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 :)
