There is tricky qualification about E=mc**2.

It does NOT strictly hold for the energy of the gravitational field.

In fact, the gravitational field is NOT like other force fields (the electromagnetic field and the force fields of the strong nuclear force and weak nuclear force). It is an emergent field from the curvature of space and it does NOT have localizable energy that itself gravitates. Albert Einstein (1879--1955) ruled that out in developing general relativity (1915) since it led to the paradox of mass-energy creating a gravitational field that had its own mass-energy that created its own gravitational field, and so on ad infinitum.

The curvature of space is what creates gravity and that curvature of space encodes a non-local gravitational field energy implicitly that does NOT in itself have a gravitating effect.

The non-local gravitational field energy does NOT appear in the energy-momentum tensor T_μν which dictates the curvature of space in the Einstein field equations (which along with the geodesic equation are the core formulae of general relativity (1915)). Hence, general relativity (1915) avoids the ad infinitum paradox discussed above.

Now by "non-local", we mean there is NOT so much energy here and so much energy there and there is NO density of energy. This is unlike the other force fields.

Consider gravitational waves again. When they are emitted from a physical system (e.g., slowly from a binary pulsar or in sudden blast from a binary black hole merger) energy (or, if you prefer, mass-energy) is lost from the physical system, but as the gravitational waves propagate they make NO contribution to the energy-momentum tensor T_μν of the space they propagate through. In fact, the energy-momentum tensor T_μν would be all zeros if there were nothing else in space. Yet when the gravitational waves interact with other physical systems, they deposit energy. Where was that energy between emission and interaction of the gravitational waves? It wasn't locally in the energy-momentum tensor T_μν. Also, as mentioned above, general relativity (1915) does NOT guarantee that the amount of energy emitted will be recovered ever from the gravitational waves (see Roger Penrose, The Road to Reality, 2004, p. 467--468).

To give an imperfect analogy, say you wanted to get a car to Reno, Nevada. You could drive it and then it was a car all the time. Or you could disassemble the car and send the parts separately to Reno and then reassemble the car. Did the car exist when in parts? Non-locally. In fact, general relativity (1915) does NOT guarantee the "car" will be the same size when reassembled: it could be smaller or bigger.

For more on the tricky point of non-local gravitational field energy, see Roger Penrose, The Road to Reality, 2004, p. 464--469.

File: Relativity file: e_mc2_1bb.html.