A Gravitational Potential Energy & the Equivalence Principle

[exmarine]

TL;DR Summary

2 questions about the MTW textbook section on why the energy of the gravitational field cannot be localized

First, in section 20.4, after listing all the things gravitational potential energy does not do, they say the equivalence principle forbids it being localized. I thought I understood the equivalence principle, but maybe I don’t. Any comments explaining that would be appreciated.

Second, they allude to previous attempts to “answer this question”. Who, what, where, and any links to reference material would be appreciated.

 

[anuttarasammyak]

Second, they allude to previous attempts to “answer this question”. Who, what, where, and any links to reference material would be appreciated.

Hello. I have once read a Einstein's paper saying two body system e.g. the sun and the Earth has mass or energy/c^2 of
M+m-\frac{GMm}{c^2r}
for kinetic property and for gravitational effect to the third body far from them in the frame of reference where space time is almost flat in great distance from the bodies. The third term is gravitational energy which reduces total mass but we can not say how it is distributed in an absolute way. I should appreciate someone may tell where in the web the paper is.

 

[PeterDonis]

they say the equivalence principle forbids it being localized. I thought I understood the equivalence principle, but maybe I don’t. Any comments explaining that would be appreciated.

One way of stating the equivalence principle is that, by an appropriate choice of coordinates, you can always make "the gravitational field" vanish in a small, localized patch of spacetime. Making "the gravitational field" vanish means making any localized "energy stored in the gravitational field" vanish as well--but if "energy stored in the gravitational field" were something localizable, it would be impossible to make it vanish by any choice of coordinates. All other kinds of energy are contained in the stress-energy tensor, and you can't make the stress-energy tensor vanish just by choosing coordinates. So there can't be any localized "energy stored in the gravitational field", because if there were, it would have to be contained in something like the stress-energy tensor, which could not be made to vanish just by choosing coordinates.

 

[PeterDonis]

they allude to previous attempts to “answer this question”

They are alluding to the various pseudotensors described in the previous section. A reference to Landau & Lifschitz is given there; the pseudotensor they defined is the one most commonly encountered in discussions of this topic.