Gravitational potential energy in GR

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I read that General Relativity reduces to Newtonian when v<<c as well as when Newtonian gravitational potential energies are small compared to mc^2.

What is the GR version of gravitational potential energy?
 

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Jonathan Scott
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I read that General Relativity reduces to Newtonian when v<<c as well as when Newtonian gravitational potential energies are small compared to mc^2.

What is the GR version of gravitational potential energy?
In GR, the local time rate near massive objects varies compared with the time rate of a distant observer. This effectively causes a fractional change in the rest energy of a test body when it is moved into that potential, which corresponds to the Newtonian potential energy. For the potential at distance r from a central mass M, the relative time rate is approximately (1-GM/rc2) so the potential energy is a fraction -GM/rc2 of the rest energy, or a fraction -GM/r of the test body rest mass, as in Newtonian theory.

This approximate scheme doesn't however work when the potential energy of the source is considered as well as that of a test object, because a simple approximation to GR would suggest that their potential energy due to each other is equal, which means that when the test mass was brought near to the source mass the total energy of the system has been decreased by twice the Newtonian potential energy. In Newtonian theory, we understand that potential energy is a property of the system, so we should only count it for one or the other object, but it is not clear (at least to me) how to correct the GR approximation to allow for this.
 
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Thanks very much for your input. I can see what is meant now by the GR version of gravitational potential energy.

It is interesting how ‘clock rates’ can be directly translated into potential energy.
Rest energy [tex]E = (1-\frac{GM}{rc^2})mc^2=mc^2-\frac{GMm}{r}[/tex]

But [tex](1-\frac{GM}{rc^2})[/tex] is a clock rate ratio. Can multiplying a clock rate ratio by mc^2 be justified?

Regarding your last point about GR and twice the potential energy, doesn’t one choose a reference frame in GR? Taking a system wide approach sounds more like a Newtonian approach.
 
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Jonathan Scott
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But [tex](1-\frac{GM}{rc^2})[/tex] is a clock rate ratio. Can multiplying a clock rate ratio by mc^2 be justified?
Energy is related to frequency by Planck's constant: [itex]E = h\nu = \hbar \omega[/itex]; If time is running slower in one place relative to another, the energy of objects at that place is decreased in the same proportion. For the simplest comparison with Newton's theory, we can take the reference point to be somewhere "distant". As a first order approximation, we can also take the ratio of the clock rate at two points within the potential to get a Newtonian potential difference:

[tex]
\frac{1-Gm/r_1\, c^2}{1-Gm/r_2\, c^2} \approx {1-Gm/{r_1\, c^2} + Gm/{r_2\,c^2}}
[/tex]

Regarding your last point about GR and twice the potential energy, doesn’t one choose a reference frame in GR? Taking a system wide approach sounds more like a Newtonian approach.
In both Newtonian theory and GR, energy is relative to a particular observer frame. Making it add up in a sensible way in GR is an extremely advanced and somewhat speculative topic.
 

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