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Does General Relativity Violate Conservation of Energy?

Does General Relativity spacetime curvature conserve gravitational potential energy?

Meaning, if you added an object with mass M to a system, would the total increase of gravitational potential energy (mgh) of the system always equal mass energy (E = mc2)?

It’s hard to see how energy could be conserved, since the system you’re introducing mass into could have zero objects in it, in which case the increase in gravitational potential energy would be zero; or it could have tons of massive objects in it that are very far away, in which case the gravitational potential energy would need to be very large.

But this seems like it would be a major problem to General Relativity, since energy is always conserved in physics. It would seem, the only way to balance this would be to tweak General Relativity.

Either:

  1. The amount of spacetime curvature should depend on the total mgh of the surrounding massive objects, OR
  2. The amount of mass (for the new object) should depend on the total mgh of the surrounding massive objects, OR
  3. The force of gravity constant G is variable, and somehow locally related to the total mh of the surrounding massive objects.

If either of these were true, it would mean that standard General Relativity is an approximation. Perhaps the G factor is what we measure in our sector of the Universe, and it can vary in other places. That would seem like the best factor to tweak, as it shows up in Newtonian gravity:

as well as General Relavity, in Einstein’s Field Equations:

I don’t know the answer, but I do know that energy must be conserved in physics. It would seem that solving this question might give new insights into understanding dark matter, blackholes, and the fabric of spacetime.

Any thoughts? Feel free to share them below.

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