Covariant derivative of stress-energy tensor

[solveforX]

hi, I understand that T^ab_,b=0 because the change in density equals the negative divergence, but why do the christoffel symbols vanish for T^ab_;b=0?

 

[Ben Niehoff]

They don't. Why would you think that?

I think your first equation came from flat space, because it is not true in curved space.

 

[WannabeNewton]

$\triangledown _{\mu }T^{\mu \nu } = 0$ can be gotten from $\triangledown_{\mu }G^{\mu \nu } = 0$ which is a consequence of the second bianchi identity. You also know that $T^{\mu \nu }, _{\mu } = 0$ but what you can do is say that locally this is the same thing as $T^{\mu \nu }; _{\mu } = 0$ and if this is true for some coordinate system it will be true for all coordinate systems.

 

[solveforX]

thank you