Ratbay Myrzakulov

(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.
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