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In the perfect fluid solution for general relativity, you get

$$T_{ab} = u_a u_b (\mu + p) - g_{ab} \, p$$

I've seen varying descriptions of what $\mu$ is, and some places describe it as the local energy density. But I also thought that $T_{00}$ was the energy density.

So, what is the physical interpretation and difference between these two quantities? Because they are clearly not equal.

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  • $\begingroup$ For the record, Wikipedia appears to claim that it's the 'matter density'. $\endgroup$
    – HDE 226868
    Dec 3, 2014 at 3:15
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    $\begingroup$ MTW refers to both as "mass-energy density". $\endgroup$
    – XYZT
    Dec 3, 2014 at 3:25
  • $\begingroup$ I'd take MTW over Wikipedia, though they seem similar. $\endgroup$
    – HDE 226868
    Dec 3, 2014 at 3:29
  • $\begingroup$ But that doesn't help! They are not equal quantities :( $\endgroup$
    – XYZT
    Dec 3, 2014 at 3:33

1 Answer 1

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If we work in the rest frame of the fluid then $u_0 = 1$. In that case the formula for $T_{00}$ gives:

$$ T_{00} = (\mu + p) + \eta_{00}p = (\mu + p) - p = \mu $$

So $T_{00}$ is just the energy density $\mu$ as you say.

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