What seems to be the common consensus in physics is that a gravitational field does not have a stress energy tensor due to the equivalence principle, but rather a psuedotensor.
A more modern (at least since 1982 work of Penrose) perspective is that gravitational energy–momentum is a quasi-local quantity rather than simply local, i.e. associated to a closed 2-surface. Classical pseudotensors are incorporated into this picture through the integration of their components over the spatial volume that have this 2-surface as a boundary. Through the use of superpotentials defined for each pseudotensor the volume integral is expressed as a boundary term only, and thus is independent of the choice of coordinates and space-time slicing inside the volume. Such quasi-local energy–momentum(s) of the gravitational field are covariant expressions but they do depend on the choice of boundary conditions on the 2-surface. Consequently there are several choices for these quasi-local quantities depending on the physics of the problem. This situation is somewhat analogous to the notion of energy in thermodynamics, where the choice of variables and boundary conditions leads to many “energies”: internal, Gibbs, enthalpy, etc.
For more information on quasi-local approach to gravitational energy–momentum see the review:
- Szabados, L. B. (2009). Quasi-local energy-momentum and angular momentum in general relativity. Living Reviews in Relativity, 12(1), 4, doi:10.12942/lrr-2009-4.
Is this pseudotensor frame dependent?
Yes, pseudotensors are frame dependent.
Can I choose a frame in a gravitational field that doesn't experience gravitational stress energy?
This is a more complicated question. Through the choice of coordinates one could indeed make stress–energy pseudotensors zero at any given point of spacetime, but not at all the points. It boils down to what meaning we give to “experience gravitational stress energy”. If we concentrate on experimentally measurable (at least in principle) quantities, and also consider the fact that any experimental apparatus must have at least nonzero physical dimensions, then the answer would be no.
As an example of experiencing gravitational stress energy one could consider tidal heating of Io. This physical situation is a good illustration of quasi-local energy-momentum balance, with the surface of a body being the natural 2-surface boundary. State of motion of a body is frame dependent but no choice of reference frames/coordinate systems could remove the conversion of gravitational energy into the energy of geological processes.