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Assume that the universe is homogenous and isotropic, and the following equation holds:

\begin{equation}R_{00}-\frac{1}{2}g_{00}R=8\pi GT_{00}; \space \space \nabla_{\mu}T^{\mu 0}=0.\end{equation}

How do I prove that the following equations are identically satisfied provided that the above two are satisfied?

\begin{equation}R_{0i}-\frac{1}{2}g_{0i}R=8\pi GT_{0i}; \space \space R_{ij}-\frac{1}{2}g_{ij}R=8\pi GT_{ij}; \space \space \nabla_{\mu}T^{\mu i}=0.\end{equation}

My approach was to write $g_{00}=1$ and $g_{ij}=-a^2\gamma_{ij}$ and evaluate the Ricci tensors and so on, but I know this is not the way to do it. Can anyone suggest me the way?

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closed as off-topic by Brandon Enright, Chris White, Kyle Kanos, Kyle Oman, Jim Jun 20 '14 at 14:50

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Brandon Enright, Chris White, Kyle Kanos, Kyle Oman, Jim
If this question can be reworded to fit the rules in the help center, please edit the question.

Here is a simple approach that might work. Start by defining

$$F_{\mu\nu} \equiv G_{\mu\nu} - T_{\mu\nu}$$

where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor. Now from what you know we have $$F_{00} = 0$$

$$\nabla^{\mu}F_{\mu0} = 0$$

You must show that $F_{\mu\nu} = 0$. Writing out the last equation gives

$$0 = \partial_{i} F^{i 0} + \Gamma^{\mu}_{\mu \alpha}F^{\alpha 0} + \Gamma^{0}_{\mu \alpha}F^{\mu \alpha}$$

Homogenous and isotropic implies that gradients vanish and that $F^{11}=F^{22}=F^{33}$ so

$$0 = a^2 H \delta_{ij} F^{i j}$$

This shows that $F_{00} = F_{11} = F_{22} = F_{33} = 0$. Now to show $F_{ij} = 0$ for $i\not =j$ you might need some additional assumptions on the energy momentum tensor $T^{\mu\nu}$, for example $T^{ij} = 0$.

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First what we know: $G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2}g_{\mu \nu} R$ and $T_{\mu \nu}$ are tensors in that they transform properly under coordinate transformations ($G_{\mu \nu}$ by construction and $T_{\mu \nu}$ because of the EFEs), so it doesn't matter which frame we do our measurements in, this tensor equation will always hold.

Suppose that a comoving observer takes careful measurements in his frame and finds the first equation to be true. This is a special case of how we determine what an observer with an arbitrary four-velocity will measure, which is the contraction with that four-velocity $$G_{\mu \nu} u^{\mu} u^{\nu} = 8\pi T_{\mu \nu} u^{\mu} u^{\nu}$$ Then imagine other observers with four velocities of the form $u_i^{\alpha} = Ae_0^{\alpha} + B e_i^{\alpha}$, where $e_0$ denotes the unit vector in the time direction, $e_1$ denotes the unit vector in the $1/x/r/$whathaveyou direction, etc., and $A$ and $B$ are normalization factors. The above equation is an invariant scalar equation and from this fact and a plethora of observers we can build up the rest of the relations. The same procedure can be applied to the energy conservation equation, only now we are contracting $u_{\nu}\nabla_{\mu} T^{\mu \nu}$

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