Speed of light not an invariant in GR

[PeroK]

I've checked out the proper time definition on Wikipedia and there are indeed two different definitions for SR and GR. What do you know, the definition for GR:
$$\Delta\tau = \int_P \, d\tau = \int_P \frac{1}{c}\sqrt{g_{\mu\nu} \; dx^\mu \; dx^\nu}$$
is claimed to be invariant and also contains c.
So we may safely assume, according to Wikipedia, that the postulate of c invariance also holds for GR.

The underlying postulate of both is that spacetime is a 4D manifold, characterised by some metric tensor, $g_{\mu\nu}$. The (mathematical) theory of manifolds tells you that the above integral is independent of your choice of coordinates. I.e. it is an invariant quantity.

We physically interpret that quantity as the proper time (along a timelike curve in the manifold). And we expect that quantity to be the time recorded by a clock that moves along that timelike curve. This is something we can test - especially for SR where we postulate that the spacetime is (at least approximately) flat.

Note that when we say the time measured by a clock, we really mean the time that passes for physical processes to evolve. So, we have a theory that tells us how much physical change will take place along any given timelike path. And when we test this (e.g. lifetime of high-speed particles as measured in the lab) we find the experiment matches the theory.

We have a further postulate that light moves on null curves. This can be tested (e.g. by measuring the deflection of light during a solar eclipse). This postulate, moreover, implies that a) in flat spacetime the speed of light is invariant - i.e. always measured as $c$ in an IRF; and b) in curved spacetime, for sufficiently local trajectories, the speed of light is measured locally as $c$.

Note that b) is easy to show for an observer at rest and a static, diagonal metric (which is the exercise I suggested you undertake).

 

[Pyter]

b) in curved spacetime, for sufficiently local trajectories, the speed of light is measured locally as .

What could be a non-local experiment where the light speed is not measured as c?

 

[PeroK]

What could be a non-local experiment where the light speed is not measured as c?

Consider a radial light ray in Schwarzschild coordinates: starting from some coordinate $R_0$ moving out to $R_1$, off a mirror and back again.

We can measure the proper distance from $R_0$ to $R_1$ by integrating the line element. This will gives some length $l$. The speed of light, as measured by an observer at $R_0$, will be $\frac{2l}{\Delta \tau_0}$.

To save the calculations, we can argue that this is not equal to $1$ as follows. The locally measured speed of light at each point is $1$, which means the rate of change $\frac{dl}{d\tau_r} = 1$. But, by relating $\tau_0$ to $\tau_r$ using the coordinate time $t$, we can see that $d\tau_r > d\tau_0$ for every point $r > R_0$ and hence $\frac{dl}{d\tau_0} > 1$. And we must end up with $\frac{2l}{\Delta \tau_0} > 1$.

An alternative argument is to compare the measured (round trip) speed of light for observers at $R_0$ and $R_1$. The proper length is the same and the coordinate time interval must be the same, but $\Delta \tau_0 \ne \Delta \tau_1$, so the measured speed of light must be different for those two observers.

Note: the previous calculations have argued that $$\lim_{l \rightarrow 0} \frac{2l}{\Delta \tau_0} = \lim_{l \rightarrow 0} \frac{2l}{\Delta \tau_1} = 1$$ And, in the example above, the measured speeds will approach the limit from above and below $1$ for $R_0$ and $R_1$ respectively - and only the limits are equal to $1$.

 

[Pyter]

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

 

[PeroK]

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

One of the things that GR requires is an ability to define things properly and organise your calculations.

For example, someone at rest in Schwarzschild coordinates is not moving inertially. GR is a subject that requires solid prerequisites, focus and disciplined thinking. If you don't have those, then you cannot make progress. You'll just end up wandering around the subject in a state of permanent confusion.

 

[Pyter]

@PeroK what did I say?

 

[PeroK]

@PeroK what did I say?

You said this:

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

Three statements, three errors.

 

[Pyter]

@PeroK you wrote earlier that in the radial case, on the long distance you measure c > 1. I assumed that's because the space is "contracted" radially and the light appears to move faster.
And what the c in the Wikipedia's proper time equation would be if not the one measured in a local IFR?

 

[PeroK]

@PeroK you wrote earlier that in the radial case, on the long distance you measure c > 1. I assumed that's because the space is "contracted" radially and the light appears to move faster.
And what the c in the Wikipedia's proper time equation would be if not the one measured in a local IFR?

I never wrote $c > 1$. I said that the measured (non-local) speed of light $> 1$. In what I did $c = 1$ by definition. In GR, $c$ is not really the speed of light but the conversion factor between units of length and time. The locally measured speed of light $= c$. Which is a consequence of light traveling on null paths.

If you are studying GR, you need to stop thinking in terms of IRF's all the time. Think coordinates, Schwarzschild or otherwise.

Also, you are doing none of the calculations yourself - and are misunderstanding all our posts and calculations.

You need to get a textbook and start posting the exercises as homework if you get stuck.

 

[Pyter]

@PeroK, you're right. Please read "the measured speed of light" instead of c in my previous post.
And no, actually it's not warp speed, because if it was the case, the observer in $R_1$ would observe it too, but he measures a speed of light < 1 instead.

 

[Pyter]

Thank you all for all the pointers and explanations.

 

[PeterDonis]

That's what I meant. How can it be that: $$\mathrm{d}\tau^ \triangleq \mathrm{d}x^0$$ (by definition)

There is no such definition. This statement is simply wrong whenever $g_{00} \neq 1$.

 

[PeterDonis]

I assumed that's because the space is "contracted" radially

You should not be assuming anything. You should be learning how to do the math and understand the results yourself. Trying to make assumptions based on ordinary language statements when you don't understand the underlying math is not a good idea.

what the c in the Wikipedia's proper time equation would be if not the one measured in a local IFR?

You should not be trying to learn physics from Wikipedia. You need to learn it from a textbook. Sean Carroll's online lecture notes on GR would be a good place to start.

The short answer to the question you pose here is the one @PeroK gave: $c$ in GR is a conversion factor between units of length and units of time. Choosing $c = 1$ means choosing units in which length and time have the same units (e.g., seconds and light-seconds, or years and light-years, or nanoseconds and feet). It has nothing to do with anything anybody measures.

 

[Pyter]

You should not be trying to learn physics from Wikipedia. You need to learn it from a textbook.

I did study more than one book, but they didn't cover the topic of the invariance of velocity of light in GR. Even Einstein in his papers and divulgative book is rather reticent about that.

 

[PeterDonis]

I did study more than one book, but they didn't cover the topic of the invariance of velocity of light in GR.

Which books have you studied?

 

[Pyter]

@PeterDonis

• Intro to Differential Geometry and General Relativity - S. Waner
• Generalized Relativity - P. A. M. Dirac (which I sort of consider a "cram sheet", only 77 pages but very densely packed)
• Special Relativity and Classical Field Theory - Leonard Susskind et al. (SR only)

 

[PeterDonis]

Intro to Differential Geometry and General Relativity - S. Waner

This one doesn't discuss the invariant light cone structure of spacetime? That is what "invariance of velocity of light" translates to in GR.

The other two I can understand, IIRC Dirac doesn't discuss geometry much at all, and the Susskind one, since you say it focuses on SR, wouldn't be expected to discuss generalizations of concepts to GR. (Also it seems more focused on field theory.)

 

[Pyter]

I forgot to add: also the Einstein original papers and (partly) his "divulgation" book.
After all this, if it wasn't for your informative answers, I never could've guessed that the measured velocity of light could be > 1.

 

[PeroK]

After all this, if it wasn't for your informative answers, I never couldn't have guessed that the measured velocity of light could be > 1.

Take a look at this:

https://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

 

[Pyter]

@PeroK

Einstein talked about the speed of light changing in his new theory. In the English translation of his 1920 book "Relativity: the special and general theory" he wrote: "according to the general theory of relativity, the law of the constancy of the velocity [—Either Einstein or his translator obviously mean "speed" here, since velocity (a vector) is not in keeping with the rest of his sentence. People often say "velocity" when they clearly mean "speed".] of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [...] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity [speed] of propagation of light varies with position." This difference in speeds is precisely that referred to above by ceiling and floor observers.

That's the very excerpt I cited earlier in this thread. And that's pretty much all Einstein says about it.

When all is said and done, to insist that a non-c speed of light is nothing more than an artifact of a "nonphysical" choice of coordinates is to make a wrong over-simplification.

Interesting.

 

[anuttarasammyak]

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

For example in rotating frame of reference anybody at $r >\frac {c}{\omega}$ from Origin cannot stop but move with coordinate speed > c. But we cannot invent the warp from it.

 

[Pyter]

Here's an infinitesimal Minkowski diagram defined by the observer's tetrad. The derivation is from Landau&Lifshitz vol. 2. There you find a very clear discussion of spacetime geometry/tensor analysis.

View attachment 277490

@vanhees71 I've finally found the time to look into the textbook you cited. But if you're transcribing the discussion starting from equation 84.1 on, it's just to determine the $dl$ from the $d\tau$, and the authors assume that c is an invariant. It's not used to show that the measured speed of light is an invariant in GR.
I'm quoting from the textbook, just before eq. 84.4 (sorry for the bad formatting):

SupposealightsignalisdirectedfromsomepointBinspace(withcoordinatesxa+dxa)toapointAinfinitelyneartoit(andhavingcoordinatesxa)andthenbackoverthesamepath.Obviously,thetime(asobservedfromtheonepointB)requiredforthis,whenmultipliedbyc,istwicethedistancebetweenthetwopoints.

 

[PeterDonis]

It's not used to show that the measured speed of light is an invariant in GR.

Perhaps it will help to rephrase exactly what is invariant in relativity. The invariant is the light cone structure of spacetime. For any given event, call it event E, if you imagine light signals emitted from that event in all possible directions and never being absorbed, the set of events that those light signals will reach is the future light cone of event E. Similarly, if you imagine light signals arriving at event E from all possible directions, all of which were emitted infinitely far in the past, the set of events that those light signals passed through before reaching event E is the past light cone of event E. And relativity (special or general) says that those light cones--those sets of points in spacetime--are invariant for any given event E; they are the same no matter what coordinates you choose, or what system of units you choose, or what conventions or methods you adopt for measuring the speed of light.

To put it another way: you can't change what events can send light signals to what other events. That will stay the same no matter what coordinates you choose, or what system of units you choose, or what conventions or methods you adopt for measuring the speed of light.

 

[vanhees71]

@vanhees71 I've finally found the time to look into the textbook you cited. But if you're transcribing the discussion starting from equation 84.1 on, it's just to determine the $dl$ from the $d\tau$, and the authors assume that c is an invariant. It's not used to show that the measured speed of light is an invariant in GR.
I'm quoting from the textbook, just before eq. 84.4 (sorry for the bad formatting):

Of course, famously special relativity has been derived by Einstein from the assumption (sic!) that the (two-way) speed of light is an invariant and this is of course inherited by general relativity, which is just an extension obtained by "gauging" Lorentz invariance, i.e., making the proper orthochronous Lorentz group a local symmetry. Indeed this implies, what @PeterDonis called the "lightcone structure" of spacetime. It's however a bit more than that, because in addition it also provides the (pseudo-)metrical structure of spacetime, i.e., it's the Lorentz group which is the local symmetry group of GR not the larger group including scale invariance of pure electromagnetics (i.e., Maxwell theory without charges and currents).

 

[Pyter]

The invariant is the light cone structure of spacetime.

I've always struggled to visualize it in 4D, and also in 3D. What's its shape exactly?
If the metric changes with time, the light speed also changes, so you might say that the cone is not "static" (in time) and is not a "cone", in the sense that its surface is not regular but bent here and there by the massive bodies?

 

[PeroK]

I've always struggled to visualize it in 4D, and also in 3D. What's its shape exactly?
If the metric changes with time, the light speed also changes, so you might say that the cone is not "static" (in time) and is not a "cone", in the sense that its surface is not regular but bent here and there by the massive bodies?

If you don't like the phrase "light cone", then we could call it the past "causal container".

More importantly, we are dealing with spacetime, not space. A past light cone at a point in spacetime has no concept of being "static".

Also, there is no sense in which the speed of light changes with time.

I'm not sure how you get to the point where you can grasp these concepts. Maybe it goes back to a point I made earlier that you are reading material on SR/GR, but you are not doing any of the problems, so you've never tested that you've understood anything.

I'd like to see you post some problems as homework instead of just talking about the subject.

 

[Pyter]

@Pyter a good exercise for you would be to extend the above for the case of an arbitrary diagonal metric.

That was already transcribed by @vanhees71 in this thread, for an arbitrary metric, from the Landau-Lifshitz textbook. Anyway, the very informative link you posted from the Physics FAQ convinced me that the speed of light is c only if measured in an inertial frame or if the observer is near the light beam, and thus generally <> c in a non-IFR and at a great distance.

More importantly, we are dealing with spacetime, not space. A past light cone at a point in spacetime has no concept of being "static".

I have a hard time visualizing it as a "cone" in the 4D spacetime with an arbitrary metric. Moreover, in GR, there's no such a thing as a "global" flat 4D spacetime with Minkowski ($\eta_{\mu\nu}$) metric.

Also, there is no sense in which the speed of light changes with time.

The measured speed of a faraway light beam depends on the metric, and the metric generally depends on time.
I'm citing from said textbook, just before eq. 84.8:

 

[PeroK]

The measured speed of a faraway light beam depends on the metric, and the metric generally depends on time.
I'm citing from said textbook, just before eq. 84.8:

View attachment 278279

Well, no you're not "citing from said textbook". The textbook is saying one thing and you're saying something entirely different!

 

[Pyter]

@PeroK I meant that the speed of light depends on the metric, and thus time, through the $\int dl$.

 

[PeroK]

@PeroK I meant that the speed of light depends on the metric, and thus time, through the $\int dl$.

It's still wrong and not at all what that text is saying. That text does not even mention the speed of light, let alone say it "changes with time".

 

[Pyter]

Take a look at this:

https://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

Then I'm citing from your link (emphasis mine):

None of the preceding discussion actually depends on the distances being large; it's just easier to visualise if we use such large distances. So now transfer that discussion to a rocket you are sitting in, far from any gravity and uniformly accelerated, meaning you feel a constant weight pulling you to the floor. "Above" you (in the direction of your acceleration), time speeds up and light travels faster than c, arbitrarily faster the higher up you want to consider. Now use the Equivalence Principle to infer that in the room you are sitting in right now on Earth, where real gravity is present and you aren't really accelerating (we'll neglect Earth's rotation!), light and time must behave in the same way to a high approximation: light speeds up as it ascends from floor to ceiling, and it slows down as it descends from ceiling to floor; it's not like a ball that slows on the way up and goes faster on the way down. Light travels faster near the ceiling than near the floor. But where you are, you always measure it to travel at c; no matter where you place yourself, the mechanism that runs the clock you're using to measure the light's speed will speed up or slow down precisely in step with what the light is doing. If you're fixed to the ceiling, you measure light that is right next to you to travel at c. And if you're fixed to the floor, you measure light that is right next to you to travel at c. But if you are on the floor, you maintain that light travels faster than c near the ceiling. And if you're on the ceiling, you maintain that light travels slower than c near the floor.

The observed speed of a faraway light beam depends on the metric, and if the latter depends on time, also the light speed does.

 

[PeroK]

Then I'm citing from your link (emphasis mine):
The observed speed of a faraway light beam depends on the metric, and if the latter depends on time, also the light speed does.

Okay, but it's impossible to keep up with this. You posted a completely different text previously (claiming you were referencing that). Now you're posting a different text altogether. The key point is that you are still paraphrasing and extrapolating what Baez says.

There is a huge difference between quoting Baez directly (a citation) and taking something he said and applying your own logical reasoning to it and claiming it's something Baez says.

To put it in simple terms: Baez does not say that the speed of light changes with time. He simply does not say that.

 

[Pyter]

@PeroK Baez says that the speed changes with the metric.
I'm arguing that if the metric changes with time, also the speed does.

 

[PeterDonis]

What's its shape exactly?

Topologically, it's a double cone--two conical null surfaces (one future, one past) joined at the apex, which is the chosen event.

Null surfaces don't really have an invariant "shape" in the ordinary geometric sense. How they are represented in a diagram depends on what coordinates you choose.

If the metric changes with time, the light speed also changes, so you might say that the cone is not "static" (in time) and is not a "cone", in the sense that its surface is not regular but bent here and there by the massive bodies?

None of this is correct, no.

Remember that there is a light cone attached to every event--every point in spacetime. Each point has its own, and they are all different.

It makes no sense to say the light cone of a particular event "changes with time" or is "static". The light cones are invariant features of the 4-dimensional spacetime geometry; there is no "change" in the 4-dimensional geometry, it just is.

Which events are in the light cone for a given event is of course determined by the spacetime geometry, which in turn is determined by the presence of stress-energy; but to say the cone's surface is "bent" by the presence of massive bodies is an oversimplification.

Also, the term "cone", as my statement at the start of this post indicates, refers to the topology of the light cone, not its geometric shape; as I noted, null surfaces don't really have a geometric "shape" in the ordinary sense.

 

[PeterDonis]

The measured speed of a faraway light beam

There is no such thing. You can't directly measure the speed of something that is far away from you. You can calculate various "speeds", but those calculations will depend on your choice of coordinates.

Baez says that the speed changes with the metric

He's talking about the coordinate speed of light, not anything that is directly measured. The coordinate speed he is talking about is a calculation and depends on your choice of coordinates.