What limits the speed of a gravitational wave to c?

[Rochelobe]

The speed of light (in the vacuum) is a function of the permeability and permittivity of the vacuum. In other mediums the phase velocity will be different. It is assumed (by me) that the speed of a gravitational wave does not change depending on the medium i.e. a gravitational wave would not slow down while passing through the sun. This would cause diffraction and presumably dispersion of the wave. Why then is the speed of a gravitational wave limited by the speed of light in a vacuum? How did Einstein come to this conclusion?

 

[Nugatory]

The speed of light (in the vacuum) is a function of the permeability and permittivity of the vacuum.

Actually it's the other way around - the numerical values that we assign to the permeability and permittivity of the vacuum are determined by the speed of light. (That's not how these ideas were developed historically; it is only with the befit of hindsight that we could properly understand the role of the speed of light).

Why then is the speed of a gravitational wave limited by the speed of light in a vacuum? How did Einstein come to this conclusion?

Einstein came to this conclusion the same way that Maxwell came to his conclusion about electromagnrtic radiation: he solved a differential equation and the answer came out to be waves traveling at speed $c$. It just about has be that way because gravitational waves must have the same property that light does, of having the same speed for all inertial observers regardless of their relative speed; and it is mathematically impossible for there to be more than one such speed. It's a historical accident that we call that one invariant speed "the speed of light"; it would be more accurate to call $c$ "the invariant speed that both light and gravitational radiation travels at". (The historical accident is that we knew the speed of light long before we knew what light was or that we were measuring the invariant speed limit of the universe, so we had already formed the habit of calling that particular speed "the speed of light").

 

[Orodruin]

he solved a differential equation and the answer came out to be waves traveling at speed c

To be fair, he only needed to find the appropriate differential equation and see that it is what we usually refer to as the wave equation. From there you already know the phenomenology and can identify the speed of the waves.

 

[vanhees71]

Actually it's the other way around - the numerical values that we assign to the permeability and permittivity of the vacuum are determined by the speed of light. (That's not how these ideas were developed historically; it is only with the befit of hindsight that we could properly understand the role of the speed of light).
Einstein came to this conclusion the same way that Maxwell came to his conclusion about electromagnrtic radiation: he solved a differential equation and the answer came out to be waves traveling at speed $c$. It just about has be that way because gravitational waves must have the same property that light does, of having the same speed for all inertial observers regardless of their relative speed; and it is mathematically impossible for there to be more than one such speed. It's a historical accident that we call that one invariant speed "the speed of light"; it would be more accurate to call $c$ "the invariant speed that both light and gravitational radiation travels at". (The historical accident is that we knew the speed of light long before we knew what light was or that we were measuring the invariant speed limit of the universe, so we had already formed the habit of calling that particular speed "the speed of light").

Permeability and permittivity are just an artifact of our men-made SI units. They are conversion factors to the natural units.

 

[Rochelobe Reference]

The speed of light (in the vacuum) is a function of the permeability and permittivity of the vacuum. In other mediums the phase velocity will be different. It is assumed (by me) that the speed of a gravitational wave does not change depending on the medium i.e. a gravitational wave would not slow down while passing through the sun. This would cause diffraction and presumably dispersion of the wave. Why then is the speed of a gravitational wave limited by the speed of light in a vacuum? How did Einstein come to this conclusion?

OK, then this begs the question, the speed of light is not constant but depends on the medium it is passing through. Does the speed of gravity also depend on the medium? Both travel at the speed of causality in free space but what happens when a gravity wave passes through a star?

 

[renormalize]

Does the speed of gravity also depend on the medium? Both travel at the speed of causality in free space but what happens when a gravity wave passes through a star?

Yes, the propagation of gravitational radiation depends on the medium. See, e.g.:
Gravitational Waves in Matter
R. L. Ingraham
General Relativity and Gravitation volume 29, pages117–140 (1997).
Abstract
The theory of gravitational waves in matter is given. This covers the questions of constitutive relation, number of independent polarizations, index of refraction, reflection and refraction at an interface, etc. The theory parallels the familiar optics of electromagnetic waves in material media, but there are some striking differences. The use of the Campbell-Morgan formalism in which the gauge-invariant tidal force dyads E and B rather than the gauge-dependent metric perturbations are the unknowns is essential. The main justification of the theory at the moment is as a theoretical exercise worth doing. The assumption: size L of the medium ≫ gravitational wave length λ (“infinite medium”) rules out application to the already well-understood detection problem, but there may be an application to gravitational wave propagation through molecular gas clouds of galactic or inter-galactic size.

Paywalled at: https://link.springer.com/article/10.1023/A:1010208315587

 

[berkeman]

The speed of light

OK

Hey, are you two related?

 

[Mister T]

Does the speed of gravity also depend on the medium?

Well, you're replying to a thread from 2017; but it would be best to ask this question in the relativity forum.

 

[DaveC426913]

If the speed of the propagation of gravity were not limited the same way the speed of light were, all sorts of causality issues could arise. We could send superluminal signals for one.

 

[berkeman]

but it would be best to ask this question in the relativity forum.

Yep, that's where we are...

If the speed of the propagation of gravity were not limited the same way the speed of light were, all sorts of causality issues could arise. We could send superluminal signals for one.

I don't think that's what the latest incarnation of this user is asking. They are asking if a change in the gravitational equivalent of $\epsilon_r$ could slow down gravitational waves...

 

[PeterDonis]

The speed of light (in the vacuum) is a function of the permeability and permittivity of the vacuum.

No, it isn't. It's an inherent property of the spacetime geometry. Vacuum permeability and permittivity are artifacts of particular choices of units; they don't cause anything.

It is assumed (by me) that the speed of a gravitational wave does not change depending on the medium

Your assumption is not quite correct. What is correct is that a given medium will have much less of an effect on the speed of a gravitational wave than on the speed of an electromagnetic wave, because the latter interacts much more strongly with the medium than the former. But there is no reason why the effect of the medium on a gravitational wave must be exactly zero.

Why then is the speed of a gravitational wave limited by the speed of light in a vacuum?

Because of the inherent property of spacetime geometry described above.

 

[Rochelobe Reference]

Yes, the propagation of gravitational radiation depends on the medium. See, e.g.:
Gravitational Waves in Matter
R. L. Ingraham
General Relativity and Gravitation volume 29, pages117–140 (1997).
Abstract
The theory of gravitational waves in matter is given. This covers the questions of constitutive relation, number of independent polarizations, index of refraction, reflection and refraction at an interface, etc. The theory parallels the familiar optics of electromagnetic waves in material media, but there are some striking differences. The use of the Campbell-Morgan formalism in which the gauge-invariant tidal force dyads E and B rather than the gauge-dependent metric perturbations are the unknowns is essential. The main justification of the theory at the moment is as a theoretical exercise worth doing. The assumption: size L of the medium ≫ gravitational wave length λ (“infinite medium”) rules out application to the already well-understood detection problem, but there may be an application to gravitational wave propagation through molecular gas clouds of galactic or inter-galactic size.

Paywalled at: https://link.springer.com/article/10.1023/A:1010208315587

Thank you. This paper is exactly what I needed. Someone has tried (is tyring) to anwser the question.

 

[Rochelobe Reference]

Hey, are you two related?

Yes. I uncovered the original post and realized I never really got the answer to the question I was thinking. So, I thought it would be fun to stir the pot. And sure enough, this time I got my answer!

 

[berkeman]

Yes. I uncovered the original post and realized I never really got the answer to the question I was thinking.

It looks like that initial account has been dormant since 2017. Should I just ban it so you can go forward with this new account? We only allow one account per person at PF. Thanks.