arXiv:1408.2689v1 [gr-qc] 12 Aug 2014
Bianchi type-I, type-III and Kantowski-Sachs solutions in f (T ) gravity M. E. Rodrigues (a,b) 1 , A. V. Kpadonou (b,e) 2 , F. Rahaman P. J. Oliveira (f ) 4 and M. J. S. Houndjo (b,c) 5
(d) 3
,
(a) Faculdade de Ciˆencias Exatas e Tecnologia, Universidade Federal do Par´ a - Campus Universit´ ario de Abaetetuba, CEP 68440-000, Abaetetuba, Par´ a, Brazil (b) Institut de Math´ematiques et de Sciences Physiques (IMSP) - 01 BP 613 Porto-Novo, B´enin (c) Facult´e des Sciences et Techniques de Natitingou - Universit´e de Parakou - Natitingou - B´enin (d) Department of Mathematics, Jadavpur University, Kolkata - 700032, India (e)Ecole Normale Sup´erieure de Natitingou - Universit´e de Parakou - B´enin (f) Instituto Federal de Educa¸c˜ao, Ciˆencia e Tecnologia do Esp´ırito Santo, Campus Cachoeiro de Itapemirim - CEP 29.300-970 — Caixa Postal: 527 Cachoeiro de Itapemirim - ES - Brazil Abstract In the context of modified tele-parallel theory of gravity, we undertake cosmological anisotropic models and search for their solutions. Within a suitable choice of non-diagonal tetrads, the decoupled equations of motion are obtained for Bianchi-I, Bianchi-III and KantowskiSachs models, from which we obtain the correspondent solutions. By the way, energy density and pressures are also obtained, showing, as an important result, that our universe may live a quintessence like universe even still anisotropic models are considered.
1
Introduction
It is well known nowadays that modifying the law of gravity may lead to possible explanations for the acceleration mechanism of the our universe [1]. Various theories based on the modification of the law of gravity have been 1
E-mail E-mail 3 E-mail 4 E-mail 5 E-mail 2
address: address: address: address: address:
[email protected] [email protected] [email protected] [email protected] [email protected]
1
performed, among those, the modified version of the Tele-Parallel theory of gravity (TT), the so-called f (T ) gravity, where T denotes the torsion scalar. The teleparallel gravity is a theory equivalent to the general relativity, but does not depend on the Levi-Civita’s connection, as is the case of the GR. The TT and its modified version are based on the Weitzenbock connection where the curvature scalar vanishes whereas the torsion is different from zero. Within this theory, various works have been developed and interesting results have been found, see [15, 44]. Note that the observational data are based on the assumption that the universe is homogeneous and isotropic for large scales. Indeed, the matter distribution in the universe is rigorously non-homogeneous and the propagation of the light may not be isotropic. In this way, it reasonable to look for at models able to approach this feature of the universe. This is the goal of this paper. There are several metric models that traduce the anisotropic feature of the universe but we focus our attention on three of them, the Bianchi-I, Bianchi-III and Kantowski-Sachs. The motif of using these models is the fact that there their topologies, for large scales, may lead to the well known trivial FRW model within physical assumptions. An important point to be put out here is that the geometrical objects as the torsion and contorsion, which are indispensable for any calculus depend on the so-called tetrad. The tetrads play an important role and depending on their form, the theory may present some constraints on the gravitational action. In fact, for diagonal tetrads, constraints appears and the action is reduced to the TT one, with cosmological constant. However, when the non-diagonal tetrads are considered, the constraints disappear letting free the choice of the action. Our goal is to search for the solutions of these models, say, finding the expression of the scale factor in the three space directions, within the modified TT. As consequence, through the dependence of the content of the universe of the geometrical part of the field equations, solutions are found for the energy density and the pressure, showing, as an interesting and important result, that still using anisotropic metric models, our universe may live a quintessence-like phase. The paper is organised as follows: at the section 2, we general equation of motion and write down the expression of the energy-momentum tensor within the assumption of inhomogeneous content of the universe. The section 3 is devoted to the decoupling of the generalized equations following the pressures and the energy density. Still in this section solutions and comments are performed. The conclusion is presented at the section 4.
2
2
General equations of motion We first define a metric for the geometry of the Weitzenbok spacetime
as dS 2 = gµν dxµ dxν ,
(1)
where gµν are the components of the metric which is symmetric and possess 10 degrees of freedom. One can describe the theory in the spacetime or in the tangent space, which allows to rewrite the line element (1) as follows dS 2 = gµν dxµ dxν = ηij θ i θ j , dx
µ
µ i
i
i
µ
= ei θ , θ = e µ dx ,
(2) (3)
where ηij = diag[1, −1, −1, −1] and ei µ ei ν = δνµ or ei µ ej µ = δij . The square √ root of the metric determinant is given by −g = det ei µ = e and the matrix eaµ are called tetrads and represent the dynamic fields of the theory. Note that the tetrads have a degree of freedom greater than the one of the metric, and this is showed in [28], and we still have a possibility of non unequivocal choice [21]. We can define the Weitzenbock’s connection as Γαµν = ei α ∂ν ei µ = −ei µ ∂ν ei α .
(4)
The main geometrical objects of the spacetime are constructed from this connection. The components of the tensor torsion are defined by the antisymmetric part of this connection T αµν = Γανµ − Γαµν = ei α ∂µ ei ν − ∂ν ei µ . (5)
We also define the components of the so-called contorsion tensor as 1 K µνα = − (T µνα − T νµα − Tα µν ) . 2
(6)
In order to make more clear the definition of the scalar equivalent to the curvature scalar of RG, we first define a new tensor Sαµν , constructed from the components of the torsion and contorsion tensors as 1 µν (7) K α + δαµ T βνβ − δαν T βµβ . Sαµν = 2 Now, we are able to construct a contraction which is equivalent to the scalar curvature in GR. We define then the torsion scalar as T = T αµν Sαµν . 3
(8)
As f (R) gravity generalizes the GR, we also define the action of f (T ) gravity as the generalization of the teleparallel, being the continue sum Z S = d4 x e [f (T ) + LM atter ] , (9) where f (T ) is an algebraic function of the torsion scalar T . Making the functional variation of the action (9) with respect to the tetrads, we get the following equations of motion [30, 27] i h 1 Sµνρ ∂ρ T fT T + e−1 eiµ ∂ρ (eei α Sανρ ) + T αλµ Sανλ fT + δµν f = 4πTµν , (10) 4
where Tµν is the energy momentum tensor, fT = df (T )/dT and fT T = d2 f (T )/dT 2 . By setting f (T ) = T − 2Λ, the equations of motion (10) are the same as that of the Teleparallel theory with a cosmological constant, and this is dynamically equivalent to the GR. These equations clearly depend on the choice made for the set of tetrads [31]. The contribution of the interaction with the matter fields is given by the energy momentum tensor which, is this case, is defined as Tµν = (ρ + pt ) uµ uν − pt δµν + (pr − pt ) vµ v ν ,
(11)
where uµ is the four-velocity, v µ a unitary space-like vector in the radial direction, ρ the energy density, pr the pressure in the direction of v µ (radial pressure) and pt the pressure orthogonal to vµ (tangential pressure). This tensor characterizes an anisotropic fluid.
3
Anisotropic models
In this section, we will study the equations of motion coming from the metric of Bianchi type-I, type-III and Kantowski-Sachs, first in a unified way and with a choice of two sets of tetrads, the diagonal and the non-diagonal.
3.1
Diagonal tetrads
In this subsection we will study the case of a choice of a set of diagonal tetrads for some inhomogeneous models. We can describe the models of Bianchi type-I, type-III and Kantowski-Sachs (KS) through the metric dS 2 = dt2 − A2 (t)dr 2 − B 2 (t) dθ 2 + Kj2 (θ)dφ2 , (12) 4
where j = 1, 2, 3, K1 (θ) = θ, K2 (θ) = sin θ and K3 (θ) = sinh θ are the models of Bianchi type-I, Kantowski-Sachs and Bianchi type-III, respectively. Let us start choosing a set of diagonal tetrads a e µ = diag [1, A, B, BKj (θ)] . (13) The determinant of this matrix is e = AB 2 Kj (θ). The components of the torsion tensor (5), for the choice of tetrads (13), are given by T 101 =
B˙ 1 dKj A˙ , T 202 = T 303 = , T 323 = , A B Kj dθ
(14)
those the correspond the contorsion (6) components are K 011 =
A˙ B˙ 1 dKj , K 022 = K 033 = , K 323 = 2 , A B B Kj dθ
(15)
and those of the tensor Sαµν , in (7), are given by S1
10
B˙ 1 dKj 1 = , S0 20 = S1 21 = , S2 20 = S3 30 = B 2B 2 Kj dθ 2
A˙ B˙ + A B
!
. (16)
Making use of the components (14) and (16), the torsion scalar (8) is given by !2 A˙ B˙ B˙ −4 . (17) T = −2 B AB The equations of motion (10) are !2 ˙ B˙ ˙ A k B f , +2 + fT + (18) 4πρ = 4 B AB 2B 2 !2 ˙ B˙ ˙ ¨ B˙ ˙ A f k B B , −4πpr = + T fT T + + fT + + (19) B 4 B B AB 2B 2 ! !2 ˙ ˙ ˙ ˙ ¨ ˙ ¨ f 1 A B ˙ AB fT A B B T fT T + + −4πpt = +3 + + + (20), 2 A B 4 2 A B B AB
T˙ (dKj /dθ) fT T 2B 2 Kj
= 0,
(21) 5
where k = −Kj−1 (d2 Kj /dθ 2 ). For the Kantowski-Sachs model k = +1, for Bianchi type-I k = 0, and for Bianchi type-III k = −1. This classification stems from the scalar curvature of the spatial hyper-surface (3) R = 2k/B 2 (t), in the case of the GR [32]. The constraint equation (21) imposes to the function f (T ) to be linear, as in the Teleparallel Theory (TT), and we can choose it as f (T ) = T − 2Λ, where Λ is the cosmological constant. Hence, the equations (18)-(20) become B˙ B 2
!2
A˙ B˙ + AB !2 B˙ + B
k B2
= Λ + 8πρ ,
(22)
k B2
= Λ − 8πpr ,
(23)
¨ A¨ B A˙ B˙ + + A B AB
= Λ − 8πpt .
(24)
+2
¨ B + B
These equations are the same as those of the GR [33], and then, all the known solutions for these models can be regained. Once again, this is not surprising, since the TT is dynamically equivalent to the GR and the equations of motion must be identical. We can also observe that the particular case A(t) = B(t) = a(t) and pr = pt , for the model of Bianchi type-I (k = 0), we regain the flat FLRW universe in (22)-(24), with a(t) being the scale factor. In the next section, we will study a choice of a set of non-diagonal tetrads.
3.2
Non-diagonal tetrads
In this subsection we will study the inhomogeneous models, with a set of non-diagonal tetrads, similar to the static case [21], or in the cosmological case [27]. Our goal here is to get some slight modification in the equations of motion, coming from a set of non-diagonal tetrads, and compare the results with the diagonal case. This can lead in general to some understanding for the use of anisotropic models. By choosing the set of non-diagonal tetrads, 1 0 0 0 a 0 A sin θ cos φ B cos θ cos φ −BKj (θ) sin φ e µ = (25) 0 A sin θ sin φ B cos θ sin φ BKj (θ) cos φ , 0 A cos θ −B sin θ 0
we get the determinant e = AB 2 Kj . The components of the torsion (5), 6
contorsion (6) and tensor Sαµν (7), are given by (dKj /dθ) − cos θ A˙ B˙ A A sin θ , T 202 = T 303 = , T 221 = , T 331 = , T 323 = , (26) A B B BKj Kj A˙ B˙ 1 sin θ (dKj /dθ) − cos θ = , K 022 = K 033 = , K 122 = , K 133 = , K 323 = (27), A B AB ABKj B 2 Kj θ B˙ S 01 = Kj +sin θ , S 20 = S 21 = (dKj /dθ)−cos , S1 10 = B , 0 2ABKj 2B 2 Kj 0 1 (28) ˙ ˙ B 1 sin θ A 1 31 21 30 20 S2 = S3 = = 2ABKj , S3 = 2AB . 2 A + B , S2
T 101 = K 011
The torsion scalar (8), from the components (26) and (28), is given by T = −2
B˙ B
!2
−4
A˙ B˙ 2 sin θ + 2 AB B Kj
(29)
The equations of motion (10) become ˙ ˙ (dKj /dθ) − cos θ dT AB k − (sin θ/Kj ) 4πρ = − fT T + (30), +2 2 2B Kj dθ AB 2B 2 !2 ˙ ˙ ˙ ˙ ¨ B ˙ f AB k − (sin θ/Kj ) B B −4πpr = T fT T + + fT + + + , (31) B 4 B B AB 2B 2 ! !2 ˙ ¨ ˙ ¨ ˙ ˙ ˙ f B fT A B sin θ 1 A B ˙ AB + T fT T + + + + − , (32) −4πpt = +3 2 A B 4 2 A B B AB B 2 Kj
f B˙ + + fT 4 B
!2
Kj + sin θ ˙ T fT T = 0 , 2ABKj
(33)
(dKj /dθ) − cos θ ˙ T fT T = 0 , 2B 2 Kj sin θ dT fT T = 0 , 2ABKj dθ ! B˙ dT 1 A˙ + fT T = 0 . 2 A B dθ
(34) (35) (36)
Also here, the constraint equations (33)-(36) impose to the function f (T ) to be linear or constant torsion. 7
By choosing f (T ) = T − 2Λ, the equations of motion (30)-(32) turn into !2 B˙ A˙ B˙ k +2 + 2 = Λ + 8πρ , (37) B AB B !2 ¨ B k B˙ 2 + + 2 = Λ − 8πpr , (38) B B B ¨ A˙ B˙ A¨ B + + A B AB
= Λ − 8πpt .
(39)
We can define the average volume as V (t) = A(t)B 2 (t). Supposing first an exponential expansion V (t) = v0 e3H0 t , with v0 , H0 ∈ ℜ+ , one gets A(t) =
v0 e3H0 t . B 2 (t)
(40)
Now, defining the average Hubble’s parameter as 1 V˙ 1 H(t) = = 3V 3
B˙ A˙ +2 A B
!
,
(41)
one has H(t) = H0 . By defining a deceleration factor q = [(d/dt)(1/H(t)) − 1], one gets q = −1 which yields an accelerated universe. We also define ˙ ˙ H1 = A/A, H2,3 = B/B, and the anisotropic parameter of the expansion as !2 2 3 1 X Hi 2 B˙ A˙ ∆(t) = . (42) −1 = − 3 H 9H 2 A B i=1
Making similar considerations to those of Adhav et al [29], we can simplify the equations, using pr = ω(t)ρ(t) and pt = [ω(t) + δ(t)]ρ(t), with ρ(t) =
k . 8πδ(t)B 2 (t)
(43)
Then, we can solve the equations of motion (37)-(39), setting B(t) = exp [b(t)], which yields ke−2b − Λ = 0, 6H0 b˙ + ke−2b − 3b˙ 2 − δ ke−2b 3b˙ 2 + 2¨b + ke−2b + ω − Λ = 0, δ ke−2b 9H02 + 3b˙ 2 − 9H0 b˙ − ¨b + ke−2b + ω − Λ = 0. δ 8
(44) (45) (46)
Subtracting (45) from (46) one gets 3¨b + 9H0 b˙ − 9H02 = 0
(47)
leading to b(t) = H0 t − (b0 e−3H0 t /3H0 ) + b1 , with b0 ∈ ℜ. The solution is then given by v0 b0 −3H0 t 2b0 −3H0 t A(t) = 2 exp H0 t + e e , B(t) = b2 exp H0 t − .(48) 3H0 3H0 b2 The anisotropic parameter of the expansion (42) is given by ∆(t) =
2c22 −6kt e . k2
We can now isolate δ(t) in the equation (44), getting 2 −3kt exp 4kt + 2s 3k e δ(t) = 2 2 6kt . c3 (3k e − 3c22 − Λe6kt )
(49)
(50)
We also isolate ω(t) in (45), obtaining ω(t) = −
3k2 e6kt + 3c22 − Λe6kt . 3k2 e6kt − 3c22 − Λe6kt
(51)
We present the curve that traduces the evolution of this parameter versus t for some values of the input parameters. Hence, we see that, setting k2 = 1, Λ = 0.01 and c22 = 0.1, the parameter ω tends to −1; see Fig. 1. The density (43) becomes ρ(t) =
3k2 − 3c22 e−6kt − Λ . 8π
(52)
Now we can test the physical criteria for this solution, as we have in [34]. The following criteria have to be obeyed: a) density ρ(t) always positive ; b) volume V (t) going to infinity when t → ∞; c) the anisotropic parameter of the expansion ∆(t) must tend to zero when t → ∞. For k2 > c22 + Λ/3 the density (52) is always positive, then, satisfies the criterion a). In the limit t → ∞, one gets ∆(t) → 0 , V (t) → ∞ , ρ(r) →
3k2 − Λ , δ(t) → 0 , ω(t) → −1 . 8π
(53)
Note that the solution is physical, the anisotropy disappears, the density is always positive, the volume tends to infinity and the model tends to the ΛCDM one, with ω = −1. 9
We also consider a power type expansion, i.e, V (t) = c1 t3n , with c1 , n ∈ Then, we have A(t) = c1 t3n /B 2 (t) and H(t) = n/t. In this case, considering (43) and B = exp(b), the equations (37)-(39), for Λ = 0, yield
ℜ+ .
6n ˙ e−2b = 0 b+ t δ e−2b 2¨b + 3b˙ 2 + ω = 0 δ 9n ˙ 3n(1 − 3n) e−2b −¨b + 3b˙ 2 − + ω = 0. b− t t2 δ 3b˙ 2 −
(54) (55) (56)
Subtracting (55) from (56), one gets t2¨b + 3ntb˙ + n(1 − 3n) = 0 ,
(57)
which, after integration, yields b(t) = log(tn ) + [c2 t1−3n /(1 − 3n)] + b0 . The solution is given by 1−3n 2t −2t1−3n c1 n n , B(t) = c2 t exp (58) A(t) = 2 t exp 1 − 3n 1 − 3n c2 The anisotropic parameter of the expansion (42) is given by ∆(t) =
2t2−3n . n2
We can now isolate δ(t) in the equation (54), getting 1−3n t2+4n exp −2t 1−3n δ(t) = . 2 2 6n 3c2 (n t − t2 )
(59)
(60)
Also, isolating ω(t) in the equation (55), one gets ω(t) =
3t2 + n(3n − 2)t6n . 3(t2 − n2 t6n )
(61)
In this case, we plot the evolution of ω versus time for two different cases. The first one, Fig. 2, shows that the parameter ω tends to −1/3 only when n is very closed to 1. On other hand, we see that, for n > 1, more precisely n = 2 (as an example), ω does not tend to −1/3, but rather 0.66; see Fig. 3. Therefore, we conclude that the parameter n is responsible for quintessence era. 10
The density (43) becomes 2
3( n2 − t−6n ) . ρ(t) = t 8π
(62)
Let us now test the physical criteria for this solution. For t > n ≥ 1 the density (62) is always positive, and then, satisfies the criterion a). For t → ∞, there is a dependence on the value of n. There is a quintessence-like behaviour, with ω ∈ (−1, 0). In the particular case, n = 1, we have 1 ∆(t) → 0 , V (t) → ∞ , ρ(r) → 0 , δ(t) → 0 , ω(t) → − . 3
(63)
Note that the solution is physical, the anisotropy disappears, the energy density is always positive, the volume tends to infinity and the model tends to a quintessence-like one.
4
Conclusion
In this paper, we undertake the modified tele-parallel theory, f (T ) gravity, where T denotes the torsion scalar. In order to approach the real feature of the universe, due to the fact that the universe is not rigorously isotropic and homogeneous, we look for three interesting and realistic metric models, the Bianchi-I, Bianchi-III and Kantowski-Sachs models, and tried to solve the equations of motion. We focus our attention on non-diagonal tetrads where the gravitational action is free from the constraint of recovering the TT. In this way, we found solutions for the three space directions, and through the equations of motion, the expressions of the energy density and pressures are obtained. An analysis of these expressions showed, as an interesting result that, even considering that our universe is isotropic, it may live in a quintessence-like phase.
11
Figure 1: This figure point out the evolution of the parameter ω in terms of t for k2 = 1, Λ = 0.01 and c22 = 0.1.
12
Figure 2: The figure shows the evolution of the parameter ω as the time evolves, for n = 1.
13
Figure 3: This figure presents two curves, one for n = 1 and the second for n = 2, both pointing out evolutions of the parameter ω.
14
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