How is the stress energy tensor obtained? In most textbooks, it's simply stated as
$$T^\mu{}_\nu=(\rho+P)U^\mu U_\nu-P\delta^\mu{}_\nu$$
I can see why this makes sense for a comoving observer at rest wrt. the perfect fluid. But I don't understand how the general case is arrived at. Would someone kindly explain?
The answer to this depends on what you're starting from. If you know the Einstein tensor, then you can find the stress-energy tensor from the Einstein field equations. If you know the Lagrangian density, then you can find the stress-energy tensor by variation with respect to the metric. If you know the rate at which energy-momentum is being transported along four orthogonal axes, then that corresponds to the stress-energy tensor.
