Gravitational Waves: Expansion of Spacetime?

[alantheastronomer]

Gravitational waves are quadrupoles, they bend and stretch spacetime. Does the expansion of spacetime due to gravitational waves exceed flat (Minkowski) spacetime or is flat spacetime it's limit?

 

[Orodruin]

What do you mean by "exceed" flat spacetime?

If you perturb the Minkowski metric (or any other underlying metric) by a small amount, you end up with the wave equation for the perturbation. This is in essence why you have gravitational waves. Just like any wave equation, this wave equation can have several different solutions, including infinitely extended plane waves and wave packets. For an actual gravitational wave from a real object such a black hole merger, the pulse is clearly a finite wave packet and not a plane wave.

 

[alantheastronomer]

Does the perturbation "stretch" spacetime beyond asymptotic flatness? ...Causing an expansion?

 

[pervect]

Does the perturbation "stretch" spacetime beyond asymptotic flatness? ...Causing an expansion?

I think there is some confusion behind this question, but I'm not sure what it is or how to resolve it.

If you look at the usual picture of a linearized plane gravitational wave in a vacuum, such as https://en.wikipedia.org/wiki/Gravitational_wave#/media/File:GravitationalWave_PlusPolarization.gif you'll see that in this particular description, at any instant in time the wave stretches space in one direction, compresses it in the other, and has no effect on time. This description is coordinate dependent, it based on the tranverse-traceless (TT) gauge, so it is a coordiante and gauge dependent description of a linearized plane GW.

I believe the expansion scalar for this wave is zero, the net effect on a spatial volume element of expansion in one direction and compression in the other is zero.

Asymptotic flatness may or may not be satisfied by a GW. It's certainly possible to have gravitational waves in an asymptotically flat space time, for instance those emitted by a binary inspiral in an asymptotically flat space-time. Those waves won't be the global plane-wave solution I referred to previously, however. The waves emitted from an inspiral will get weaker as they get further away from the source. Eventually they will become undetectable. At this point they won't perturb the background, and if the solution had a flat background without the GW's, it will have a flat background with the GW's. However, the plane-wave solution I referred to earlier won't get weaker with distance, so they won't be asymptotically flat.

GW's in asymptotically flat space-time can be regarded as contributing energy to the space-time. Thus a vacuum space-time with GW's can have a non-zero ADM mass due to the presence of the GW's. The ADM mass is a sort of conserved mass or energy that is associated with asymptotically flat space-times. This can be loosely described by saying that the GW carries energy if one is careful to define what one means by energy, which isn't easy to do in GR. The "sticky bead argument" also suggests that GW's contain energy, in spite of them being vacuum solutions. However, It's also possible to confuse oneself by thinking that GW's carrry energy with the wrong interpretation of what one means by energy.

I'm not sure if any of this will help, but it's my best shot at an answer at this time.

 

[PeterDonis]

I believe the expansion scalar for this wave is zero

That's correct, it is. Heuristically, the reason for this is that the wave is entirely "made" of Weyl curvature (which is the only kind of spacetime curvature you can have in the absence of stress-energy), and Weyl curvature is inherently volume-preserving.

a vacuum space-time with GW's can have a non-zero ADM mass due to the presence of the GW's

Moreover, in a non-vacuum asymptotically flat spacetime with GWs being produced by a source, we can measure the energy carried away from the source by GWs by looking at the difference between the ADM mass and the Bondi mass.

 

[PeterDonis]

Does the perturbation "stretch" spacetime beyond asymptotic flatness? ...Causing an expansion?

You appear to be trying to look at this as though the GWs "change" spacetime in some way; but that's not how spacetime works. Any given spacetime is a 4-dimensional geometry that already contains all the information about how everything in it, GWs or otherwise, "changes" with "time". Asymptotic flatness is a property that a 4-dimensional spacetime either has or doesn't have; it can't "change" based on the presence of GWs or anything else.

A different question that you might be trying to ask is, can a spacetime with nothing but GWs in it be "expanding" in the sense that, for example, the FRW spacetime that describes our universe is expanding? I think the answer to that is no, for the reason I gave in response to @pervect's post just now: a spacetime containing only GWs only has Weyl curvature, and Weyl curvature is inherently volume-preserving. However, I don't know if there is an actual theorem to this effect in GR.

 

[alantheastronomer]

I should have rather said," in comparison to asymptotic flatness", but your answer is the answer I was looking for. Thank you.