Let us say that we have small object of mass ##m## at some location far away from the earth (with zero velocity compared to the earth). The energy of this object is according to relativity ##E=mc^2##.(adsbygoogle = window.adsbygoogle || []).push({});

Now we drop this object and it starts falling towards the earth, transforming potential energy to kinetic. Finally the objects hits the Earth. It may heat up from the impact but finally it cools down again.

What is the energy of the object now? It still contains the same number of atoms as initially but energy has been converted into heat that has radiated away. It can not really be ##E=mc^2## again if the mass has remained contant and by ##c## we mean velocity of light as measured locally which is invariant. Is ##E=mc^2## only supposed to hold in a non-gravitational setting?

If we assume something like:

##E=mc^2\sqrt{1-\frac{2GM}{rc^2}}##

Then at least the energy of an object at rest infinitelly close to the Schwarzschild radius of a black hole (using Schwarzschild coordinates) have zero energy, which seems logical? What is the correct interpretation of the energy formula ##E=mc^2## when the objects whose energy we are interested in is sunk into the gravitational field of (for instance) a planet?

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# E=mc^2 in a gravitational field

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