Understanding the Constant Speed of Light: Mathematical and Logical Explanations

[Alex_Sanders]

Mathematically you can prove the speed of light is a constant, but I also read from a book by Brian Greene that seemingly there is a rather logical, other than mathematical understanding of it, an explanation regarding time scale change of an observer in motion.

Does anyone else know it?

 

[Alex_Sanders]

It goes like if you move, "your time" runs "slower", so it kinda "payoff"... something like that.

 

[I_am_learning]

You can't mathematically prove that, Thats theoretically assumed because it is experimentally verified.

 

[bcrowell]

FAQ: Why is the speed of light the same in all frames of reference?

The first thing to worry about here is that when you ask someone for a satisfying answer to a "why" question, you have to define what you think would be satisfying. If you ask Euclid why the Pythagorean theorem is true, he'll show you a proof based on his five postulates. But it's also possible to form a logically equivalent system by replacing his parallel postulate with one that asserts the Pythagorean theorem to be true; in this case, we would say that the reason the "parallel theorem" is true is that we can prove it based on the "Pythagorean postulate."

Einstein's original 1905 postulates for special relativity went like this:

P1 - "The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion."

P2 - "Any ray of light moves in the 'stationary' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body."

From the modern point of view, it was a mistake for Einstein to single out light for special treatment, and we imagine that the mistake was made because in 1905 the electromagnetic field was the only known fundamental field. Really, relativity is about space and time, not light. We could therefore replace P2 with:

P2* - "There exists a velocity c such that when something has that velocity, all observers agree on it."

And finally, there are completely different systems of axioms that are logically equivalent to Einstein's, and that do not take the frame-independence of c as a postulate (Rindler 1979).

For someone who likes axioms P1+P2, the frame-independence of the speed of light is a postulate, so it can't be proved. The reason we pick it as a postulate is that it appears to be true based on observations such as the Michelson-Morley experiment.

If we prefer P1+P2* instead, then we actually don't know whether the speed of light is frame-independent. What we do know is that the empirical upper bound on the mass of the photon is extremely small (Lakes 1998), and we can prove that massless particles must move at the universal velocity c.

In a system such as Rindler's, the existence of a universal velocity c is proved rather than assumed, and the behavior of photons is related empirically to c in the same way as for P1+P2*. We then have a satisfying answer to the "why" question, which is that existence of a universal speed c is a property of spacetime that must exist because spacetime has certain other properties (basically, it has some symmetries, and it doesn't have universal simultaneity).

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

R.S. Lakes, "Experimental limits on the photon mass and cosmic magnetic vector potential", Physical Review Letters 80 (1998) 1826, http://silver.neep.wisc.edu/~lakes/mu.html

 

[Alex_Sanders]

Apologies for a late reply, and thank you for your patient response, especially
bcrowell

Why I got this feeling that Einstein was quite unscientific when raising his theory?

And I still do not understand why can't you mathematically prove, or at least verify the invariable speed of light?

You found the orbit of Uranus is rather grotesque, then you postulate, under the premise that Newtonian laws are correct, there is another body influencing Uranus and calculate its position, later by focusing on the calculation result suggested area, you found the Neptune. Shouldn't science work that way?

 

[Fredrik]

I think the main motivation for the development of GR was that Newtonian gravity in Minkowski spacetime is logically inconsistent. The observed orbits of the planets probably had very little to do with it.

A question about something in the real world can only be answered by a theory. When you're asking "why is the speed of light invariant?", you're probably wondering how SR and GR answers that question. The answer is that they don't. The invariant speed of light is part of the definitions of those theories. You could ask the follow-up question "why do those theories predict results of experiments so accurately?", but the only thing that can answer that is another theory, and what theory would that be? It would have to be a better theory of gravity than GR. There are a few candidates (e.g. string theory), but none of them is developed to the point where they can make predictions about experiments.

I'm not a fan of Brian Greene's presentation of SR. What you have in mind is probably the claim that "everything travels through spacetime at speed c".

The description of motion that you're asking about in the first sentence has been brought up here many times, because Brian Greene used it in the "The elegant universe". I think that explanation is really bad. I wrote some comments about it in another thread recently. These are the relevant posts: 64, 65. The words "earlier in this thread" in the second one refers to 17.

 

[Alex_Sanders]

Thanks Fredrik.

Uh, this is a bit stupid but, I do not want to know why the speed of light is a constant, I want to know if there is mathematical approach that actually verifies this theory, or in other words, if there is calculation done showing the speed of light should be a constant.

And if there is a more non-physicist friendly understanding of this phenomenon that do not have fundamental errors.

 

[Alex_Sanders]

If there is no math can be done here... then it's just odd, I mean c'mon, even biology can pull a lot decent probability and statistics trick. :/

 

[Janus]

If there is no math can be done here... then it's just odd, I mean c'mon, even biology can pull a lot decent probability and statistics trick. :/

The mathematics involved would be Maxwell's equations, which shows that the speed of light is independent of the velocity of the source. What Einstein did was to to combine this with Galileo's principle of relativity, which held that there was no absolute motion, only relative motion, making the speed of light independent of any relative motion between source and measurer.

 

[Fredrik]

I do not want to know why the speed of light is a constant, I want to know if there is mathematical approach that actually verifies this theory, or in other words, if there is calculation done showing the speed of light should be a constant.

Are you asking for a calculation that shows that the theory that has "c is invariant" as a part of its definition predicts that c is invariant? What would be the point of that?

I suppose you could do a Lorentz transformation of the coordinates of an event where light is emitted, and another event where that same light is detected, and then verify that Δx/Δt=c. But I don't see the point of such a calculation, since an operator that gives us a different result than that wouldn't be considered a Lorentz transformation.

What Janus is describing in the first sentence is another calculation that ends with the result "the speed of light is c", but classical electrodynamics is just the theory of a classical spin-1 field in Minkowski spacetime, so that calculation is circular too. Edit: That doesn't mean that it's worthless. As you may know, Maxwell's equations were known before SR. They were clever guesses based on the results of experiments. So this calculation shows how it's possible to take something that's known (that Maxwell's equations make accurate predictions about the results of experiments) and use it to guess an important part of a new theory.

 

[bcrowell]

I want to know [...] if there is calculation done showing the speed of light should be a constant.

The Rindler reference in #4 is such a calculation. If that's not sufficiently non-physicist friendly, you could try this: http://www.lightandmatter.com/area1book6.html But keep in mind that any such calculation can only prove this fact if you assume certain other things are valid to start with, because experiments show them to be true.

 

[DaveC426913]

Alex: "the speed of light is invariant" is a postulate.

i.e. if this postulate were true, what would be see?

What we see, verified countless times in countless ways over decades, matches what we would expect to see if that postulate were true.

It is one of the most thoroughly verified experiments in all science.

We cannot prove a postulate.

 

[bcrowell]

We cannot prove a postulate.

Unless you adopt a different set of postulates, which is IMO actually the best choice here. See #4.

Einstein's axiomatization of SR is 105 years old, and it shows its age. From the modern point of view, it's absolutely the wrong way to approach SR.

In a similar spirit, no modern mathematician would approach Euclidean geometry using Euclid's axiomatization. It has all kinds of disadvantages. It isn't up to modern standards or rigor. It doesn't give a clear separation the various cases (2 dimensions versus 3, discrete versus continuous, Euclidean versus noneuclidean).

 

[Passionflower]

FAQ: Why is the speed of light the same in all frames of reference?

The speed of light is the same in all inertial frames of reference. The speed of light is also the same for all observers measured locally. The speed of light differs when measured between two locations or measured at a single remote location in a non-inertial frame of reference.

 

[bcrowell]

The speed of light is the same in all inertial frames of reference. The speed of light is also the same for all observers measured locally. The speed of light differs when measured between two locations or measured at a single remote location in a non-inertial frame of reference.

You're correct about the restriction to locality, but that's a trivial restriction because GR doesn't even have an unambiguously defined notion of speed measured non-locally. You're incorrect about the restriction to inertial frames; once you restrict to locality, the restriction to inertial frames becomes unnecessary.

FAQ: Is the speed of light equal to c even in an accelerating frame of reference?

The short answer is "yes."

The long answer is that it depends on what you mean by measuring the speed of light.

In the SI, the speed of light has a defined value of 299,792,458 m/s, because the meter is defined in terms of the speed of light. In the system of units commonly used by relativists, it has a defined value of 1. Obviously we can't do an experiment that will remeasure 1 to greater precision. However, it could turn out to have been a bad idea to give the speed of light a defined value, if that results in a distance unit whose length depends on extraneous variables.

One such extraneous variable might be the direction in which the light travels, as in the Sagnac effect, which was first observed experimentally in 1913. In the Sagnac effect, a beam of light is split, and the partial beams are sent clockwise and counterclockwise around an interferometer. If the interferometer is rotating in the plane of the beams' path, then a shift is observed in their interference, revealing that the time it takes light to go around the apparatus clockwise is different from the time it takes to go around counterclockwise. An observer in a nonrotating frame explains the observation by saying that the beams went at equal speeds, but their times of flight were unequal because while they were in flight, the apparatus accelerated. An observer in the frame rotating along with the apparatus says that clearly the beams could not have always had the same speed c, since they took unequal times to travel the same path. If we insist on letting c have a defined value, then the rotating observer is forced to say that the same closed path has a different length depending on whether the length is measured clockwise or counterclockwise. This is equivalent to saying that the distance unit has a length that depends on whether length is measured clockwise or counterclockwise.

Silly conclusions like this one can be eliminated by specifying that c has a defined value not in all experiments but in local experiments. The Sagnac effect is nonlocal because the apparatus has a finite size. The observed effect is proportional to the area enclosed by the beam-path. "Local" is actually very difficult to define rigorously [Sotiriou 2007], but basically the idea is that if your apparatus is of size L, any discrepancy in its measurement of c will approach zero in the limit as L approaches zero.

General relativity is not needed in order to understand examples like the Sagnac effect, which occurs in flat spacetime, but GR does help to clarify some of the issues. The fact that we can give c a certain value by definition is a specific example of a broader property of GR, which is that it is coordinate-independent. For example, we can subject our coordinates to a transformation x->x'=x*exp(-t/k), which is like making all the meter-sticks in the universe shrink exponentially with time. According to GR, all the laws of physics are obeyed in the x' coordinates just as they were in the original ones. This shows that we can never determine whether a fundamental "constant" is really constant unless it is something like the fine-structure constant that has the same value in all systems of units.[Webb 1999],[Chand 2004]

In a curved spacetime, it is theoretically possible for electromagnetic waves in a vacuum to undergo phenomena like refraction and partial reflection. Such effects are far too weak to be detected by any foreseeable technology. Assuming that they do really exist, they could be seen as analogous to what one sees in a dispersive medium. The question is then whether this constitutes a local effect or a nonlocal one. Only if it's a local effect would it violate the equivalence principle. This is closely related to the famous question of whether falling electric charges violate the equivalence principle. The best known paper on this is DeWitt and DeWitt (1964). A treatment that's easier to access online is Gron and Naess (2008). You can find many, many papers on this topic going back over the decades, with roughly half saying that such effects are local and violate the e.p., and half saying they're nonlocal and don't.

J.K. Webb et al. (1999). "Search for Time Variation of the Fine Structure Constant". Physical Review Letters 82 (5): 884–887.

H. Chand et al. (2004). "Probing the cosmological variation of the fine-structure constant: Results based on VLT-UVES sample". Astron. Astrophys. 417: 853.

Sotiriou, Faraoni, and Liberati, arxiv.org/abs/0707.2748

Cecile and Bryce DeWitt, "Falling Charges," Physics 1 (1964) 3

Gron and Naess, arxiv.org/abs/0806.0464v1

 

[Passionflower]

You're incorrect about the restriction to inertial frames; once you restrict to locality, the restriction to inertial frames becomes unnecessary.

Read again:

I wrote:

The speed of light is the same in all inertial frames of reference. The speed of light is also the same for all observers measured locally.

Notice the word also?

 

[bcrowell]

Read again:

I wrote:

The speed of light is the same in all inertial frames of reference. The speed of light is also the same for all observers measured locally.

Notice the word also?

Sorry, I misread your post.

 

[Saw]

In a system such as Rindler's, the existence of a universal velocity c is proved rather than assumed

We then have a satisfying answer to the "why" question, which is that existence of a universal speed c is a property of spacetime that must exist because spacetime has certain other properties (basically, it has some symmetries, and it doesn't have universal simultaneity).

Hmm… We discussed this in another thread and, although the discussion was very enrichening, I am afraid still unhappy with this sort of explanations.

But please let me try yo understand what is meant by them. The “universal speed c” that you mention (“universal” meaning “invariant”, I presume)… is it a pure logical consequence of the 1st postulate (the PoR), as apparently was said in that thread, or is it a consequence of an extra-postulate, namely that there is no universal (invariant) simultaneity, as you seem to affirm here?

 

[bcrowell]

The “universal speed c” that you mention (“universal” meaning “invariant”, I presume)… is it a pure logical consequence of the 1st postulate (the PoR), as apparently was said in that thread, or is it a consequence of an extra-postulate, namely that there is no universal (invariant) simultaneity, as you seem to affirm here?

It's a consequence of a completely different set of postulates, which mostly deal with the symmetry of spacetime. Rindler doesn't explicitly list a set of axioms. Based on the symmetry principles, you can prove that there are only two possibilities: SR or Galilean relativity. You do need some other postulate to distinguish between those, and non-simultaneity is one postulate that will do that. Another way of putting it is that the symmetry principles imply the existence of an invariant speed, c, but they are consistent with arbitrarily large values of c, and in that limit SR becomes indistinguishable from Galilean relativity.

 

[Saw]

Based on the symmetry principles, you can prove that there are only two possibilities: SR or Galilean relativity. You do need some other postulate to distinguish between those, and non-simultaneity is one postulate that will do that. Another way of putting it is that the symmetry principles imply the existence of an invariant speed, c, but they are consistent with arbitrarily large values of c, and in that limit SR becomes indistinguishable from Galilean relativity.

Yes, then this is the same that you explained in the other thread. Apart from Rindler, there was a paper by Palash in the same line, wasn’t there?

I admit the beauty of that explanation and it may be right, although I don’t share it. In the other thread the discussion followed with JesseM and it got a little heated, for which I also blame myself. I suppose it can be continued in a milder tone.

My problem with your explanation is the following:

It says that the Principle of Relativity (do you mean the same by “symmetry principles”?) necessarily implies that there is an invariant speed, which may be infinite (Galilean Relativity, in which case there is a frame-invariant time) or finite (Special Relativity, in which case time is frame-variant). (Well, there was also a “Causality Violation” case but we can leave that aside.)

However, I think that if we find in the reasoning a finite invariant speed, that is because we are already in the domain of SR. And logically in that domain, GR is still admissible albeit only as a very special or “degenerate” case (i.e. if the invariant speed is infinite).

The proof is that you could always take the reverse logical approach. With the same mindset but changing camps, I could say that the Principle of Relativity (or symmetry principles) necessarily imply that all speeds are variant, except in the case where the speed under consideration is infinite…

The idea is only half-born, so bear with me, but here it is, expressed in math. The background is the light clock thought experiment. Hope it is self-explanatory.

$${(ct)^2} = {(c't')^2} + {(vt)^2}$$

SR’s route: If t ≠ t’ but c = c’ --> $${\rm{t' = t}}\gamma$$

$$\begin{array}{l} {(ct)^2} = {(ct\gamma )^2} + {(vt)^2}\\ {c^2} = {(c\gamma )^2} + {v^2}\\ {\gamma ^2} = \frac{{{c^2} - {v^2}}}{{{c^2}}} = 1 - \frac{{{v^2}}}{{{c^2}}}\\ \gamma = \sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \to t' = t\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \end{array}$$

This reduces to t = t’ if c = ∞

GR’s route: If c ≠ c’ but t = t’ --> $${\rm{c' = c}}\gamma$$

$$\begin{array}{l} {(ct)^2} = {(c\gamma t)^2} + {(vt)^2}\\ {c^2} = {(c\gamma )^2} + {v^2}\\ {\gamma ^2} = \frac{{{c^2} - {v^2}}}{{{c^2}}} = 1 - \frac{{{v^2}}}{{{c^2}}}\\ \gamma = \sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \to c' = c\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \end{array}$$

This reduces to c = c’ if c = ∞

 

[starthaus]

$$\begin{array}{l} {(ct)^2} = {(c\gamma t)^2} + {(vt)^2}\\ {c^2} = {(c\gamma )^2} + {v^2}\\ {\gamma ^2} = \frac{{{c^2} - {v^2}}}{{{c^2}}} = 1 - \frac{{{v^2}}}{{{c^2}}}\\ \gamma = \sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \to c' = c\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \end{array}$$

This reduces to c = c’ if c = ∞

It is an interesting line of argument, it has been tried before. Both Tolman and Pauli (and later on, Mermin and Rindler) tried to reduce the second postulate to a theorem that could be derived from the fPoR. The argument is much more complicated than the one you sketched above. I can see two problems with your approach:

1. c is not ∞
2. Even if you admitted that c=∞ all that the above would show is that c'=∞, not that c=c'.

An interestting approach nevertheless.

 

[bcrowell]

It is an interesting line of argument, it has been tried before. Both Tolman and Pauli (and later on, Mermin and Rindler) tried to reduce the second postulate to a theorem that could be derived from the fPoR.

In the Rindler reference given in #4, he isn't claiming to deduce the second postulate from the first postulate. He's deducing it from a different set of axioms. If you think he's wrong, I suggest you write him a letter and tell him so. Maybe he'll thank you for pointing out his error. I don't think there is any such error, but I suppose it's always possible that there are glaring errors, even in a standard textbook, written by a well-known expert in the field, that has been through several editions.

 

[starthaus]

In the Rindler reference given in #4, he isn't claiming to deduce the second postulate from the first postulate. He's deducing it from a different set of axioms.

This is not what I posted. I posted. In his second edition he clearly uses the second postulate. I do not have your 1979 edition , so I never argued about that.

If you think he's wrong, I suggest you write him a letter and tell him so. Maybe he'll thank you for pointing out his error. I don't think there is any such error, but I suppose it's always possible that there are glaring errors, even in a standard textbook, written by a well-known expert in the field, that has been through several editions.

Interestingly, I wrote to him about a different error, in deriving the formulas for accelerated motion, he wasn't too keen on admitting and changing the text.

 

[Saw]

It is an interesting line of argument, it has been tried before. Both Tolman and Pauli (and later on, Mermin and Rindler) tried to reduce the second postulate to a theorem that could be derived from the fPoR. .

Well, actually, what I was trying to prove is that Rindler’s approach is not correct.

Rindler claims that from the 1st postulate alone, you can derive “an” invariant speed (c=c’=c’’’….) and then you only need a 2nd postulate to attribute a value to c: if your 2nd postulate is that c = infinite, you get the Galilean Transformation, whereas if your 2nd postulate is c = finite, you get the Lorentz Transformation.

Instead, what I argue is that on the basis of the 1st postulate alone you only get, logically, the 1st postulate. There is no invariant speed implicit in the 1st postulate. If you want to make progress, you need a second axiom: if it is that t = t’, you get the Galilean Transformation, whereas if it is c = c’, you get the Lorentz Transformation.

 

[starthaus]

Well, actually, what I was trying to prove is that Rindler’s approach is not correct.

Rindler claims that from the 1st postulate alone, you can derive “an” invariant speed (c=c’=c’’’….) and then you only need a 2nd postulate to attribute a value to c: if your 2nd postulate is that c = infinite, you get the Galilean Transformation, whereas if your 2nd postulate is c = finite, you get the Lorentz Transformation.

Instead, what I argue is that on the basis of the 1st postulate alone you only get, logically, the 1st postulate. There is no invariant speed implicit in the 1st postulate. If you want to make progress, you need a second axiom: if it is that t = t’, you get the Galilean Transformation, whereas if it is c = c’, you get the Lorentz Transformation.

I see, I misunderstood you. Perhaps this may be one of the reasons Rindler dropped the proof in the second edition of his book.