The ADM formalism is supposed to prove conservation of the energy of a closed system when observed from "infinity".
But there is a conceptual problem in this. Suppose that we initially have a system S which is static, and we have a static solution of the Einstein equations. Let M be the ADM mass of S at infinity.
Let us then make some change to S. Some masses inside S move and produce gravitational waves. Conservation of energy requires that the total mass of S plus the energy of the waves stays the same.
Let us then calculate the new ADM mass as the limit at infinity. Since the speed of light is finite, no information about the change in S has yet reached infinity. The metric is the same as before, and the ADM mass is trivially the same old M!
What we would like to have is the conservation law:
(*) The total energy of the system S plus the energy of the emitted gravitational waves is conserved. We calculate the energy of the waves using some approximation method.
Almost nothing is known about the existence of solutions for the Einstein equations. Therefore, (*) is an open problem.
We need to check the original papers about the ADM formalism. What do the authors state about conservation laws?
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