Is Gravity Affected by its Own Potential Energy?

[Sam Gralla]

And why do you think that there is any "gravitational energy" at all?

If you can attribute all of the total energy of the system to "matter energy" plus radiation energy without anything left then there is no reason to think that there anything like "field energy".

There is such a thing as "gravitational energy" because gravitational waves carry it. You can precisely characterize the rate at which an isolated system is losing energy to gravitational-wave emission. What you can't do is say that so much of the energy was at one place in spacetime and so much of the energy was elsewhere. or whether the energy "came from" matter or field energy. (Of course, you could try to make such a statement using a "psuedotensor" as somebody brought up, but somebody else could come along with another inequivalent pseudotensor and claim that *that* one was the true gravitational energy. What everyone will agree about is the total energy in the system as well as its rate of change.) People do use "stress-energy pseudotensors" in computations, but that's mainly for convenience in performing some specific computation. All claims about a "true" local gravitational energy density have disappeared from the literature at this point. (There is a community still looking for "quasi-local" gravitational energy, but that doesn't seem to work very well either.)

You may be trying to call this sort of energy "radiation energy" so you don't need "field energy". But there are reasons for saying that even a non-radiating system has "gravitational binding energy" (that can't be localized). For example, in the Newtonian limit of GR the energy/mass of an isolated system can be written as an integral over the mass density of the matter. But as soon as you go back to GR (or just a post-Newtonian correction), this property is lost. What is the other contribution? It makes sense to think of it as energy in the field. But there's no way to write it as an integral over matter plus an integral over field, so again it makes sense to say "gravitational energy exists but can't be localized".

 

[pervect]

Do you happen to have MTW's textbook, "Graitation"? I know they have a good section on the topic of why you can't localize the energy of a gravitational field.

I'm not sure how much more clear I can be, but I'll try saying it again.

Covariance is an important physical principle. It boils down to saying that measurements made by different observers represnt the same underlying reality.

The sort of covariance we need for relativity is Lorentz covariance. Any four-vector, regardless of whether it is (time, distance) or (energy, momentum) must transform via the lorentz transforms to have a physical meaning that's independent of the coordinate system.

If you don't have covariance, your quantity cannot be defined in an observer independent way.

Pseudotensors, in the sense used in General Relatiavity (i.e. the energy pseudotensor you refer to) do not define energy in a way that's independent of the observer.

Some people have remarked, with some merit, that the "pseudotensors" in GR are really just non-tensors.

The thing that makes energy pseudotensors useful at all is that while they don't offer an observer-compatible definition of energy, the total energy computed via them will transform properly given the proper conditions (usually asymptotic flatness).

So the pseudotensors themselves do not offer any physically meaningful way to localize energy because different observers will, as other posters have remarked, not have compatible views of how the energy is distributed.

They do allow you to come up with a total energy that everyone agrees on, however. (And they aren't the only method of doing it, there's a proof I think that the pseudotensor definition of energy matches the Bondi defiition).

 

[zonde]

There is such a thing as "gravitational energy" because gravitational waves carry it. You can precisely characterize the rate at which an isolated system is losing energy to gravitational-wave emission.

Isolated system can't loose energy.
Actually we don't have truly isolated systems so in order to model such a system we can imagine system in environment that mirrors all the outward interactions of the system with equivalent inward interactions.
This means that if the system is emitting gravitational waves then equivalent waves are directed toward the system from environment. And I would say that if gravitating system can emit these waves then it can absorb the same waves unless you have some strong arguments why this shouldn't be so.

What you can't do is say that so much of the energy was at one place in spacetime and so much of the energy was elsewhere. or whether the energy "came from" matter or field energy. (Of course, you could try to make such a statement using a "psuedotensor" as somebody brought up, but somebody else could come along with another inequivalent pseudotensor and claim that *that* one was the true gravitational energy. What everyone will agree about is the total energy in the system as well as its rate of change.) People do use "stress-energy pseudotensors" in computations, but that's mainly for convenience in performing some specific computation. All claims about a "true" local gravitational energy density have disappeared from the literature at this point. (There is a community still looking for "quasi-local" gravitational energy, but that doesn't seem to work very well either.)

What I say is that mass is moving down in gravitational potential when the system emits gravitational waves. So it's mass that is loosing this energy. The same way if you would try to reverse this process it's mass where you have to put this energy to move it upwards in gravitational potential.

You may be trying to call this sort of energy "radiation energy" so you don't need "field energy". But there are reasons for saying that even a non-radiating system has "gravitational binding energy" (that can't be localized). For example, in the Newtonian limit of GR the energy/mass of an isolated system can be written as an integral over the mass density of the matter. But as soon as you go back to GR (or just a post-Newtonian correction), this property is lost. What is the other contribution? It makes sense to think of it as energy in the field. But there's no way to write it as an integral over matter plus an integral over field, so again it makes sense to say "gravitational energy exists but can't be localized".

I have problems with that. "Gravitational binding energy" is negative energy. So we have missing energy not excess energy.

 

[jfy4]

Thanks pervect,

your helpful as always. I do have that book and I take a look at that section.