I want a good reference to a discussion of the meaning of "the speed of light is constant"

[gnnmartin]

I am interested in making an observation that involves the speed of light. It is widely stated that the speed of light is constant, but without saying what that means. I need to be precise, and would like a reference to an acknowledgement of the problem.

When people talk about the speed of light being constant, they usually mean (in effect) that when calculating the units of a constant, the units may be multiplied by -(d^2x/dt^2) at any point in space/time, but in that context dx/dt is not really a ‘speed’. A speed is a rate of change of position, and that can only take place along a line in space time.

I wish to consider a chart of a space time, and treat space/time as a 3 space varying with time. Along any line we can construct a metric with line element ds^2=-g(t,t)dt^2+g(x,x)dx^2, and define the speed of light along the line as sqrt[-g(x,x)/g(t,t)].

By that definition, the speed of light is not constant along a timelike line in an expanding universe, but the variation is generally assumed to be negligible. I read a paper in the last few months which mentioned this observation, and annoyingly I have forgotten the name of both paper and author.

Is there a good reference that expands on the above, ideally including a discussion of the implication of the speed of light (by that definition) changing so fast that the change can not be ignored?

 

[PeroK]

In GR, light moves on null worldlines, which means that the speed of light, as measured locally, is a universal invariant. Measured globally, speed is coordinate dependent. There is no unambiguous way to define speed across a region of curved spacetime.

 

[Nugatory]

It is widely stated that the speed of light is constant, but without saying what that means.

Presuming that “g(t,t)” and “g(x,x)” represent $g_{tt}$ and $g_{xx}$, components of the metric tensor written in some coordinate system in which the metric is diagonal…

Then $\sqrt{-g_{xx}/g_{tt}}$ is the coordinate speed of light, and is only constant if these are the coordinates of a local inertial frame. If curvature effects are relevant, as they will be when we’re considering an expanding universe, the frame is not inertial and coordinate speeds have no physical meaning.

Thus in general relativity we do not start with the premise that “light speed is constant”. Instead we start from the premise that “light moves on null worldlines”. From there we can calculate the coordinate speed of light using whatever coordinates we please, and that that coordinate speed will be the same in all locally inertial frames.

It may not be possible to say much more unless you can dig up the paper you’re thinking of.

 

[Dale]

The important thing about $c$ isn’t that it is constant. The important thing is that it is invariant.

 

[PeterDonis]

The important thing about $c$ isn’t that it is constant. The important thing is that it is invariant.

But what is invariant is the light cones, not the numerical value of $c$. One can always find a coordinate transformation that changes the coordinate speed of light. But one cannot find a coordinate transformation that changes the light cones.

 

[cianfa72]

I would add that in the context of SR what is constant/invariant is the two-way speed of light along any closed path (of given length). It is the invariant $c$ and physically measurable since its measurement involves only one clock (no simultaneity convention involved at all).

Edit: MMX experiment shows that the two-way speed of light is invariant and isotropic.

 

[PeterDonis]

I would add that in the context of SR what is constant/invariant is the two-way speed of light along any closed path (of given length). It is the invariant $c$ and physically measurable since its measurement involves only one clock (no simultaneity convention involved at all).

Careful. The measurement involves only one clock, so the round-trip travel time of the light is invariant. But converting that to a speed requires knowing the distance that the light traveled, and depending on how you interpret "distance", that can depend on your choice of coordinates. Also, even if you have agreed on a definition of "distance" that gives you an invariant, unless the round-trip is between free-falling worldlines at rest relative to each other in flat spacetime, the numerical value of the resulting calculated speed will not, in general, be $c$.

MMX experiment shows that the two-way speed of light is invariant and isotropic.

Strictly speaking, no, it only shows that the round-trip travel time of light is invariant and isotropic. See above.

 

[cianfa72]

Also, even if you have agreed on a definition of "distance" that gives you an invariant, unless the round-trip is between free-falling worldlines at rest relative to each other in flat spacetime, the numerical value of the resulting calculated speed will not, in general, be $c$.

In the context of SR (no gravity, no spacetime curvature, flat spacetime) what does mean that two free-falling timelike worldlines (zero path curvature, zero proper acceleration) are at rest each other ?

 

[PeterDonis]

In the context of SR (no gravity, no spacetime curvature, flat spacetime) what does mean that two free-falling timelike worldlines (zero path curvature, zero proper acceleration) are at rest each other ?

That the round-trip light travel time between them is constant, as measured by an observer following each worldline.

 

[cianfa72]

That the round-trip light travel time between them is constant, as measured by an observer following each worldline.

Then, let me say, the fact that the resulting calculated two-way speed of light is constant/invariant is actually "tautologically" true (i.e. it is true by definition).

 

[PeterDonis]

the fact that the resulting calculated two-way speed of light is constant/invariant is actually "tautologically" true (i.e. it is true by definition).

For that particular case, yes, you can define the "distance" between the worldlines so that it is constant, and so that the numerical value of the calculated two-way speed of light is $c$.

 

[cianfa72]

For that particular case, yes, you can define the "distance" between the worldlines so that it is constant, and so that the numerical value of the calculated two-way speed of light is $c$.

So your point is: in flat spacetime in any global inertial frame/chart the "representatives" of the two timelike geodesics having constant "distance" (constant as defined from round-trip light travel time between them, as measured by an observer following each worldline) are "parallel" straight lines (i.e. their coordinates in the chart have linear dependence on the proper time $\tau$ along each of them). Therefore it exists a global inertial chart in which the two timelike geodesics are "at rest" (and their distance is the Euclidean distance). Since in such a chart the metric is in the standard Minkowski form, that means that the (invariant, i.e. frame-independent) calculated two-way speed of light along closed paths of round-trip light pulses results to be $c$.

 

[PeterDonis]

So your point is

Yes, for that particular case, all of the things you say are correct.

 

[cianfa72]

In case of GR, instead, (spacetime curvature) do not exist in general two timelike geodesics (zero proper acceleration) with constant "distance" (as defined from round-trip light travel time between them, as measured by an observer following each worldline) since the geodesic deviation is not null, I believe.

 

[PeterDonis]

In case of GR, instead, (spacetime curvature) do not exist in general two timelike geodesics (zero proper acceleration) with constant "distance" (as defined from round-trip light travel time between them, as measured by an observer following each worldline) since the geodesic deviation is not null, I believe.

Yes, in a general curved spacetime that is correct.

 

[gnnmartin]

Many thanks for all your replies. I'm struggling to decide whether I just expressed myself badly, or am simply confusing myself. If I decide after all that I do have something I wish to clarify, I will post again.

 

[cianfa72]

Yes, in a general curved spacetime that is correct.

If in the given spacetime do exist two timelike paths that are "Born rigid" then their "distance" by definition does not change in time. Therefore the calculated invariant two-way speed of light of bouncing radar/light pulses (as measured by observers following each worldline) will be constant, however it might be not numerically equal to $c$.

 

[PeterDonis]

If in the given spacetime do exist two timelike paths that are "Born rigid" then their "distance" by definition does not change in time.

Yes, but in general such timelike worldlines will not be geodesics in a curved spacetime.

Therefore the calculated invariant two-way speed of light of bouncing radar/light pulses (as measured by observers following each worldline) will be constant, however it might be not numerically equal to $c$.

Yes.

 

[cianfa72]

Yes, but in general such timelike worldlines will not be geodesics in a curved spacetime.

Yes, however at least one of the two worldlines can possibly be.

 

[PeterDonis]

at least one of the two worldlines can possibly be.

In general at most one can possibly be.

 

[cianfa72]

Strictly speaking, no, it only shows that the round-trip travel time of light is invariant and isotropic. See above.

So the point is that in MMX experiment one cannot assume that the "distance" along each of the arms of the interferometer stays unchanged. Hence from the invariance and isotropy of round-trip travel time of light one cannot conclude that the two-way speed of light is isotropic and invariant.

 

[PeterDonis]

So the point is that in MMX experiment one cannot assume that the "distance" along each of the arms of the interferometer stays unchanged. Hence from the invariance and isotropy of round-trip travel time of light one cannot conclude that the two-way speed of light is isotropic and invariant.

No, that's not the point. One can conclude that the "distance" stays unchanged from the fact that the round-trip light travel time is unchanged. (At least, one can conclude that if one puts reasonable limits on what counts as a valid definition of "distance" between a pair of worldlines.) But that, in itself, does not tell you what the distance is.

The point is that you have to define what "distance" means, and that requires a choice of coordinates/simultaneity convention. Even in the simplest case, parallel free-falling worldlines in flat spacetime, defining the "distance" between them requires defining which pairs of events, one on each worldline, are "at the same time", so that the spacelike interval between those can be defined as the "distance" between the worldlines. The natural such definition, of course, is the standard one used in the Minkowski inertial chart, which makes the distance work out such that the round-trip speed of light is $c$. But that's still a definition, a choice of coordinates and simultaneity convention. There's no way around that.

 

[cianfa72]

No, that's not the point. One can conclude that the "distance" stays unchanged from the fact that the round-trip light travel time is unchanged. (At least, one can conclude that if one puts reasonable limits on what counts as a valid definition of "distance" between a pair of worldlines.) But that, in itself, does not tell you what the distance is.

Ok, so the "distance" stays unchanged however its value is basically undefined until one picks a simultaneity convention/coordinates.

Even in the simplest case, parallel free-falling worldlines in flat spacetime, defining the "distance" between them requires defining which pairs of events, one on each worldline, are "at the same time", so that the spacelike interval between those can be defined as the "distance" between the worldlines.

Ok, here in MMX experiment one pair of timelike worldlines are the worldlines of the endpoints of each arm of the interferometer. The only restriction to define a "distance" between them is to pick events one along each of them that are spacelike separated.

 

[PeterDonis]

the "distance" stays unchanged however its value is basically undefined until one picks a simultaneity convention/coordinates.

Yes.

here in MMX experiment one pair of timelike worldlines are the worldlines of the endpoints of each arm of the interferometer. The only restriction to define a "distance" between them is to pick events one along each of them that are spacelike separated.

Yes, but there are an infinite number of possible definitions that meet that restriction.

 

[cianfa72]

Therefore in MMX experiment if one assumes flat spacetime (no gravity) & the apparatus without any force acting on it (i.e. all its parts and arms actually follow geodesic paths through spacetime) then from the isotropy and invariance of round-trip light travel time (as measured by any single clock along its worldline) and from the "natural distance" definition given by the global inertial chart (and its implied simultaneity convention) in which it is "at rest", it follows that the two-way speed of light is isotropic, invariant with value $c$.

 

[PeterDonis]

Therefore in MMX experiment if one assumes flat spacetime (no gravity) & the apparatus without any force acting on it (i.e. all its parts and arms actually follow geodesic paths through spacetime) then from the isotropy and invariance of round-trip light travel time (as measured by any single clock along its worldline) and from the "natural distance" definition given by the global inertial chart (and its implied simultaneity convention) in which it is "at rest", it follows that the two-way speed of light is isotropic, invariant with value $c$.

You keep restating the same thing.

 

[cianfa72]

You keep restating the same thing.

Yes, the above was just a summary of my understanding and I believe the logical conclusion in those hypothesis actually makes sense.

 

[bhobba]

These discussions are always interesting.

I like the approach where an invariant speed follows from the POR in inertial frames:

http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf.

That the constant must be the speed of light follows from all sorts of experimental and theoretical reasons.

One can even start with SR and an undetermined constant, derive Maxwell's equations, and hence that the constant is the speed of light:
http://richardhaskell.com/files/Special Relativity and Maxwells Equations.pdf

Which approach you prefer is a matter of taste.

Thanks
Bill

 

[cianfa72]

I like the approach where an invariant speed follows from the POR in inertial frames:

Sorry, what do you mean with POR ?

 

[DrGreg]

Sorry, what do you mean with POR ?

I presume bhobba means "Principle of Relativity" a.k.a. "Einstein's 1st Postulate".

 

[cianfa72]

I like the approach where an invariant speed follows from the POR in inertial frames:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf.

That the constant must be the speed of light follows from all sorts of experimental and theoretical reasons.

Very interesting. The referenced paper leverages on Principle of Relativity (PoR) for inertial frames and symmetries of space and time

However, the most general transformation of space and time coordinates can be derived using only the equivalence of all inertial reference frames and the symmetries of space and time

One is left only with three possibilities: standard Lorentz transformations for the underlying Minkowski spacetime, Galilei transformations and orthogonal transformation that preserve the euclidean metric of the underlying "Euclidean spacetime".

 

[gnnmartin]

These discussions are always interesting.

I like the approach where an invariant speed follows from the POR in inertial frames:

http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf.
...
Thanks
Bill

Bill comes closest to my concern. I note that Wikipedia https://en.wikipedia.org/wiki/Variable_speed_of_light
describes a number of papers exploring the possibility that the speed of light is not constant. I am not so interested in these theories per se, but rather in the implication for measurement in GR of a variable speed of light.

My reasoning is as follows:
1: there are notions of (physical) length and of duration which underly physical theories.
2: The metric reflects the magnitude of lengths and durations, enabling the size of a coordinate interval to be related to the size of a physical interval, and vice versa.
3: If we have two points in the same chart, then if an interval at one point has the same physical value as another at the second point, the coordinate intervals predicted by the metric should also be the same.
4: The 'speed of light' is the ratio of a physical length and a physical interval. A change of this ratio is conceivable, but given how the intervals are used almost interchangeably in the definitions of equations and constants in physics, it is not obvious what that change implies.
5: An illustration of the problem is the question often asked "What is the local effect of the expansion of the universe". The answer often given (including by me) is that the local effect is so small that it can be ignored, but that ducks the real issue.
6: I feel that at least we can say the speed of light must only vary extremely slowly along a line in space/time, so that for most purposes the variation can be ignored.
7: The expansion of the universe, as it is generally described, assumes the speed of light varies along the time axis of a typical coordinate system representing the cosmic universe.

So, my question was/is: is there a better treatment of the implications described in point 4 above. And/or, at which point above have I taken an illogical turn?

 

[PeterDonis]

an interval at one point has the same physical value as another at the second point

Intervals aren't "at" points. They are between two points. If you want to consider quantities at one point, you really should be considering vectors.

Also, in a general curved spacetime, there is no unique way to transport vectors (or more generally tensor quantities) from one point to another; the result is path-dependent. So the comparison you are making between quantities at one point and quantities at another point doesn't actually work.

The 'speed of light' is the ratio of a physical length and a physical interval.

Not the coordinate speed of light, which is what you appear to be interested. That is the ratio of a coordinate length to a coordinate time.

As far as "physical" quantities are concerned, since in a general curved spacetime you can only compare them, take ratios, etc., at one point, the "physical speed of light" is just a choice of units: the null vectors at a given point are picked out by the spacetime geometry, and the "physical speed of light" is just the ratio of length units to time units for null vectors--which makes it obvious that the most natural choice is $1$, i.e., use the same units for length and time.

 

[PeterDonis]

The expansion of the universe, as it is generally described, assumes the speed of light varies along the time axis of a typical coordinate system representing the cosmic universe.

What does this even mean?

 

[PeterDonis]

I note that Wikipedia https://en.wikipedia.org/wiki/Variable_speed_of_light
describes a number of papers exploring the possibility that the speed of light is not constant. I am not so interested in these theories per se, but rather in the implication for measurement in GR of a variable speed of light.

Unfortunately, Wikipedia, as is often the case, does a poor job of describing the actual physics involved. The actual physical quantity whose variation is tested is the fine structure constant. That is the dimensionless quantity that describes the electromagnetic interaction and its consequences, of which the propagation of light is one. But one can make such variation appear as variation in the speed of light, or variation in the charge on the electron, or variation in Planck's constant, or any mixture of those, by an appropriate choice of units. The choice of units has nothing to do with the physics.

(In alternative gravity theories to GR, some of which are mentioned in the Wikipedia article, the variation can also be made to appear in Newton's gravitational constant by choice of units.)