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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?" $\endgroup$ – David Butler 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? $\endgroup$ – Duncan X Simpson 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. $\endgroup$ – David Butler May 9 '16 at 17:04
  • $\begingroup$ hey Bob. can you learn to use $\LaTeX$ so that your answer makes sense? $\endgroup$ – robert bristow-johnson 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. $\endgroup$ – David Butler 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

protected by Qmechanic May 8 '16 at 9:15

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