# The invariance vs constancy of the speed of light in vacuum

This is perhaps as much a question of semantics as of physics but it is something I have been thinking about recently and was wondering if anyone else had a perspective on this. Now, it could be that it turns out that this is something that is obvious in theory and I am just being stupid.

We are told that the invariance of the speed of light in vacuum is an axiom, the way there are Axioms in Euclid's geometry. It is a given and there is no point in asking the question, why is the speed of light invariant and constant across frames of reference.

My point is that the invariance of the speed of light and the constancy of the speed of light are different kind of things. And while it is philosophically acceptable to just "know" that the speed of light is constant but it not to just "know" that it is invariant. Fixed constant values such as the mass of an electron or the spin set of an electron are things one can accept as given. That's just the starting condition. Similarly it is understandable that the speed of light is some constant value or as you do in field equations, just say that $c=1$.

However the invariance of the speed of light across reference frames is different. It seems like an obtuse boundary hiding some physics that we cannot yet understand. (Perhaps there are some theories in particle physics I am not aware of).

So my question is this.... is this a semantic game, is saying that the speed of light is fixed the same as saying that it is invariant across reference frames? Because perhaps, something can be constant but appear to be variant (changing) across reference frames because of some co-ordinate transformation. The way gravity does. The way we differentiate real gravity from accelerated reference frames by potentially trying to look for gravitational waves (theoretically). I know this may sound silly but I hope you see what I am trying to say.

And while it is philosophically acceptable to just "know" that the speed of light is constant but it not to just "know" that it is invariant. Fixed constant values such as the mass of an electron or the spin set of an electron are things one can accept as given.

Kind of: be careful. We can fix some of the fundamental constants by choosing covarying unit systems with those constants, just as the speed of light is fixed at the numeric value of exactly 299,792,458 meters per second by the definition of the meter.

With that said, the speed of light gets rolled into several dimensionless parameters like the fine-structure constant, and we could very meaningfully discuss what would happen if the fine-structure constant varied in time. In Planck units where $c = \hbar = k_\text{e} = 1$, the fine-structure constant $k_e e^2 / (\hbar c)$ becomes simply $e^2$ and we would interpret this as a time-variation of the electron's mass.

So what you're accepting as "invariant" need not be accepted as "given", if you use the right units.

So my question is this.... is this a semantic game, is saying that the speed of light is fixed the same as saying that it is invariant across reference frames?

It depends what you mean by "semantic game". If you mean, "does this trivially have no predictive value?" then the answer is no. (Prediction number one: no continuous acceleration can outrun a light beam due to Zeno's paradox: to outrun it you need to go half its speed, but in that reference frame, it's still moving away from you at speed $c$.)

If you mean, "can we never see light appear to slow down from a distance?" then the answer is similarly no. (General relativity contains things like black holes which can trap light.)

But if you mean, as you seem to clarify, "does 'the speed of light is fixed' just mean that the speed of light is the same for all of the inertial reference frames tangent to a point in spacetime?' then that answer is yes. Locally, if you start to examine some reference frames and how they look at the expanding bubble that is the light emitted from a supernova into vacuum, assuming that gravitational distortions are negligible so that one viewer thinks that it's a sphere moving outward from a stationary point at speed $c$, then all of those reference frames see it as a sphere moving outward from a stationary point at speed $c$, even if they do not think that the event which caused it (the moving star that collapsed to create the supernova) is stationary.

• That is awesome Chris. Exactly what I was looking for. I had forgotten this c=ℏ=ke=1 relationship and the fact that variance in the speed of light across different time or space transformation would also been time or space variation of the electron's mass as you mention. Yes that is a much deeper understanding of the invariance. Jun 18, 2015 at 14:35
• I am little confused about your supernova example though. Can you elaborate a little? Jun 18, 2015 at 14:38
• So when any event happens, we see it because it casts out an expanding bubble of light in all directions, informing the universe that it happened: we see the light from the event and know that it happened. A supernova is just a particularly sudden event (in stellar terms) which is mostly defined by everyone seeing this one point of space flash super-brightly: so I can get you to think about how you model that light in some coordinates (a thin bubble $|\vec r(t) - \vec r_0| = c t$) and point out that every Lorentz transform maps such bubbles (often called light cones) to other such bubbles. Jun 18, 2015 at 14:44
• Thanks. Yes, I get it mathematically and I see that bubbles would be invariant across Lorentz transformations. I see what you meant in your previous comment too. Sorry I seem to have a lot of trouble connecting the mathematics to the experience/language Jun 18, 2015 at 14:54
• Don't fret -- even experienced physicists have had trouble connecting the mathematics of relativity to the experience. Let's take the most mundane thing: the perceived shape of a marble moving at speed $c/2$. To this day, even among physicists, it is still not all that widely known that its relativistic length contraction is "invisible", appearing instead as a Terrell rotation of the marble itself. Jun 18, 2015 at 15:27

I'm not sure I understand your question, but I'll try anyways. It's true that constancy (not changing in time) and invariance (same in all reference frames) are different.

SR and GR assume a constant speed of light, but lets imagine a theory where the speed of light changes in time. The passage of time is different for different observers. If my clock and your clock run at different rates, then we've broken the invariance of the speed of light.

In order to fix this we may want to use some "universal clock" to keep track of the speed of light. This introduces a preferred reference frame. In short there is no way to keep Special and General Relativity intact while having a non-constant speed of light.

On philosophical grounds we might think that relativity is a special case of some quantum theory of gravity. That theory may break the constancy of the speed of light in some way we don't yet understand, but that's pure speculation.

There are alternative theories that don't include the invariance of the speed of light. Some of them are interesting, but most are very well constrained by experiments. In general these theories are called "Lorentz violating" because they break the Lorentz invariance of Special Relativity.

Because Lorentz invariance is an axiom, it's important to have strong experimental justification for it. You can read more about experimental tests of Lorentz invariance on wikipedia or your source of choice.

• I am not sure I understand you though. The passage of time is already different for different observers. If you move closer and closer to the speed of light then time will move slower and slower for you compared to someone who is slower. A theory where the speed of light is variant will have far more serious consequences. Jun 18, 2015 at 14:16
• Yes, breaking invariance would have a bigger effect, but the invariance of the speed of light is very well constrained by experiments. I'll edit my answer. Jun 18, 2015 at 14:22
• Yes Paul, I agree that the invariance is very well proven by experiments. This is exactly what is puzzling me. I am not questioning the law, I just think it feels less acceptable as an axiom than just a value such as the speed of light itself or the cosmological constant or mass of electron etc. Jun 18, 2015 at 14:27
• "The passage of time is already different for different observers. If you move closer and closer to the speed of light then time will move slower and slower for you compared to someone who is slower." Be VERY careful with this. If you pass me in a train, it's not just that I see your clocks "ticking slowly", but also that you see my clocks "ticking slowly". (Here "ticking slowly" means, "after naively accounting for the Doppler shift.") Jun 18, 2015 at 14:29