Does Spacetime Absorb Energy in General Relativity?

[Suekdccia]

TL;DR Summary

If the conservation of energy is not well defined in General Relativity, can energy be created (or transferred to matter)?

Some physicists prefer to explain the problem of conservation of energy in General Relativity by considering the gravitational potential energy of the universe that would cancel all the other energies and therefore the energy in the universe would be conserved this way.

However, many other physicists opt to just say that energy is not conserved [1]. If we take this explanation, we can conclude that energy can be created or destroyed in cosmological scales, as in [1] says:


"*In general relativity spacetime can give energy to matter, or absorb it from matter, so that the total energy simply isn’t conserved*"

The photon redshifting due to the expansion of the universe is usually given to indicate that energy can be lost in General Relativity. But I cannot find anything that would be an example of spacetime creating energy or giving it to matter (as it says in the reference [1] that I quoted). Therefore, if the conservation of energy is not well defined in General Relativity, can energy be created?


[1]: https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[Ibix]

Therefore, if the conservation of energy is not well defined in General Relativity, can energy be created?

Feynman's sticky beads is a classic. They use gravitational waves to generate heat - but it's not at all clear where (if anywhere) the energy was beforehand.

Also, note that a contracting universe is a possible solution to the field equations, although not of much real world interest because we seem to be in an expanding one. In that case you get photon blue shift.

 

[PeterDonis]

the problem of global conservation of energy in General Relativity

See the bolded qualifier I added. It is very important. There is no problem whatever with local conservation of energy in GR; the covariant divergence of the stress-energy tensor must be zero by the Einstein Field Equation (since the covariant divergence of the Einstein tensor is zero as a geometric identity, and the EFE equates the stress-energy tensor to a constant times the Einstein tensor), and that is all that is required for local energy conservation.

The problem of global energy conservation in GR, in a nutshell, is that there is no such thing as an invariant global energy in a general curved spacetime. There are such global invariants only for particular special cases (ADM energy or Bondi energy for asymptotically flat spacetimes, Komar energy for stationary spacetimes). So any definition of "global energy" for spacetimes that are not one of the special cases will not be an invariant.

The various approaches you describe are simply different attempts to handwave around the above fact, by coming up with some kind of definition of "energy" that, while not invariant, at least has some kind of desirable property. But an even simpler approach is to not handwave at all and just accept the fact that there is no global invariant "energy" in a general curved spacetime, period. Then there is no problem.

 

[PeterDonis]

if the conservation of energy is not well defined in General Relativity, can energy be created?

It depends on what you mean by "created". Stress-energy cannot be created or destroyed; that is what local energy conservation in GR (i.e., the covariant divergence of the stress-energy tensor being zero) means. That is the only invariant way of formulating the question.

 

[Nugatory]

If the conservation of energy is not well defined in General Relativity...

It's well-defined, but stating the accurate definition is more difficult than in classical mechanics.

 

[PeterDonis]

It's well-defined

Local energy conservation is (I gave the definition in post #3). But global energy conservation is not, except for the special classes of spacetimes I gave in post #3.

 

[robphy]

One problem with "global conservation laws" in a general spacetime is that
it lacks the symmetry (that Minkowski spacetime has)
to compare tensors at different events [different spacetime-points]
and to add up such quantities from different regions of spacetime.

Transporting quantities from different regions (in order to carry out a "sum") is, in general, "path-dependent".
So, there is no sensible invariant way to carry out this sum,
and thus no invariant definition as @PeterDonis said above in #3.

 

[Suekdccia]

Feynman's sticky beads is a classic. They use gravitational waves to generate heat - but it's not at all clear where (if anywhere) the energy was beforehand.

Also, note that a contracting universe is a possible solution to the field equations, although not of much real world interest because we seem to be in an expanding one. In that case you get photon blue shift.

But as I understand it, the gravitational waves would come from an external source like a couple of astronomical bodies orbiting each other, but this energy would be dissipated with time ending when the two bodies lose that energy and crash onto each other, so the energy would come from a source and would be conserved, isn't it?

 

[Suekdccia]

It depends on what you mean by "created". Stress-energy cannot be created or destroyed; that is what local energy conservation in GR (i.e., the covariant divergence of the stress-energy tensor being zero) means. That is the only invariant way of formulating the question.

Can you think of an specific and realistic example in the sense that can be observed by our telescopes that spacetime would give energy? (for example something like the redshift that photons suffer when traveling through space but a process that would give energy instead of erasing it)

 

[PeterDonis]

Can you think of an specific and realistic example in the sense that can be observed by our telescopes that spacetime would give energy? (for example something like the redshift that photons suffer when traveling through space but a process that would give energy instead of erasing it)

A contracting universe, as @Ibix suggested in post #2.

 

[PeterDonis]

as I understand it, the gravitational waves would come from an external source like a couple of astronomical bodies orbiting each other, but this energy would be dissipated with time ending when the two bodies lose that energy and crash onto each other, so the energy would come from a source and would be conserved, isn't it?

The energy ultimately comes from a source, yes, but:

First, not all of the energy radiated as gravitational waves from the source will get absorbed somewhere; some of it just keeps propagating forever and never gets absorbed. So there is no way in general to guarantee "energy conservation" in the sense that all of the energy radiated from the source gets absorbed somewhere.

Second, while the gravitational waves are propagating, the energy they carry cannot be localized; this is different from non-gravitational radiation, such as light, for which the energy carried can be localized. To put it another way, any non-gravitational energy, like light, has a nonzero stress-energy tensor; but in a region of spacetime where the only "energy" is gravitational waves, the stress-energy tensor is zero.

This last fact led many researchers in the 1950s and 1960s to doubt whether gravitational waves could carry energy at all. Feynman's bead thought experiment was intended to show how they could indeed carry energy, even if the energy could not be localized while it was being carried by gravitational waves.

 

[Suekdccia]

A contracting universe, as @Ibix suggested in post #2.

And is there any way where an expanding space can give energy?

 

[Ibix]

And is there any way where an expanding space can give energy?

Gravitational waves in an expanding spacetime.

 

[Suekdccia]

Gravitational waves in an expanding spacetime.

But would these gravitational waves come from spacetime itself or from celestial bodies (like stars,planets, black holes)...?

 

[PeterDonis]

Gravitational waves in an expanding spacetime.

How would those correspond to "giving energy"?

 

[Ibix]

How would those correspond to "giving energy"?

Via Feynman's sticky beads, as above. Still works fine in an expanding FLRW spacetime.

 

[PeterDonis]

Via Feynman's sticky beads, as above. Still works fine in an expanding FLRW spacetime.

I don't think that's the sort of thing the OP was asking about. I think the OP was asking whether there is any process in an expanding spacetime that "gives" energy in the way that the expansion "takes" energy from photons (by redshifting them). The energy "taken" from photons by expansion in this way doesn't go anywhere; similarly, energy "given" by spacetime to photons in a contracting universe (where photons blueshift) doesn't come from anywhere. I'm not aware of any process in an expanding universe that "gives" energy the way spacetime "gives" energy to photons in a contracting universe.

GWs aren't quite the same because they can carry energy regardless of whether the universe is expanding or contracting (or neither). Whereas the photon redshift/blueshift referred to above only happens in an expanding/contracting universe.

 

[PeterDonis]

would these gravitational waves come from spacetime itself or from celestial bodies (like stars,planets, black holes)...?

Note that black holes, although they are "celestial bodies", have zero stress-energy: they are made of "spacetime itself", unlike stars and planets. So when gravitational waves are emitted from a black hole merger, there is no "source" in the sense of stress-energy. It's all just spacetime curvature.