Using the metric $$ds^2=-V^2 dt^2 +g_{ab} dx^a dx^b$$, where $$g_{ij}$$ is the Riemannian metric of the constant t-surfaces, and V is the gravitational potential, show that Einstein's equation $$G_{ij}=8\pi T_{ij}$$ for perfect fluids reduces to the system(adsbygoogle = window.adsbygoogle || []).push({});

$$D^a D_a V= 4 \pi V( \rho +3p)\\

R_{ab}=V^{-1}D_a D_b V+4 \pi ( \rho -p) g_{ab}$$

Where $$D_a$$ and $$R_{ab}$$ are the three dimensional covariant derivative and Ricci curvature tensor associated with $$g_{ab}$$.

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# How to reduce Einstein's equation for perfect fluids

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