Ratbay Myrzakulov
Ratbay Myrzakulov Ratbay Myrzakulov
The action (3.8) can be written as where the point-like Lagrangian reads ∫ S T = dtL T , (2.14) The equations of F(T) gravity look like L T = a 3 (F − F T T) − 6F T aȧ 2 − a 3 L m . (2.15) 12H 2 F T + F = ρ, (2.16) ) 48H 2 F TT Ḣ − F T (12H 2 + 4Ḣ − F = p, (2.17) ˙ρ + 3H(ρ + p) = 0. (2.18) 2.3 F(G) gravity The action of F(G) theory is given by ∫ S G = d 4 xe[F(G) + L m ], (2.19) where the Gauss – Bonnet scalar G for the FRW metric is G = G s = 24H 2 (Ḣ + H2 ). (2.20) 3 Geometrical roots of F(R, T) gravity We start from the M 43 - model (about our notations, see e.g. Refs. [10]-[11]). This model is one of the representatives of F(R,T) gravity. The action of the M 43 - model reads as ∫ S 43 = d 4 x √ −g[F(R,T) + L m ], R = R s = ɛ 1 g µν R µν , (3.1) T = T s = ɛ 2 S ρ µν T ρ µν, where L m is the matter Lagrangian, ɛ i = ±1 (signature), R is the curvature scalar, T is the torsion scalar (about our notation see below). In this section we try to give one of the possible geometric formulations of M 43 - model. Note that we have different cases related with the signature: (1) ɛ 1 = 1,ɛ 2 = 1; (2) ɛ 1 = 1,ɛ 2 = −1; (3) ɛ 1 = −1,ɛ 2 = 1; (4) ɛ 1 = −1,ɛ 2 = −1. Also note that M 43 - model is a particular case of M 37 - model having the form ∫ S 37 = d 4 x √ −g[F(R,T) + L m ], where R = u + R s = u + ɛ 1 g µν R µν , (3.2) T = v + T s = v + ɛ 2 S ρ µν T ρ µν, are the standard forms of the curvature and torsion scalars. 3.1 General case R s = ɛ 1 g µν R µν , T s = ɛ 2 S ρ µν T ρ µν (3.3) To understand the geometry of the M 43 - model, we consider some spacetime with the curvature and torsion so that its connection G λ µν is a sum of the curvature and torsion parts. In this paper, the Greek alphabets (µ, ν, ρ, ... = 0,1,2,3) are related to spacetime, and the Latin alphabets 4
(i,j,k,... = 0,1,2,3) denote indices, which are raised and lowered with the Minkowski metric η ij = diag (−1,+1,+1,+1). For our spacetime the connection G λ µν has the form G λ µν = e i λ ∂ µ e i ν + e j λ e i νω j iµ = Γ λ µν + K λ µν. (3.4) Here Γ j iµ is the Levi-Civita connection and Kj iµ is the contorsion. Let the metric has the form ds 2 = g ij dx i dx j . (3.5) Then the orthonormal tetrad components e i (x µ ) are related to the metric through so that the orthonormal condition reads as Here η ij = diag(−1,1,1,1), where we used the relation The quantities Γ j iµ and Kj iµ are defined as g µν = η ij e i µe j ν, (3.6) η ij = g µν e µ i eν j . (3.7) e i µe µ j = δi j. (3.8) Γ l jk = 1 2 glr {∂ k g rj + ∂ j g rk − ∂ r g jk } (3.9) and K λ µν = − 1 2 ( T λ µν + T µν λ + T νµ λ ) , (3.10) respectively. Here the components of the torsion tensor are given by The curvature R ρ σµν is defined as T λ µν = e i λ T i µν = G λ µν − G λ νµ, (3.11) T i µν = ∂ µ e i ν − ∂ ν e i µ + G i jµe j ν − G i jνe j µ. (3.12) R ρ σµν = e i ρ e j σR i jµν = ∂ µ G ρ σν − ∂ ν G ρ σµ + G ρ λµG λ σν − G ρ λνG λ σµ = ¯R ρ σµν + ∂ µ K ρ σν − ∂ ν K ρ σµ + K ρ λµK λ σν − K ρ λνK λ σµ +Γ ρ λµ Kλ σν − Γ ρ λν Kλ σµ + Γ λ σνK ρ λµ − Γ λ σµK ρ λν, (3.13) where the Riemann curvature of the Levi-Civita connection is defined in the standard way ¯R ρ σµν = ∂ µ Γ ρ σν − ∂ ν Γ ρ σµ + Γ ρ λµ Γλ σν − Γ ρ λν Γλ σµ. (3.14) Now we introduce two important quantities namely the curvature (R) and torsion (T) scalars as R = g ij R ij , (3.15) T = S ρ µν Tµν, ρ (3.16) where S ρ µν = 1 2 ( Kρ µν + δ µ ρT θ θν − δ ν ρT θ θµ ) . (3.17) Then the M 43 - model we write in the form (3.1). To conclude this subsection, we note that in GR, it is postulated that T λ µν = 0 and such 4-dimensional spacetime manifolds with metric and without torsion are labelled as V 4 . At the same time, it is a general convention to call U 4 , the manifolds endowed with metric and torsion. 5
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The action (3.8) can be written as<br />
where the point-like Lagrangian reads<br />
∫<br />
S T =<br />
dtL T , (2.14)<br />
The equations of F(T) gravity look like<br />
L T = a 3 (F − F T T) − 6F T aȧ 2 − a 3 L m . (2.15)<br />
12H 2 F T + F = ρ, (2.16)<br />
)<br />
48H 2 F TT Ḣ − F T<br />
(12H 2 + 4Ḣ − F = p, (2.17)<br />
˙ρ + 3H(ρ + p) = 0. (2.18)<br />
2.3 F(G) gravity<br />
The action of F(G) theory is given by<br />
∫<br />
S G =<br />
d 4 xe[F(G) + L m ], (2.19)<br />
where the Gauss – Bonnet scalar G for the FRW metric is<br />
G = G s = 24H 2 (Ḣ + H2 ). (2.20)<br />
3 Geometrical roots of F(R, T) gravity<br />
We start from the M 43 - model (about our notations, see e.g. Refs. [10]-[11]). This model is one<br />
of the representatives of F(R,T) gravity. The action of the M 43 - model reads as<br />
∫<br />
S 43 = d 4 x √ −g[F(R,T) + L m ],<br />
R = R s = ɛ 1 g µν R µν , (3.1)<br />
T = T s = ɛ 2 S ρ µν T ρ µν,<br />
where L m is the matter Lagrangian, ɛ i = ±1 (signature), R is the curvature scalar, T is the torsion<br />
scalar (about our notation see below). In this section we try to give one of the possible geometric<br />
formulations of M 43 - model. Note that we have different cases related with the signature: (1)<br />
ɛ 1 = 1,ɛ 2 = 1; (2) ɛ 1 = 1,ɛ 2 = −1; (3) ɛ 1 = −1,ɛ 2 = 1; (4) ɛ 1 = −1,ɛ 2 = −1. Also note that M 43 -<br />
model is a particular case of M 37 - model having the form<br />
∫<br />
S 37 = d 4 x √ −g[F(R,T) + L m ],<br />
where<br />
R = u + R s = u + ɛ 1 g µν R µν , (3.2)<br />
T = v + T s = v + ɛ 2 S ρ µν T ρ µν,<br />
are the standard forms of the curvature and torsion scalars.<br />
3.1 General case<br />
R s = ɛ 1 g µν R µν , T s = ɛ 2 S ρ µν T ρ µν (3.3)<br />
To understand the geometry of the M 43 - model, we consider some spacetime with the curvature<br />
and torsion so that its connection G λ µν is a sum of the curvature and torsion parts. In this paper,<br />
the Greek alphabets (µ, ν, ρ, ... = 0,1,2,3) are related to spacetime, and the Latin alphabets<br />
4
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