) gravityrdquo TheEuropean Physical Journal C vol 75 article 77 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
