Vacuum Solutions: General Relativity Explained

[MeJennifer]

From Wikipedia http://en.wikipedia.org/wiki/Vacuum_solution_%28general_relativity%29" we can read:

" Since T^ab = 0 in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no energy. But the gravitational field can do work, so we must expect the gravitational field itself to possesses energy, and it does. However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest". "

Does this make any sense, or should we consider this some wild and confused interpretation?

 

[robphy]

From Wikipedia http://en.wikipedia.org/wiki/Vacuum_solution_%28general_relativity%29" we can read:

" Since T^ab = 0 in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no energy. But the gravitational field can do work, so we must expect the gravitational field itself to possesses energy, and it does. However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest". "

Does this make any sense, or should we consider this some wild and confused interpretation?

One lesson I have learned in trying to understand relativity is
to make clear definitions of terms, often formulated mathematically with the precision needed to prove a theorem or formulated operationally in terms of an experiment. The old classical definitions of terms, like "energy" and "time", etc... , need to be better defined, and most probably refined and given important adjectives or completely renamed if necessary.

 

[pervect]

From Wikipedia http://en.wikipedia.org/wiki/Vacuum_solution_%28general_relativity%29" we can read:

" Since T^ab = 0 in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no energy. But the gravitational field can do work, so we must expect the gravitational field itself to possesses energy, and it does. However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest". "

Does this make any sense, or should we consider this some wild and confused interpretation?

This makes sense - if it's a bit murky, a clear explanation would be rather long. The section could probably use a reference for the interested reader, MTW for instance has a section about some of these issues. I could look up the page if there's any interest.

One can also approach this from a different point of view, Noether's theorem provides the most insight. This is discussed for instance in http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

As robphy mentions, to be unambiguous a writer has to specify the particular "bookkeepping" system being used when talking about energy by using the specific name for that sort of energy - for specific examples, there are ADM energy, Komar energy (usually called Komar mass), Bondi energy, and these are all different than the SR defintion. The difference can be important, but only matters for non-isolated systems. All of the schemes give the same energy for isolated systems (AFAIK, note that some systems may lack the requirements needed for some of the defintions to apply), but in general they assign the energy to different locations.

Example: the Komar energy of the interior of a pressure vessel filled with hot gas (i.e. ignoring the shell) will be different and larger than the energy one computes from special relativity.

 

[MeJennifer]

The section could probably use a reference for the interested reader, MTW for instance has a section about some of these issues. I could look up the page if there's any interest.

Sure, if you could mention the paragraph(s).

 

[Jonathan Scott]

The Wikipedia "Vacuum solutions" statement seems consistent with the usual way of describing things in GR. Basically, if you describe things from a local inertial viewpoint, you can't even see the first-order effects of gravity locally, so it's not totally surprising that in the tensor view you can't see gravitational energy.

MTW section 20.4 "Why the energy of the gravitational field cannot be localized" argues fiercely that energy cannot be localized, but in my opinion rather spoils it with an analogy in the last paragraph of being unable to localize the curvature on the surface of a potato, since on such a surface the "total angular deficit" is well-defined locally and globally.

Example: the Komar energy of the interior of a pressure vessel filled with hot gas (i.e. ignoring the shell) will be different and larger than the energy one computes from special relativity.

This continues to bother me, and definitely doesn't seem to provide a physically plausible explanation of where energy is actually located.

The Komar energy is based on a Gaussian integral approach to the energy within a large volume (as is used for example to show the equivalence in electrostatics between the energy of a system of charges and the energy of the field) plus the assumption from Einstein's vacuum equations that the energy density in a vacuum is zero.

For the simple spherically symmetrical case it is equivalent to adding in 3 times the integral of the pressure (essentially once for each dimension of space, corresponding to the diagonal terms of the energy momentum tensor) to the total energy as in special relativity, calculated from the proper mass times the time dilation. For a fluid sphere in Newtonian physics, this term gives an amount which happens to be exactly equal to the potential energy of formation of the sphere, and a similar effect occurs in GR.

In GR, the time-dilation loss of energy of a mass due to its own potential is exactly twice the potential energy, and the Komar pressure term compensates for this so that the overall Komar energy is equal to the original energy minus the potential energy of formation, matching Newtonian concepts of conservation of energy.

In classical physics, building up pressure stores energy which is equal to the integral of the pressure times the change in volume. However, within Special Relativity, any energy due to such pressure has already been included in the energy density, so this is extra. Also, any actual practical energy stored due to pressure depends on the compressibility of the material, and is much less than even the product of the pressure and the volume, let alone three times that.

On top of that, if you consider the Newtonian case where two objects are present, the 3 times pressure correction no longer adds up to the overall potential energy.

I have seen some statements that the Komar pressure term is like the case where twice the thermal kinetic energy of an ideal monatomic gas is equal to its potential energy, as in the virial theorem. However, I can't pin that down to a physical interpretation, especially since it apparently relies on the implausible assumption that all objects being described by GR are ideal monatomic gases.

It therefore seems to me very doubtful that the pressure term in the Komar energy expression represents a real physical presence of energy within the object.

As I've been saying in a separate thread "Effect of gravitational potential on energy" I've not yet found a satisfactory explanation of exactly where the gravitational energy can be located in a situation involving multiple objects (using Newtonian approximation to GR because GR itself gets too complicated), but as far as I can see, the simplest physical solution would be that relative to a flat background coordinate system there is in fact gravitational energy distributed within the field, with a positive energy density equal to half of the square of the field (as in a Maxwell model but with the positive sign).

This would mean that the total field energy would have exactly the same magnitude as the potential energy, so for any configuration of masses the total rest energy would be the time-dilated energy of the masses themselves plus the field energy, which would be equal to the original energy of the masses minus the potential energy. (Any kinetic energy would be the same on both sides of this equation).

This scheme says that there is "energy" in the field, where GR says there is no "energy" in the field, so this idea appears to conflict with GR, but I think that the meaning of "energy" is sufficiently different in these two contexts that I do not yet know whether there is necessarily any conflict at all.