Gravitational wave interactions and the equivalence principle

[sf1001]

TL;DR

Theoretically, couldn’t gravitational waves interact to produce a photon pair, and couldn't such an interaction have more than one outcome given the same initial conditions for the two gravity waves. If so would this violate the strong equivalence principle.

According to wikipedia, the strong equivalence principle states “the gravitational motion of a small test body depends only on its initial position in space time and velocity, and not on its constitution, and the outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in space time.

Would the possibility of interacting gravity waves producing photon pairs with different orientations or other properties with the same initial conditions for the gravitational waves, or the reverse of such a process (non deterministic production of gravity waves from two or more colliding photons) violate the principle that the inertial motion of a body depends only on its initial position in spacetime and velocity.

Also would the “fact”, assuming my comprehension of the wikipedia article on vacuum solutions of the Einstein Field Equations is correct, that some vacuum solutions to EFEs are not locally rotationally invariant or more generally not locally the same as Minkowski space mean that a local experiment in a laboratory at a point in spacetime where a non-minkowski vacuum solution applies have a different outcome then the same experiment in a laboratory where spacetime is locally described as a Minkowski space?

 

[Vanadium 50]

If so would this violate the strong equivalence principle.

Why would it do this?

 

[PeterDonis]

some vacuum solutions to EFEs are not locally rotationally invariant or more generally not locally the same as Minkowski space

These are two different statements. There are plenty of vacuum solutions that are not rotationally invariant about every point; a simple example is Schwarzschild spacetime.

However, every solution of the EFE, vacuum or not, is locally Minkowskian. I don't know what Wikipedia article you are reading that makes you think otherwise (a link would be helpful), but either you are misinterpreting what it says or it's wrong (which is unfortunately a non-negligible possibility with Wikipedia).

 

[sf1001]

These are two different statements. There are plenty of vacuum solutions that are not rotationally invariant about every point; a simple example is Schwarzschild spacetime.

However, every solution of the EFE, vacuum or not, is locally Minkowskian. I don't know what Wikipedia article you are reading that makes you think otherwise (a link would be helpful), but either you are misinterpreting what it says or it's wrong (which is unfortunately a non-negligible possibility with Wikipedia).

https://en.m.wikipedia.org/wiki/Ozsváth–Schücking_metric
The article says this metric is not isometric to Minkowski space; but maybe this doesn't imply that it’s not locally the same as Minkowski space? Also maybe I misconstrued the metric to think that it implied a local rotational asymmetry when it may have only implied a non-local rotational asymmetry.

 

[sf1001]

Why would it do this?

I thought the possibility that a resulting photon pair could propagate outward on differing alignments with the same initial colliding gravitational waves would mean that a test particle’s inertial path (eg one of the resulting photons) could be different with the same initial location in spacetime (where the gravitational waves collided) and the gravitational waves from which it arrived having a known velocity. What I thought is that two (or more) gravitational wave fronts could collide (at a specific location in space time with a specific velocity) to produce a pair of photons with each photon having many directions in which it could propagate, so long as they canceled out each other’s momentum. But maybe this wouldn't contradict the equivalence principle since the test particle I should be considering is the center of mass of all the non-gravitational particles created in the interaction of the gravitational waves.

 

[timmdeeg]

Theoretically, couldn’t gravitational waves interact to produce a photon pair

Why do you think that a photon pair can be produced by gravitational waves?

 

[PAllen]

https://en.m.wikipedia.org/wiki/Ozsváth–Schücking_metric
The article says this metric is not isometric to Minkowski space; but maybe this doesn't imply that it’s not locally the same as Minkowski space? Also maybe I misconstrued the metric to think that it implied a local rotational asymmetry when it may have only implied a non-local rotational asymmetry.

They are talking about global isometry not local Minkowski character. The latter is part of the formal definition of psuedo-Riemannian manifold, which is the mathematical framework for GR. Talking about a solution violating this would be as absurd as talking about fractional integers.

 

[PeterDonis]

The article says this metric is not isometric to Minkowski space; but maybe this doesn't imply that it’s not locally the same as Minkowski space?

That's right; it's talking about the global geometry of the spacetime. As I said, every spacetime in GR is locally Minkowskian.

maybe I misconstrued the metric to think that it implied a local rotational asymmetry when it may have only implied a non-local rotational asymmetry

"Rotational asymmetry" is not the best way to describe the property of the spacetime you are talking about. As the Wikipedia article you linked to notes, this is a pp-wave spacetime, which means it can be viewed as a type of gravitational wave. The Wikipedia article on vacuum solutions in GR more generally [1] calls it a "circularly polarized sinusoidal gravitational wave". At any rate, this property does not in any way prevent the spacetime from being locally Minkowskian.

[1] https://en.wikipedia.org/wiki/Vacuum_solution_(general_relativity)

 

[sf1001]

Why do you think that a photon pair can be produced by gravitational waves?

I’m aware gravitational energy and momentum can be difficult to define in general relativity, but I thought gravitational waves might be able to carry energy & momentum in some sense & that when they collide they could create a photon pair (or other particle pair if energetic enough) the same way a colliding photon pair could give rise to other particle pairs.

Wikipedia says that the covariant divergence of the stress energy tensor is zero (ie energy-momentum is conserved in some sense) but suggests that non-gravitational energy momentum may not be absolutely conserved (ie cartesian divergence of stress-energy tensor may not be zero when gravity is significant), and that opens up the possibility that gravitational and non gravitational energy-momentum can be exchanged. However, another commentor has mentioned that every solution to EFEs is locally the same as Minkowski space, which I think would mean non-gravitational energy-momentum is always locally conserved, so I guess gravitational waves would have to interact with already existing non-gravitational energy-momentum to convert gravitational energy-momentum to non-gravitational energy-momentum, and the scenario I proposed earlier is physically impossible.

 

[sf1001]

That's right; it's talking about the global geometry of the spacetime. As I said, every spacetime in GR is locally Minkowskian.
"Rotational asymmetry" is not the best way to describe the property of the spacetime you are talking about. As the Wikipedia article you linked to notes, this is a pp-wave spacetime, which means it can be viewed as a type of gravitational wave. The Wikipedia article on vacuum solutions in GR more generally [1] calls it a "circularly polarized sinusoidal gravitational wave". At any rate, this property does not in any way prevent the spacetime from being locally Minkowskian.

[1] https://en.wikipedia.org/wiki/Vacuum_solution_(general_relativity)

Thanks for the clarification

 

[PAllen]

I’m aware gravitational energy and momentum can be difficult to define in general relativity, but I thought gravitational waves might be able to carry energy & momentum in some sense & that when they collide they could create a photon pair (or other particle pair if energetic enough) the same way a colliding photon pair could give rise to other particle pairs.

Wikipedia says that the covariant divergence of the stress energy tensor is zero (ie energy-momentum is conserved in some sense) but suggests that non-gravitational energy momentum may not be absolutely conserved (ie cartesian divergence of stress-energy tensor may not be zero when gravity is significant), and that opens up the possibility that gravitational and non gravitational energy-momentum can be exchanged. However, another commentor has mentioned that every solution to EFEs is locally the same as Minkowski space, which I think would mean non-gravitational energy-momentum is always locally conserved, so I guess gravitational waves would have to interact with already existing non-gravitational energy-momentum to convert gravitational energy-momentum to non-gravitational energy-momentum, and the scenario I proposed earlier is physically impossible.

GR is classical, so mixing in photon production is going outside it’s strict range of validity. There are semiclassical gravity approaches, but their validity is unclear absent a consistent approach to quantum gravity.

However, it is well known in GR that colliding gravitational waves can, in principle, produce singularities and black holes. There is extensive literature on this. It may be argued that this suggests a quantum gravity theory would include particle production by colliding gravitational waves. I don’t see what this has to do with strong equivalence principle, though.

A reference: https://www.jstor.org/stable/2398160?seq=1#page_scan_tab_contents

 

[timmdeeg]

I’m aware gravitational energy and momentum can be difficult to define in general relativity, but I thought gravitational waves might be able to carry energy & momentum in some sense & that when they collide they could create a photon pair (or other particle pair if energetic enough) the same way a colliding photon pair could give rise to other particle pairs.

Gravitational waves can loose some energy if they deform a body by exerting tidal forces. But I'm not aware of any process other that annihilation to produce two photons.

 

[Nugatory]

But I'm not aware of any process other that annihilation to produce two photons.

Photons are produced in just about any sufficiently energetic interaction.

The gravitational waves we’ve observed are many orders of magnitude too weak to produce photons.

 

[sf1001]

Photons are produced in just about any sufficiently energetic interaction.

The gravitational waves we’ve observed are many orders of magnitude too weak to produce photons.

I thought photons could have arbitrarily low energy as wavelength increases.

 

[Nugatory]

I thought photons could have arbitrarily low energy as wavelength increases.

They can, so if we were being pedantic we would say that the gravitational waves we’ve observed are many decimal orders of magnitude too weak to produce observable photons.

In practice, no one is ever that pedantic in normal conversational English. Would you say “a jumping flea does not strike sparks when it lands” or “a jumping flea does not strike observable sparks when it lands”?

 

[pervect]

Someone correct me if I'm wrong, but I believe that as a purely classical theory, two uncharged gravitational waves impacting wouldn't produce electromagnetic radiation.

So, to get a scenario where we did get EM waves, I believe we'd need to invoke quantum mechanics and/or quantum gravity of some sort. We don't have a full theory of quantum gravity, but we do have some approximations that allow us to do quantum mechanics in curved space-time. I am not very familiar with these approximations, though they are discussed in one of my textbooks (Wald). I don't know if these approximations would allow us to reliably answer the Orignal Poster's question or not. My main point is that I don't think the question falls in the well-tested realm of classical general relativity.

 

[PeterDonis]

I believe that as a purely classical theory, two uncharged gravitational waves impacting wouldn't produce electromagnetic radiation.

I agree, in a classical theory, gravitational waves are electrically neutral and so cannot serve as a source of EM waves.

 

[Paul Colby]

Gravitational waves can loose some energy if they deform a body by exerting tidal forces.

Just a note in passing, piezoelectric materials, such as quartz, when strained produce currents which then radiate.