Constancy of Speed of Light: Postulate or Assumption?

[vanhees71]

The Galilean realization of the special principle of relativity (indistinguishability of inertial frames) is simply observed to be inaccurate and the Lorentzian/Minkowskian realization is closer to the observed phenomena. Within GR it's even refined to be valid only locally, and that's the hitherto most comprehensive spacetime model, which is consistent with all observations, and there are some very accurate ones (pulsar timing, gravitational wave shapes, motion of stars around the black hole in our galaxy,...) in favor of GR (e.g., when tested against post-Newtonian parametrizations).

 

[strangerep]

The versions I'm familiar with get you as far as "it's either Galileo, or Einstein with an unknown constant", and I think it was the rejecting Galileo bit that @A.T. was objecting to. Unless you know a way to reject Galilean relativity with a one-postulate approach? Or are you rejecting it as just being the $c\rightarrow\infty$ version of Einsteinian relativity?

Well, near the end of the derivation one arrives at a boost transformation formula like this:
$$t' ~=~ \gamma(t + \lambda_a v x) ~,~~~
x' ~=~ \gamma (x - vt) ~,~~~
y' = y ~,~~ z' = z ~,~~~~~~ \left[ \gamma := \frac{1}{\sqrt{1 + \lambda_a v^2}} \right] ~,$$with a velocity addition formula of the form:$$v'' ~=~ \frac{v + v'}{1 - \lambda_a v v'} ~.$$In the above, $\lambda_a$ is a (real-valued) constant with dimensions $T^2/L^2$, i.e., inverse speed squared.

For the next step, one examines the possible cases: $\,\lambda_a < 0,~$ $\,\lambda_a = 0\,~$ and $\,\lambda_a > 0\,$.

For the $\lambda_a = 0$ case, one finds the Galilean boost formula.

For the $\lambda_a > 0$ case, one finds that it doesn't satisfy the principle of physical regularity. The velocity addition formula in that case means one can go from rest to infinite velocity by 2 applications of the transformation with parameter $\zeta = \lambda_a^{-1/2}$. To banish this type of embarrassment, one must reduce the domain of the parameter $v$ to the trivial set $\{0\}$. This is easily enough to dismiss $\lambda_a > 0$ on very simple physical grounds, since distinct inertial observers with nonzero relative speed are definitely known to exist.

That leaves $\lambda_a < 0$, at which point one introduces a new parameter called $c := (-\lambda_a)^{-1/2}$, which gives the familiar Lorentz formulas. For $v/c \ll 1$, we get an approximation of the Galilean formula.

So we're not just rejecting Galilean relativity arbitrarily, but rather recognizing it as a low speed approximation feature of the $\lambda_a < 0$ case.

 

[vanhees71]

Another argument against the case $\lambda_a>0$ is the fact that all the boosts (in one direction) should form a group and with $\lambda_a>0$ these transformations are rotations. The consequence of this is that there is no causality structure possible, which is the case for $\lambda_a<0$, where you get, of course, the Lorentz group, for whose part that's smoothly connected with the identity (the proper orthochronous Lorentz transformations) for the time-like vectors the time components' sign doesn't change.

 

[strangerep]

$\lambda_a>0$ [...] no causality structure possible, [...]

It occurred to me overnight that we can actually do a bit better. In fact, no additional physical postulate or argument, such as causality or my "physical regularity", are needed. We can determine the sign of $\lambda_a$ simply because the case $\lambda_a > 0$ does not yield a mathematically well-defined nontrivial group of velocity boosts.

To see this in more detail, note first that a nontrivial 1-parameter group of velocity boosts must have a parameter space $V$ which is at least an open set containing 0. Elementary group properties require that for any two velocities $v, v' \in V$, the composition of those velocities must also be in $V$, else we do not have a good group. Now consider the velocity addition formula in post #37, specialized to the case where$0 < \lambda_a =: \zeta^{-2}$, for some real $\zeta > 0$. The velocity addition formula becomes $$v'' ~=~ \frac{v + v'}{1 - v'v/\zeta^2} ~.$$The value $\,v = \zeta\,$ then cannot be an allowed parameter value in $V$, since composition with itself gives $v''\sim\infty$, i.e., undefined. For convenience, let us introduce a new variable $\omega := v/\zeta$. Then the velocity addition formula can be written as $$\omega'' ~=~ \frac{\omega + \omega'}{1 - \omega'\omega} ~.$$Any pair $\omega'$ and $\omega$ satisfying $\,\omega\,\omega' = 1\,$ yields an undefined $\omega'' \sim\infty$. Moreover, we cannot solve this problem by restricting to $|\omega| < W$, for some constant $W$, since multiple boosts can eventually yield a resultant velocity greater than $W$, contradicting our attempted restriction. Only the trivial case $w \in \{0\}$ remains mathematically valid, but this is useless for physics.

Therefore we can discard $\lambda_a > 0$ simply because on a nontrivial velocity domain it gives a mathematically invalid group. There is no need to invoke any additional physical postulate or argument. Merely requiring mathematical consistency is sufficient.

So if $\lambda_a \ne 0$, only 1 possibility remains: $$\boxed{~ \lambda_a ~<~ 0 \;,~}$$ with velocity addition formula: $$v'' ~=~ \frac{v + v'}{1 + v'v \, |\lambda_a|} ~.$$This gives a mathematically well-defined group, with $V$ containing only those $v$ such that $\,|v|^2 \le -\lambda_a^{-1}$.

 

[Sagittarius A-Star]

Any pair $\omega'$ and $\omega$ satisfying $\,\omega\,\omega' = 1\,$ yields an undefined $\omega'' \sim\infty$.

I don't think this argument alone is sufficient. In the normal relativistic velocity composition, any pair $\omega'$ and $\omega$ satisfying $\,\omega\,\omega' = -1\,$ yields an undefined $\omega''$.

 

[strangerep]

I don't think this argument alone is sufficient. In the normal relativistic velocity composition, any pair $\omega'$ and $\omega$ satisfying $\,\omega\,\omega' = -1\,$ yields an undefined $\omega''$.

In the normal relativistic case, we also have the restriction $|w| < 1$. So $\omega\,\omega' = -1\,$ is not possible in that case.

However, if we try to "fix" the $\lambda_a > 0$ case by imposing a similar restriction, it doesn't work. E.g., for $w = w' = 0.9$ we find $$ \omega'' ~=~ \frac{0.9 + 0.9}{1 - 0.9^2} ~\approx~ 9.47 ~,$$which violates the condition $|w|<1$, hence it's not a valid group.

 

[Sagittarius A-Star]

However, if we try to "fix" the $\lambda_a > 0$ case by imposing a similar restriction, it doesn't work.

Yes, that's correct.

 

[vanhees71]

It occurred to me overnight that we can actually do a bit better. In fact, no additional physical postulate or argument, such as causality or my "physical regularity", are needed. We can determine the sign of $\lambda_a$ simply because the case $\lambda_a > 0$ does not yield a mathematically well-defined nontrivial group of velocity boosts.

To see this in more detail, note first that a nontrivial 1-parameter group of velocity boosts must have a parameter space $V$ which is at least an open set containing 0. Elementary group properties require that for any two velocities $v, v' \in V$, the composition of those velocities must also be in $V$, else we do not have a good group. Now consider the velocity addition formula in post #37, specialized to the case where$0 < \lambda_a =: \zeta^{-2}$, for some real $\zeta > 0$. The velocity addition formula becomes $$v'' ~=~ \frac{v + v'}{1 - v'v/\zeta^2} ~.$$The value $\,v = \zeta\,$ then cannot be an allowed parameter value in $V$, since composition with itself gives $v''\sim\infty$, i.e., undefined. For convenience, let us introduce a new variable $\omega := v/\zeta$. Then the velocity addition formula can be written as $$\omega'' ~=~ \frac{\omega + \omega'}{1 - \omega'\omega} ~.$$Any pair $\omega'$ and $\omega$ satisfying $\,\omega\,\omega' = 1\,$ yields an undefined $\omega'' \sim\infty$. Moreover, we cannot solve this problem by restricting to $|\omega| < W$, for some constant $W$, since multiple boosts can eventually yield a resultant velocity greater than $W$, contradicting our attempted restriction. Only the trivial case $w \in \{0\}$ remains mathematically valid, but this is useless for physics.

Therefore we can discard $\lambda_a > 0$ simply because on a nontrivial velocity domain it gives a mathematically invalid group. There is no need to invoke any additional physical postulate or argument. Merely requiring mathematical consistency is sufficient.

So if $\lambda_a \ne 0$, only 1 possibility remains: $$\boxed{~ \lambda_a ~<~ 0 \;,~}$$ with velocity addition formula: $$v'' ~=~ \frac{v + v'}{1 + v'v \, |\lambda_a|} ~.$$This gives a mathematically well-defined group, with $V$ containing only those $v$ such that $\,|v|^2 \le -\lambda_a^{-1}$.

That's an interesting argument, but indeed the transformations for $\lambda_a>0$ are simply the rotations in the $t$-$x$ plane, building the group O(2). The trouble with it is the argument with the causality structure, i.e., to have a spacetime model which allows a notion of causality or "time direction" you need the indefinite fundamental form a la Minkowski rather than the usual Euclidean positive definite scalar product. The case $\lambda_a=0$ is the limiting case also allowing a causality structure, which is simply the oriented absolute time of Newtonian mechanics.

 

[Sagittarius A-Star]

The case $\lambda_a>0$ has several unphysically properties. Time-reversal is one. But @strangerep is right in posting #39: The group has an infinite velocity discontinuity at $\Theta = \pi/2$ (at the normal coordinate rotation) which is also unphysical.

Source: W. Rinder, book "Essential Relativity" 2nd edition, chapter 2.17 "Special Relativity without the Second Postulate"

 

[Sagittarius A-Star]

Well, near the end of the derivation one arrives at a boost transformation formula like this:

The calculation can arrive there i.e. by demanding, that the velocity composition is commutative.

https://www.physicsforums.com/threa...rom-commutative-velocity-composition.1017275/

 

[vanhees71]

Well, velocity composition is commutative only in (1+1) dimensions. If you "add" two velocities in different direction in (1+3) dimensions, it's not. See Eq. (1.5.4) in

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[robphy]

These “1-postulate” approaches are likely analogous to the Cayley-Klein geometry approach. Interestingly, the 1-postulate approach seems to be complementary to Klein’s (initial?) rejection of the affine geometries that would become the Galilean and Minkowski spacetime geometries because he felt that angle measure (which would later to be identified with rapidities) needed to be periodic.

See the bottom of my old post
https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6647528

 

[strangerep]

The calculation can arrive there i.e. by demanding, that the velocity composition is commutative.
https://www.physicsforums.com/threa...rom-commutative-velocity-composition.1017275/

Yes, in my private workfile on this I use that technique, i.e., that a 1-parameter Lie group is necessarily commutative. (Btw, I'm pleased to hear that someone else has actually noticed that section of Rindler.)

 

[strangerep]

These “1-postulate” approaches are likely analogous to the Cayley-Klein geometry approach. [...]

The dS and AdS parts of the CK approach do seem related to the "Possible Kinematics" paper of Bacry & Levy-Leblond. J. Math. Phys., vol 9, no 10, 1605, (1968), where they try to generalize the Lie algebra consisting of (generators of) rotations, boosts, and spatiotemporal displacements. But, AFAICT, both dS and AdS suffer from similar mathematical invalidity (in either space- or time-displacement subgroups) as in the $\lambda_a > 0$ case above for boosts.

 

[DanMP]

If the speed of light were relative instead of absolute would you be seeking an explanation for that?

Absolutely I want to understand the nature as much as I can.

The question "why is the speed of light what it is" is the same question as "why is a nautical mile 1013 fathoms?"

No, the fact that is a constant, not the particular number.

Well, you can "explain" the reason for the constancy of the speed of light by saying that it is a consequence of the fine structure constant. BUT ... that really just pushes the problem off to having to explain why the fine structure constant has the value it has. Try doing THAT in your spare time !

Interesting idea. Maybe the short answer regarding the reason for the constancy of the speed of light is that the instruments used to measure it are made of atoms/molecules, held together by electromagnetic forces, and that the EM force carrier is also travelling with the speed of light? When you measure something with instruments directly affected by that something you should be surprised that you get the exact same result?

 

[Peter Strohmayer]

The axiom (postulate, assumption etc.) of the constancy of the "speed of light" is misleading in my opinion, as far as it is based on the Newtonian concept of "velocity" (resp. his concept of space and time).

If one derives new concepts of space and time (and thus also of velocity) from the mentioned axiom of the constancy of the „speed of light", which was actually the case so far, one changes the most important premise of his former axiom. You are sawing off the branch you are sitting on.

If, on the other hand, one bases the "postulate" on a "relativistic" concept of velocity, then the "postulate" becomes a matter of course.

Not the given (Newtonian) terms of space, time and velocity describe the propagation of the light, but the propagation of the light describes the terms of space, time and velocity. Not rigid scales and ticking clocks are the basis for the concepts of time, space and velocity, but the length of the propagation of a light pulse from the point of view of the observer who has emitted this light pulse (in other words: the light clock). Time is what passes when a light pulse propagates from event E1 of its emission to event E2 of its arrival. Space is what is bridged when a light pulse propagates from event E1 of its emission to event E2 of its arrival.

From the point of view of the particular observer, the length of propagation of a light pulse from its start from a light source at rest with him (event E1) to its arrival at a target (event E2) is both the length of time and the space that lies between these two events from his point of view (events E1 and E2 have a "light-like distance" from each other). The speed of the light pulse as a ratio of the space covered to the time required for it must by definition always be "1". This is not a postulate, but follows from the concepts of space and time.

 

[phinds]

Maybe the short answer regarding the reason for the constancy of the speed of light is that the instruments used to measure it are made of atoms/molecules, held together by electromagnetic forces, and that the EM force carrier is also travelling with the speed of light?

Is that intended to be a joke? I hope so.

 

[DanMP]

Is that intended to be a joke? I hope so.

No, it is an honest question. See here this:

A photon (from Ancient Greek φῶς, φωτός (phôs, phōtós) 'light') is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless,[a] so they always move at the speed of light in vacuum, 299792458 m/s

 

[strangerep]

If one derives new concepts of space and time (and thus also of velocity) from the mentioned axiom of the constancy of the „speed of light", which was actually the case so far, one changes the most important premise of his former axiom.

Blind people can set up coordinate systems in space and time, at least in their local vicinity. They don't need light propagation to accomplish that. Similarly, they can sense whether they're accelerating, regardless of whether they can sense their surroundings.

My point is that physical equivalence of inertial frames is a more fundamental principle than the tool of light propagation. [One doesn't have to be "sighted" to be an "observer". ]

 

[robphy]

In modern formulations of relativity, the emphasis is on causality and causal structures based on a finite upper limit of signal speeds, which was historically motivated by “light” and “electromagnetic phenomena”.

If the photon were found to have a nonzero invariant mass, then relativity would survive… with many references to light being replaced by maximum-speed signals.

 

[Sagittarius A-Star]

Not rigid scales and ticking clocks are the basis for the concepts of time, space and velocity, but the length of the propagation of a light pulse from the point of view of the observer who has emitted this light pulse (in other words: the light clock).

But the light clock contains a ruler (=rigid scale).

 

[Peter Strohmayer]

"But the light clock contains a ruler, what, I think, you mean with ‚rigid scale‘.“

The unit with which to measure is determined by a material basis in the form of a period of time found in nature that can be reproduced as accurately as possible (e.g. the decay of a caesium atom, the rotation of a pulsar, etc.).

From this arbitrarily determined unit of the time "1", during which a light pulse spreads out temporally between two events (start and arrival), follows as spatial length of this spreading out between the two events necessarily the unit of the space "1".

This could be used to construct a light clock with the length "1". But this "rigid scale" is not the actual material basis of the system of units.

 

[PeterDonis]

The unit with which to measure is determined by a material basis in the form of a period of time found in nature that can be reproduced as accurately as possible (e.g. the decay of a caesium atom, the rotation of a pulsar, etc.).

No, that's not what a light clock is. A cesium clock is not a light clock.

A light clock is a clock whose "ticks" are the bouncing of a light pulse between two mirrors that are held rigidly a fixed distance apart. In the context of SR, this works fine because spacetime is flat and the mirrors can simply be placed in free fall at rest relative to each other, and they will then stay at rest relative to each other, the same distance apart, forever.

However, as soon as you either allow spacetime to be curved (GR), or accelerate the clock, it is no longer a simple matter to keep the mirrors the same distance apart. But that is what is required for the light clock to work properly.

This could be used to construct a light clock with the length "1". But this "rigid scale" is not the actual material basis of the system of units.

Yes, it is. See above.

 

[Peter Strohmayer]

Two mirrors in free fall are a nice and useful idea, but not a "rigid scale" in the conventional sense ("primal meter") in which I used it above.

 

[PeterDonis]

Two mirrors in free fall are a nice and useful idea

Yes, one which goes by the name "light clock".

but not a "rigid scale" in the conventional sense ("primal meter") in which I used it above.

Then you're going to need to explain what, exactly, you think "rigid scale" means. Note that you weren't the first to use that term in this thread, as far as I can tell: @Sagittarius A-Star was. He is welcome to correct me if I'm wrong, but I took him to mean by "rigid scale" the fact that, as I described, the two mirrors of a light lock must remain the same distance apart at all times for the light clock to work properly.

 

[Peter Strohmayer]

In #51 I replaced the "rigid scale" (conventionally of solid matter) with a light clock. If you also call the spatial distance in an ideal light clock with two free-falling mirrors a "rigid scale", then we agree not in the terms used, but in the matter.

The underlying question is still the same: What could be the material basis of the unit system found in nature: a spatial distance between two simultaneous events (= the existence of two resting points of matter) or a temporal distance between two events at the same place?

I think this question is self-explanatory.

In this sense I agree with Sagittarius A-Star in #56: There is a "rigid scale" in the light clock, but originally only in the form of the time distance between two events in the same place.

The term "rigid scale" for a reproducible constant time interval at the same place does not seem appropriate to me. Perhaps "constant scale" would be better.

 

[Sagittarius A-Star]

In #51 I replaced the "rigid scale" (conventionally of solid matter) with a light clock.

The purpose of a light clock is not to measure a spatial distance, but to measure it's proper time.

 

[Peter Strohmayer]

Since a light clock connects the temporal distance "1" between two light-like distant events and the spatial distance "1" between these events, it can be used not only to measure the proper time, but also to measure a spatial distance between two mass points resting on top of each other (laser measuring device), and also to measure a possible relative speed between the mass points (radar).

 

[Sagittarius A-Star]

Since a light clock connects ...

Usually, time is measured with clock(s) and distance with a ruler.
You can measure distance also with the combination of a clock and light propagation (radar).
You can measure time also with the combination of a ruler and light propagation (light clock).

 

[Peter Strohmayer]

Ok, thank you.

 

[bhobba]

This is an interesting thread.

As a proponent of the so called one postulate version of SR, I must point out for it to work you need to be careful in defining what an inertial frame is. Most just say it's a frame where Newtons first law holds without mentioning its symmetry properties. I dont know why, but Landau is the only author I know that does that - and it is not even in his relativity book, its in his Mechanics textbook.

Thanks
Bill

 

[vanhees71]

Of course, you can't even formulate Newton's Lex I, if you don't have a minimal spacetime model to begin with. The usual one-postulate derivations of the Poincare group assume that there's a class of inertial frames, where Lex I holds ("special principle of relativity"), where there is a time which is assumed to be described by the oriented $\mathbb{R}$ with the usual topology (the Archimedian ordered Cauchy-complete field of real numberes) and that for any inertial observer at any moment of time space is described as a 3D Euclidean affine manifold. Taking then all the symmetries of this "preliminary spacetime model" (i.e., translation invariance of time + ISO(3) of euclidean space) you are lead to either the Galilei group and thus Newtonian spacetime with the notion of an absolute time, the Poincare group and Minkowski spacetime or ISO(4), but the latter possibility is ruled out by the additional assumption of the possibility to establish a "causality structure". See, e.g.,

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz
Transformations, Jour. Math. Phys. 10, 1518 (1969),
https://doi.org/10.1063/1.1665000

 

[DanMP]

Well, you can "explain" the reason for the constancy of the speed of light by saying that it is a consequence of the fine structure constant.

When you wrote this, did you ponder a moment on why the constancy of the speed of light through vacuum can be explained using the fine structure constant? Do you have any explanation, other than the fact that the instruments we use to measure it (clocks, rulers) are structures made of atoms/molecules, held together by the electromagnetic force, and that the force carrier for the electromagnetic force is a photon, moving with the same speed as the light we measure?

If you measure the speed of light on a mountain top using a clock on sea level, you won't get the same value as you get using a local clock.

Another thing, how this speed of light can be "enforced"? Are the photons carrying speedometers? And/or the photons are acting like that because they were programmed to do so? Don't you think that this constancy of the speed of light is simply a consequence of the fact that we measure it using instruments affected by it?

 

[phinds]

If you measure the speed of light on a mountain top using a clock on sea level, you won't get the same value as you get using a local clock

If you make ANY local measurement regarding time but using a non-local clock, you would likely get the wrong results. This is what is called a "vacuous truth". It's true but it doesn't tell you anything meaningful or useful.

Don't you think that this constancy of the speed of light is simply a consequence of the fact that we measure it using instruments affected by it?

Absolutely not.
(1) We measure distance with a ruler --- are there any distances that are forced to be constant by virtue of the fact that we measure them with rulers?
(2) You are not required to use a light clock to measure the speed of light, you could use a mechanical clock.
(3) How would the fact that we use a clock to measure the speed of light explain the fact that if an object is approaching us slowly and shines a light at us, we see the speed of that light as c, and if it is approaching very fast and shines a light at us, we also see the speed of that light as c?

 

[PeterDonis]

how this speed of light can be "enforced"?

By the geometry of spacetime. The geometry of spacetime is such that any massless object will move at what we have been calling "the speed of light". But that term is really a misnomer as it's a property of spacetime geometry and not of light; it should be called "the invariant speed" or something like that.

Another way of looking at it is that what we have been calling "the speed of light" is really just a unit conversion factor: it converts from time units to distance units (or, if you use its reciprocal, it converts from distance units to time units). It's no different from, for example, sailors measuring horizontal distance in nautical miles but depth of water in fathoms, and having to convert between them sometimes. The conversion is just a matter of geometry.

Don't you think that this constancy of the speed of light is simply a consequence of the fact that we measure it using instruments affected by it?

The fact that all of our current instruments are made of atoms whose structure and behavior is primarily determined by the electromagnetic interaction, and hence depends on the fine structure constant, is an artifact of our current technology, not a law of physics. We can imagine measuring devices made out of something else, whose behavior would not depend on the fine structure constant, but such devices would still have to measure the same invariant speed.

The relevance of the fine structure constant to measuring the speed of light, taken literally--i.e., we are actually measuring the speed of light beams, not trying to theoretically understand why there is a particular speed that is invariant based on the geometry of spacetime--is that light is electromagnetic radiation and the fine structure constant is the relevant physical constant for electromagnetic behavior. But there could be some other field that is also massless (for example neutrinos were believed to be massless until a few decades ago), and its speed would also be the invariant speed even though it would not be electromagnetic. Understanding why such a field would have the same invariant speed as light would require understanding the geometry of spacetime.