# If c changes w/ time, what happens with laws of SR?

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Howdy,

Was wondering if you guys would mind critiquing, correcting, or adding any good insights to the following which was posted elsewhere by me, thanks!

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.... remember a ways back there was a finding that went viral about the speed of light not being constant? I forget the details now, but it had everybody up in arms that it would be the downfall of Special Relativity.

But, I don't think it really would have been (as most physicists were saying at the time), but it shows the subtleties involved. In Special Relativity, you have the all-important metric at the heart of the theory:

d_tau^2 = -c^2*dt^2 + dx^2

This equation outlines the geometric structure of space-time as it measures distances in space-time, in a similar fashion to how Pythagorean Theorem measures distances in Euclidean Space (ds^2 = dx^2 + dy^2). What's special about c, is not any role as a Universal "speed limit", but that it actually helps define this unified geometric structure of space-time and converts between time and space in such a way as to keep the interval (distances in space-time) invariant. Again, this is the same thing as distances staying invariant under a Euclidean Space, or regular space, as we normally think about it

But, if the magnitude of the speed of light, or c, changes, the form of the Lorentz Transformation equations do not necessarily have to change. The underlying symmetry which preserves the invariance of the interval under Lorentz Transformations, and the form of all the equations would stay the same. The form of the equation above for the metric stays the same. But, it seems to me something about the geometric structure of space-time would change as the speed of light changes. If the speed of light in the equation above changed slowly over time, the geometric structure of space-time would slowly change with time. Indeed, I think equivalent experiments done at different times might give different answers, since the measured space-time distance between events would change. Perhaps, it would be somewhat analogous to an n-sided polygon slowly growing/shrinking, the symmetries wouldn't change in that under certain rotations (of 2*pi/n) the appearance of the n-sided polygon would still not change at any given instant of time, but length magnitudes are indeed changing over time. Then again, if you're stuck on the side of the polygon (or in space-time), perhaps from your vantage point nothing does change, as you (and your experimental apparatus) would be changing along with the environment!?!?

Orodruin
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If c changes with time in the definition of the metric, you are no longer dealing with SR. You will generally be dealing with a curved space-time, i.e., GR.

In SR, c is just an artefact of using different units for time and space. You can get a similar effect in Eulidean geometry by using different units for the x and y directions.

If c changes with time in the definition of the metric, you are no longer dealing with SR. You will generally be dealing with a curved space-time, i.e., GR.

Good point, that does make space-time dynamic, but it's still not exactly GR because c is constant in GR. I wanted to think about what it means ONLY if c is changing with time. It seems to me all the equations of SR are still valid, as well as Lorentz Transformations. In addition, at any one instant of time all observers still agree on the length of the interval.

So, has the underlying symmetries of the theory changed? I guess THAT's what I am really after.

In SR, c is just an artefact of using different units for time and space. You can get a similar effect in Eulidean geometry by using different units for the x and y directions.

Artifact doesn't seem quiet the right term to me, because I mean, yeah, it does act as a conversion factor between units, but "c" is also a physically measurable quantity. I did this in one my undergrad labs, best lab ever. The agreement between students using such a crude method was pretty amazing

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The more general question I'm after is what is the relationship between the constants we find in nature, c, hbar, etc and the underlying symmetries we find in nature.

It seems to me that symmetries changing in time, imply regularities in nature changing in time, which potentially means laws of physics changing in time. Constants changing don't seem quite as dramatic in their effect. Does anybody agree/disagree?

Orodruin
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Good point, that does make space-time dynamic, but it's still not exactly GR because c is constant in GR.
No, this is incorrect, it is exactly GR, but probably violating the Einstein equations. The c that appears in the metric is just an arbitrary coordinate velocity. You can pick your coordinates in such a way that it takes any value. What matters is the space-time geometry, which is the same regardless of what coordinates you pick. In GR, it is very common to pick coordinates such that the coordinate speed of light is larger than the speed of light. This does not correspond to any physical velocity and local measurements will still display local Lorentz invariance.

So, has the underlying symmetries of the theory changed? I guess THAT's what I am really after.
Since your basic conception is flawed, this question makes little change. If you change c with t, then your space-time is curved. This directly leads to a curved manifold so it is obvious that this changes the space-time geometry.

but "c" is also a physically measurable quantity
No it is not. It is a defined quantity, used to define the length unit one meter.

I did this in one my undergrad labs, best lab ever.
No you did not. As I told my professor in the corresponding lab when he enthusiastically asked what we had done: "we have calibrated your ruler"

The more general question I'm after is what is the relationship between the constants we find in nature, c, hbar, etc and the underlying symmetries we find in nature.
These are both arbitrary constants that depend on the choice of units. In natural units they are both equal to one. Physically they are not much more than unit conversion factors.

No, this is incorrect, it is exactly GR, but probably violating the Einstein equations. The c that appears in the metric is just an arbitrary coordinate velocity. You can pick your coordinates in such a way that it takes any value. What matters is the space-time geometry, which is the same regardless of what coordinates you pick. In GR, it is very common to pick coordinates such that the coordinate speed of light is larger than the speed of light. This does not correspond to any physical velocity and local measurements will still display local Lorentz invariance.

Since your basic conception is flawed, this question makes little change. If you change c with t, then your space-time is curved. This directly leads to a curved manifold so it is obvious that this changes the space-time geometry.

No it is not. It is a defined quantity, used to define the length unit one meter.

No you did not. As I told my professor in the corresponding lab when he enthusiastically asked what we had done: "we have calibrated your ruler"

These are both arbitrary constants that depend on the choice of units. In natural units they are both equal to one. Physically they are not much more than unit conversion factors.

You're stuck on semantics and definitions and not really seeing what I am saying. Actual light, or EM waves, can care less how we define a meter, and it sure doesn't tailor its behavior around that.

Symmetries in nature are not 'arbitrary constants'. Not sure where else to take this conversation with you if that's what you think. The constants of nature aren't even arbitrary, just the values we proscribe to them.

Whether, or not, you're dealing with a curved manifold doesn't answer my question. I already agreed making c time dependent makes space-time dynamic. My question was does this invalidate the Lorentz Transformations, or does it change the form of the equations in Special Relativity?

Ibix
I think Orodruin's point is that you couldn't apply Special Relativity to a spacetime where c is changing. You could only apply it as an approximation to a small patch (a 4d patch, so a volume with a duration) that is small enough that the variation of c is negligible.

Orodruin
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Symmetries in nature are not 'arbitrary constants'.
No they are not, so it is unclear why you want to make a connection between the two. SR in itself is based on a flat Minkowski space-time. If you change c you will no longer have that. Clearly you cannot expect that Lorentz transformations will still preserve the form of the metric just as you cannot expect that a Euclidean structure will be preserved if you change the geometry of a Riemannian manifold.

Actual light, or EM waves, can care less how we define a meter, and it sure doesn't tailor its behavior around that.
Physics do not care about how you define any units. Actual light has nothing to do with this. However, in order to measure a meter, you ultimately have to calibrate using light (or something else travelling at light speed).

My question was does this invalidate the Lorentz Transformations, or does it change the form of the equations in Special Relativity?

Come to think of it, that manifold actually is the same. If c is a function of t, just define a new time variable ##T## such that ##c(t) dt = dT## and you are back in Minkowski space with c = 1. The coordinate t would likely seem very unnatural because it will not constitute an affine time parameter for an observer at rest. If instead you put the time dependence in front of dx, you have a FRW type metric which is different from Minkowski space (for most scale factors).

Come to think of it, that manifold actually is the same. If c is a function of t, just define a new time variable T such that c(t) dt = dT and you are back in Minkowski space with c = 1

Thanks, that's pretty much what I expected, but you stated it more precisely than I was able. This is more what I was after.

No they are not, so it is unclear why you want to make a connection between the two.

I didn't. My original contention/suspicion was that symmetries changing with time would have fundamentally different ramifications than constants changing with time.

PAllen
Well, the simplest way to encode time varying c on SR would be a metric of form:

ds2 = c(t)2dt2 - dx2 - dy2 - dz2

which is just a slightly funny way of writing the class of FLRW open universe solutions with Euclidean spatial slices! In other words, the cosmology our universe is believed to be very close to, but not exactly.

Orodruin
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Well, the simplest way to encode time varying c on SR would be a metric of form:

ds2 = c(t)2dt2 - dx2 - dy2 - dz2

which is just a slightly funny way of writing the class of FLRW open universe solutions with Euclidean spatial slices! In other words, the cosmology our universe is believed to be very close to, but not exactly.
It is not a FLRW universe. In fact, it is just a reparametrisation of Minkowski space. Just use a time parameter such that ##dT = c(t) dt## and the line element becomes the Minkowski one.

You could only apply it as an approximation to a small patch (a 4d patch, so a volume with a duration) that is small enough that the variation of c is negligible.

This is a good point, thanks. We know our Universe isn't Minkowskian flat space, which is just an approximation, albeit a pretty good one over rather significant domains of validity when it comes to human activity. So, seems like this also implies that, rather than invalidate the laws/symmetries of Special Relativity, a changing speed of light would just further constrain the space-time region they're valid over (based on some arbitrary degree of desired accuracy). Again, I think this also demonstrates that a time-evolving "fundamental" constant does not necessarily violate the underlying symmetries of the theory. Is there any cases where this does not hold true (anywhere in physics)?

I guess this would also mean that the symmetries found in Special Relativity (or perhaps ALL of our theories for that matter) are also just approximations to whatever reality is truly like. They may be mathematical truths, but they may only correspond to certain domains when it comes to physical reality, in general

It is not a FLRW universe. In fact, it is just a reparametrisation of Minkowski space. Just use a time parameter such that ##dT = c(t) dt## and the line element becomes the Minkowski one.

Yes, for simple FLRW metrics, isn't the scale factor on the position coordinates, NOT the time coordinate

PAllen
It is not a FLRW universe. In fact, it is just a reparametrisation of Minkowski space. Just use a time parameter such that ##dT = c(t) dt## and the line element becomes the Minkowski one.
Oops, you are right. What would be needed is:

c(t)2d##\tau##2 = c(t)2dt2 - dx2 - dy2 - dz2

which can still be claimed as a simple way to 'encoding' varying light speed that produces FLRW.

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PAllen
Yes, for simple FLRW metrics, isn't the scale factor on the position coordinates, NOT the time coordinate
That doesn't matter, just a re-arrangement. I just forgot to replace implicit c on the left side as well. Then you do get just an alternate form of FLRW.

That doesn't matter, just a re-arrangement. I just forgot to replace implicit c on the left side as well

Based on the correction in your latest post, I totally agree, the two are equivalent.

Orodruin
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Based on the correction in your latest post, I totally agree, the two are equivalent.
It should be noted that the FLRW universe is (usually) not equivalent to Minkowski space. The two versions we have discussed here are different space-times.

PAllen
It should be noted that the FLRW universe is (usually) not equivalent to Minkowski space. The two versions we have discussed here are different space-times.
Right, the version with Euclidean spatial slices has space-time curvature.

pervect
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If c changes with time in the definition of the metric, you are no longer dealing with SR. You will generally be dealing with a curved space-time, i.e., GR.

In SR, c is just an artefact of using different units for time and space. You can get a similar effect in Eulidean geometry by using different units for the x and y directions.

I think a good first step in answering the question "what happens when c varies with time" is to clarify the question. Since we only discuss peer reviewed literature here on PF, a link to a peer reviewed paper would provide the needed context. If the intent is to discuss personal theories, well, the answer becomes that we don't discuss personal theories on PF. Robphy already posted some links to variable speed of light papers.

Duffy wrote a rather strong paper suggesting that it's never meaningful to ask what happens when one varies a quantity with dimension. Unfortunately I don't recall the exact name of the paper, or have a link to it. I do believe Duffy's paper was in response to various papers about "variable speed of light". Perhaps Duffy goes a bit far (there were published responses disagreeing with his position, as I recall). But certainly, if one can rephrase the question in such a way that one talks about varying dimensionless quantities, rather than quantities with dimensio, the objections raised by Duffy disappear. This is a good thing if it is possible, it silences some important criticism and reduces non-productive debate. If it's not possible in some specific case, it would be good to know why it's not possible in that specific case.

I think a good first step in answering the question "what happens when c varies with time" is to clarify the question. Since we only discuss peer reviewed literature here on PF, a link to a peer reviewed paper would provide the needed context. If the intent is to discuss personal theories, well, the answer becomes that we don't discuss personal theories on PF. Robphy already posted some links to variable speed of light papers.

Not so much a personal theory, but rather asking what are the ramifications if it could happen. I think it's pretty much been answered at this point. Thanks to all.

Duffy wrote a rather strong paper suggesting that it's never meaningful to ask what happens when one varies a quantity with dimension

Any idea where I could track down this article? I think it sounds pretty valid to my original question.

robphy
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dm4b
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.... remember a ways back there was a finding that went viral about the speed of light not being constant? I forget the details now, but it had everybody up in arms that it would be the downfall of Special Relativity.

It was a cherry-picking of the measured values of c over the centuries that showed the value decreasing with each measurement. What was left out was a bunch of other measurements taken at times in between these that don't fit the pattern. The point was that since c is decreasing, estimates of the age of the universe are off. Instead of the currently-accepted age of 13.7 billion years it might be only the 6000 to 10 000 years consistent with Old Earth Creationists' interpretation of the Book of Genesis.

If that analysis were accurate it wouldn't be a downfall of Special Relativity so much as a downfall of cosmology.

Now, a few years ago there was a measurement of faster-than-light neutrinos. That could have been the downfall of Special Relativity. It turned out, instead, to be an honest experimental error.

It was the second thing, I remember that now. This was all related to discussions spurring up around that (false) result.

I hadn't actually heard of the first scenario you mentioned before now.

Thanks for the background! Not sure how I could forget about the "faster-than-light" neutrinos! ;-)