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I own an educational YouTube channel on physics and astronomy. I am currently working on a gravitational waves video extension to my "How Fast Is It" video book on relativity theory. I have a question on the speed of gravitational waves. I understand that the field equations show that it is equal to the speed of light. My question goes one level deeper. My audience knows that the speed of light is fixed by two key characteristics of 'empty space' namely permittivity and permeability. The speed of a gravitational wave would be related to the elasticity of 'empty space'. Is it just a coincidence that these give the same result, or is there a deeper physics in play here?

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Not really. The "speed of light" has very little to do with light; it is built into the actual geometry of spacetime independent of what matter fills it.

In particular, $\epsilon_0$ and $\mu_0$ don't tell us anything physical about the vacuum; looking at the (simplified) expressions $$E = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2}, \quad B = \frac{\mu_0}{4\pi} \frac{I \times \hat{r}}{r^2}$$ we see that $\epsilon_0$ and $\mu_0$ just define the units of the electric and magnetic fields. We can (and often do) change their definitions; for example, in Gaussian units, we set $1/4\pi \epsilon_0 \to 1$.


An edit to address the comment: light and gravitational waves travel at the "speed of light" because they obey the relativistic wave equation, $$\partial^2 \phi = (\partial_t^2 - \partial_x^2) \phi = 0.$$ You can't write this second-order differential equation in terms of two first-order differential equations in a natural way; you have to make an arbitrary choice. For example, let's consider the simpler case of the harmonic oscillator, $\partial_t^2 x = -\omega^2 x$. We can rewrite this equation as $$y = \alpha \partial_t x, \quad x = -\frac{\omega^2}{\alpha} \partial_t y$$ by introducing the intermediate quantity $y$. Then you could say $\alpha$ is the "resistance to motion" while $\omega^2/\alpha$ is the "restoring force". But these quantities are totally meaningless because $\alpha$ is arbitrary. Splitting the electromagnetic field into electric and magnetic fields and introducing the constants $\epsilon_0$ and $\mu_0$ is exactly the same.

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  • $\begingroup$ I have seen equations where c is set to one. That does not wipe out the basic physical property that light travels fast. Setting ϵ0 and μ0 for unit purposes does not change the fact that the speed of light is what it is because space resists the formation of electric and magnetic fields. By the same token, gravitational waves depend on the speed that a disturbed volume of space will return to a pre-disturbance volume (elasticity). My question is "Why would resistance to field creation set the speed of light to exactly the same value as elasticity sets the speed of gravity?" $\endgroup$ May 9 '16 at 16:49
  • $\begingroup$ @DavidButler I edited to address this. $\endgroup$
    – knzhou
    May 9 '16 at 19:28
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A better way to think of it is "speed of causality". That's the fastest any cause-and-effect will spread over space.

With nothing to cause it to go slower, changes to electric and magnetic fields will occur at that speed. No coincidence that changes to spacetime (causing gravity) propigate at the same speed.

You really need to show how Minkowski spacetime results in such a speed limit as a basic principle. It's not a speed limit in the usual sense; it's a deep principle of what speed is.

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  • $\begingroup$ What about the theory of tachyons, stemming from the idea that C is not a maximum but simply a barrier? $\endgroup$ May 8 '16 at 16:16
  • $\begingroup$ Tachyons are discussed on other questions on this site. They still would not provide for cause-and-effect to travel faster than light. $\endgroup$
    – JDługosz
    May 8 '16 at 17:50
  • $\begingroup$ There is no such thing as a deep principle of what speed is. Read all of modern or old physics and a principle of sore does not exist. It is special relativity and that c I constant in all reference frames that led to Lorentz invariance and then making all other laws of physics locally in rest frames Lorentz invariant is the way you get to nothing traveling faster than light. It is cause and effect because of that, not the other way around. It is important to be precise in imprecise question or you add to the confusion $\endgroup$
    – Bob Bee
    May 8 '16 at 23:51
  • $\begingroup$ I meant a principle of speed. Sore was Microsoft fixing something. Then cause and effect is simply possible if they could have communicated at speed of light or less, technically if inside or on the past light cone of the effect. Plenty of effects slower than c also $\endgroup$
    – Bob Bee
    May 9 '16 at 1:12
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Nonsense. Maxwell derived his electromagnetic equations, with $\epsilon_0$ and $\mu_0$, and those quantities were known. The fact that his equations led to the speed of electromagnetic waves to be, in terms of $\epsilon_0$ and $\mu_0$, equal to the approximately then known speed of light is a big part of what led Maxwell to conclude that light is electromagnetic.

No coincidence, light is electromagnetic and those entities define the speed of propagation of electromagnetic waves.

See Jackson or any other good electromagnetism textbook for the derivations.

Btw, it's not about units. $\mathbf{E}$ and $\mathbf{B}$ are simply used to define forces, and the values of those were known, so $\epsilon_0$ and $\mu_0$ were also approximately known.

Finally, it's gravitational wave speed also because Einstein got GR (general relativity) through a (mind blowing) generalization of special relativity to an arbitrary frame of reference, with gravitation equivalent to acceleration (equivalence principle). SR (special relativity) included $c$, the speed of light, as the maximum speed possible, achieved by zero mass particles. GR had to reduce to SR in a local inertial frame, so it also had to include the same $c$. GR waves reduce to a Lorentzian wave equation with $c$, in the weak field limit. Also in a local inertial frame.

Theoretically it all adds up, there's isn't any other way if GR is true. The way it might not be totally true, with respect to gravitational waves going at a speed different (and necessarily slower than ) $c$ is if the graviton (the presumed quanta carrying the gravitational radiation or force) is a non-zero mass particle. Based on measurements of gravitational effects in the solar system it is known the mass of the presumed graviton is zero to about 1 part in (and here I am not sure I have the correct number, but it is to great accuracy) maybe about $10^{15}$ or $10^{18}$. The eLISA satellites to be launched in a few (2-3?, see Wikipedia on it) years will measure it even better by seeing if there is any delays between different frequencies of the gravitational waves they will see - it'll have orders of magnitude more accuracy, it's interferometer baselegs are 1 million Kms compared to the 5 Kms of LiGO which recently detected gravitational waves.

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  • $\begingroup$ I understand the theory. The equations add up and measurements confirm. I am asking a question about the properties of empty space that made this true. For example, in Newtonian mechanics, inertial mass was equal to gravitational mass. That was the theory and the equations added up and measurements confirmed the equality. Then Einstein pointed out that such an equality could not stand until the underlying physics was understood to be equal. The rest is history. I am looking for insights that explain why resistance to field creation and resistance to deformation would produce the same speed c. $\endgroup$ May 9 '16 at 17:04
  • $\begingroup$ hey Bob. can you learn to use $\LaTeX$ so that your answer makes sense? $\endgroup$ May 10 '16 at 0:57
  • $\begingroup$ Resistance to electromagnetic field radiation is permitivity and permeability which gives c. Resistance to deformation radiation also led to c because there was no choice due to the equivalence principle and special relativity. So the property of empty space that makes it possible is that it is arranged as spacetime, its geometry. Any finite c would have done it. That's the deep meaning, with c the invariant. If the neutrino was massless (almost is) it'd travel at c (almost does), and if the weak force was stronger maybe we would have called c the neutrino speed. You'd ask why the same c. $\endgroup$
    – Bob Bee
    May 10 '16 at 4:07
  • $\begingroup$ Thanks Bob. You are correct. I would and still do ask "why the same c". I think the answer may exist in the realm of string theory. $\endgroup$ May 10 '16 at 17:54
  • $\begingroup$ Maybe. I believe string theory still has c as the speed limit, and graviton a come up naturally also at speed c. Still not sure how speeds even get reflected at the Planck scale when spacetime doesn't really exist, and comes about as an emergent property. Maybe even then some version of causality would still be limited by c.There are mods that argue that c is variable, but there s lots of constraints from astrophysical measurements on even very small variations. Yes, you you'd have to explore those scales to get anything different, and c is not even the most interesting, eg, 10 dimensions $\endgroup$
    – Bob Bee
    May 10 '16 at 20:00
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Existing answers correctly bring in the fact that the speed commonly indicated by the symbol $c$ is primarily about geometry of spacetime and the notion of causality, or what is the difference between temporal and spatial separation. Only secondarily does it have anything to do with electromagnetic phenomena. But nevertheless, electromagnetic influences do propagate at $c$ in vacuum (relative to anything nearby), so one might ask whether this is a coincidence, before even coming to the issue of gravitational waves. In other words we have three speeds to consider:

  1. $c_{\rm s}$ the speed appearing in the formula for invariant spacetime interval: $$ ds^2 = -c_{\rm s}^2 dt^2 + dx^2 + dy^2 + dz^2 $$
  2. $c_{\rm L}$ the speed appearing in Maxwell's equations: $$ c_{\rm L}^2 \nabla \times {\bf B} = \epsilon_0^{-1} {\bf j} + \frac{\partial {\bf E}}{\partial t} $$
  3. $c_{\rm gw}$ the speed appearing in the Einstein field equation of general relativity, which, in the weak field limit and Lorenz gauge, can be written $$ \left(-\frac{1}{c_{\rm gw}^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right) \bar{h}_{ab} = -\kappa\left( T_{ab} + t_{ab} \right) $$ where $\kappa$ is a constant proportional to $G$.

So now the question is, is it a coincidence that $c_{\rm s} = c_{\rm L} = c_{\rm gw}$, and what does this tell us about the properties of spacetime?

First, it is no coincidence that $c_{\rm s} = c_{\rm gw}$. These are precisely the same quantity, the same aspect of the geometry of spacetime.

Next, to explain why it is that $c_{\rm L} = c_{\rm s}$ one cannot really do much better than saying that this is the way the equations come out. But one can add a little. In any field theory we have learned to expect that the equations do not bring in unnecessary baggage, and Maxwell's equations are about as simple as they could possibly be while respecting the principle of relativity and the kind of spacetime geometry we have (i.e. one with a finite $c_{\rm s}$). If they brought in some other speed then they would have to become quite a lot more complicated, and there would have to be a new physical quantity associated with the fields---something like mass, for example.

An observation which is relevant to some astronomical observations is that since the region between stars and galaxies is not a perfect vacuum (it has a few hydrogen atoms and other stuff floating around), it has a refractive index very very slightly different from 1, so light propagates across the universe at a speed very very slightly different from $c_{\rm s}$. Gravitational waves, on the other hand, interact with this very diffuse gas much less, so their speed is in practice closer still to $c_{\rm s}$ (if someone would care to place a number on this so as to improve this answer, please do).

Finally, what does this tell us about spacetime. I guess one thing one might like to note is that it is hard in practice to do stuff that introduces significant amounts of curvature into spacetime: very large masses and motions are required to generate gravitational waves of significant strain amplitude. In this sense spacetime is sometimes said to be very "stiff".

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Let us begin with an article: https://doi.org/10.1007/s11467-019-0913-4

This is one among the articles which refer to a time delay of 1.7 seconds in observing a gamma ray burst following an observation for gravitational waves. That is, gravitational waves were observed first and then light rays (gamma-ray-burst) were observed after 1.7 seconds from a collision of two neutron stars which are about some billion light years distance from our earth. Even if we begin with the assumption that incidents for both observations happened simultaneously at stars, then we find that speed of gravitational waves should be greater than the speed of light rays of gamma-ray-burst. Since no energy wave in “vacuum” can exceed speed of light in “vacuum”, then we can conclude that the gravitational waves were released first and the light waves were released second. Elementary numerical calculations also show that the difference 1.7 seconds in reaching earth from a billion light years distance place does not make any significant difference between speed of gravitational waves and speed of light rays. So, the speed of gravitational waves should be equal to the speed of light rays, and the event of occurrence for gravitational waves happened first and the event of occurrence of light rays happened second (at the place of stars). So, no deep physics and no deep mathematics are involved in the conclusion that the speed of gravitational waves is equal to the speed of light rays in vacuum. I like to add some more things without giving a reference for basic principles (to avoid “self-promotion”).

  1. When two self rotating neutron stars come closer the magnetic fields of the stars are disturbed and energy waves are released from the magnetic fields. These waves are gravitational waves. Since they are from magnetic fields, they are of electromagnetic wave type. So, speed of gravitational waves is equal to the speed of electromagnetic waves in vacuum.
  2. After 1.7 seconds collision happens. During this collision, gamma rays, X-rays, ultra-violet rays, visible light rays and infra red rays may be released.

These two things are also acceptable reasons for calculating the speed of gravitational waves and for the delay 1.7 seconds. These reasons mentioned above are consequences of a fundamental question: Why should we accept the earlier “standard” reasons for creation of gravitational waves?

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