How are mass and density treated in general relativity? [duplicate]

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?

marked as duplicate by John Rennie general-relativity StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 22 '15 at 9:59
• Apparently the 4-current comes from continuity equation for $T^{\mu \nu}$. But I guess this is also assuming a perfect fluid. – Otto Oct 22 '15 at 9:01