Question

This problem is " Spacetime and Geometry, An Introduction to General Relativity by Sean Carroll"
Chapter 4 - Problem 2

Please explain every thing.
Please write in the paper and then take a photo.

Answer 1

R_uv - 12 Rg_uv = 8 pi GT_uv R_uv = 0 we have delta (squareroot - g) = delta [(- g^-1)^-1/2] = -1/2 squareroot - g g_uv delta g^- uv delta s/delta g^uv = squareroot - g (R_uv - 1/2 Rg_uv) = 0 S = 1/8 pi G S_H + S_M T_uv = -1/squareroot g delta S_M/delta g^uv so we have T_uv V^u V^v greaterthanorequalto 0 thus we have c_uv^alpha vanish \r\n we have already derived d^n. instein equation by varying Hilbert action, Therefore S = integral d^4 times squareroot - g g^- uv R_uv (-1) Ricci tensor is constructed purely from connection. Therefore Chirsttoffel symbols F_uv^lambda tensor C_uv^lambda Gamma_uv^lambda = Gamma_uv^u lambda + C_uv^lambda we have Gamma_00^u = 12 g^u lambda (delta_0 g_lambda 0 + delta_0 g_0 lambda - delta_lambda g_00) g_uv = n_uv + h_vv g^uv = n^uv - h^vu For g_00 = --(1 + 2 Phi) we have Delta_u T^uv = 0 also we have Delta^u T_uv = 0 R_uv = K T_uv R_uv = K(T_uv - T2 g_uv)