I Stress-Energy Tensor

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My understanding is that this tensor contains sources for spacetime curvature, analogous to how charge and current are sources for electric and magnetic fields. In other words, the elements of this tensor, such as mass density, are specified and used in the Einstein equation to solve for the metric.

In the derivation of the interior solution for a spherically symmetric static mass distribution, however, there appears the Tolman–Oppenheimer–Volkoff equation, a differential equation for the pressure of the material as a function of radial distance from the center. In other words, pressure is something to solve for rather than something that is specified (as the mass density is in this example). Any clarification would be helpful. Are the other elements specified or calculated?


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This is no different from electromagnetism. The equations of motion for the charges depend on the EM fields and the EM fields depend on the charges and their motion.
Are the other elements specified or calculated?
It depends on the particular case. But in general, "specified" does not always (or even often) mean "specify an exact value at every point in spacetime". Usually it just means giving some general properties.

In the case of the TOV equation, for example, the underlying general property is spherical symmetry: that constrains both the metric and the stress-energy tensor. Given spherical symmetry, it turns out that the energy density (which is a better term for the 0-0 component than mass density) satisfies an equation all by itself, independent of any other stress-energy tensor components, so we can pick any solution of that equation we like. The TOV equation then gives you an equation for the pressure given that you've picked a particular solution for the energy density. So neither one is really specified in advance; it's just that the mass density can be specified earlier in the process.


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Another example is the Friedmann-Lemaitre-Robertson-Walker metric, which constrains automatically (due to its large symmetry) the energy-stress tensor of "matter and radiation" to the form of an ideal fluid. The motion of the fluid thus is automatically adiabatic! Note that on the right-hand side of the Einstein-Hilbert field equation,
$$G_{\mu \nu}=-\frac{8 \pi G}{c^4} T^{\mu \nu},$$
there's no contribution from the gravitational field/metric.

This is analogous to the Maxwell equations, where also the only local sources of the field are charge and current distributions and no parts of the electromagnetic field (particularly the socalled "displacement current" does NOT belong to the local sources but is part of the left-hand side):
$$\partial_{\mu} F^{\mu \nu} = \frac{1}{c} j^{\mu}.$$

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