Speed of gravitational waves and light

We all know that speed of gravitational waves (GW) and that of light in space are exactly the same (= $c$).

We also know that space is medium for GW.

Does that mean space is also the medium for light?

Because both of them are waves, and speed of a wave is property of the medium.

Is there any other way to justify the identical speed of the two?

• I don't know that. I suspect it, but there is no conclusive measurement that motivates with very high precision that gravitational waves are traveling at exactly the speed of light. It's not a very good use of physical terminology to call spacetime a "medium". It doesn't behave like a medium and one can not assign a rest system to it, which one could, if it was an actual medium. – CuriousOne May 11 '16 at 21:15
• The Hulse-Taylor constrains GW velocity to something like within 1% of speed of light. – DilithiumMatrix May 11 '16 at 23:45

As described in this article, whenever we are talking about gravitational waves in cosmology, we actually assume the underlying spacetime to be flat (so just Minkowski metric $\eta^{\mu\nu}$) and a small perturbation $h^{\mu\nu}$ causing ripples in the metric is the graviton field:

$$g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}+O(h^2)$$

Truncating this expansion after the linear term as written above simplifies Einsteins Equations, so that after a few variable transformations and field redefinitions Einsteins Equations in vacuum can be written as

$$\Box \bar h^{\mu\nu}=0$$

Note that here $\bar h^{\mu\nu}$ is related to $h^{\mu\nu}$, and even though it originated as part of the metric - it used to describe spacetime curvature itself - it is now treated as a field propagating in flat spacetime. Therefore, we reduced the picture to special relativity with an extra field.

Interestingly, the photon field $A^\mu$ satisfies the same equation of motion in Lorentz gauge

$$\Box A^\mu=0$$

The fact that both fields satisfy the same Wave equations of motion on the Minkowski background tells us that both excitations travel with the same speed through flat spacetime. It is not appropriate to think of spacetime as a medium, it is just the background space. There is a difference. Look up Aether for more details on that.

EDIT:

If you have doubt about the approximations used, take a look at this publication, which constraints graviton mass to $<10^{-32}eV$. This is 32 orders of magnitude smaller than i.e. electron neutrino mass, and those particles are already moving at effectively the speed of light.

• We don't know with high precision what wave equation gravitational waves satisfy. We believe that general relativity holds, but that's a poorly tested hypothesis in this regime at this point. – CuriousOne May 12 '16 at 2:54
• @CuriousOne I believe with the same level of uncertainly one can doubt the equation of motion of the Photons (since it is just a classical approximation, derived from Maxwells equations). In that sense, taking both theories at the same level of approximation we can make a definite statement that both velocities should agree. Their corrections might differ, yet it would be a challenge to find an observable for which this slight difference would be of relevance. – Kagaratsch May 12 '16 at 22:05
• That is just not so. We have numerous precision experiments on electromagnetic waves spanning 20+ decades of frequencies/energy. There is but one published example of a gravitational wave detection from an unknown source at an unknown distance that happens to have a bandwidth of a couple kHz or so. There is absolutely no comparison between how much we know about electromagnetism vs. how little we still know about gravity. Physics does not start with theory. Physics starts with experiments and our experiments with gravity are, so far, extremely limited. – CuriousOne May 12 '16 at 22:47
• @CuriousOne OK, then, please take a look at the following todays publication, constraining Graviton mass to $<10^{-32}eV$, which is 32 orders of magnitude smaller than i.e. electron neutrino mass. And everyone pretty much agrees that neutrinos move at speeds effectively indistinguishable from the speed of light. arxiv.org/pdf/1605.05928.pdf – Kagaratsch May 20 '16 at 12:50
• @Kagratsch: One can't constrain the mass of something that hasn't even been observed, yet. Logically that is equivalent of saying that no more than four angels can dance on the pin of a needle. Having said that, that's an extremely weak constraint based on exactly the classical phenomenology of gravity (plus a few implicit and untested model assumptions) that I pointed out. – CuriousOne May 20 '16 at 22:13

You are using the wrong scientific terminology but you are right in a sense. The speed of light is indeed determined by the environment it travels in. We don't call it medium in relativity or physics, we call it spacetime. Spacetime has a geometry, a 4 dimensional (ignoring quantum gravity) manifold that has 1 time dimension and 3 spatial ones. The simplest spacetime is Minkowski, which is flat. The metric of it, that which describes its geometry (though not its global topology) is:

ds^2 = - c^2 * dt^2 + dx^2 + dy^2 + dz^2

That spacetime means that anyone in an inertial reference frame on it will see any massless particle travel at speed c.

For General Relativity (GR)it is curved spacetime, and the metric is similar to the above but each term can have functions of space and time in front, and mixed terms (also with functions in front) like dxdt dxdy and so on. The functions in front are determined by Einsteins gravitational equations which relate the metric (i.e., the spacetime) to the matter content in the spacetime. The interesting thing about GR for your question is that the speed of massless particles, in a LOCAL reference frame is also c in ALL those cases. It is said that they locally obey special relativity.

So you see, it is the spacetime which determines the speed of light, gravitons, and any other massless particle. And it is always c. So in a sense you are right. But it is even more, spacetime also determines the motion of any test particle (i.e., small enough so you can ignore its own gravitational field), and in fact all test particles move exactly the same way in the gravitational field. They move as free falling particles, in spacetime curves determined by the spacetime metric, and INDEPENDENT of their mass. This is the so called principle of equivalence, the spacetime determines the motions of freely falling particles. It's the same reason in Newtonian gravity astronauts float exactly the same inside their space station orbiting the earth, regardless of how heavy or small they are. GR says the same. (But there's plenty other differences).

Your idea of the medium determining the speed of the waves sort of means in physics that a material medium is involved. It really is not, it is just spacetime. A wireless or electromagnetic medium is a standard term used in wireless and optical communications, an electronic engineering view of electromagnetic propagation. In that case it means the medium, whether air or clouds or glass or fiber, or for cellular wireless the air plus the terrain and objects around from which the waves diffract, reflect, scatter and get absorbed. Those are different concerns and physical media. In physics we are talking mostly, when we talk about c, about empty space, and with gravity possibly other matter around creating that gravity (for cosmological distances plenty of mostly empty space, which however can be curved due to the other matter in the universe. Similarly for the spacetime outside stars, or outside black holes). In all of those the local speed of massless particles is c.

So you are partially right. It is all about the environment in which they travel. Because of the equivalence principle nothing else can make a difference.

• Bob Bee: What a material medium provide is certain properties. If those same properties were present without the material, the wave would still propagate at the same speed as it did through material medium. So, it is the properties of the medium, not necessarily the matter. Empty space does possess certain properties, so why it can not be considered a medium - a medium made of the properties, until we discover what causes these properties. In material medium, we know that the material causes the properties, but in case of space, we do not know that. – kpv May 13 '16 at 22:10
• Well, just words. We know a lot about spacetime, and that it is its curvature that propagates as a wave in gravitational waves. We know it's symmetries, and how it responds to matter. We just don't know yet how it arose, i.e., once we go to Planckian sizes and energies we don't know what spacetime is. When and if we get a quantum gravity theory that we get convinced is true and have enough evidence for, well know. But is is known that 'empty space', at a classical level has no matter, and at the quantum level we need that theory. We know a little more, of quantum fields in classical spacetime. – Bob Bee May 14 '16 at 3:14
• I agree on things we know about space. What I was referring to is for example, - stress energy tensor is a property of space, and we do not know what causes the stress energy tensor to exist. And I think such properties alone should be able to act as medium. – kpv May 14 '16 at 4:49
• No, the stress energy tensor depends on the matter fields that are present, nothing to do with the spacetime (other than to keep the tensor covariant). In cosmology we put in a gas-fluid, electromagnetic fields, some model for dark matter (zero pressure dust), and then add the dark energy arbitrarily to the equation. We can add scalar and other fields. The only spacetime constraint on the energy momentum tensor is covariance.Only in quantum gravity Tryingto unify gravity and the other 'forces' fields such as in string theory is matter tried to be described as eg strings, and spacetime too – Bob Bee May 14 '16 at 23:44

The general equation of an electromagnetic wave can be written as $$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu} A_{\sigma}=0$$ This can be derived from the vacuum solution of Maxwell equations. The metric for empty space-time can be written as $$g_{\mu\nu} = \eta_{\mu\nu}+ h_{\mu\nu}$$ The Ricci tensor in first order can be reduced to $$R_{\mu\nu} = \partial^{\alpha}\partial_{\alpha}h_{\mu\nu}$$ The vacuum field equations would become $$\partial_{\alpha}\partial^{\alpha}h_{\mu\nu}= 0$$ This is the same almost the same as the previous equation. The solutions to both the equations is $$h_{\mu\nu}= \epsilon_{\mu\nu}e^{i\bf{kx}}$$ $$A_{\mu} = \epsilon_{\mu}e^{i\bf{kx}}$$ Observing both the equations one can say that both will travel at the same speed $$c$$.