Modelling wormholes in f(R, T ) gravity

P.H.R.S. Moraes∗, P.K. Sahoo†∗ITA - Instituto Tecnologico de Aeronautica - Departamento de Fısica,

12228-900, Sao Jose dos Campos, Sao Paulo, Brazil ∗ and†Department of Mathematics,

Birla Institute of Technology and Science-Pilani,Hyderabad Campus, Hyderabad-500078, India †

In this work we propose the modelling of static wormholes within the f(R, T ) extended theoryof gravity perspective. We present some models of wormholes, which are constructed from differenthypothesis for their matter content, i.e., different relations for their pressure components (radialand lateral) and different equations of state. The solutions obtained for the shape function of thewormholes obey the necessary metric conditions. They show a behaviour similar to those found inprevious references about wormholes, which also happens to our solutions for the energy density ofsuch objects. We also apply the energy conditions for the wormholes physical content.

PACS numbers: 04.50.kd.

I. INTRODUCTION

Extended theories of gravity have been proposed aim-ing the explanation of some observable phenomena thathardly can be explained through General Relativity the-ory. As examples for some of these phenomena, we canquote dark energy [1, 2], dark matter [3, 4], massivepulsars [5, 6], super-Chandrasekhar white dwarfs [7, 8],among others, for which, gravity theories such as f(R)[9] and f(T ) [10, 11], with R and T being, respectively,the Ricci and torsion scalars, may propose some expla-nation. For a review on extended theories of gravity, wequote Reference [12].

Not only upon geometrical (or torsion) aspects ex-tended theories of gravity are constructed, but also fromthe consideration of extra material contributions, as inthe f(R,Lm) [13] and f(R, T ) theories [14], with Lm be-ing the matter lagrangian and T the trace of the energy-momentum tensor. The material corrections are ex-pected to come from the existence of imperfect fluidsor quantum effects, such as particle production (check[15, 16]).

Particularly, the f(R, T ) gravity theories have alreadyshown to provide good alternatives for the issues quotedabove, as it can be seen in References [17–21], for exam-ple.

In the present article, we are concerned in modellingwormholes (WHs) in the f(R, T ) gravity. WHs are hy-pothetical passages which associate two different regionsof the space-time and their matter content is describedby an anisotropic energy-momentum tensor.

They still have not been observed although attemptsto achieve so were proposed [22–30]. Anyhow, they weretheoretically predicted a long time ago. A. Einstein andN. Rosen were the first to investigate WH solutions withevent horizon [31]. M.S. Morris and K.S. Thorne manydecades later have shown that WHs can be traversable iffilled with exotic matter, which violates the energy con-ditions [32].

∗Email: moraes.phrs@gmail.com†Email: pksahoo@hyderabad.bits-pilani.ac.in

The motivation for working with WHs in the f(R, T )gravity context can be seen as coming from the factthat WHs matter content is described by an imperfectanisotropic fluid while the T−dependence of the f(R, T )theory, as mentioned above, may be related with the exis-tence of imperfect fluids in the universe. Also, the quan-tum effects contained in T , or terms proportional to it,may be related to the mechanism of particle production,since those terms prevent the energy-momentum tensorof the theory to conserve. In this regard, the possibilityof particle creation in WHs has already been analysed in[33, 34].

Our article is organized as follows: in Section II wepresent a review of the f(R, T ) gravity. In Section III,we present the WH metric and the conditions that mustbe satisfied by the shape function. In Section 4 we applythe WH metric in the f(R, T ) formalism. In Section 5 weconstruct different WH models from different hypothesisfor their matter content. Our results are discussed inSection 6.

II. f(R, T ) GRAVITY

The f(R, T ) theory of gravity starts from the followingtotal action [14]

S =1

16π

∫d4x√−gf(R, T ) +

∫d4x√−gLm, (1)

where f(R, T ) is an arbitrary function of the Ricci scalarR and trace of the energy-momentum tensor T , g is themetric determinant and Lm is the matter Lagrangiandensity, related to the energy-momentum tensor as [35]

Tij = − 2√−g

[∂(√−gLm)

∂gij− ∂

∂xk∂(√−gLm)

∂(∂gij/∂xk)

]. (2)

Moreover, we will work with units such that c = 1 = G.Following the steps of Refs.[13, 14], we assume Lm de-

pends only on the metric components gij and not on itsderivatives, such that we obtain

Tij = gijLm − 2∂Lm

∂gij. (3)

arX

iv:1

707.

0696

8v2

[gr

-qc]

31

Aug

201

7

2

By varying the action S given in Eq.(1) with respectto the metric gij provides the f(R, T ) field equations [14]

fR(R, T )

(Rij −

1

3Rgij

)+

1

6f(R, T )gij

= 8πG

(Tij −

1

3Tgij

)− fT (R, T )

(Tij −

1

3Tgij

)− fT (R, T )

(θij −

1

3θgij

)+∇i∇jfR(R, T ), (4)

with fR(R, T ) = ∂f(R, T )/∂R, fT (R, T ) = ∂f(R, T )/∂Tand

θij = gij∂Tij∂gij

. (5)

The f(R, T ) (as well as the f(R,Lm)) gravity theoriesare able to describe a non-minimal matter-geometry cou-pling in their formalism. This implies that in such theo-ries, test particles moving in a gravitational field will notfollow geodesic lines. The coupling between matter andgeometry induces an extra force acting on the particles,which is perpendicular to the 4-velocity.

It is interesting to note that the extra force depends onthe form of the matter lagrangian [36]. It was shown in[37] that by considering Lm = p, with p being the totalpressure, the extra force vanishes. On the other hand,more natural forms for Lm, such as Lm = −ρ, with ρbeing the energy density, are more generic, in the sensethat they do not imply the vanishing of the extra force.

In this paper, we assume the matter Lagrangian isLm = −ρ. Hence, Equation (5) can be written as

θij = −2Tij − ρgij . (6)

Let us assume the function f(R, T ) = R+2f(T ), wheref(T ) is an arbitrary function of T . The f(R, T ) gravityfield equations (4) with (6) take the form

Rij −1

2Rgij = 8πTij + 2f†(T )Tij + [2ρf†(T ) + f(T )]gij ,

(7)with Rij being the Ricci tensor and f†(T ) = df(T )/dT .Assuming f(T ) = λT , with λ being a constant, the aboveequations can be rewritten as

Gij = (8π + 2λ)Tij + λ(2ρ+ T )gij , (8)

with Gij being the usual Einstein tensor. Such an as-sumption for the f(R, T ) functional form was originallyproposed by the f(R, T ) gravity authors themselves in[14]. It has been applied to a high number of papers,such as [17, 19, 38–43], among many others.

III. WORMHOLE METRIC AND ITSCONDITIONS

The static spherically symmetric WH metric inSchwarzschild coordinates (t, r, θ, φ) is [32, 44]

ds2 = −U(r)dt2 +dr2

V+ r2dΩ2, (9)

where dΩ2 = dθ2+sin2 θdφ2 and V = 1−b(r)/r. The red-shift function U(r) and the shape function b(r) in metric(9) must obey the following conditions [32, 44]:

1. The radial coordinate r lies between r0 ≤ r < ∞,where r0 is known as the throat radius.

2. At the throat r = r0, b(r) must obey the condition

b(r0) = r0 (10)

and for r > r0, i.e., out of the throat,

1− b(r)

r> 0. (11)

3. b(r) has to obey the flaring out condition at thethroat, i.e.,

b′(r0) < 1, (12)

with ′ = d/dr.

4. For asymptotically flatness of the spacetime geom-etry, the limit

b(r)

r→ 0 as |r| → ∞ (13)

is required.

5. U(r) must be finite and non-vanishing at the throatr0.

One can consider U(r) = constant to achieve the de Sit-ter and anti-de Sitter asymptotic behaviour. Since a con-stant redshift function can be absorbed in rescaled timecoordinate, we consider U(r) = 1, such as Refs.[45, 46],among others.

IV. FIELD EQUATIONS FOR WORMHOLES INf(R, T ) GRAVITY

We consider an anisotropic fluid satisfying the mattercontent of the form

T ij = diag(−ρ, pr, pl, pl) (14)

where ρ = ρ(r) is the energy density, pr = pr(r) and pl =pl(r) are respectively the radial and lateral (measuredorthogonally to the radial direction) pressures. The traceT of the energy momentum tensor (14) turns out to beT = −ρ+ pr + 2pl.

The components of the field equations (8) for the met-ric (9) with (14) are

b′

r2= (8π + λ)ρ− λ(pr + 2pl), (15)

− b

r3= λρ+ (8π + 3λ)pr + 2λpl, (16)

b− b′r2r3

= λρ+ λpr + (8π + 4λ)pl. (17)

3

2 4 6 8 101.0

1.1

1.2

1.3

1.4

1.5

1.6

r

b(r)

FIG. 1: The shape function b(r), with A = 1 and n = −0.4.

The above set of field equations admits the solutions

ρ =b′

r2(8π + 2λ), (18)

pr = − b

r3(8π + 2λ), (19)

pl =b− b′r

2r3(8π + 2λ). (20)

V. MODELS OF WORMHOLES

In this Section, we will obtain models of WHs whichare constructed from different hypothesis for their mattercontent.

A. Model 1

For our first model we assume the pressures pl and prare related as

pl = npr, (21)

where n is an arbitrary constant. Such a relation wasassumed in [46, 47], for instance.

Using (21) in (19) and (20), we can obtain

b(r) = Ar1+2n, (22)

where A is an integration constant.Since the metric is asymptotically flat, Equation (22)

above is consistent when A is positive and n is negative.

This also satisfies the flaring out condition, i.e., b−b′rb2 >

0.The shape function b(r) is plotted versus r in Figure 1

above for the values A = 1 and n = −0.4. From such afigure, the fundamental WH condition, i.e., b(r) < r, isobeyed.

The throat of the WH occurs at r = r0 so that Eq.(10)yields

A =1

r2n0. (23)

The energy density will be

ρ =A(1 + 2n)

8π + 2λr2(n−1), (24)

while the radial and lateral pressures are

pr =−A

8π + 2λr2(n−1) (25)

and

pl =−nA

8π + 2λr2(n−1). (26)

From Equations (24)-(26), we have

ρ+ pr =nA

4π + λr2(n−1), (27)

ρ+ pl =A(1 + n)

8π + 2λr2(n−1). (28)

The existence of exotic matter in WHs violates theenergy conditions. Specifically, the energy-momentumtensor violates the null energy condition (NEC) at theWH throat [48, 49]. One can observe from Eq.(27) theviolation of NEC, i.e., ρ+pr ≤ 0, which implies λ > −4π.NEC is presented in Fig.2 below. The validity region ofρ+ pl ≥ 0 is presented in Fig.3.

FIG. 2: Violation of NEC, ρ + pr ≤ 0, with A = 1 andλ = −12.

FIG. 3: NEC, ρ+ pl ≥ 0, with A = 1 and λ = −12.

4

The dominant energy condition (DEC), ρ ≥ |pr| andρ ≥ |pl|, for this model is obtained from

ρ− pr =A(1 + n)

4π + λr2(n−1), (29)

ρ− pl =A(1 + 3n)

8π + 2λr2(n−1). (30)

From Eqs.(29)-(30), DEC is plotted in Figs.4-5.

FIG. 4: DEC, ρ ≥ |pr|, with A = 1 and λ = −12.

FIG. 5: DEC, ρ ≥ |pl|, with A = 1 and λ = −12.

From Eqs.(18)-(20), the strong energy condition (SEC)yields ρ+ pr + 2pl = 0.

B. Model 2

Let us consider that matter with equation of state(EoS)

pr + ω(r)ρ = 0 (31)

is filling the WH, for which ω(r) is a positive functionof the radial coordinate. The same EoS with varyingparameter ω(r) was assumed in Reference [50] by F. Ra-haman and collaborators, for example.

Taking (31) into account, from Equations (18) and (19)we are able to obtain

ω(r) =b

rb′. (32)

Next, we will check two cases for Eq.(32) above, as itwas previously made in [50].

Case I - ω(r) = ω = constant

Let us take, as a first case, ω(r) = ω = constant. Thisimplies, from Eq.(32), that

b = b0r1/ω, (33)

where b0 is an integrating constant. Since the metricis asymptotically flat, Equation (33) is consistent whenω > 1.

Using some particular values of the parameters, we plotb(r) in Fig.6 below. It shows that for r > r0, b(r)− r <0, an essential condition for the shape function to obey.b(r) − r is also a decreasing function for r > r0, whichsatisfies the flaring out condition b′(r0) < 1.

2 4 6 8 101.00

1.05

1.10

1.15

1.20

1.25

1.30

r

b(r)

FIG. 6: b(r) with b0 = 1 and ω = 10.

The throat of the WH occurs at

r0 = bω

ω−1

0 . (34)

Using Eq.(33) in Eqs.(18)-(20), we have

ρ =b0r

1−3ωω

ω(8π + 2λ), (35)

pr = − b0r1−3ω

ω

8π + 2λ, (36)

pl =b0(ω − 1)r

1−3ωω

2ω(8π + 2λ). (37)

NEC for this model is obtained as

ρ+ pr =b0(1− ω)r

1−3ωω

ω(8π + 2λ), (38)

ρ+ pl =b0(1 + ω)r

1−3ωω

2ω(8π + 2λ). (39)

From (38) the violation of NEC happens for b0 > 0 andλ > −4π. NEC for this model is presented in Figs.7-8below.

5

FIG. 7: Violation of NEC, ρ + pr ≤ 0, with b0 = 1 andλ = −12.

FIG. 8: Validity region of NEC, ρ+ pl ≥ 0, with b0 = 1 andλ = −12.

DEC for this model is

ρ− pr =b0(1 + ω)r

1−3ωω

ω(8π + 2λ), (40)

ρ− pl =b0(3− ω)r

1−3ωω

2ω(8π + 2λ). (41)

Those can be seen in Figs.9-10 below.

FIG. 9: Validity of DEC, ρ ≥ |pr|, with b0 = 1 and λ = −12.

FIG. 10: Violation of DEC, ρ ≥ |pl|, with b0 = 1 andλ = −12.

Case II - ω(r) = Brm

Here, we consider ω(r) = Brm, where B and m arepositive constants. From Eq.(32), the shape function nowreads

b = exp

(C − 1

Bmrm

), (42)

where C is a constant of integration.At the throat of the WH we can obtain

C = ln r0 +1

Bmrm0, (43)

so that

b = exp

[ln r0 +

1

Bm

(1

rm0− 1

rm

)]. (44)

Since ω > 1 and m > 0, we have r > r0 >(1B

) 1m , which

satisfies the asymptotically flat condition for the metric.Hence the considered EoS is justified.

From Fig.11 below, one can note that when r > r0,

b(r) − r < 0, which implies b(r)r < 1 when r > r0. We

can observe that b(r) − r is a decreasing function of rfor r ≥ r0 and b′(r0) < 1, which satisfies the flaring outcondition. Hence, the obtained shape function indeedrepresents a WH structure.

2 4 6 8 101.06

1.07

1.08

1.09

1.10

1.11

1.12

1.13

1.14

r

b(r)

FIG. 11: b(r) with r0 = 1, B = 4 and m = 2.

Using Eq.(42) into Eqs.(18)-(20), we have

ρ =exp

(C − 1

Bmrm

)(8π + 2λ)Brm+3

, (45)

6

pr = −exp

(C − 1

Bmrm

)(8π + 2λ)r3

, (46)

pl =(Brm+1 − 1) exp

(C − 1

Bmrm

)2(8π + 2λ)Brm+3

. (47)

NEC for this model reads

ρ+ pr =(1−Brm) exp

(C − 1

Bmrm

)(8π + 2λ)Brm+3

, (48)

ρ+ pl =(Brm+1 + 1) exp

(C − 1

Bmrm

)2(8π + 2λ)Brm+3

. (49)

Those are plotted in Figs.12-13 below.

FIG. 12: Violation of NEC, ρ + pr ≤ 0, with r0 = 1, B = 4and λ = −12.

FIG. 13: NEC, ρ+ pl ≥ 0, with r0 = 1, B = 4 and λ = −12.

DEC for this model is

ρ− pr =(1 +Brm) exp

(C − 1

Bmrm

)(8π + 2λ)Brm+3

, (50)

ρ− pl =(3−Brm+1) exp

(C − 1

Bmrm

)2(8π + 2λ)Brm+3

. (51)

Those are plotted in Figs.14-15 below.Moreover, for both Cases of Model 2, SEC yields ρ +

pr + 2pl = 0.

FIG. 14: Validity of DEC, ρ ≥ |pr|, with r0 = 1, B = 4 andλ = −12.

FIG. 15: Violation of DEC, ρ ≥ |pl|, with r0 = 1, B = 4 andλ = −12.

VI. DISCUSSION

We have constructed, in the present article, differentmodels of static WHs within the f(R, T ) theory of grav-itation. In this section we will discuss the results we ob-tained for the geometrical and material content of thoseWHs.

As we have mentioned in the previous sections, thecondition b′(r0) < 1, which is required to describe a WHsolution, is satisfied in all the constructed models. ForModel 1, b′(r0) = 1+2n, which is < 1, since n is negative,corresponding to Eqs.(22)-(23). For Case I of Model 2,b′(r0) = 1

ω < 1, since ω > 1 and for Case II, b′(r0) < 1

implies in r0 >(1B

) 1m .

Furthermore, we have assumed the redshift functionis a constant (U(r) = constant), which implies that thetidal gravitational force experienced by a hypotheticaltraveller is null.

In view of the seminal paper by M.S. Morris and K.S.Thorne [32], in such a reference, the authors have foundthat in a static WH, ρ ∼ r−2. We can see that oursolution for ρ in Case I of Model 2 predicts the sameproportionality for r when ω → 1. Still in this case,our solution for ρ agrees with those found for Morris-Thorne WHs with a cosmological constant [51] and WHsminimally violating NEC [52].

7

In General Relativity, the analysis of the geometry oftraversable WHs has shown that NEC is violated at theWH throat [32, 44]. In our models, at the WH throat, theweaker inequality ρ(r0)+pr(r0) ≤ 0 holds, which impliesthat NEC is violated. The authors in [53] obtained thevalidity of NEC, ρ + pr ≥ 0, by considering negativeenergy density. SEC, on the other side, is satisfied in allof our models, i.e., ρ + pr + 2pl = 0, as one can checkEqs.(18)-(20).

It is interesting to remark that further analysis off(R, T ) WHs may come from the implementation ofhigher order terms of R and/or T in the f(R, T ) function.Different forms for f(R) have already been considered inWH analysis in [54–56] for instance. Those can be addedto functions of T in order to get more generic f(R, T )WHs.

Another valuable functional forms for the f(R, T )function that can be used to construct WH models arethose that predict a matter-geometry non-minimal cou-pling, such as f(R, T ) = R + αRT , which has alreadybeen applied in cosmological models, yielding observa-tional acceptable results [57].

Acknowledgments

PHRSM would like to thank Sao Paulo Research Foun-dation (FAPESP), grant 2015/08476-0, for financial sup-port. PKS acknowledges the support of CERN, inGeneva, during an academic visit, where a part of thiswork was done.

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