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Background:

I am confused by how mass relates to the equations in general relativity. For example, given a certain mass density distribution, I am unsure how to express a system in terms of GR.

Example

Einstein field equation:

$G_{\mu \nu} = 8\pi T_{\mu \nu}$

$T_{\mu \nu}$ encodes both mass and energy, but I am unsure how. For example, even if I know the whole phase space density of the system $f(x^\mu,p^\mu)$, I do not know how to express $T_{\mu \nu}$, except in some special cases such as when dealing with a perfect fluid.

Question(s)

  • Is there any general way to express mass / density in GR?
  • Additional/Bonus question: How would one express e.g. mass flow near a black hole, when most of the gravity contribution is from the black hole metric?

Any sources / additional reading is appreciated :)

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marked as duplicate by John Rennie general-relativity Oct 22 '15 at 9:59

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Apparently the 4-current comes from continuity equation for $T^{\mu \nu}$. But I guess this is also assuming a perfect fluid. $\endgroup$ – Otto Oct 22 '15 at 9:01
  • $\begingroup$ The stress-energy tensor isn't a description of mass/density. It is an object that (as usually written) has the energy density as one of its ten components. Likewise the current has the density as one of its four components. It isn't clear to me what you are asking. $\endgroup$ – John Rennie Oct 22 '15 at 9:08
  • $\begingroup$ Let me revise my question a little. $\endgroup$ – Otto Oct 22 '15 at 9:15
  • $\begingroup$ @JohnRennie Cheers for the comment; I tried clearing up my question a little bit. Basically, I am confused how to relate mass to the equations in general relativity. $\endgroup$ – Otto Oct 22 '15 at 9:27
  • $\begingroup$ @JohnRennie Thanks for the link, yeah it's a duplicate. Sorry about that. Cheers! Edit: Now I got it; now I'm confused by what I was told by someone else about not being able to construct the stress-energy tensor in a general case. $\endgroup$ – Otto Oct 22 '15 at 9:56