Research Article Einstein Geometrization …downloads.hindawi.com › journals › amp › 2016 ›...
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Research Article Einstein Geometrization Philosophy and Differential Identities in PAP-Geometry M. I. Wanas, 1,2 Nabil L. Youssef, 2,3 W. El Hanafy, 2,4 and S. N. Osman 1,2 1 Astronomy Department, Faculty of Science, Cairo University, Giza, Egypt 2 Egyptian Relativity Group, Cairo University, Giza 12613, Egypt 3 Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt 4 Centre for eoretical Physics, e British University in Egypt, P.O. Box 43, Cairo 11837, Egypt Correspondence should be addressed to M. I. Wanas; [email protected] Received 8 May 2016; Accepted 11 October 2016 Academic Editor: Manuel De Le´ on Copyright © 2016 M. I. Wanas et al. 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 importance of Einstein’s geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. e role of differential identities in this philosophy is clarified. e use of Bianchi identity to write the field equations of GR is shown. Another similar identity in the absolute parallelism geometry is given. A more general differential identity in the parameterized absolute parallelism geometry is derived. Comparison and interrelationships between the above mentioned identities and their role in constructing field theories are discussed. 1. Introduction In the second decade of the twentieth century, Einstein constructed a successful theory for gravity, the general theory of relativity (GR) [1]. Although an action principle has been used aſterwards to derive the equations of the theory, Einstein has used the geometrization philosophy in constructing his theory. Among other principles of this philosophy is that laws of nature are just differential identities in the chosen geometry. e geometry used by Einstein at that time was Riemannian geometry, in which second Bianchi identity is a differential one. is identity can be written in its contracted form as ⋅ ; ≡ 0, (1) where is the Einstein tensor and the semicolon denotes the covariant differentiation using Christoffel symbols (the “dot” in the leſt hand side of (1) means that when we lower the upper index, it occupies the place of the “dot”). Einstein has considered identity (1) as a geometric representation of the law of conservation of matter and energy which is written as ⋅ , = 0, (2) where is a symmetric second-order tensor describing the material-energy distribution. e comma in (2) denotes partial differentiation with respect to . In order for (2) to represent a tensor, Einstein has replaced this comma by a semicolon and wrote the field equations of his theory as ] = − ] , (3) where is a conversion constant. is shows the importance of differential identities in constructing GR. In the third decade of the twentieth century, Einstein [2, 3] used another type of geometry, absolute parallelism (AP-) geometry, in his attempt to construct a field theory uni- fying gravity and electromagnetism. It is characterized by a nonsymmetric linear connection, the Weitzenb¨ ock connec- tion, with vanishing curvature. Aſter long discussions and correspondence between A. Einstein and E. Cartan [4] for more than three years, Einstein stopped using AP-geometry claiming that his attempt is not a successful one. is geom- etry has been neglected for about two decades until McCrea and Mikhail have reconsidered it to treat continuous creation of matter [5] and to build a steady state world model [6]. In the sixties of the past century, Dolan and McCrea (1963, private communication to the first author in 1973) have Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2016, Article ID 1037849, 8 pages http://dx.doi.org/10.1155/2016/1037849
Transcript of Research Article Einstein Geometrization …downloads.hindawi.com › journals › amp › 2016 ›...
Research ArticleEinstein Geometrization Philosophy and Differential Identitiesin PAP-Geometry
M I Wanas12 Nabil L Youssef23 W El Hanafy24 and S N Osman12
1Astronomy Department Faculty of Science Cairo University Giza Egypt2Egyptian Relativity Group Cairo University Giza 12613 Egypt3Mathematics Department Faculty of Science Cairo University Giza Egypt4Centre for Theoretical Physics The British University in Egypt PO Box 43 Cairo 11837 Egypt
Correspondence should be addressed to M I Wanas miwanasscicuedueg
Received 8 May 2016 Accepted 11 October 2016
Academic Editor Manuel De Leon
Copyright copy 2016 M I Wanas et al This 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
The importance of Einsteinrsquos geometrization philosophy as an alternative to the least action principle in constructing generalrelativity (GR) is illuminated The role of differential identities in this philosophy is clarified The use of Bianchi identity to writethe field equations ofGR is shownAnother similar identity in the absolute parallelism geometry is given Amore general differentialidentity in the parameterized absolute parallelism geometry is derived Comparison and interrelationships between the abovementioned identities and their role in constructing field theories are discussed
1 Introduction
In the second decade of the twentieth century Einsteinconstructed a successful theory for gravity the general theoryof relativity (GR) [1] Although an action principle has beenused afterwards to derive the equations of the theory Einsteinhas used the geometrization philosophy in constructing histheory Among other principles of this philosophy is thatlaws of nature are just differential identities in the chosengeometry The geometry used by Einstein at that time wasRiemannian geometry in which second Bianchi identity is adifferential one This identity can be written in its contractedform as 119866120572sdot 120573120572 equiv 0 (1)
where 119866120572120573 is the Einstein tensor and the semicolon denotesthe covariant differentiation using Christoffel symbols (theldquodotrdquo in the left hand side of (1) means that when we lowerthe upper index it occupies the place of the ldquodotrdquo) Einsteinhas considered identity (1) as a geometric representation ofthe law of conservation of matter and energy which is writtenas 119879120572sdot 120573120572 = 0 (2)
where 119879120572120573 is a symmetric second-order tensor describingthe material-energy distribution The comma in (2) denotespartial differentiation with respect to 119909120572 In order for (2) torepresent a tensor Einstein has replaced this comma by asemicolon and wrote the field equations of his theory as119866120583] = minus120581119879120583] (3)
where 120581 is a conversion constant This shows the importanceof differential identities in constructing GR
In the third decade of the twentieth century Einstein[2 3] used another type of geometry absolute parallelism(AP-) geometry in his attempt to construct a field theory uni-fying gravity and electromagnetism It is characterized by anonsymmetric linear connection the Weitzenbock connec-tion with vanishing curvature After long discussions andcorrespondence between A Einstein and E Cartan [4] formore than three years Einstein stopped using AP-geometryclaiming that his attempt is not a successful one This geom-etry has been neglected for about two decades until McCreaandMikhail have reconsidered it to treat continuous creationof matter [5] and to build a steady state world model [6]
In the sixties of the past century Dolan and McCrea(1963 private communication to the first author in 1973) have
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 1037849 8 pageshttpdxdoiorg10115520161037849
2 Advances in Mathematical Physics
developed a variational method to get fundamental identitiesin Riemannian geometry without using an action principleUnfortunately this method is not published although it isshown to be of importance when applied in geometries otherthan the Riemannian one
In the seventies of the twentieth century two importantresults have been obtained by developing AP-geometry ina certain direction [7 8] The first is the use of the dualconnection (the transposed Weitzenbock connection) whichhas simultaneously nonvanishing torsion and nonvanishingcurvature The second result is a differential identity inAP-geometry obtained by developing the Dolan-McCreamethod to suit AP-space with its dual connection Theidentity obtained can be written in the form119864120583]|120583
minus
equiv 0 (4)
where the stroke ldquo|rdquo and the ldquominusrdquo sign are used to denotecovariant differentiation using the dual connection and119864120583sdot ] isa second-order nonsymmetric tensor defined in the contextof AP-geometry by [7]
120582119864120583sdot ] def= 120575L120575120582120583119894
120582]119894
(5)
where 120575L120575120582120583119894
is theHamiltonian derivative of the Lagrangi-
an density L with respect to the building blocks (BB) 120582120583119894
of
AP-geometry and 120582 is the determinant of (120582119894
120583) (more detailsabout AP-geometry are given in the next section)
It is to be noted that identity (4) is a generalization of theBianchi identity (1) to AP-geometry Einstein tensor 119866120572sdot 120573 isuniquely defined in Riemannian geometry while 119864120572sdot 120573 is notuniquely defined in the AP-geometry The latter depends onthe choice of the Lagrangian densityLwhichhasmany formsin AP-geometry Several field theories have been constructed[7 9 10] using the geometrization philosophy and theidentity (4) Many successful applications have been obtainedwithin these theories [11ndash17]
In the last decade of the twentieth century anotherimportant result has been obtained as a consequence ofanother development in AP-geometry It has been discoveredthat AP-geometry admits a hidden parameter [18] It was theBazanski approach [19] that when applied to AP-geometryhas led to the above mentioned hidden parameter Thegeneralization of this hidden parameter has given rise toa modified version of AP-geometry called parameterizedabsolute parallelism (PAP-) geometry [20 21]
In the past ten years or so AP-geometry has gained a lotof attention in constructing field equations for what is called119891(119879) theories and their applications (cf [22ndash24]) The scalar119879 is a torsion scalar characterizing the teleparallel equivalentof GR It has been shown that many other scalars can bedefined in AP-geometry and its different versions Thesescalars have been used to construct several field theories withGR limits [7 9 10 25ndash27]
It is the aim of the present work to find out differentialidentities in PAP-geometry andto study their relations to
similar identities in AP-geometry (4) and in Riemanniangeometry (1) For this reason we give a brief review of PAP-geometry in the next section In Section 3 we apply a modi-fied version of theDolan-McCrea variationalmethod in PAP-geometry to get the general form of the identities admittedIn Section 4 we present the interrelationships between theidentities in Riemannian geometry AP-geometry and PAP-geometry We discuss the results obtained and give someconcluding remarks in Section 5
2 Brief Review of PAP-Geometry
In the present sectionwe are going to give basicmathematicalmachinery and formulae of PAP-geometry necessary for thepresent work For more details the reader is referred to [2021] and the references therein
As mentioned in Section 1 it has been shown [18] thatthe conventional AP-geometry admits a hidden jumpingparameter This parameter has been discovered when usingthe Bazanski approach [19] to derive path and path deviationequations The importance of this parameter is that amongother things its value jumps by a step of one half whichhas been tempted to show its relation to some quantumphenomena when used in application This parameter hasno explicit appearance in the conventional AP-geometryThis motivates authors to give the parameter an explicitappearance in a modified version of AP-geometry known inthe literature as PAP-geometry
An AP-space is a pair (119872 120582119894) where119872 is a smoothmani-
fold and 120582119894are 119899 independent global vector fields 120582
119894(119894 = 1 119899) on119872 Formore details see [28ndash32] AnAP-space admitted
at least four natural (ldquonaturalrdquo means that the geometricobject under consideration is constructed from the buildingblocks 120582
119894only) linear connections the Weitzenbock connec-
tion Γ120572120583] def= 120582119894
120572120582120583]119894
(6)
its dual (or transposed) connection
Γ120572120583] def= Γ120572]120583 (7)
the symmetric part of (6)
Γ120572(120583]) def= 12 (Γ120572120583] + Γ120572]120583) (8)
and the Levi-Civita connection
120572120583 ] def= 12119892120572120590 (119892120583120590] + 119892]120590120583 minus 119892120583]120590) (9)
associated with the metric 119892120583] def= 120582120583119894
120582]119894
It is to be noted that all these linear connections have non-vanishing curvature except for (6) Among these connectionsthe only connection that has simultaneously nonvanishingtorsion and nonvanishing curvature is the dual connection(7) In order to generalize these connections and to give anexplicit appearance of the above mentioned jumping param-
Advances in Mathematical Physics 3
eter these connections have been linearly combined to givethe object (1198861 1198862 1198863 and 1198864 are parameters)
nabla120572120583] = 1198861Γ120572120583] + 1198862Γ120572120583] + 1198863Γ120572(120583]) + 1198864 120572120583 ] (10)
Imposing the general coordinate transformation of a linearconnection on (10) using (6)ndash(9) the above 4 parametersreduce to one parameter 119887 which leads to
nabla120572120583] = 120572120583 ] + 119887120574120572120583] (11)
where 120574120572120583] is the contortion of AP-geometry Connection (11)will be called the parameterized canonical connection It isnonsymmetric with nonvanishing curvature that is it is ofRiemann-Cartan type Moreover it is a metric connection[20]
The parameter 119887 is a dimensionless parameter For somegeometric and physical reasons [21] this parameter is sug-gested to have the form
119887 = 1198732 120572120574 (12)
where 119873 is a natural number taking the value 0 1 2 120572 is the fine structure constant and 120574 is a dimensionlessparameter to be fixed by experiment or observation for thesystem under consideration The explicit appearance of theparameter 119887 in (11) gives rise to the following advantages toPAP-geometry
(1) In the case of 119887 = 1 (11) reduces to the Weitzenbockconnection (6) of the conventional AP-geometry
(2) In the case of 119887 = 0 connection (11) reduces to theLevi-Civita connection (9)
(3) Between the limits 119887 = 0 and 119887 = 1 there is a discretespectrum of spaces (due to the presence of 1198732 in(12)) each of which has simultaneously nonvanishingcurvature and torsion
In what follows we will decorate the tensors containingthe parameter 119887 by a star Also these tensors will retain thesame name as in the conventional AP-geometry precededby the adjective ldquoparameterizedrdquo For example the parame-terized contortion is given by
lowast120574120572120583] = 119887120574120572120583] (13)
The torsion of the parameterized connection (11) is given bylowastΛ120572120583] = 119887 (120574120572120583] minus 120574120572]120583) = 119887Λ120572120583] (14)
where Λ120572120583] is the torsion of the AP-geometry Also theparameterized basic form can be obtained by contraction of(13) or (14)
lowast119862120583 def= lowastΛ120572120583120572 = lowast120574120572120583120572 = 119887119862120583 (15)
It is to be noted that starred objects have the same symmetryproperties as the corresponding unstarred objects defined inthe conventional AP-geometry
Now since the parameterized canonical connection nabla120572120583]is nonsymmetric one can define two other linear connec-tions the parameterized dual (transposed) connection
nabla120572120583] def= nabla120572]120583 (16)
and the parameterized symmetric connection
nabla120572(120583]) def= 12 (nabla120572120583] + nabla120572]120583) (17)
Consequently using (11) (16) and (17) we can define respec-tively the covariant derivatives
119860120583]+
def= 119860120583] minus 119860120572nabla120572sdot 120583]119860120583]minus
def= 119860120583] minus 119860120572nabla120572sdot 120583]119860120583] def= 119860120583] minus 119860120572nabla120572sdot (120583])
(18)
for any arbitrary vector 119860120583It is of importance to note that the BB of PAP-geometry
are the same as those of AP-geometry In other words theparameter 119887 has no effect on the BB of the geometry
3 Dolan-McCrea Variational Method inPAP-Geometry
We are going to apply the Dolan-McCrea variationalmethod to obtain differential identities in PAP-geometryThis method is an alternative to the least action method Itwas originally suggested in 1963 to derive Bianchi and otheridentities in the context of Riemannian geometry It has beengeneralized [7] to AP-geometry and used to derive identity(4) Here we generalize this method to PAP-geometry givingsome details since it is not widely known
Since as stated above the BB of PAP-geometry are thesame as those of AP-geometry let us define the Lagrangianfunction
lowast119871 = lowast119871(120582120583119894
120582120583]119894
120582120583]120590119894
) (19)
constructed from the BB of PAP 120582120583119894
and its first and second
derivatives Consequently we can write the scalar density asfollows
lowast
L0 = lowast
L0 (120582120583119894
120582120583]119894
120582120583]120590119894
) = 120582 lowast119871 (20)
wherelowast
L0 is a scalar lagrangian density and 120582 fl det(120582119894
120583) = 0Now let us assume that the following integral defined oversome arbitrary 119899-dimensional domainΩ
lowast1198680 def= intΩ
lowast
L0 (119909) 119889119899119909 (21)
is invariant under the infinitesimal transformation120582119894(119909) 997888rarr 120582
119894(119909) + 120598ℎ
119894(119909) (22)
4 Advances in Mathematical Physics
where 120598 is an infinitesimal parameter and ℎ119894are 119899 1-forms
defined on119872 Let Σ be the (119899minus1)-space such that the 119899-spaceregionΩ is enclosed within Σ Thus
lowast119868120598 def= intΩ
lowast
L120598119889119899119909 (23)
where119889119899119909 = 11988911990911198891199092 sdot sdot sdot 119889119909119899 (24)
lowast
L120598 = lowast
L(120582120583119894
+ 120598ℎ120583119894
120582120583]119894
+ 120598ℎ120583]119894
120582120583]120590119894
+ 120598ℎ120583]120590119894
) (25)
Assuming that ℎ120583119894
and ℎ120583]119894
vanish at all points of Σ thenlowast119868120598 minus lowast1198680 = int
Ω( lowastL120598 minus lowastL0) 119889119899119909 (26)
wherelowast1198680 is the value of lowast119868120598 at 120598 = 0 Now lowastL120598 can be written
using Taylor expansion in the form
lowast
L120598 = lowast
L0 + 120597 lowastL0120597120582120583119894
120598ℎ120583119894
+ 120597 lowastL0120597120582120583]119894
120598ℎ120583]119894
+ 120597 lowastL0120597120582120583]120590119894
120598ℎ120583]120590119894
+ 119874 (1205982) (27)
Using (26) and (27) and by integration by parts it can beshown after some manipulation that (26) can be written as
lowast119868120598 minus lowast1198680 = 120598intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 + 119874 (1205982) (28)
where 120575 lowast119871120575120582120583119894
is the Hamiltonian derivative defined by
120575 lowast119871120575120582120573119894
def= 1120582 [[[120597 lowastL0120597120582120573119894
minus 120597120597119909120574 ( 120597 lowastL0120597120582120573120574119894
)
+ 1205972120597119909120574120597119909120590 ( 120597 lowastL0120597120582120573120574120590119894
)]]] (29)
Consider a new set of coordinates 119909 related to the 119909-coordinate system by the transformation119909120583 = 119909120583 minus 120598119911120583 (119909) (30)
where 119911120583 is a vector field vanishing at all points of Σ and 120598 isindependent of 119909 We can write120597119909120583120597119909] = 120575120583] = 120597119909120583120597119909] minus 120598120597119911120583120597119909120590 120597119909120590120597119909] (31)
120597119909120583120597119909] = 120575120583] + 120598119911120583120590 120597119909120590120597119909] (32)
Since 119911120583 is a vector its transformation law is
119911120583 (119909) = 120597119909120583120597119909] 119911] (119909) (33)
Substituting from (32) into the above equation we get
119911120583 (119909) = 120575120583] 119911] (119909) + 120598119911] (119909) 119911120583120590 120597119909120590120597119909] (34)
Substituting from (32) again into the above equation gives
119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583120590 (120575120590] + 120598119911120590120572 120597119909120572120597119909] ) (35)
Finally this equation can be written as119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583] + 119874 (1205982) (36)
Substituting from (36) into (30) we get119909120583 = 119909120583 + 120598119911120583 (119909) + 119874 (1205982) (37)
Consequently we can write120597119909120583120597119909] = 120575120583] minus 120598119911120583]120597119909120583120597119909] = 120575120583] + 120598119911120583] + 119874 (1205982) (38)
Then the Jacobian expressions of the above matrices can begiven as 120597119909120597119909 = 1 minus 120598119911120572120572 + 119874 (1205982) 120597119909120597119909 = 1 + 120598119911120572120572 + 119874 (1205982)
(39)
since the difference between 119911120583(119909) and 119911120583(119909) would be oforder 1205982 as shown by (36)
Now we express 120582120583119894
(119909) in terms of 120582120583119894
(119909) in two different
ways(i) Using the vector transformation law
120582120583119894
(119909) = 120597119909]120597119909120583 120582]119894 (119909)= 120575]120583120582]119894
(119909) + 120598120582]119894
(119909) 119911] 120583 + 119874 (1205982)= 120582120583119894
(119909) + 120598120582]119894
119911] 120583 + 119874 (1205982)(40)
(ii) Using Taylor expansion120582120583119894
(119909) = 120582120583119894
(119909 minus 120598119911) = 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982)= 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982) (41)
Wehave dropped the bar in the second termon the right handside of this equation as the difference would be of order 1205982Comparing (40) and (41) and using (22) we get120582120583
119894
(119909) minus 120582120583119894
(119909) = 120598ℎ120583119894
+ 119874 (1205982) (42)
where ℎ120583119894
= 120582]119894
119911] 120583 + 120582120583]119894
119911] = 120582]119894
119911]120583minus
+ 120582120583]+
119894
119911] (43)
and the double stroke operator ldquordquo is defined by (18)
Advances in Mathematical Physics 5
Now we treat the Lagrangian density in the two differentways mentioned above
(a) Using Taylor expansionlowast
L (119909) = lowast
L (119909 minus 120598119911)= lowast
L (119909) minus 120598 lowastL (119909)120572 119911120572 + 119874 (1205982) (44)
also we have dropped the bar in the second term on the righthand side of this equation as the difference would be of order1205982
The Lagrangian densitylowast
L(119909) in (44) meanslowast
L (119909) equiv lowast
L(120582120583119894
(119909) 120582120583]119894
(119909) 120582120583]120590119894
(119909)) (45)
(b) Using scalar transformationlowast119871 (119909) = lowast119871 (119909) 120582 lowast119871 (119909) = 120582 lowast119871 (119909) = 120582 lowast119871 (119909) 120597119909120597119909
(46)
then using (20) and (39) we getlowast
L (119909) = lowast
L (119909) + 120598 lowastL (119909) 119911120572120572 + 119874 (1205982) (47)
Comparing (44) and (47) we havelowast
L (119909) minus lowastL (119909) = 120598 ( lowastL (119909) 119911120572)120572+ 119874 (1205982) (48)
By (46) we get
intΩ
lowast119871119889119899119909 minus intΩ
lowast119871119889119899119909 = 0 (49)
intΩ( lowastL (119909) minus lowastL (119909)) 119889119899119909= intΩ[120598 ( lowastL (119909) 119911120572)
120572+ 119874 (1205982)] 119889119899119909 (50)
Applying Gaussrsquos theorem to convert the volume integral onΩ (119899-space) to a surface integral over Σ ((119899 minus 1)-space) theintegral vanishes as 119911120572 vanishes at all points of Σ Now wecan writeint
Ω( lowastL (119909) 119911120572)
120572119889119899119909 = int
Σ
lowast
L (119909) 119911120572119899120572119889119899minus1119909= ∮Σ
lowast119871 (119909) 119911120572119899120572119889Σ equiv 0 (51)
where 119899120572 is a unit vector normal to ΣSo (50) shows that
intΩ[ lowastL (119909) minus lowastL (119909)] 119889119899119909 = 119874 (1205982) (52)
Substituting from (42) into (45) we getlowast
L (119909) = lowast
L(120582120583119894
(119909) + 120598ℎ120583119894
+ 119874 (1205982) 120582120583]119894
(119909) + 120598ℎ120583]119894
+ 119874 (1205982) 120582120583]120590119894
(119909) + 120598ℎ120583]120590119894
+ 119874 (1205982)) (53)
Using (25)lowast
L (119909) = lowast
L120598 (119909) + 119874 (1205982) (54)
then we can write (52) as
intΩ[ lowastL120598 (119909) minus lowastL0 (119909)] 119889119899119909 = 119874 (1205982) (55)
Comparing (26) (28) and (55) implies that
intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 equiv 0 (56)
Substituting from (43) into (56) we write
intΩ120582[[[
120575 lowast119871120575120582120583119894
120582]119894
119911]120583minus
+ 120575 lowast119871120575120582120583119894
120582120583]+
119894
119911]]]]119889119899119909 equiv 0 (57)
Let
lowast119864120583] def= 120575 lowast119871120575120582120583119894
120582]119894
= minuslowast119869119894
120583120582]119894
lowast119869119894
120583 def= minus 120575 lowast119871120575120582120583119894
(58)
The integral identity (57) then becomes
intΩ120582[[lowast119864120583]119911]120583
minus
minus lowast119869119894
120583120582120583]+
119894
119911]]]119889119899119909 equiv 0 (59)
Now the integral of the first term can be treated as follows
intΩ120582 lowast119864120583]119911]minus120583119889119899119909 = int
Ω120582 ( lowast119864120583]119911])
120583minus
119889119899119909minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= intΩ(120582 lowast119864120583]119911])
120583119889119899119909
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= ∮Σ120582 lowast119864120583]119911]119899120583119889Σ
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= minusintΩ120582 lowast119864120583]120583
minus
119911]119889119899119909
(60)
Thus the integral identity (59) reads
intΩ120582( lowast119864120583]120583
minus
+ 120582120583]+
119894
lowast119869119894
120583)119911]119889119899119909 equiv 0 (61)
6 Advances in Mathematical Physics
since 120582 = 0 and 119911] is an arbitrary vector field Using the fun-damental lemma of the calculus of variation [33] the expres-sion in the brackets vanishes identically Hence we have
lowast119864120583]120583minus
+ 120582120583]+
119894
lowast119869119894
120583 equiv 0 (62)
Using (11) (18) and (58) after some manipulations identity(62) can be written as
lowast119864120583]120583minus
= 119887 (1 minus 119887) 119864120583120572120574120572120583] (63)
This is the differential identity characterizing PAP-geometry
4 Relation between Differential Identities
The three differential identities (1) of Riemannian geometry(4) of AP-geometry and (63) of PAP-geometry are to berelated This idea comes from the fact that the general linearconnection (11) reduces to the Weitzenbock connection (6)of the AP-space for 119887 = 1 and to the Live-Civita connection(9) of the Riemannian space for 119887 = 0 Note that thethree identities mentioned are derivatives in which linearconnections are used The following two results relate theseidentities
Theorem 1 Iflowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582
119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]120583 + 119887 lowast119864120583120572120574120583120572] (64)
Consequently if 119887 = 0 (the Riemannian case) orlowast119864120583] is symmet-
ric one getslowast119864120583]120583minus
= lowast119864120583]120583 (65)
Proof Using the definition of the parameterized dual connec-tion nabla120572120583] we have
lowast119864120583]120583minus
= lowast119864120583]120583 minus lowast119864120583120572nabla120572120583] + lowast119864120572]nabla120583120583120572= lowast119864120583]120583 minus lowast119864120583120572 120572120583 ]
minus 119887 lowast119864120583120572120574120572120583]+ lowast119864120572] 120583120583 120572 + 119887 lowast119864120572]120574120583120583120572
(66)
As 120574120583]120590 is skew-symmetric in the first pair of indices the lastterm on the right vanishes Hence we get
lowast119864120583]120583minus
= lowast119864120583]120583 minus 119887 lowast119864120583120572120574120572120583] (67)
Now consider the last term on the right of the above identitylowast119864120583120572120574120572120583] = lowast119864120583120572120575120572120598 120574120598120583] = lowast119864120583120572119892120572120573119892120598120573120574120598120583] = lowast119864120583120573120574120573120583]
= minus lowast119864120583120573120574120583120573] (68)
from which (64) follows
Finally it is clear that if 119887 = 0 then lowast119864120583]120583minus
= 119864120583]120583 On the
other hand iflowast119864120583] is symmetric then the term
lowast119864120583120572120574120583120572] vanish-es since 120574120583120572] is skew-symmetric in 120583 and 120572Theorem2 If
lowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]|120583minus
+ (119887 minus 1) lowast119864120583120572120574120583120572] (69)
Consequently if 119887 = 1 (the AP case) orlowast119864120583] is symmetric one
getslowast119864120583]120583minus
= lowast119864120583]|120583minus
(70)
The above two theorems imply the following result
Corollary 3 (a) For a symmetric tensor119864120583] one has from (64)and (69)
lowast119864120583]120583minus
= lowast119864120583]|120583minus
= lowast119864120583]120583 (71)
(b) In the Riemannian case the tensor119864120583] defined by (5) (usingthe Ricci scalar in the Lagrangian function) coincides with theEinstein tensor then 119864120583]120583 equiv 0 (72)
and consequently for any symmetric tensor defined by (5) onehas 119864120583]|120583
minus
equiv 0lowast119864120583]120583minus
equiv 0 (73)
in AP- and PAP-spaces respectively
5 Discussion and Concluding Remarks
In the present work we have used the Dolan-McCrea vari-ational method to derive possible differential identities inPAP-geometry The importance of this work for physicalapplications can be discussed in the following points
(1) It is well known that the field equations of GR can beobtained using either one of the following approaches
(a) The Einstein approach in which the geometri-zation philosophy plays the main role in con-structing the field equations of the theory Oneof the principles of this philosophy is that ldquolawsof nature are just differential identities in anappropriate geometryrdquo Einstein has used thisprinciple to write his field equations In otherwords he has used the second Bianchi identityof Riemannian geometry to construct the fieldequations of his theory
(b) Hilbert approach inwhich the standardmethodof theoretical physics the action principle hasbeen used to derive the field equations of thetheory
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
developed a variational method to get fundamental identitiesin Riemannian geometry without using an action principleUnfortunately this method is not published although it isshown to be of importance when applied in geometries otherthan the Riemannian one
In the seventies of the twentieth century two importantresults have been obtained by developing AP-geometry ina certain direction [7 8] The first is the use of the dualconnection (the transposed Weitzenbock connection) whichhas simultaneously nonvanishing torsion and nonvanishingcurvature The second result is a differential identity inAP-geometry obtained by developing the Dolan-McCreamethod to suit AP-space with its dual connection Theidentity obtained can be written in the form119864120583]|120583
minus
equiv 0 (4)
where the stroke ldquo|rdquo and the ldquominusrdquo sign are used to denotecovariant differentiation using the dual connection and119864120583sdot ] isa second-order nonsymmetric tensor defined in the contextof AP-geometry by [7]
120582119864120583sdot ] def= 120575L120575120582120583119894
120582]119894
(5)
where 120575L120575120582120583119894
is theHamiltonian derivative of the Lagrangi-
an density L with respect to the building blocks (BB) 120582120583119894
of
AP-geometry and 120582 is the determinant of (120582119894
120583) (more detailsabout AP-geometry are given in the next section)
It is to be noted that identity (4) is a generalization of theBianchi identity (1) to AP-geometry Einstein tensor 119866120572sdot 120573 isuniquely defined in Riemannian geometry while 119864120572sdot 120573 is notuniquely defined in the AP-geometry The latter depends onthe choice of the Lagrangian densityLwhichhasmany formsin AP-geometry Several field theories have been constructed[7 9 10] using the geometrization philosophy and theidentity (4) Many successful applications have been obtainedwithin these theories [11ndash17]
In the last decade of the twentieth century anotherimportant result has been obtained as a consequence ofanother development in AP-geometry It has been discoveredthat AP-geometry admits a hidden parameter [18] It was theBazanski approach [19] that when applied to AP-geometryhas led to the above mentioned hidden parameter Thegeneralization of this hidden parameter has given rise toa modified version of AP-geometry called parameterizedabsolute parallelism (PAP-) geometry [20 21]
In the past ten years or so AP-geometry has gained a lotof attention in constructing field equations for what is called119891(119879) theories and their applications (cf [22ndash24]) The scalar119879 is a torsion scalar characterizing the teleparallel equivalentof GR It has been shown that many other scalars can bedefined in AP-geometry and its different versions Thesescalars have been used to construct several field theories withGR limits [7 9 10 25ndash27]
It is the aim of the present work to find out differentialidentities in PAP-geometry andto study their relations to
similar identities in AP-geometry (4) and in Riemanniangeometry (1) For this reason we give a brief review of PAP-geometry in the next section In Section 3 we apply a modi-fied version of theDolan-McCrea variationalmethod in PAP-geometry to get the general form of the identities admittedIn Section 4 we present the interrelationships between theidentities in Riemannian geometry AP-geometry and PAP-geometry We discuss the results obtained and give someconcluding remarks in Section 5
2 Brief Review of PAP-Geometry
In the present sectionwe are going to give basicmathematicalmachinery and formulae of PAP-geometry necessary for thepresent work For more details the reader is referred to [2021] and the references therein
As mentioned in Section 1 it has been shown [18] thatthe conventional AP-geometry admits a hidden jumpingparameter This parameter has been discovered when usingthe Bazanski approach [19] to derive path and path deviationequations The importance of this parameter is that amongother things its value jumps by a step of one half whichhas been tempted to show its relation to some quantumphenomena when used in application This parameter hasno explicit appearance in the conventional AP-geometryThis motivates authors to give the parameter an explicitappearance in a modified version of AP-geometry known inthe literature as PAP-geometry
An AP-space is a pair (119872 120582119894) where119872 is a smoothmani-
fold and 120582119894are 119899 independent global vector fields 120582
119894(119894 = 1 119899) on119872 Formore details see [28ndash32] AnAP-space admitted
at least four natural (ldquonaturalrdquo means that the geometricobject under consideration is constructed from the buildingblocks 120582
119894only) linear connections the Weitzenbock connec-
tion Γ120572120583] def= 120582119894
120572120582120583]119894
(6)
its dual (or transposed) connection
Γ120572120583] def= Γ120572]120583 (7)
the symmetric part of (6)
Γ120572(120583]) def= 12 (Γ120572120583] + Γ120572]120583) (8)
and the Levi-Civita connection
120572120583 ] def= 12119892120572120590 (119892120583120590] + 119892]120590120583 minus 119892120583]120590) (9)
associated with the metric 119892120583] def= 120582120583119894
120582]119894
It is to be noted that all these linear connections have non-vanishing curvature except for (6) Among these connectionsthe only connection that has simultaneously nonvanishingtorsion and nonvanishing curvature is the dual connection(7) In order to generalize these connections and to give anexplicit appearance of the above mentioned jumping param-
Advances in Mathematical Physics 3
eter these connections have been linearly combined to givethe object (1198861 1198862 1198863 and 1198864 are parameters)
nabla120572120583] = 1198861Γ120572120583] + 1198862Γ120572120583] + 1198863Γ120572(120583]) + 1198864 120572120583 ] (10)
Imposing the general coordinate transformation of a linearconnection on (10) using (6)ndash(9) the above 4 parametersreduce to one parameter 119887 which leads to
nabla120572120583] = 120572120583 ] + 119887120574120572120583] (11)
where 120574120572120583] is the contortion of AP-geometry Connection (11)will be called the parameterized canonical connection It isnonsymmetric with nonvanishing curvature that is it is ofRiemann-Cartan type Moreover it is a metric connection[20]
The parameter 119887 is a dimensionless parameter For somegeometric and physical reasons [21] this parameter is sug-gested to have the form
119887 = 1198732 120572120574 (12)
where 119873 is a natural number taking the value 0 1 2 120572 is the fine structure constant and 120574 is a dimensionlessparameter to be fixed by experiment or observation for thesystem under consideration The explicit appearance of theparameter 119887 in (11) gives rise to the following advantages toPAP-geometry
(1) In the case of 119887 = 1 (11) reduces to the Weitzenbockconnection (6) of the conventional AP-geometry
(2) In the case of 119887 = 0 connection (11) reduces to theLevi-Civita connection (9)
(3) Between the limits 119887 = 0 and 119887 = 1 there is a discretespectrum of spaces (due to the presence of 1198732 in(12)) each of which has simultaneously nonvanishingcurvature and torsion
In what follows we will decorate the tensors containingthe parameter 119887 by a star Also these tensors will retain thesame name as in the conventional AP-geometry precededby the adjective ldquoparameterizedrdquo For example the parame-terized contortion is given by
lowast120574120572120583] = 119887120574120572120583] (13)
The torsion of the parameterized connection (11) is given bylowastΛ120572120583] = 119887 (120574120572120583] minus 120574120572]120583) = 119887Λ120572120583] (14)
where Λ120572120583] is the torsion of the AP-geometry Also theparameterized basic form can be obtained by contraction of(13) or (14)
lowast119862120583 def= lowastΛ120572120583120572 = lowast120574120572120583120572 = 119887119862120583 (15)
It is to be noted that starred objects have the same symmetryproperties as the corresponding unstarred objects defined inthe conventional AP-geometry
Now since the parameterized canonical connection nabla120572120583]is nonsymmetric one can define two other linear connec-tions the parameterized dual (transposed) connection
nabla120572120583] def= nabla120572]120583 (16)
and the parameterized symmetric connection
nabla120572(120583]) def= 12 (nabla120572120583] + nabla120572]120583) (17)
Consequently using (11) (16) and (17) we can define respec-tively the covariant derivatives
119860120583]+
def= 119860120583] minus 119860120572nabla120572sdot 120583]119860120583]minus
def= 119860120583] minus 119860120572nabla120572sdot 120583]119860120583] def= 119860120583] minus 119860120572nabla120572sdot (120583])
(18)
for any arbitrary vector 119860120583It is of importance to note that the BB of PAP-geometry
are the same as those of AP-geometry In other words theparameter 119887 has no effect on the BB of the geometry
3 Dolan-McCrea Variational Method inPAP-Geometry
We are going to apply the Dolan-McCrea variationalmethod to obtain differential identities in PAP-geometryThis method is an alternative to the least action method Itwas originally suggested in 1963 to derive Bianchi and otheridentities in the context of Riemannian geometry It has beengeneralized [7] to AP-geometry and used to derive identity(4) Here we generalize this method to PAP-geometry givingsome details since it is not widely known
Since as stated above the BB of PAP-geometry are thesame as those of AP-geometry let us define the Lagrangianfunction
lowast119871 = lowast119871(120582120583119894
120582120583]119894
120582120583]120590119894
) (19)
constructed from the BB of PAP 120582120583119894
and its first and second
derivatives Consequently we can write the scalar density asfollows
lowast
L0 = lowast
L0 (120582120583119894
120582120583]119894
120582120583]120590119894
) = 120582 lowast119871 (20)
wherelowast
L0 is a scalar lagrangian density and 120582 fl det(120582119894
120583) = 0Now let us assume that the following integral defined oversome arbitrary 119899-dimensional domainΩ
lowast1198680 def= intΩ
lowast
L0 (119909) 119889119899119909 (21)
is invariant under the infinitesimal transformation120582119894(119909) 997888rarr 120582
119894(119909) + 120598ℎ
119894(119909) (22)
4 Advances in Mathematical Physics
where 120598 is an infinitesimal parameter and ℎ119894are 119899 1-forms
defined on119872 Let Σ be the (119899minus1)-space such that the 119899-spaceregionΩ is enclosed within Σ Thus
lowast119868120598 def= intΩ
lowast
L120598119889119899119909 (23)
where119889119899119909 = 11988911990911198891199092 sdot sdot sdot 119889119909119899 (24)
lowast
L120598 = lowast
L(120582120583119894
+ 120598ℎ120583119894
120582120583]119894
+ 120598ℎ120583]119894
120582120583]120590119894
+ 120598ℎ120583]120590119894
) (25)
Assuming that ℎ120583119894
and ℎ120583]119894
vanish at all points of Σ thenlowast119868120598 minus lowast1198680 = int
Ω( lowastL120598 minus lowastL0) 119889119899119909 (26)
wherelowast1198680 is the value of lowast119868120598 at 120598 = 0 Now lowastL120598 can be written
using Taylor expansion in the form
lowast
L120598 = lowast
L0 + 120597 lowastL0120597120582120583119894
120598ℎ120583119894
+ 120597 lowastL0120597120582120583]119894
120598ℎ120583]119894
+ 120597 lowastL0120597120582120583]120590119894
120598ℎ120583]120590119894
+ 119874 (1205982) (27)
Using (26) and (27) and by integration by parts it can beshown after some manipulation that (26) can be written as
lowast119868120598 minus lowast1198680 = 120598intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 + 119874 (1205982) (28)
where 120575 lowast119871120575120582120583119894
is the Hamiltonian derivative defined by
120575 lowast119871120575120582120573119894
def= 1120582 [[[120597 lowastL0120597120582120573119894
minus 120597120597119909120574 ( 120597 lowastL0120597120582120573120574119894
)
+ 1205972120597119909120574120597119909120590 ( 120597 lowastL0120597120582120573120574120590119894
)]]] (29)
Consider a new set of coordinates 119909 related to the 119909-coordinate system by the transformation119909120583 = 119909120583 minus 120598119911120583 (119909) (30)
where 119911120583 is a vector field vanishing at all points of Σ and 120598 isindependent of 119909 We can write120597119909120583120597119909] = 120575120583] = 120597119909120583120597119909] minus 120598120597119911120583120597119909120590 120597119909120590120597119909] (31)
120597119909120583120597119909] = 120575120583] + 120598119911120583120590 120597119909120590120597119909] (32)
Since 119911120583 is a vector its transformation law is
119911120583 (119909) = 120597119909120583120597119909] 119911] (119909) (33)
Substituting from (32) into the above equation we get
119911120583 (119909) = 120575120583] 119911] (119909) + 120598119911] (119909) 119911120583120590 120597119909120590120597119909] (34)
Substituting from (32) again into the above equation gives
119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583120590 (120575120590] + 120598119911120590120572 120597119909120572120597119909] ) (35)
Finally this equation can be written as119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583] + 119874 (1205982) (36)
Substituting from (36) into (30) we get119909120583 = 119909120583 + 120598119911120583 (119909) + 119874 (1205982) (37)
Consequently we can write120597119909120583120597119909] = 120575120583] minus 120598119911120583]120597119909120583120597119909] = 120575120583] + 120598119911120583] + 119874 (1205982) (38)
Then the Jacobian expressions of the above matrices can begiven as 120597119909120597119909 = 1 minus 120598119911120572120572 + 119874 (1205982) 120597119909120597119909 = 1 + 120598119911120572120572 + 119874 (1205982)
(39)
since the difference between 119911120583(119909) and 119911120583(119909) would be oforder 1205982 as shown by (36)
Now we express 120582120583119894
(119909) in terms of 120582120583119894
(119909) in two different
ways(i) Using the vector transformation law
120582120583119894
(119909) = 120597119909]120597119909120583 120582]119894 (119909)= 120575]120583120582]119894
(119909) + 120598120582]119894
(119909) 119911] 120583 + 119874 (1205982)= 120582120583119894
(119909) + 120598120582]119894
119911] 120583 + 119874 (1205982)(40)
(ii) Using Taylor expansion120582120583119894
(119909) = 120582120583119894
(119909 minus 120598119911) = 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982)= 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982) (41)
Wehave dropped the bar in the second termon the right handside of this equation as the difference would be of order 1205982Comparing (40) and (41) and using (22) we get120582120583
119894
(119909) minus 120582120583119894
(119909) = 120598ℎ120583119894
+ 119874 (1205982) (42)
where ℎ120583119894
= 120582]119894
119911] 120583 + 120582120583]119894
119911] = 120582]119894
119911]120583minus
+ 120582120583]+
119894
119911] (43)
and the double stroke operator ldquordquo is defined by (18)
Advances in Mathematical Physics 5
Now we treat the Lagrangian density in the two differentways mentioned above
(a) Using Taylor expansionlowast
L (119909) = lowast
L (119909 minus 120598119911)= lowast
L (119909) minus 120598 lowastL (119909)120572 119911120572 + 119874 (1205982) (44)
also we have dropped the bar in the second term on the righthand side of this equation as the difference would be of order1205982
The Lagrangian densitylowast
L(119909) in (44) meanslowast
L (119909) equiv lowast
L(120582120583119894
(119909) 120582120583]119894
(119909) 120582120583]120590119894
(119909)) (45)
(b) Using scalar transformationlowast119871 (119909) = lowast119871 (119909) 120582 lowast119871 (119909) = 120582 lowast119871 (119909) = 120582 lowast119871 (119909) 120597119909120597119909
(46)
then using (20) and (39) we getlowast
L (119909) = lowast
L (119909) + 120598 lowastL (119909) 119911120572120572 + 119874 (1205982) (47)
Comparing (44) and (47) we havelowast
L (119909) minus lowastL (119909) = 120598 ( lowastL (119909) 119911120572)120572+ 119874 (1205982) (48)
By (46) we get
intΩ
lowast119871119889119899119909 minus intΩ
lowast119871119889119899119909 = 0 (49)
intΩ( lowastL (119909) minus lowastL (119909)) 119889119899119909= intΩ[120598 ( lowastL (119909) 119911120572)
120572+ 119874 (1205982)] 119889119899119909 (50)
Applying Gaussrsquos theorem to convert the volume integral onΩ (119899-space) to a surface integral over Σ ((119899 minus 1)-space) theintegral vanishes as 119911120572 vanishes at all points of Σ Now wecan writeint
Ω( lowastL (119909) 119911120572)
120572119889119899119909 = int
Σ
lowast
L (119909) 119911120572119899120572119889119899minus1119909= ∮Σ
lowast119871 (119909) 119911120572119899120572119889Σ equiv 0 (51)
where 119899120572 is a unit vector normal to ΣSo (50) shows that
intΩ[ lowastL (119909) minus lowastL (119909)] 119889119899119909 = 119874 (1205982) (52)
Substituting from (42) into (45) we getlowast
L (119909) = lowast
L(120582120583119894
(119909) + 120598ℎ120583119894
+ 119874 (1205982) 120582120583]119894
(119909) + 120598ℎ120583]119894
+ 119874 (1205982) 120582120583]120590119894
(119909) + 120598ℎ120583]120590119894
+ 119874 (1205982)) (53)
Using (25)lowast
L (119909) = lowast
L120598 (119909) + 119874 (1205982) (54)
then we can write (52) as
intΩ[ lowastL120598 (119909) minus lowastL0 (119909)] 119889119899119909 = 119874 (1205982) (55)
Comparing (26) (28) and (55) implies that
intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 equiv 0 (56)
Substituting from (43) into (56) we write
intΩ120582[[[
120575 lowast119871120575120582120583119894
120582]119894
119911]120583minus
+ 120575 lowast119871120575120582120583119894
120582120583]+
119894
119911]]]]119889119899119909 equiv 0 (57)
Let
lowast119864120583] def= 120575 lowast119871120575120582120583119894
120582]119894
= minuslowast119869119894
120583120582]119894
lowast119869119894
120583 def= minus 120575 lowast119871120575120582120583119894
(58)
The integral identity (57) then becomes
intΩ120582[[lowast119864120583]119911]120583
minus
minus lowast119869119894
120583120582120583]+
119894
119911]]]119889119899119909 equiv 0 (59)
Now the integral of the first term can be treated as follows
intΩ120582 lowast119864120583]119911]minus120583119889119899119909 = int
Ω120582 ( lowast119864120583]119911])
120583minus
119889119899119909minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= intΩ(120582 lowast119864120583]119911])
120583119889119899119909
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= ∮Σ120582 lowast119864120583]119911]119899120583119889Σ
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= minusintΩ120582 lowast119864120583]120583
minus
119911]119889119899119909
(60)
Thus the integral identity (59) reads
intΩ120582( lowast119864120583]120583
minus
+ 120582120583]+
119894
lowast119869119894
120583)119911]119889119899119909 equiv 0 (61)
6 Advances in Mathematical Physics
since 120582 = 0 and 119911] is an arbitrary vector field Using the fun-damental lemma of the calculus of variation [33] the expres-sion in the brackets vanishes identically Hence we have
lowast119864120583]120583minus
+ 120582120583]+
119894
lowast119869119894
120583 equiv 0 (62)
Using (11) (18) and (58) after some manipulations identity(62) can be written as
lowast119864120583]120583minus
= 119887 (1 minus 119887) 119864120583120572120574120572120583] (63)
This is the differential identity characterizing PAP-geometry
4 Relation between Differential Identities
The three differential identities (1) of Riemannian geometry(4) of AP-geometry and (63) of PAP-geometry are to berelated This idea comes from the fact that the general linearconnection (11) reduces to the Weitzenbock connection (6)of the AP-space for 119887 = 1 and to the Live-Civita connection(9) of the Riemannian space for 119887 = 0 Note that thethree identities mentioned are derivatives in which linearconnections are used The following two results relate theseidentities
Theorem 1 Iflowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582
119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]120583 + 119887 lowast119864120583120572120574120583120572] (64)
Consequently if 119887 = 0 (the Riemannian case) orlowast119864120583] is symmet-
ric one getslowast119864120583]120583minus
= lowast119864120583]120583 (65)
Proof Using the definition of the parameterized dual connec-tion nabla120572120583] we have
lowast119864120583]120583minus
= lowast119864120583]120583 minus lowast119864120583120572nabla120572120583] + lowast119864120572]nabla120583120583120572= lowast119864120583]120583 minus lowast119864120583120572 120572120583 ]
minus 119887 lowast119864120583120572120574120572120583]+ lowast119864120572] 120583120583 120572 + 119887 lowast119864120572]120574120583120583120572
(66)
As 120574120583]120590 is skew-symmetric in the first pair of indices the lastterm on the right vanishes Hence we get
lowast119864120583]120583minus
= lowast119864120583]120583 minus 119887 lowast119864120583120572120574120572120583] (67)
Now consider the last term on the right of the above identitylowast119864120583120572120574120572120583] = lowast119864120583120572120575120572120598 120574120598120583] = lowast119864120583120572119892120572120573119892120598120573120574120598120583] = lowast119864120583120573120574120573120583]
= minus lowast119864120583120573120574120583120573] (68)
from which (64) follows
Finally it is clear that if 119887 = 0 then lowast119864120583]120583minus
= 119864120583]120583 On the
other hand iflowast119864120583] is symmetric then the term
lowast119864120583120572120574120583120572] vanish-es since 120574120583120572] is skew-symmetric in 120583 and 120572Theorem2 If
lowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]|120583minus
+ (119887 minus 1) lowast119864120583120572120574120583120572] (69)
Consequently if 119887 = 1 (the AP case) orlowast119864120583] is symmetric one
getslowast119864120583]120583minus
= lowast119864120583]|120583minus
(70)
The above two theorems imply the following result
Corollary 3 (a) For a symmetric tensor119864120583] one has from (64)and (69)
lowast119864120583]120583minus
= lowast119864120583]|120583minus
= lowast119864120583]120583 (71)
(b) In the Riemannian case the tensor119864120583] defined by (5) (usingthe Ricci scalar in the Lagrangian function) coincides with theEinstein tensor then 119864120583]120583 equiv 0 (72)
and consequently for any symmetric tensor defined by (5) onehas 119864120583]|120583
minus
equiv 0lowast119864120583]120583minus
equiv 0 (73)
in AP- and PAP-spaces respectively
5 Discussion and Concluding Remarks
In the present work we have used the Dolan-McCrea vari-ational method to derive possible differential identities inPAP-geometry The importance of this work for physicalapplications can be discussed in the following points
(1) It is well known that the field equations of GR can beobtained using either one of the following approaches
(a) The Einstein approach in which the geometri-zation philosophy plays the main role in con-structing the field equations of the theory Oneof the principles of this philosophy is that ldquolawsof nature are just differential identities in anappropriate geometryrdquo Einstein has used thisprinciple to write his field equations In otherwords he has used the second Bianchi identityof Riemannian geometry to construct the fieldequations of his theory
(b) Hilbert approach inwhich the standardmethodof theoretical physics the action principle hasbeen used to derive the field equations of thetheory
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3
eter these connections have been linearly combined to givethe object (1198861 1198862 1198863 and 1198864 are parameters)
nabla120572120583] = 1198861Γ120572120583] + 1198862Γ120572120583] + 1198863Γ120572(120583]) + 1198864 120572120583 ] (10)
Imposing the general coordinate transformation of a linearconnection on (10) using (6)ndash(9) the above 4 parametersreduce to one parameter 119887 which leads to
nabla120572120583] = 120572120583 ] + 119887120574120572120583] (11)
where 120574120572120583] is the contortion of AP-geometry Connection (11)will be called the parameterized canonical connection It isnonsymmetric with nonvanishing curvature that is it is ofRiemann-Cartan type Moreover it is a metric connection[20]
The parameter 119887 is a dimensionless parameter For somegeometric and physical reasons [21] this parameter is sug-gested to have the form
119887 = 1198732 120572120574 (12)
where 119873 is a natural number taking the value 0 1 2 120572 is the fine structure constant and 120574 is a dimensionlessparameter to be fixed by experiment or observation for thesystem under consideration The explicit appearance of theparameter 119887 in (11) gives rise to the following advantages toPAP-geometry
(1) In the case of 119887 = 1 (11) reduces to the Weitzenbockconnection (6) of the conventional AP-geometry
(2) In the case of 119887 = 0 connection (11) reduces to theLevi-Civita connection (9)
(3) Between the limits 119887 = 0 and 119887 = 1 there is a discretespectrum of spaces (due to the presence of 1198732 in(12)) each of which has simultaneously nonvanishingcurvature and torsion
In what follows we will decorate the tensors containingthe parameter 119887 by a star Also these tensors will retain thesame name as in the conventional AP-geometry precededby the adjective ldquoparameterizedrdquo For example the parame-terized contortion is given by
lowast120574120572120583] = 119887120574120572120583] (13)
The torsion of the parameterized connection (11) is given bylowastΛ120572120583] = 119887 (120574120572120583] minus 120574120572]120583) = 119887Λ120572120583] (14)
where Λ120572120583] is the torsion of the AP-geometry Also theparameterized basic form can be obtained by contraction of(13) or (14)
lowast119862120583 def= lowastΛ120572120583120572 = lowast120574120572120583120572 = 119887119862120583 (15)
It is to be noted that starred objects have the same symmetryproperties as the corresponding unstarred objects defined inthe conventional AP-geometry
Now since the parameterized canonical connection nabla120572120583]is nonsymmetric one can define two other linear connec-tions the parameterized dual (transposed) connection
nabla120572120583] def= nabla120572]120583 (16)
and the parameterized symmetric connection
nabla120572(120583]) def= 12 (nabla120572120583] + nabla120572]120583) (17)
Consequently using (11) (16) and (17) we can define respec-tively the covariant derivatives
119860120583]+
def= 119860120583] minus 119860120572nabla120572sdot 120583]119860120583]minus
def= 119860120583] minus 119860120572nabla120572sdot 120583]119860120583] def= 119860120583] minus 119860120572nabla120572sdot (120583])
(18)
for any arbitrary vector 119860120583It is of importance to note that the BB of PAP-geometry
are the same as those of AP-geometry In other words theparameter 119887 has no effect on the BB of the geometry
3 Dolan-McCrea Variational Method inPAP-Geometry
We are going to apply the Dolan-McCrea variationalmethod to obtain differential identities in PAP-geometryThis method is an alternative to the least action method Itwas originally suggested in 1963 to derive Bianchi and otheridentities in the context of Riemannian geometry It has beengeneralized [7] to AP-geometry and used to derive identity(4) Here we generalize this method to PAP-geometry givingsome details since it is not widely known
Since as stated above the BB of PAP-geometry are thesame as those of AP-geometry let us define the Lagrangianfunction
lowast119871 = lowast119871(120582120583119894
120582120583]119894
120582120583]120590119894
) (19)
constructed from the BB of PAP 120582120583119894
and its first and second
derivatives Consequently we can write the scalar density asfollows
lowast
L0 = lowast
L0 (120582120583119894
120582120583]119894
120582120583]120590119894
) = 120582 lowast119871 (20)
wherelowast
L0 is a scalar lagrangian density and 120582 fl det(120582119894
120583) = 0Now let us assume that the following integral defined oversome arbitrary 119899-dimensional domainΩ
lowast1198680 def= intΩ
lowast
L0 (119909) 119889119899119909 (21)
is invariant under the infinitesimal transformation120582119894(119909) 997888rarr 120582
119894(119909) + 120598ℎ
119894(119909) (22)
4 Advances in Mathematical Physics
where 120598 is an infinitesimal parameter and ℎ119894are 119899 1-forms
defined on119872 Let Σ be the (119899minus1)-space such that the 119899-spaceregionΩ is enclosed within Σ Thus
lowast119868120598 def= intΩ
lowast
L120598119889119899119909 (23)
where119889119899119909 = 11988911990911198891199092 sdot sdot sdot 119889119909119899 (24)
lowast
L120598 = lowast
L(120582120583119894
+ 120598ℎ120583119894
120582120583]119894
+ 120598ℎ120583]119894
120582120583]120590119894
+ 120598ℎ120583]120590119894
) (25)
Assuming that ℎ120583119894
and ℎ120583]119894
vanish at all points of Σ thenlowast119868120598 minus lowast1198680 = int
Ω( lowastL120598 minus lowastL0) 119889119899119909 (26)
wherelowast1198680 is the value of lowast119868120598 at 120598 = 0 Now lowastL120598 can be written
using Taylor expansion in the form
lowast
L120598 = lowast
L0 + 120597 lowastL0120597120582120583119894
120598ℎ120583119894
+ 120597 lowastL0120597120582120583]119894
120598ℎ120583]119894
+ 120597 lowastL0120597120582120583]120590119894
120598ℎ120583]120590119894
+ 119874 (1205982) (27)
Using (26) and (27) and by integration by parts it can beshown after some manipulation that (26) can be written as
lowast119868120598 minus lowast1198680 = 120598intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 + 119874 (1205982) (28)
where 120575 lowast119871120575120582120583119894
is the Hamiltonian derivative defined by
120575 lowast119871120575120582120573119894
def= 1120582 [[[120597 lowastL0120597120582120573119894
minus 120597120597119909120574 ( 120597 lowastL0120597120582120573120574119894
)
+ 1205972120597119909120574120597119909120590 ( 120597 lowastL0120597120582120573120574120590119894
)]]] (29)
Consider a new set of coordinates 119909 related to the 119909-coordinate system by the transformation119909120583 = 119909120583 minus 120598119911120583 (119909) (30)
where 119911120583 is a vector field vanishing at all points of Σ and 120598 isindependent of 119909 We can write120597119909120583120597119909] = 120575120583] = 120597119909120583120597119909] minus 120598120597119911120583120597119909120590 120597119909120590120597119909] (31)
120597119909120583120597119909] = 120575120583] + 120598119911120583120590 120597119909120590120597119909] (32)
Since 119911120583 is a vector its transformation law is
119911120583 (119909) = 120597119909120583120597119909] 119911] (119909) (33)
Substituting from (32) into the above equation we get
119911120583 (119909) = 120575120583] 119911] (119909) + 120598119911] (119909) 119911120583120590 120597119909120590120597119909] (34)
Substituting from (32) again into the above equation gives
119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583120590 (120575120590] + 120598119911120590120572 120597119909120572120597119909] ) (35)
Finally this equation can be written as119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583] + 119874 (1205982) (36)
Substituting from (36) into (30) we get119909120583 = 119909120583 + 120598119911120583 (119909) + 119874 (1205982) (37)
Consequently we can write120597119909120583120597119909] = 120575120583] minus 120598119911120583]120597119909120583120597119909] = 120575120583] + 120598119911120583] + 119874 (1205982) (38)
Then the Jacobian expressions of the above matrices can begiven as 120597119909120597119909 = 1 minus 120598119911120572120572 + 119874 (1205982) 120597119909120597119909 = 1 + 120598119911120572120572 + 119874 (1205982)
(39)
since the difference between 119911120583(119909) and 119911120583(119909) would be oforder 1205982 as shown by (36)
Now we express 120582120583119894
(119909) in terms of 120582120583119894
(119909) in two different
ways(i) Using the vector transformation law
120582120583119894
(119909) = 120597119909]120597119909120583 120582]119894 (119909)= 120575]120583120582]119894
(119909) + 120598120582]119894
(119909) 119911] 120583 + 119874 (1205982)= 120582120583119894
(119909) + 120598120582]119894
119911] 120583 + 119874 (1205982)(40)
(ii) Using Taylor expansion120582120583119894
(119909) = 120582120583119894
(119909 minus 120598119911) = 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982)= 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982) (41)
Wehave dropped the bar in the second termon the right handside of this equation as the difference would be of order 1205982Comparing (40) and (41) and using (22) we get120582120583
119894
(119909) minus 120582120583119894
(119909) = 120598ℎ120583119894
+ 119874 (1205982) (42)
where ℎ120583119894
= 120582]119894
119911] 120583 + 120582120583]119894
119911] = 120582]119894
119911]120583minus
+ 120582120583]+
119894
119911] (43)
and the double stroke operator ldquordquo is defined by (18)
Advances in Mathematical Physics 5
Now we treat the Lagrangian density in the two differentways mentioned above
(a) Using Taylor expansionlowast
L (119909) = lowast
L (119909 minus 120598119911)= lowast
L (119909) minus 120598 lowastL (119909)120572 119911120572 + 119874 (1205982) (44)
also we have dropped the bar in the second term on the righthand side of this equation as the difference would be of order1205982
The Lagrangian densitylowast
L(119909) in (44) meanslowast
L (119909) equiv lowast
L(120582120583119894
(119909) 120582120583]119894
(119909) 120582120583]120590119894
(119909)) (45)
(b) Using scalar transformationlowast119871 (119909) = lowast119871 (119909) 120582 lowast119871 (119909) = 120582 lowast119871 (119909) = 120582 lowast119871 (119909) 120597119909120597119909
(46)
then using (20) and (39) we getlowast
L (119909) = lowast
L (119909) + 120598 lowastL (119909) 119911120572120572 + 119874 (1205982) (47)
Comparing (44) and (47) we havelowast
L (119909) minus lowastL (119909) = 120598 ( lowastL (119909) 119911120572)120572+ 119874 (1205982) (48)
By (46) we get
intΩ
lowast119871119889119899119909 minus intΩ
lowast119871119889119899119909 = 0 (49)
intΩ( lowastL (119909) minus lowastL (119909)) 119889119899119909= intΩ[120598 ( lowastL (119909) 119911120572)
120572+ 119874 (1205982)] 119889119899119909 (50)
Applying Gaussrsquos theorem to convert the volume integral onΩ (119899-space) to a surface integral over Σ ((119899 minus 1)-space) theintegral vanishes as 119911120572 vanishes at all points of Σ Now wecan writeint
Ω( lowastL (119909) 119911120572)
120572119889119899119909 = int
Σ
lowast
L (119909) 119911120572119899120572119889119899minus1119909= ∮Σ
lowast119871 (119909) 119911120572119899120572119889Σ equiv 0 (51)
where 119899120572 is a unit vector normal to ΣSo (50) shows that
intΩ[ lowastL (119909) minus lowastL (119909)] 119889119899119909 = 119874 (1205982) (52)
Substituting from (42) into (45) we getlowast
L (119909) = lowast
L(120582120583119894
(119909) + 120598ℎ120583119894
+ 119874 (1205982) 120582120583]119894
(119909) + 120598ℎ120583]119894
+ 119874 (1205982) 120582120583]120590119894
(119909) + 120598ℎ120583]120590119894
+ 119874 (1205982)) (53)
Using (25)lowast
L (119909) = lowast
L120598 (119909) + 119874 (1205982) (54)
then we can write (52) as
intΩ[ lowastL120598 (119909) minus lowastL0 (119909)] 119889119899119909 = 119874 (1205982) (55)
Comparing (26) (28) and (55) implies that
intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 equiv 0 (56)
Substituting from (43) into (56) we write
intΩ120582[[[
120575 lowast119871120575120582120583119894
120582]119894
119911]120583minus
+ 120575 lowast119871120575120582120583119894
120582120583]+
119894
119911]]]]119889119899119909 equiv 0 (57)
Let
lowast119864120583] def= 120575 lowast119871120575120582120583119894
120582]119894
= minuslowast119869119894
120583120582]119894
lowast119869119894
120583 def= minus 120575 lowast119871120575120582120583119894
(58)
The integral identity (57) then becomes
intΩ120582[[lowast119864120583]119911]120583
minus
minus lowast119869119894
120583120582120583]+
119894
119911]]]119889119899119909 equiv 0 (59)
Now the integral of the first term can be treated as follows
intΩ120582 lowast119864120583]119911]minus120583119889119899119909 = int
Ω120582 ( lowast119864120583]119911])
120583minus
119889119899119909minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= intΩ(120582 lowast119864120583]119911])
120583119889119899119909
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= ∮Σ120582 lowast119864120583]119911]119899120583119889Σ
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= minusintΩ120582 lowast119864120583]120583
minus
119911]119889119899119909
(60)
Thus the integral identity (59) reads
intΩ120582( lowast119864120583]120583
minus
+ 120582120583]+
119894
lowast119869119894
120583)119911]119889119899119909 equiv 0 (61)
6 Advances in Mathematical Physics
since 120582 = 0 and 119911] is an arbitrary vector field Using the fun-damental lemma of the calculus of variation [33] the expres-sion in the brackets vanishes identically Hence we have
lowast119864120583]120583minus
+ 120582120583]+
119894
lowast119869119894
120583 equiv 0 (62)
Using (11) (18) and (58) after some manipulations identity(62) can be written as
lowast119864120583]120583minus
= 119887 (1 minus 119887) 119864120583120572120574120572120583] (63)
This is the differential identity characterizing PAP-geometry
4 Relation between Differential Identities
The three differential identities (1) of Riemannian geometry(4) of AP-geometry and (63) of PAP-geometry are to berelated This idea comes from the fact that the general linearconnection (11) reduces to the Weitzenbock connection (6)of the AP-space for 119887 = 1 and to the Live-Civita connection(9) of the Riemannian space for 119887 = 0 Note that thethree identities mentioned are derivatives in which linearconnections are used The following two results relate theseidentities
Theorem 1 Iflowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582
119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]120583 + 119887 lowast119864120583120572120574120583120572] (64)
Consequently if 119887 = 0 (the Riemannian case) orlowast119864120583] is symmet-
ric one getslowast119864120583]120583minus
= lowast119864120583]120583 (65)
Proof Using the definition of the parameterized dual connec-tion nabla120572120583] we have
lowast119864120583]120583minus
= lowast119864120583]120583 minus lowast119864120583120572nabla120572120583] + lowast119864120572]nabla120583120583120572= lowast119864120583]120583 minus lowast119864120583120572 120572120583 ]
minus 119887 lowast119864120583120572120574120572120583]+ lowast119864120572] 120583120583 120572 + 119887 lowast119864120572]120574120583120583120572
(66)
As 120574120583]120590 is skew-symmetric in the first pair of indices the lastterm on the right vanishes Hence we get
lowast119864120583]120583minus
= lowast119864120583]120583 minus 119887 lowast119864120583120572120574120572120583] (67)
Now consider the last term on the right of the above identitylowast119864120583120572120574120572120583] = lowast119864120583120572120575120572120598 120574120598120583] = lowast119864120583120572119892120572120573119892120598120573120574120598120583] = lowast119864120583120573120574120573120583]
= minus lowast119864120583120573120574120583120573] (68)
from which (64) follows
Finally it is clear that if 119887 = 0 then lowast119864120583]120583minus
= 119864120583]120583 On the
other hand iflowast119864120583] is symmetric then the term
lowast119864120583120572120574120583120572] vanish-es since 120574120583120572] is skew-symmetric in 120583 and 120572Theorem2 If
lowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]|120583minus
+ (119887 minus 1) lowast119864120583120572120574120583120572] (69)
Consequently if 119887 = 1 (the AP case) orlowast119864120583] is symmetric one
getslowast119864120583]120583minus
= lowast119864120583]|120583minus
(70)
The above two theorems imply the following result
Corollary 3 (a) For a symmetric tensor119864120583] one has from (64)and (69)
lowast119864120583]120583minus
= lowast119864120583]|120583minus
= lowast119864120583]120583 (71)
(b) In the Riemannian case the tensor119864120583] defined by (5) (usingthe Ricci scalar in the Lagrangian function) coincides with theEinstein tensor then 119864120583]120583 equiv 0 (72)
and consequently for any symmetric tensor defined by (5) onehas 119864120583]|120583
minus
equiv 0lowast119864120583]120583minus
equiv 0 (73)
in AP- and PAP-spaces respectively
5 Discussion and Concluding Remarks
In the present work we have used the Dolan-McCrea vari-ational method to derive possible differential identities inPAP-geometry The importance of this work for physicalapplications can be discussed in the following points
(1) It is well known that the field equations of GR can beobtained using either one of the following approaches
(a) The Einstein approach in which the geometri-zation philosophy plays the main role in con-structing the field equations of the theory Oneof the principles of this philosophy is that ldquolawsof nature are just differential identities in anappropriate geometryrdquo Einstein has used thisprinciple to write his field equations In otherwords he has used the second Bianchi identityof Riemannian geometry to construct the fieldequations of his theory
(b) Hilbert approach inwhich the standardmethodof theoretical physics the action principle hasbeen used to derive the field equations of thetheory
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
where 120598 is an infinitesimal parameter and ℎ119894are 119899 1-forms
defined on119872 Let Σ be the (119899minus1)-space such that the 119899-spaceregionΩ is enclosed within Σ Thus
lowast119868120598 def= intΩ
lowast
L120598119889119899119909 (23)
where119889119899119909 = 11988911990911198891199092 sdot sdot sdot 119889119909119899 (24)
lowast
L120598 = lowast
L(120582120583119894
+ 120598ℎ120583119894
120582120583]119894
+ 120598ℎ120583]119894
120582120583]120590119894
+ 120598ℎ120583]120590119894
) (25)
Assuming that ℎ120583119894
and ℎ120583]119894
vanish at all points of Σ thenlowast119868120598 minus lowast1198680 = int
Ω( lowastL120598 minus lowastL0) 119889119899119909 (26)
wherelowast1198680 is the value of lowast119868120598 at 120598 = 0 Now lowastL120598 can be written
using Taylor expansion in the form
lowast
L120598 = lowast
L0 + 120597 lowastL0120597120582120583119894
120598ℎ120583119894
+ 120597 lowastL0120597120582120583]119894
120598ℎ120583]119894
+ 120597 lowastL0120597120582120583]120590119894
120598ℎ120583]120590119894
+ 119874 (1205982) (27)
Using (26) and (27) and by integration by parts it can beshown after some manipulation that (26) can be written as
lowast119868120598 minus lowast1198680 = 120598intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 + 119874 (1205982) (28)
where 120575 lowast119871120575120582120583119894
is the Hamiltonian derivative defined by
120575 lowast119871120575120582120573119894
def= 1120582 [[[120597 lowastL0120597120582120573119894
minus 120597120597119909120574 ( 120597 lowastL0120597120582120573120574119894
)
+ 1205972120597119909120574120597119909120590 ( 120597 lowastL0120597120582120573120574120590119894
)]]] (29)
Consider a new set of coordinates 119909 related to the 119909-coordinate system by the transformation119909120583 = 119909120583 minus 120598119911120583 (119909) (30)
where 119911120583 is a vector field vanishing at all points of Σ and 120598 isindependent of 119909 We can write120597119909120583120597119909] = 120575120583] = 120597119909120583120597119909] minus 120598120597119911120583120597119909120590 120597119909120590120597119909] (31)
120597119909120583120597119909] = 120575120583] + 120598119911120583120590 120597119909120590120597119909] (32)
Since 119911120583 is a vector its transformation law is
119911120583 (119909) = 120597119909120583120597119909] 119911] (119909) (33)
Substituting from (32) into the above equation we get
119911120583 (119909) = 120575120583] 119911] (119909) + 120598119911] (119909) 119911120583120590 120597119909120590120597119909] (34)
Substituting from (32) again into the above equation gives
119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583120590 (120575120590] + 120598119911120590120572 120597119909120572120597119909] ) (35)
Finally this equation can be written as119911120583 (119909) = 119911120583 (119909) + 120598119911] (119909) 119911120583] + 119874 (1205982) (36)
Substituting from (36) into (30) we get119909120583 = 119909120583 + 120598119911120583 (119909) + 119874 (1205982) (37)
Consequently we can write120597119909120583120597119909] = 120575120583] minus 120598119911120583]120597119909120583120597119909] = 120575120583] + 120598119911120583] + 119874 (1205982) (38)
Then the Jacobian expressions of the above matrices can begiven as 120597119909120597119909 = 1 minus 120598119911120572120572 + 119874 (1205982) 120597119909120597119909 = 1 + 120598119911120572120572 + 119874 (1205982)
(39)
since the difference between 119911120583(119909) and 119911120583(119909) would be oforder 1205982 as shown by (36)
Now we express 120582120583119894
(119909) in terms of 120582120583119894
(119909) in two different
ways(i) Using the vector transformation law
120582120583119894
(119909) = 120597119909]120597119909120583 120582]119894 (119909)= 120575]120583120582]119894
(119909) + 120598120582]119894
(119909) 119911] 120583 + 119874 (1205982)= 120582120583119894
(119909) + 120598120582]119894
119911] 120583 + 119874 (1205982)(40)
(ii) Using Taylor expansion120582120583119894
(119909) = 120582120583119894
(119909 minus 120598119911) = 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982)= 120582120583119894
(119909) minus 120598120582120583]119894
(119909) 119911] + 119874 (1205982) (41)
Wehave dropped the bar in the second termon the right handside of this equation as the difference would be of order 1205982Comparing (40) and (41) and using (22) we get120582120583
119894
(119909) minus 120582120583119894
(119909) = 120598ℎ120583119894
+ 119874 (1205982) (42)
where ℎ120583119894
= 120582]119894
119911] 120583 + 120582120583]119894
119911] = 120582]119894
119911]120583minus
+ 120582120583]+
119894
119911] (43)
and the double stroke operator ldquordquo is defined by (18)
Advances in Mathematical Physics 5
Now we treat the Lagrangian density in the two differentways mentioned above
(a) Using Taylor expansionlowast
L (119909) = lowast
L (119909 minus 120598119911)= lowast
L (119909) minus 120598 lowastL (119909)120572 119911120572 + 119874 (1205982) (44)
also we have dropped the bar in the second term on the righthand side of this equation as the difference would be of order1205982
The Lagrangian densitylowast
L(119909) in (44) meanslowast
L (119909) equiv lowast
L(120582120583119894
(119909) 120582120583]119894
(119909) 120582120583]120590119894
(119909)) (45)
(b) Using scalar transformationlowast119871 (119909) = lowast119871 (119909) 120582 lowast119871 (119909) = 120582 lowast119871 (119909) = 120582 lowast119871 (119909) 120597119909120597119909
(46)
then using (20) and (39) we getlowast
L (119909) = lowast
L (119909) + 120598 lowastL (119909) 119911120572120572 + 119874 (1205982) (47)
Comparing (44) and (47) we havelowast
L (119909) minus lowastL (119909) = 120598 ( lowastL (119909) 119911120572)120572+ 119874 (1205982) (48)
By (46) we get
intΩ
lowast119871119889119899119909 minus intΩ
lowast119871119889119899119909 = 0 (49)
intΩ( lowastL (119909) minus lowastL (119909)) 119889119899119909= intΩ[120598 ( lowastL (119909) 119911120572)
120572+ 119874 (1205982)] 119889119899119909 (50)
Applying Gaussrsquos theorem to convert the volume integral onΩ (119899-space) to a surface integral over Σ ((119899 minus 1)-space) theintegral vanishes as 119911120572 vanishes at all points of Σ Now wecan writeint
Ω( lowastL (119909) 119911120572)
120572119889119899119909 = int
Σ
lowast
L (119909) 119911120572119899120572119889119899minus1119909= ∮Σ
lowast119871 (119909) 119911120572119899120572119889Σ equiv 0 (51)
where 119899120572 is a unit vector normal to ΣSo (50) shows that
intΩ[ lowastL (119909) minus lowastL (119909)] 119889119899119909 = 119874 (1205982) (52)
Substituting from (42) into (45) we getlowast
L (119909) = lowast
L(120582120583119894
(119909) + 120598ℎ120583119894
+ 119874 (1205982) 120582120583]119894
(119909) + 120598ℎ120583]119894
+ 119874 (1205982) 120582120583]120590119894
(119909) + 120598ℎ120583]120590119894
+ 119874 (1205982)) (53)
Using (25)lowast
L (119909) = lowast
L120598 (119909) + 119874 (1205982) (54)
then we can write (52) as
intΩ[ lowastL120598 (119909) minus lowastL0 (119909)] 119889119899119909 = 119874 (1205982) (55)
Comparing (26) (28) and (55) implies that
intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 equiv 0 (56)
Substituting from (43) into (56) we write
intΩ120582[[[
120575 lowast119871120575120582120583119894
120582]119894
119911]120583minus
+ 120575 lowast119871120575120582120583119894
120582120583]+
119894
119911]]]]119889119899119909 equiv 0 (57)
Let
lowast119864120583] def= 120575 lowast119871120575120582120583119894
120582]119894
= minuslowast119869119894
120583120582]119894
lowast119869119894
120583 def= minus 120575 lowast119871120575120582120583119894
(58)
The integral identity (57) then becomes
intΩ120582[[lowast119864120583]119911]120583
minus
minus lowast119869119894
120583120582120583]+
119894
119911]]]119889119899119909 equiv 0 (59)
Now the integral of the first term can be treated as follows
intΩ120582 lowast119864120583]119911]minus120583119889119899119909 = int
Ω120582 ( lowast119864120583]119911])
120583minus
119889119899119909minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= intΩ(120582 lowast119864120583]119911])
120583119889119899119909
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= ∮Σ120582 lowast119864120583]119911]119899120583119889Σ
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= minusintΩ120582 lowast119864120583]120583
minus
119911]119889119899119909
(60)
Thus the integral identity (59) reads
intΩ120582( lowast119864120583]120583
minus
+ 120582120583]+
119894
lowast119869119894
120583)119911]119889119899119909 equiv 0 (61)
6 Advances in Mathematical Physics
since 120582 = 0 and 119911] is an arbitrary vector field Using the fun-damental lemma of the calculus of variation [33] the expres-sion in the brackets vanishes identically Hence we have
lowast119864120583]120583minus
+ 120582120583]+
119894
lowast119869119894
120583 equiv 0 (62)
Using (11) (18) and (58) after some manipulations identity(62) can be written as
lowast119864120583]120583minus
= 119887 (1 minus 119887) 119864120583120572120574120572120583] (63)
This is the differential identity characterizing PAP-geometry
4 Relation between Differential Identities
The three differential identities (1) of Riemannian geometry(4) of AP-geometry and (63) of PAP-geometry are to berelated This idea comes from the fact that the general linearconnection (11) reduces to the Weitzenbock connection (6)of the AP-space for 119887 = 1 and to the Live-Civita connection(9) of the Riemannian space for 119887 = 0 Note that thethree identities mentioned are derivatives in which linearconnections are used The following two results relate theseidentities
Theorem 1 Iflowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582
119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]120583 + 119887 lowast119864120583120572120574120583120572] (64)
Consequently if 119887 = 0 (the Riemannian case) orlowast119864120583] is symmet-
ric one getslowast119864120583]120583minus
= lowast119864120583]120583 (65)
Proof Using the definition of the parameterized dual connec-tion nabla120572120583] we have
lowast119864120583]120583minus
= lowast119864120583]120583 minus lowast119864120583120572nabla120572120583] + lowast119864120572]nabla120583120583120572= lowast119864120583]120583 minus lowast119864120583120572 120572120583 ]
minus 119887 lowast119864120583120572120574120572120583]+ lowast119864120572] 120583120583 120572 + 119887 lowast119864120572]120574120583120583120572
(66)
As 120574120583]120590 is skew-symmetric in the first pair of indices the lastterm on the right vanishes Hence we get
lowast119864120583]120583minus
= lowast119864120583]120583 minus 119887 lowast119864120583120572120574120572120583] (67)
Now consider the last term on the right of the above identitylowast119864120583120572120574120572120583] = lowast119864120583120572120575120572120598 120574120598120583] = lowast119864120583120572119892120572120573119892120598120573120574120598120583] = lowast119864120583120573120574120573120583]
= minus lowast119864120583120573120574120583120573] (68)
from which (64) follows
Finally it is clear that if 119887 = 0 then lowast119864120583]120583minus
= 119864120583]120583 On the
other hand iflowast119864120583] is symmetric then the term
lowast119864120583120572120574120583120572] vanish-es since 120574120583120572] is skew-symmetric in 120583 and 120572Theorem2 If
lowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]|120583minus
+ (119887 minus 1) lowast119864120583120572120574120583120572] (69)
Consequently if 119887 = 1 (the AP case) orlowast119864120583] is symmetric one
getslowast119864120583]120583minus
= lowast119864120583]|120583minus
(70)
The above two theorems imply the following result
Corollary 3 (a) For a symmetric tensor119864120583] one has from (64)and (69)
lowast119864120583]120583minus
= lowast119864120583]|120583minus
= lowast119864120583]120583 (71)
(b) In the Riemannian case the tensor119864120583] defined by (5) (usingthe Ricci scalar in the Lagrangian function) coincides with theEinstein tensor then 119864120583]120583 equiv 0 (72)
and consequently for any symmetric tensor defined by (5) onehas 119864120583]|120583
minus
equiv 0lowast119864120583]120583minus
equiv 0 (73)
in AP- and PAP-spaces respectively
5 Discussion and Concluding Remarks
In the present work we have used the Dolan-McCrea vari-ational method to derive possible differential identities inPAP-geometry The importance of this work for physicalapplications can be discussed in the following points
(1) It is well known that the field equations of GR can beobtained using either one of the following approaches
(a) The Einstein approach in which the geometri-zation philosophy plays the main role in con-structing the field equations of the theory Oneof the principles of this philosophy is that ldquolawsof nature are just differential identities in anappropriate geometryrdquo Einstein has used thisprinciple to write his field equations In otherwords he has used the second Bianchi identityof Riemannian geometry to construct the fieldequations of his theory
(b) Hilbert approach inwhich the standardmethodof theoretical physics the action principle hasbeen used to derive the field equations of thetheory
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 5
Now we treat the Lagrangian density in the two differentways mentioned above
(a) Using Taylor expansionlowast
L (119909) = lowast
L (119909 minus 120598119911)= lowast
L (119909) minus 120598 lowastL (119909)120572 119911120572 + 119874 (1205982) (44)
also we have dropped the bar in the second term on the righthand side of this equation as the difference would be of order1205982
The Lagrangian densitylowast
L(119909) in (44) meanslowast
L (119909) equiv lowast
L(120582120583119894
(119909) 120582120583]119894
(119909) 120582120583]120590119894
(119909)) (45)
(b) Using scalar transformationlowast119871 (119909) = lowast119871 (119909) 120582 lowast119871 (119909) = 120582 lowast119871 (119909) = 120582 lowast119871 (119909) 120597119909120597119909
(46)
then using (20) and (39) we getlowast
L (119909) = lowast
L (119909) + 120598 lowastL (119909) 119911120572120572 + 119874 (1205982) (47)
Comparing (44) and (47) we havelowast
L (119909) minus lowastL (119909) = 120598 ( lowastL (119909) 119911120572)120572+ 119874 (1205982) (48)
By (46) we get
intΩ
lowast119871119889119899119909 minus intΩ
lowast119871119889119899119909 = 0 (49)
intΩ( lowastL (119909) minus lowastL (119909)) 119889119899119909= intΩ[120598 ( lowastL (119909) 119911120572)
120572+ 119874 (1205982)] 119889119899119909 (50)
Applying Gaussrsquos theorem to convert the volume integral onΩ (119899-space) to a surface integral over Σ ((119899 minus 1)-space) theintegral vanishes as 119911120572 vanishes at all points of Σ Now wecan writeint
Ω( lowastL (119909) 119911120572)
120572119889119899119909 = int
Σ
lowast
L (119909) 119911120572119899120572119889119899minus1119909= ∮Σ
lowast119871 (119909) 119911120572119899120572119889Σ equiv 0 (51)
where 119899120572 is a unit vector normal to ΣSo (50) shows that
intΩ[ lowastL (119909) minus lowastL (119909)] 119889119899119909 = 119874 (1205982) (52)
Substituting from (42) into (45) we getlowast
L (119909) = lowast
L(120582120583119894
(119909) + 120598ℎ120583119894
+ 119874 (1205982) 120582120583]119894
(119909) + 120598ℎ120583]119894
+ 119874 (1205982) 120582120583]120590119894
(119909) + 120598ℎ120583]120590119894
+ 119874 (1205982)) (53)
Using (25)lowast
L (119909) = lowast
L120598 (119909) + 119874 (1205982) (54)
then we can write (52) as
intΩ[ lowastL120598 (119909) minus lowastL0 (119909)] 119889119899119909 = 119874 (1205982) (55)
Comparing (26) (28) and (55) implies that
intΩ120582 120575 lowast119871120575120582120583119894
ℎ120583119894
119889119899119909 equiv 0 (56)
Substituting from (43) into (56) we write
intΩ120582[[[
120575 lowast119871120575120582120583119894
120582]119894
119911]120583minus
+ 120575 lowast119871120575120582120583119894
120582120583]+
119894
119911]]]]119889119899119909 equiv 0 (57)
Let
lowast119864120583] def= 120575 lowast119871120575120582120583119894
120582]119894
= minuslowast119869119894
120583120582]119894
lowast119869119894
120583 def= minus 120575 lowast119871120575120582120583119894
(58)
The integral identity (57) then becomes
intΩ120582[[lowast119864120583]119911]120583
minus
minus lowast119869119894
120583120582120583]+
119894
119911]]]119889119899119909 equiv 0 (59)
Now the integral of the first term can be treated as follows
intΩ120582 lowast119864120583]119911]minus120583119889119899119909 = int
Ω120582 ( lowast119864120583]119911])
120583minus
119889119899119909minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= intΩ(120582 lowast119864120583]119911])
120583119889119899119909
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= ∮Σ120582 lowast119864120583]119911]119899120583119889Σ
minus intΩ120582 lowast119864120583]120583
minus
119911]119889119899119909= minusintΩ120582 lowast119864120583]120583
minus
119911]119889119899119909
(60)
Thus the integral identity (59) reads
intΩ120582( lowast119864120583]120583
minus
+ 120582120583]+
119894
lowast119869119894
120583)119911]119889119899119909 equiv 0 (61)
6 Advances in Mathematical Physics
since 120582 = 0 and 119911] is an arbitrary vector field Using the fun-damental lemma of the calculus of variation [33] the expres-sion in the brackets vanishes identically Hence we have
lowast119864120583]120583minus
+ 120582120583]+
119894
lowast119869119894
120583 equiv 0 (62)
Using (11) (18) and (58) after some manipulations identity(62) can be written as
lowast119864120583]120583minus
= 119887 (1 minus 119887) 119864120583120572120574120572120583] (63)
This is the differential identity characterizing PAP-geometry
4 Relation between Differential Identities
The three differential identities (1) of Riemannian geometry(4) of AP-geometry and (63) of PAP-geometry are to berelated This idea comes from the fact that the general linearconnection (11) reduces to the Weitzenbock connection (6)of the AP-space for 119887 = 1 and to the Live-Civita connection(9) of the Riemannian space for 119887 = 0 Note that thethree identities mentioned are derivatives in which linearconnections are used The following two results relate theseidentities
Theorem 1 Iflowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582
119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]120583 + 119887 lowast119864120583120572120574120583120572] (64)
Consequently if 119887 = 0 (the Riemannian case) orlowast119864120583] is symmet-
ric one getslowast119864120583]120583minus
= lowast119864120583]120583 (65)
Proof Using the definition of the parameterized dual connec-tion nabla120572120583] we have
lowast119864120583]120583minus
= lowast119864120583]120583 minus lowast119864120583120572nabla120572120583] + lowast119864120572]nabla120583120583120572= lowast119864120583]120583 minus lowast119864120583120572 120572120583 ]
minus 119887 lowast119864120583120572120574120572120583]+ lowast119864120572] 120583120583 120572 + 119887 lowast119864120572]120574120583120583120572
(66)
As 120574120583]120590 is skew-symmetric in the first pair of indices the lastterm on the right vanishes Hence we get
lowast119864120583]120583minus
= lowast119864120583]120583 minus 119887 lowast119864120583120572120574120572120583] (67)
Now consider the last term on the right of the above identitylowast119864120583120572120574120572120583] = lowast119864120583120572120575120572120598 120574120598120583] = lowast119864120583120572119892120572120573119892120598120573120574120598120583] = lowast119864120583120573120574120573120583]
= minus lowast119864120583120573120574120583120573] (68)
from which (64) follows
Finally it is clear that if 119887 = 0 then lowast119864120583]120583minus
= 119864120583]120583 On the
other hand iflowast119864120583] is symmetric then the term
lowast119864120583120572120574120583120572] vanish-es since 120574120583120572] is skew-symmetric in 120583 and 120572Theorem2 If
lowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]|120583minus
+ (119887 minus 1) lowast119864120583120572120574120583120572] (69)
Consequently if 119887 = 1 (the AP case) orlowast119864120583] is symmetric one
getslowast119864120583]120583minus
= lowast119864120583]|120583minus
(70)
The above two theorems imply the following result
Corollary 3 (a) For a symmetric tensor119864120583] one has from (64)and (69)
lowast119864120583]120583minus
= lowast119864120583]|120583minus
= lowast119864120583]120583 (71)
(b) In the Riemannian case the tensor119864120583] defined by (5) (usingthe Ricci scalar in the Lagrangian function) coincides with theEinstein tensor then 119864120583]120583 equiv 0 (72)
and consequently for any symmetric tensor defined by (5) onehas 119864120583]|120583
minus
equiv 0lowast119864120583]120583minus
equiv 0 (73)
in AP- and PAP-spaces respectively
5 Discussion and Concluding Remarks
In the present work we have used the Dolan-McCrea vari-ational method to derive possible differential identities inPAP-geometry The importance of this work for physicalapplications can be discussed in the following points
(1) It is well known that the field equations of GR can beobtained using either one of the following approaches
(a) The Einstein approach in which the geometri-zation philosophy plays the main role in con-structing the field equations of the theory Oneof the principles of this philosophy is that ldquolawsof nature are just differential identities in anappropriate geometryrdquo Einstein has used thisprinciple to write his field equations In otherwords he has used the second Bianchi identityof Riemannian geometry to construct the fieldequations of his theory
(b) Hilbert approach inwhich the standardmethodof theoretical physics the action principle hasbeen used to derive the field equations of thetheory
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Mathematical Physics
since 120582 = 0 and 119911] is an arbitrary vector field Using the fun-damental lemma of the calculus of variation [33] the expres-sion in the brackets vanishes identically Hence we have
lowast119864120583]120583minus
+ 120582120583]+
119894
lowast119869119894
120583 equiv 0 (62)
Using (11) (18) and (58) after some manipulations identity(62) can be written as
lowast119864120583]120583minus
= 119887 (1 minus 119887) 119864120583120572120574120572120583] (63)
This is the differential identity characterizing PAP-geometry
4 Relation between Differential Identities
The three differential identities (1) of Riemannian geometry(4) of AP-geometry and (63) of PAP-geometry are to berelated This idea comes from the fact that the general linearconnection (11) reduces to the Weitzenbock connection (6)of the AP-space for 119887 = 1 and to the Live-Civita connection(9) of the Riemannian space for 119887 = 0 Note that thethree identities mentioned are derivatives in which linearconnections are used The following two results relate theseidentities
Theorem 1 Iflowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582
119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]120583 + 119887 lowast119864120583120572120574120583120572] (64)
Consequently if 119887 = 0 (the Riemannian case) orlowast119864120583] is symmet-
ric one getslowast119864120583]120583minus
= lowast119864120583]120583 (65)
Proof Using the definition of the parameterized dual connec-tion nabla120572120583] we have
lowast119864120583]120583minus
= lowast119864120583]120583 minus lowast119864120583120572nabla120572120583] + lowast119864120572]nabla120583120583120572= lowast119864120583]120583 minus lowast119864120583120572 120572120583 ]
minus 119887 lowast119864120583120572120574120572120583]+ lowast119864120572] 120583120583 120572 + 119887 lowast119864120572]120574120583120583120572
(66)
As 120574120583]120590 is skew-symmetric in the first pair of indices the lastterm on the right vanishes Hence we get
lowast119864120583]120583minus
= lowast119864120583]120583 minus 119887 lowast119864120583120572120574120572120583] (67)
Now consider the last term on the right of the above identitylowast119864120583120572120574120572120583] = lowast119864120583120572120575120572120598 120574120598120583] = lowast119864120583120572119892120572120573119892120598120573120574120598120583] = lowast119864120583120573120574120573120583]
= minus lowast119864120583120573120574120583120573] (68)
from which (64) follows
Finally it is clear that if 119887 = 0 then lowast119864120583]120583minus
= 119864120583]120583 On the
other hand iflowast119864120583] is symmetric then the term
lowast119864120583120572120574120583120572] vanish-es since 120574120583120572] is skew-symmetric in 120583 and 120572Theorem2 If
lowast119864120583] is a tensor of order 2 on a PAP-space (119872 120582119894)
then one has the identitylowast119864120583]120583minus
= lowast119864120583]|120583minus
+ (119887 minus 1) lowast119864120583120572120574120583120572] (69)
Consequently if 119887 = 1 (the AP case) orlowast119864120583] is symmetric one
getslowast119864120583]120583minus
= lowast119864120583]|120583minus
(70)
The above two theorems imply the following result
Corollary 3 (a) For a symmetric tensor119864120583] one has from (64)and (69)
lowast119864120583]120583minus
= lowast119864120583]|120583minus
= lowast119864120583]120583 (71)
(b) In the Riemannian case the tensor119864120583] defined by (5) (usingthe Ricci scalar in the Lagrangian function) coincides with theEinstein tensor then 119864120583]120583 equiv 0 (72)
and consequently for any symmetric tensor defined by (5) onehas 119864120583]|120583
minus
equiv 0lowast119864120583]120583minus
equiv 0 (73)
in AP- and PAP-spaces respectively
5 Discussion and Concluding Remarks
In the present work we have used the Dolan-McCrea vari-ational method to derive possible differential identities inPAP-geometry The importance of this work for physicalapplications can be discussed in the following points
(1) It is well known that the field equations of GR can beobtained using either one of the following approaches
(a) The Einstein approach in which the geometri-zation philosophy plays the main role in con-structing the field equations of the theory Oneof the principles of this philosophy is that ldquolawsof nature are just differential identities in anappropriate geometryrdquo Einstein has used thisprinciple to write his field equations In otherwords he has used the second Bianchi identityof Riemannian geometry to construct the fieldequations of his theory
(b) Hilbert approach inwhich the standardmethodof theoretical physics the action principle hasbeen used to derive the field equations of thetheory
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 7
It is to be noted that the first approach is capableof constructing a complete theory not only the fieldequations of the theory This will be discussed in thefollowing point
(2) Einstein geometrization philosophy can be summa-rized as follows [34]ldquoTo understand nature one has to start with geometryand end with physicsrdquoIn applying this philosophy one has to consider itsmain principles
(a) There is a one-to-one correspondence betweengeometric objects and physical quantities
(b) Curves (paths) in the chosen geometry aretrajectories of test particles
(c) Differential identities represent laws of nature
In view of the above principles Einstein has usedRiemannian geometry to construct a full theory forgravity GR in which the metric and the curva-ture tensors represent the gravitational potential andstrength respectively He has also used the geodesicequations to represent motion in gravitational fieldsFinally he has used Bianchi identity to write the fieldequations of GRThe above mentioned philosophy can be applied toany geometric structure other than the Riemannianone It has been applied successfully in the context ofconventional AP-geometry (cf [7 9 10]) using thedifferential identity (4)PAP-geometry ismore general than both Riemannianand conventional AP-geometry It is shown in Sec-tion 2 that PAP-geometry reduces to AP-geometryfor 119887 = 1 and to Riemannian geometry for 119887 = 0In other words the latter two geometries are specialcases of the PAP-geometry The importance of theparameter 119887 has been investigated in many papers (cf[35 36])The value of this parameter is extracted fromthe results of three different experiments [35 37 38]The value obtained (119887 = 10minus3) shows that space-timenear the Earth is neither Riemannian (119887 = 0) norconventional AP (119887 = 1)
(3) To use PAP-geometry as a medium for constructingfield theories the above three principles are to beconsidered The curves (paths) of the geometry havebeen derived [20] and used to describe the motionof spinning particles [37 39] Geometric objects havebeen used to represent physical quantities via theconstructed field theories [25 26]The present work is done as a step to completethe use of the geometrization philosophy in PAP-geometryThe general form of the differential identitycharacterizing this geometry is obtained in Section 3This will help in attributing physical properties togeometric objects especially conservation This willbe discussed in the next point
(4) The results obtained from the two theorems given inSection 4 are of special importance It is clear from(71) (72) and (73) that any symmetric tensor definedby (5) is subject to the differential identity of type(72) This result is independent of the values of theparameter 119887Now for any symmetric tensor defined by (5) in PAP-geometry we have
lowast119864120583] 120583 equiv 0 (74)
Due to the structure of the parameterized connection(11) the tensor
lowast119864120583] can always be written in the form
lowast119864120583] = 119866120583] + lowast119879120583] (75)
where 119866120583] is Einstein tensor defined in terms of (9)as usual and
lowast119879120583] is a second-order symmetric tensorwhich vanishes when 119887 = 0 Consequently identity(74) can be written as
119866120583] 120583 + lowast119879120583] 120583 equiv 0 (76)
As the first term vanishes due to Bianchi identitythen we get the identity
lowast119879120583] 120583 equiv 0 (77)
This implies that the physical entity represented bylowast119879120583] is conserved in any field theory constructed inPAP-geometry Identity (77) is thus an identity ofPAP-geometry independent of any such field theoryThe tensor
lowast119879120583] is usually used as a geometric repre-sentation of amaterial-energy distribution in any fieldtheory constructed in PAP-geometry [25 26]
(5) Finally the following properties are guaranteed inany field theory constructed in the context of PAP-geometry
(a) The theory of GR with all its consequences canbe obtained upon taking 119887 = 0
(b) As the parameterized connection (11) is metricthe motion along curves of PAP-geometry [20]preserves the gravitational potential
(c) One can always define a geometric material-energy tensor in terms of the BB of the geom-etry
(d) Conservation is not violated in any of suchtheories
Competing Interests
The authors declare that they have no competing interests
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Advances in Mathematical Physics
References
[1] A Einstein ldquoThe field equations of general relativityrdquo Sitzungs-berichte der Preussischen Akademie der WissenschaftenmdashPhysikalischmdashmathematische Klasse pp 844ndash847 1915
[2] A Einstein ldquoRiemannian geometry with maintaining thenotion of distant parallelismrdquo Sitzungsberichte der Preussis-chenAkademie derWissenschaften- physikalisch-mathematischeKlasse pp 217ndash221 1928
[3] A Einstein ldquoNewpossibility for a unified field theory of gravita-tion and electricityrdquo Sitzungsberichte der Preussischen Akademieder Wissenschaften- physikalisch- mathematische Klasse pp224ndash227 1928
[4] R Debever Ed Elie Cartan and Albert Einstein Letters onAbsolute Parallelism 1929ndash1932 Princeton University Press1979
[5] F I Mikhail Relativistic cosmology and some related problems ingeneral relativity [PhD thesis] University of London 1952
[6] W H McCrea and F I Mikhail ldquoVector-tetrads and thecreation of matterrdquo Proceedings of the Royal Society of LondonAMathematical Physical and Engineering Sciences vol 235 no1200 pp 11ndash22 1956
[7] F I Mikhail andM I Wanas ldquoA generalized field theory I fieldequationsrdquo Proceedings of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 356 no1687 pp 471ndash481 1977
[8] M I Wanas A generalized field theory and its applications incosmology [PhD thesis] Cairo University Cairo Egypt 1975
[9] M I Wanas and S A Ammar ldquoSpacetime structure andelectromagnetismrdquoModern Physics Letters A vol 25 no 20 pp1705ndash1721 2010
[10] M I Wanas S N Osman and R I El-Kholy ldquoUnificationprinciple and a geometric field theoryrdquoOpen Physics vol 13 no1 pp 247ndash262 2015
[11] R S de Souza and R Opher ldquoOrigin of 1015-1016 G magneticfields in the central engine of gamma ray burstsrdquo Journal ofCosmology and Astroparticle Physics vol 2010 no 2 article 222010
[12] R S de Souza and R Opher ldquoOrigin of intense magneticfields near black holes due to non-minimal gravitational-electromagnetic couplingrdquo Physics Letters B vol 705 no 4 pp292ndash293 2011
[13] A Mazumder and D Ray ldquoCharged spherically symmetricsolution in Mikhail-Wanas field theoryrdquo International Journalof Theoretical Physics vol 29 no 4 pp 431ndash434 1990
[14] F I Mikhail M I Wanas and A M Eid ldquoTheoretical inter-pretation of cosmic magnetic fieldsrdquo Astrophysics and SpaceScience vol 228 no 1-2 pp 221ndash237 1995
[15] M IWanas ldquoA generalized field theory charged spherical sym-metric solutionrdquo International Journal ofTheoretical Physics vol24 no 6 pp 639ndash651 1985
[16] M I Wanas ldquoOn the relation between mass and charge apure geometric approachrdquo International Journal of GeometricMethods in Modern Physics vol 4 no 3 pp 373ndash388 2007
[17] M I Wanas and S A Ammar ldquoA pure geometric approach tostellar structurerdquoCentral European Journal of Physics vol 11 no7 pp 936ndash948 2013
[18] M IWanasMMelek andM E Kahil ldquoNew path equations inabsolute parallelism geometryrdquo Astrophysics and Space Sciencevol 228 no 1-2 pp 273ndash276 1995
[19] S Bazanski ldquoHamilton-Jacobi formalism for geodesics andgeodesic deviationsrdquo Journal of Mathematical Physics vol 30no 5 pp 1018ndash1029 1989
[20] M I Wanas ldquoMotion of spinning particles in gravitationalfieldsrdquo Astrophysics and Space Science vol 258 no 1-2 pp 237ndash248 199798
[21] M I Wanas ldquoParameterized absolute parallelism a geometryfor physical applicationsrdquo Turkish Journal of Physics vol 24 no3 pp 473ndash488 2000
[22] GG LNashed ldquoKerr-NUTblack hole thermodynamics in f(T)gravity theoriesrdquo European Physical Journal Plus vol 130 no 7article 124 2015
[23] G G L Nashed ldquoA spherically symmetric null tetrads in non-minimal torsion-matter coupling extension of f (T) gravityrdquoAstrophysics and Space Science vol 357 no 2 pp 1ndash11 2015
[24] G G L Nashed and W El Hanafy ldquoA built-in inflation in thef (T)-cosmologyrdquo European Physical Journal C vol 74 no 10article 3099 pp 1ndash13 2014
[25] M I Wanas and M M Kamal ldquoA field theory with curvatureand anticurvaturerdquo Advances in High Energy Physics vol 2014Article ID 687103 15 pages 2014
[26] M IWanasN L Youssef andW S El-Hanafy ldquoPure geometricfield theory description of gravity and material distributionrdquohttpsarxivorgabs14042485
[27] 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
[28] F I Mikhail ldquoTetrad vector fields and generalizing the theoryof relativityrdquo Ain Shams Science Bulletin vol 6 pp 87ndash111 1962
[29] M I Wanas ldquoAbsolute parallelism geometry developmentsapplications and problemsrdquo Studii si Cercetari StiintificeMatematica University Bacau vol 10 pp 297ndash309 2001
[30] N L Youssef andW A Elsayed ldquoA global approach to absoluteparallelism geometryrdquo Reports onMathematical Physics vol 72no 1 pp 1ndash23 2013
[31] 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
[32] 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
[33] BVanBruntTheCalculus of Variation SpringerNewYorkNYUSA 2004
[34] M I Wanas ldquoThe accelerating expansion of the universe andtorsion energyrdquo International Journal of Modern Physics A vol22 no 31 pp 5709ndash5716 2007
[35] A A Sousa and J M Maluf ldquoGravitomagnetic effect and spin-torsion couplingrdquoGeneral Relativity andGravitation vol 36 no5 pp 967ndash982 2004
[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 IWanasMMelek andM E Kahil ldquoQuantum interferenceof thermal neutrons and spin-torsion interactionrdquo Gravitationand Cosmology vol 6 pp 319ndash322 2000
[38] Y Mao M Tegmark A H Guth and S Cabi ldquoConstrainingtorsion with gravity probe Brdquo Physical Review D vol 76 no 10Article ID 104029 2007
[39] M I Wanas M Melek and M E Kahil ldquoSN1987A temporalmodelsrdquo in Proceedings of the IAU Symposium vol 205 pp 396ndash397 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
