How to reduce Einstein's equation for perfect fluids

  • #1
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 $$g_{ab}$$.
 

Answers and Replies

  • #2
Matterwave
Science Advisor
Gold Member
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Is this a homework question? Have you tried it yourself?
 
  • #3
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
649
3
I'm sorry, could you write the whole line element? I don't know the components of [itex]g_{ij}[/itex] expliclity.
 

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