I Constancy of Speed of Light: Postulate or Assumption?

[Herman Trivilino]

Even better it would be to finally explain it (the reason for the constancy of the speed of light), instead of assume or postulate it.

Well, suppose it's not true. If the speed of light were relative instead of absolute would you be seeking an explanation for that?

 

[jbriggs444]

Well, suppose it's not true. If the speed of light were relative instead of absolute would you be seeking an explanation for that?

@Ibix alluded to such a possibility in #20. It is possible that light is massive and its speed is indeed relative. The idea is that it only seems to always go at $c$ because the photon mass is low. So very low that light speed is negligibly different from the maximum speed for our relativistic universe.

Experiment puts a fiendishly tight upper bound on the photon mass. ($6 \times 10^{-16} \text{ eV}$ per a quick trip to Google).

 

[Ibix]

Once you introduce "pick whatever matches experiment" into the formulation of a physical theory, then you don't have unambiguous predictions. The function of the 2nd postulate is to pick between the two choices.

Fair point. There seems to me to be a distinction in degree, however, between postulating the speed of light as a constant at the start of a derivation and postulating that option 1 is correct not option 2 at the end of it.

 

[strangerep]

Once you introduce "pick whatever matches experiment" into the formulation of a physical theory, then you don't have unambiguous predictions. The function of the 2nd postulate is to pick between the two choices.

That's a misleading representation of what actually happens in the "1-postulate" derivations of SR. Rather, one derives that the Relativity Principle implies that the most general 1-parameter velocity-boosting symmetry group contained therein admits a universal constant with dimensions of inverse speed squared. Experiment determines the value of that constant.

 

[Ibix]

That's a misleading representation of what actually happens in the "1-postulate" derivations of SR. Rather, one derives that the Relativity Principle implies that the most general 1-parameter velocity-boosting symmetry group contained therein admits a universal constant with dimensions of inverse speed squared.

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?

 

[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.