Subject: Relativity

Question : Please perform step by step explanation of equation#19 ?

1023% 1:34 AM 1704.05759.pdf Q . . . where Go = G(do) = 0, To = Po - 3Po, Po and Po are the unperturbed energy density and pressure, respectively. To explore the stability regions, we consider linear form of equation of state as P(t) = wp(t) and define linear perturbations in the scale factor and energy density as follows a(t) = an + agba(t), p(t) = po + poop(t). (16) where ba(t) and op(t) represent the perturbed scale factor and energy density, respectively. Applying the Taylor series expansion in two variables upto first order with the assumption that f(G, T) is an analytic function, we have f(9,T) = f(Go, To) + fc(Go, To)69 + fr(Go, To)ST, (17) where oG and of have the following expressions 89 = -ba, ST = Toop, (18) where da = (6a). Using Eqs.(14)-(18) in (10) and (11), we obtain the linearized perturbed field equations as follows 65a + 24po(1 + w)for(Go, To)ba + aapolk? + (1+ w)fr(Go, To) + -(1 - 3c)fr(Go. To) + Po(1 + w)(1 - 3w)far(Go. To)jop = 0, (19) 2 ba + 26a - -ofcc(Go, To)ba(") + polk2w - =(1 - 3w) fr(Go, To)jop - 4- (1 - 3w) for(Go, To)op = 0. (20) These equations show that the perturbations in a(t) are related with density perturbations. In the following subsections, we discuss the stability modes for conserved as well as non-conserved EMT. 3.1 Conserved EMT In this case, we assume that general conservation law holds in f(9, T) gravity. For this purpose, the right hand side of Eq.(13) must be zero which yields (P + - 1) fx(9.T) + ( p+ P)afr(9,T) = 0. (21) 7 O 0where a(t) is the scale factor. The energy-momentum tensor for perfect fluid is given by Two = (p + P)up,u, - Pg, (8) where p. P and up, represent the energy density, pressure and four-velocity of the matter distribution, respectively. For perfect fluid as cosmic matter distribution with Cm = -P, Eq.(5) becomes [7] (9) Using Eqs.(7)-(9) in (4), we obtain the following set of field equations 2 ( 1 +2 ) = Rep+; 1 (9. ) + (p+ P)fr (9. T) - 12-(1+%?) x fo(9, T) + 12- (1 + 23)afo(9, T), (10) -(1 + a?) - 20a = ka?P- -a'S(9.T) + 129(1+ 3)5.(9, T) - Saad, fc(9, T) - 4(1 + a3)ufc (9, T). (11) where 9 = =(1+aja, T= p-3P, (12) and dot represents the time derivative. The conservation equation (6) for perfect fluid yields p + 3-(p + P) = $2 + ST ( G. T) -1 [ ( P + $ 1 ) 1 (9. 1) + ( + P) x afr(G,T)]. (13) 3 Stability of Einstein Universe In this section, we analyze the stability of EU against linear homogeneous perturbations in the background of f(9, T) gravity. For EU, a(t) = do = constant and consequently, the field equations (10) and (11) reduce to = k' po + = f(Go. To) + (po + Po)fr(Go, To). (14) 1 = KPo - =f(Go, To). (15) 6