There are two different concepts were given by Newton and Einstein regarding gravity.
Imagine that you’ve got a particle of matter and a particle of antimatter, each with the same rest mass. You can annihilate them, and they’ll produce photons of a specific amount of energy, of the exact amount given by E = mc^2.
Now, imagine you had this particle/antiparticle pair moving rapidly, as though they had fallen from outer space, and then annihilated close to the surface of Earth. Those photons would now have extra energy: not just the E from E = mc^2, but the additional E from the amount of kinetic energy they gained by falling.
If we want to conserve energy, we have to understand that gravitational redshift (and blueshift) must be real. Newton’s gravity has no way to account for this, but in Einstein’s General Relativity, the curvature of space means that falling into a gravitational field makes you gain energy, and climbing out of a gravitational field makes you lose energy. The full and general relationship, then, for any moving object, isn’t just E = mc^2, but that E^2 = m^2.c^4 + p^2.c^2. (Where p is momentum.)
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