# How to reduce Einstein's equation for perfect fluids

1. May 14, 2012

### joshyxc1979

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
$$D^a D_a V= 4 \pi V( \rho +3p)\\ R_{ab}=V^{-1}D_a D_b V+4 \pi ( \rho -p) g_{ab}$$
Where $$D_a$$ and $$R_{ab}$$ are the three dimensional covariant derivative and Ricci curvature tensor associated with $$g_{ab}$$.

2. May 15, 2012

### Matterwave

Is this a homework question? Have you tried it yourself?

3. May 15, 2012

### joshyxc1979

No its not a homework question, I read it in an article by J. of Math Phys 29 (2) feb 1988. Static uniform density stars must be spherical in GR

4. May 15, 2012

### jfy4

I'm sorry, could you write the whole line element? I don't know the components of $g_{ij}$ expliclity.