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When is spacetime homogenous and isotropic?

For example, some metric $g_{\mu \nu}$ is homogeneous and isotropic. We now construct effective metric

$$n_{\mu \nu} ~\rightarrow~ g_{\mu \nu} + f(x^\mu)v_\mu v_\nu.$$

where $v_\mu$ is 4-vector, and $f$ is scalar. Is the effective metric $n_{\mu \nu}$ homogeneous and isotropic as well?

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Hi tensor67. Welcome to Physics.SE. This site uses an unique TeX markup style called MathJax. This markup is very useful for understanding math equations and parameters. Please have a look here for an intro or our FAQ for more info. For example, \theta results $\theta$, \omega inserts $\omega$, etc. It's quite interesting. You can revise your post if you can ;-) –  Waffle's Crazy Peanut Apr 19 '13 at 13:18
    
Thank you, I'll have a look! –  tensor67 Apr 19 '13 at 13:22
    
Essentially a duplicate of physics.stackexchange.com/q/24881/2451 –  Qmechanic Apr 19 '13 at 16:01

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