Ratbay Myrzakulov

Now we are ready to write the explicit forms of the curvature and torsion scalars. We have
R = 6(Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 − 3f 2 (3.30)
T = 6(h 2 − f 2 ). (3.31)
So finally for the FRW metric, the M 43 - model takes the form
∫
S 43 = d 4 x √ −g[F(R,T) + L m ],
R = 6(Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 − 3f 2 , (3.32)
T = 6(h 2 − f 2 ).
It (that is the M 43 - model) is one of geometrical realizations of F(R,T) gravity in the sense that
it was derived from the purely geometrical point of view.
4 Lagrangian formulation of F(R, T) gravity
Of course, we can work with the form (3.32) of F(R,T) gravity. But a more interesting and general
form of F(R,T) gravity is the so-called M 37 - model. The action of the M 37 - gravity reads as [10]
∫
S 37 = d 4 x √ −g[F(R,T) + L m ],
where
R = u + R s = u + 6ɛ 1 (Ḣ + 2H2 ), (4.1)
T = v + T s = v + 6ɛ 2 H 2 ,
R s = 6ɛ 1 (Ḣ + 2H2 ), T s = 6ɛ 2 H 2 . (4.2)
So we can see that here instead of two functions h and f in (3.32), we introduced two new functions
u and v. For example, for (3.32) these functions have the form
u = 6(1 − ɛ 1 )(Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 − 3f 2 , (4.3)
v = 6(h 2 − f 2 − ɛ 2 H 2 ) (4.4)
that again tells us that the M 43 - model is a particular case of M 37 - model [Note that if ɛ 1 = 1 = ɛ 2
we have u = 6ḣ + 18Hh + 6h2 − 3f 2 , v = 6(h 2 − f 2 − H 2 )]. But in general we think (or assume)
that u = u(t,a,ȧ,ä, ... a,...;f i ) and v = v(t,a,ȧ,ä, ... a,...;g i ), while f i and g i are some unknown
functions related with the geometry of the spacetime. So below we will work with a more general
form of F(R,T) gravity namely the M 37 - gravity (4.1). Introducing the Lagrangian multipliers
we now can rewrite the action (4.1) as
∫
S 37 = dta
{F(R,T) 3 ȧ
− λ
[T 2 ] [ (ä )] }
− v − 6ɛ 2
a 2 − γ R − u − 6ɛ 1
a + ȧ2
a 2 + L m , (4.5)
where λ and γ are Lagrange multipliers. If we take the variations with respect to T and R of this
action, we get
λ = F T , γ = F R . (4.6)
Therefore, the action (4.5) can be rewritten as
∫
S 37 =
dta 3 {
F(R,T) − F T
[
T − v − 6ɛ 2
ȧ 2
a 2 ]
− F R
[
R − u − 6ɛ 1
(ä
Then the corresponding point-like Lagrangian reads
a + ȧ2
a 2 )]
+ L m
}
. (4.7)
L 37 = a 3 [F − (T − v)F T − (R − u)F R + L m ] − 6(ɛ 1 F R − ɛ 2 F T )aȧ 2 − 6ɛ 1 (F RR Ṙ + F RT ˙ T)a 2 ȧ. (4.8)
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