Potential Energy in General Relativity

I often hear about how general relativity is very complicated because of all forms of energy are considered, including gravitation's own gravitational binding energy. I have two questions:

1. In general relativity, objects following the motion of gravitation should simply be travelling by geodesics. In such 'free fall', why would there be any 'binding energy'?


2. From Einstein's field equations, $$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R = {8 \pi G \over c^4} T_{\mu \nu},$$ isn't the curvature only coupled to the energy-momentum tensor? As far as I understand, potential energy is not included inside the energy-momentum tensor.

Answer

1) I gather you mean gravitation potential energy of the test particle. Well, any such thing is only useful in so far as it is related to a constant of motion throughout the geodesic--in the case of gravitational potential, being part of the conserved mechanical energy, kinetic + potential. (Another example could be angular momentum.)

In GTR, these constants are given by a Killing vector field, which is an infinitesimal generator of an isometry: spacetime "looks the same" in the direction of a Killing vector. Most spacetimes do not have any, but by definition, a static spacetime has a timelike Killing vector field, and can always be put in the following form: $$ds^2 = -\lambda dt^2 + d\Sigma^2,$$ where $d\Sigma^2$ is the metric for any spacelike manifold and $\lambda$ is independent of $t$. The factor $\lambda^{1/2}$ is commonly called the gravitational redshift.

For example, for the Scwarzschild spacetime in the usual Schwarzschild coordinates, $\lambda = \left(1-\frac{2GM}{c^2R}\right)$, and the following is a constant of motion representing the specific (per-mass) energy of the freefalling particle: $$e = \left(1-\frac{2GM}{c^2r}\right)\frac{dt}{d\tau}.$$ This is the natural generalization of the total mechanical energy, including also rest-mass energy; for the Schwarzschild case, the spherical symmetry allows one to build an "effective potential" quite analogous to the Newtonian case, but that approach is less useful in general.

2) Gravitational energy cannot be explicitly included in the Einstein field equations because the equivalence principle--there is always a local inertial frame (the free-falling one) in which spacetime looks like the ordinary, flat, special-relativistic one. Hence if there was a frame-independent local notion of gravitational energy, i.e., a tensor, that tensor is zero in some local frame, and hence zero in every frame.

However, one can think of the non-linearity of the Einstein field equation as caused by gravitational energy itself interacting with spacetime. In this sense, gravitational energy is "implicitly" included. Another thing one can do is try to build another notion of gravitational energy that's not necessarily both local and frame-independent, e.g., Landau-Liftshitz pseudotensor and others.