# Speed of light in NON-inertial frames

I'm just saying that the speed of light at the origin of the accelerating frame is 1 by definition of accelerating frame, if we try to measure that speed the way you (and others) have suggested, we will (almost) always get a result that's different from 1. So that way of measuring it clearly doesn't work.
What exactly 'does not work' if I am at the top of a tower with an emitter and a clock and I send out a light pulse that is reflected at the foot of that tower and take the elapsed time / (2 * the height of the tower)?

Fredrik
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What exactly 'does not work' if I am at the top of a tower with an emitter and a clock and I send out a light pulse that is reflected at the foot of that tower and take the elapsed time / (2 * the height of the tower)?
It depends on what you're trying to do. If I flap my arms, it "doesn't work" if I'm trying to fly, but it works just fine if I'm trying to look silly. And can we keep this a discussion about SR, please. Gravity is just an additional complication.

Actually conform the equivalence principle if we ignore tidal effects the above mentioned situation is identical to that of an accelerating spaceship or the proverbial elevator.

Fredrik
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The concept of "comoving inertial frame" also gets more complicated because there's more than one kind, and I'm not 100% sure that there are no extra complications regarding how to associate a coordinate system with an object's motion. I don't see a reason to introduce anything that might possibly confuse the issue.

You ignored the essential part of my post, where I explained that it's not possible to fail unless you try. I can't tell you what (if anything) is wrong with what you're doing unless you tell me what you're trying to do.

I can't tell you what (if anything) is wrong with what you're doing unless you tell me what you're trying to do.
All I am trying to do here is highlight that there is more to relativity than the 'mantras': "in the limit the speed is c" and "in the limit spacetime is flat" resp. for accelerating observers or observers in curved spacetimes. I think it is wrong to encourage people to stop thinking beyond this level.

Beyond that is where the interesting stuff is:
• What does the world look like for accelerating observers in flat spacetime?
• What does the world look like for observers in curved spacetime?
Just considering the limit is literally rather shortsighted.

Fredrik
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And all I'm doing is to say that you can't just perform this measurement and say "look, the speed of light in the accelerating frame is 0.8". This would be to ignore the definition of one of the terms in the sentence you just used, and that's never a great idea.

And all I'm doing is to say that you can't just perform this measurement and say "look, the speed of light in the accelerating frame is 0.8". This would be to ignore the definition of one of the terms in the sentence you just used, and that's never a great idea.
First of all in this topic I never even mentioned the word frame. I never said anything about the speed 'in a frame', I am talking about the measured speed of light for a particular observer.

An observer can measure the time it takes for something to go from A to B and back, if we know the distance between A and B we can calculate the speed. If this something is light it is called the speed of light if it is a rock it is the speed of a rock. Really I do not see any problem with that.

But it seems we just have to agree to disagree.

I think people make an obvious oversight when it comes to the term "speed of light". In Landau, Classical Theory of Fields, they formulate Einstein's second postulate as:

"There is a finite limit speed c with which interaction can be transmitted."

According to the First Postulate (Principle of Relativity), this limit must have the same value in every inertial reference frame.

Then, they go on to prove that the space-time interval:

$$ds^{2} \equiv c^{2} \, dt^{2} - dx^{2} - dy^{2} - dz^{2}$$

is an invariant and derive everything else from there.

In particular, they build electromagnetism "from the ground up". It turns out from Maxwell's equations then, that electromagnetic fields can exist independent from any charges and currents, but have to be time-dependent. These fields propagate as waves and their speed of propagation in vacuum (free space) is equal to the same c as above. That is why this is called speed of light in vacuum.

However, as we all know, the speed of propagation of electromagnetic waves need not always be c, as in some materials, where it is also frequency dependent (dispersion).

For the case of non-inertial reference frames (or the case where a gravitational field is present), they simply define the space-time interval as a general quadratic form:

$$ds^{2} = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu}$$

with $g_{\mu \nu} = g_{\nu \mu}$ - the metric tensor, containing all the information about the space-time and the particular coordinate system. Because at any point we may diagonalize this quadratic matrix, it must resemble the Minkowski metric tensor $g^{(0)}_{\mu \nu} = \mathrm{diag}(1, -1, -1, -1)$. In particular, for real space-time, we must have:

$$g < 0$$

The metric tensor can be used to deduce real time intervals, distances and synchronize clocks.

For example, if we are at a particular point in space ($x^{i} = \mathrm{const.} \Rightarrow dx^{i} = 0$) and consider two infinitesimally closed events, than, by analogy with SR we define the proper time interval as $ds = c \, d\tau$

$$ds^{2} = g_{0 0} \, (dx^{0})^{2} = (c \, d\tau)^{2}$$

$$g_{00} \ge 0 \Rightarrow d\tau = \frac{\sqrt{g_{0 0}}}{c} \, dx^{0}$$

The case $g_{0 0} < 0$ does not necessarily mean that that space-time is impossible, but simply that the particular coordinate system we are using is unsuitable.

Now comes the important point: Distances are defined through the same "radar procedure" using something that moves along null geodesics. Namely, let us consider two points A with space coordinates $x^{i}$ and B, which is infinitely close and with space coordinates $x^{i} + dx^{i}$. We shine a light ray from B at $x^{0} + (dx^{0})_{2}$, it propagates to A, reflects at x^{0} and arrives back at B at $x^{0} + (dx^{0})_{2}$. Since the wave is travelling along a null geodesic, we may find $(dx^{0})_{1/2}$ by equating $ds = 0$ and solving the quadratic equation:

$$g_{0 0} (dx^{0})^{2} + 2 \, g_{0 i} \, dx^{i} \, dx^{0} + g_{i j} \, dx^{i} \, dx^{j} = 0$$

where a summation from 1 to 3 over a repeated Latin superscript and subscript is implied. The solution of this equation is:

$$(dx^{0})_{1/2} = \frac{-g_{0 i} \, dx^{i} \mp \sqrt{(g_{0 i} \, g_{0 j} - g_{0 0} \, g_{i j}) \, dx^{i} \, dx^{j}}}{g_{0 0}}$$

According to what has been said above for proper time intervals, the round trip time, according to B is:

$$d\tau = \frac{\sqrt{g_{0 0}}}{c} \, \left((dx^{0})_{2} - (dx^{0})_{1}\right)$$

and this, by definition, corresponds to a distance:

$$dl = \frac{c \, d\tau}{2}$$

which gives the following:

$$dl^{2} = \gamma_{i j} \, dx^{i} \, dx^{j}, \gamma_{i j} = \frac{g_{0 i} \, g_{0 j}}{g_{0 0}} - g_{i j}$$

for the spatial distance and $\gamma_{i j} = \gamma_{j i}$ is the spatial metric tensor. For coordinate systems attainable by physical bodies, the quadratic form $dl^{2} \ge 0$ must be positive definite.

The moment of reflection of the signal at point A, according to B, corresponds to the time coordinate $x^{0} + \Delta x^{0}$, where:

$$\Delta x^{0} = \frac{(dx^{0})_{1} + (dx^{0})_{2}}{2} = g_{; i} \, dx^{i}, \; g_{; i} = -\frac{g_{0 i}}{g_{0 0}}$$

which is the synchronization offset. In this way, we can synchronize clocks along any open line in space, bu not, in general, over closed loops, since:

$$-\oint{\frac{g_{0 i}}{g_{0 0}} \, dx^{i}} \neq 0$$

in general. As a conclusion, signals that travel along null geodesics have speed c by definition.

But, if you write the equations of electromagnetism in curved spacetime, you will see that they predict propagation of electromagnetic waves at different speeds than c.

Now comes the important point: Distances are defined through the same "radar procedure" using something that moves along null geodesics. Namely, let us consider two points A with space coordinates $x^{i}$ and B, which is infinitely close and with space coordinates $x^{i} + dx^{i}$. We shine a light ray from B at $x^{0} + (dx^{0})_{2}$, it propagates to A, reflects at x^{0} and arrives back at B at $x^{0} + (dx^{0})_{2}$. Since the wave is travelling along a null geodesic, we may find $(dx^{0})_{1/2}$ by equating $ds = 0$ and solving the quadratic equation:

$$g_{0 0} (dx^{0})^{2} + 2 \, g_{0 i} \, dx^{i} \, dx^{0} + g_{i j} \, dx^{i} \, dx^{j} = 0$$

where a summation from 1 to 3 over a repeated Latin superscript and subscript is implied. The solution of this equation is:

$$(dx^{0})_{1/2} = \frac{-g_{0 i} \, dx^{i} \mp \sqrt{(g_{0 i} \, g_{0 j} - g_{0 0} \, g_{i j}) \, dx^{i} \, dx^{j}}}{g_{0 0}}$$

According to what has been said above for proper time intervals, the round trip time, according to B is:

$$d\tau = \frac{\sqrt{g_{0 0}}}{c} \, \left((dx^{0})_{2} - (dx^{0})_{1}\right)$$

and this, by definition, corresponds to a distance:

$$dl = \frac{c \, d\tau}{2}$$

which gives the following:

$$dl^{2} = \gamma_{i j} \, dx^{i} \, dx^{j}, \gamma_{i j} = \frac{g_{0 i} \, g_{0 j}}{g_{0 0}} - g_{i j}$$

for the spatial distance and $\gamma_{i j} = \gamma_{j i}$ is the spatial metric tensor. For coordinate systems attainable by physical bodies, the quadratic form $dl^{2} \ge 0$ must be positive definite.

The moment of reflection of the signal at point A, according to B, corresponds to the time coordinate $x^{0} + \Delta x^{0}$, where:

$$\Delta x^{0} = \frac{(dx^{0})_{1} + (dx^{0})_{2}}{2} = g_{; i} \, dx^{i}, \; g_{; i} = -\frac{g_{0 i}}{g_{0 0}}$$

which is the synchronization offset. In this way, we can synchronize clocks along any open line in space, bu not, in general, over closed loops, since:

$$-\oint{\frac{g_{0 i}}{g_{0 0}} \, dx^{i}} \neq 0$$

in general. As a conclusion, signals that travel along null geodesics have speed c by definition.
There are various way to define distance in the GTR. Radar distance is just one of them, ruler and optical distance are alternatives and usually they do not give the same values. Fermi coordinates also have some kind of distance.

According to you a 100 meter long Born rigid spaceship undergoing a proper acceleration is no longer 100 meter in fact its size is no longer unique.

jcsd
Gold Member
I don't see the problem, surely sometimes it is useful to use definitions of 'speed' that for light isn't always equal to c. Just need to be aware of exactly which definition you're using!

There are various way to define distance in the GTR. Radar distance is just one of them, ruler and optical distance are alternatives and usually they do not give the same values. Fermi coordinates also have some kind of distance.

According to you a 100 meter long Born rigid spaceship undergoing a proper acceleration is no longer 100 meter in fact its size is no longer unique.
Please note that, strictly speaking, it is impossible to have rigid bodies in GR as it is impossible to have fixed distance among bodies inserted in an arbitrary gravitational field. Therefore, I don't see how one can define ruler distance.

As for "optical distance", I don't know exactly what it means, so I would be greatful if you could give some more details. If you mean something involving electromagnetic waves, then that was what I was actually pointing out in my post, although it might have been lost in the rather long exposition. Electromagnetic waves are not a mathematical construct subject to random definitions, but an objective reality subject to precise dynamical laws. Therefore, claiming "photons" move along null geodesics is ad hoc. I am not saying it is incorrect, but just that it has to be proven from Maxwell's equations in curved space-time.

Also, notice that the method I had given is essentially coordinate independent because it relies on a coordinate independent quantity ds.

I have worked out a heuristic to calculate the time dilation measured by an accelerating parallel light clock, which is effectively proportional to the radar distance distance measured by a Born rigid accelerating observer. However the calculations are a bit involved and it is my experience that someone has already worked out a formula, so before I get stuck in, does anyone know of such a formula, either in a text book or worked out privately?

Just to make things clearer, consider a mirror located at the nose of a rocket. An observer at the tail of the rocket measures the to and fro time of a signal as one second when the rocket is not accelerating. What is the time measured by this observer when the rocket has Born rigid acceleration, as a function of the proper acceleration and the length of the rocket? What is the time measured when the mirror is at the back and the observer is at the nose?

Al68
An observer can measure the time it takes for something to go from A to B and back, if we know the distance between A and B we can calculate the speed. If this something is light it is called the speed of light if it is a rock it is the speed of a rock. Really I do not see any problem with that.
That's how to measure the average speed of an object between two points, not necessarily its instantaneous speed (dx/dt) at any point. Those two quantities are equal for an inertial object (or light) only when the observer is at rest in an inertial reference frame.

That's how to measure the average speed of an object between two points, not necessarily its instantaneous speed (dx/dt) at any point.
Why is for some people only the instantaneous speed of any interest?

I have worked out a heuristic to calculate the time dilation measured by an accelerating parallel light clock, which is effectively proportional to the radar distance distance measured by a Born rigid accelerating observer. However the calculations are a bit involved and it is my experience that someone has already worked out a formula, so before I get stuck in, does anyone know of such a formula, either in a text book or worked out privately?

Just to make things clearer, consider a mirror located at the nose of a rocket. An observer at the tail of the rocket measures the to and fro time of a signal as one second when the rocket is not accelerating. What is the time measured by this observer when the rocket has Born rigid acceleration, as a function of the proper acceleration and the length of the rocket? What is the time measured when the mirror is at the back and the observer is at the nose?
Kev, I'm confused. Didn't this get done in this thread #23 ? Or am I mis-reading the above ?

Fredrik
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Gold Member
Why is for some people only the instantaneous speed of any interest?
In case you consider me one of those people: It's not that it's the only speed of interest. It's that if you say that that's the one you're measuring, you shouldn't be measuring something else.

DrGreg
Gold Member
Why is for some people only the instantaneous speed of any interest?
When physicists or mathematicians say "speed" they mean "instantaneous speed" unless they explicitly say "average speed". Nothing wrong with average speed, but if that's what one is talking about, one needs to insert the word "average" otherwise one will be misunderstood.

bcrowell
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Just to make things clearer, consider a mirror located at the nose of a rocket. An observer at the tail of the rocket measures the to and fro time of a signal as one second when the rocket is not accelerating. What is the time measured by this observer when the rocket has Born rigid acceleration, as a function of the proper acceleration and the length of the rocket? What is the time measured when the mirror is at the back and the observer is at the nose?
If you want to talk about Born-rigid objects, you have to realize that the Born-rigid object requires an external set of forces being applied to it according to some prearranged plan, and that plan has to be based on criteria for what you want the object to do. When you set those criteria, you're going to say things like, "I want the distance between these two points to remain constant." Then you have to define what you mean by the distance between those two points. You can't define it using Born-rigid rulers, because that becomes circular. The best way of defining it is to use radar. But then you're taking constant c as a definition of distance, which means you can't use your Born-rigid object as a tool for determining whether c is constant. In general, virtually all arguments involving Born-rigidity that I've seen people make have problems like this. You can't just sprinkle Born-rigidity on an argument to make the argument more sound, or to get around the fundamentally slippery behavior of time and space in relativity.

Putting aside the issue of Born-rigidity, in experiments such as the one you're talking about, the deviation of the speed of light from c is always proportional to the length L of the measuring device. When we say that c is always constant, we mean the value of c in local experiments, i.e., in the limit as L approaches zero.

 By the way, if you want an example where *nonlocal* experiments give a speed of light not equal to c, it's very easy, and you don't have to invoke Born rigidity or anything like that. The Sagnac effect was first observed in 1913.

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bcrowell
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The phase velocity of EM waves in a gravitational field changes.
But, if you write the equations of electromagnetism in curved spacetime, you will see that they predict propagation of electromagnetic waves at different speeds than c.
The necessity of distinguishing between phase velocity and group velocity arises when the medium is dispersive. Experimentally, there have been searches for dispersion of the vacuum, and they've given negative results. Theoretically, I think we need to be careful about the distinction between local and nonlocal quantities. You can certainly get effects like partial reflection of an EM wave from a gravitational field, and refraction of the transmitted wave, although the predicted effect is much too small to be tested experimentally. This is to some extent analogous to phenomena that you get with dispersive materials like glass. However, you get into some very sticky issues when you try to characterize this phenomenon as local or nonlocal. If there were really a *local* dispersiveness of the vacuum, it would violate the equivalence principle. This is closely related to the famous question of whether falling electric charges violate the equivalence principle. The best known paper on this is DeWitt and DeWitt (1964). A treatment that's easier to access online is Gron and Naess (2008). You can find many, many papers on this topic going back over the decades, with roughly half saying that it violates the e.p. and half saying that it doesn't, because the effect is inherently nonlocal, whereas the e.p. only makes claims about local observations. The basic problem is that it's hard to define "local" in a rigorous way. Even if you're inclined to believe that it's a violation of the e.p., it's a good idea to keep in mind the DeWitts' caution that "The questions answered by this investigation are of conceptual interest only, since the forces involved are far too small to be detected experimentally."

Cecile and Bryce DeWitt, Falling Charges,'' Physics 1 (1964) 3
Gron and Naess, arxiv.org/abs/0806.0464v1

Fredrik
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If you want to talk about Born-rigid objects, you have to realize that the Born-rigid object requires an external set of forces being applied to it according to some prearranged plan,...
If you just want a solid object in 1+1 spacetime dimensions to approximate Born rigid motion, all you have to do is to apply a small enough force to a single point, and let the internal forces do the rest. The forces between atoms or molecules will work towards keeping the distances to their immediate neighbors (approximately) the same in all its comoving inertial frames. That approximation is only exact in the limit where the distance between neighbors go to zero.

In 3+1 dimensions, applying force to a single point would (of course) usually make the object rotate.

bcrowell
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If you just want a solid object in 1+1 spacetime dimensions to approximate Born rigid motion, all you have to do is to apply a small enough force to a single point, and let the internal forces do the rest. The forces between atoms or molecules will work towards keeping the distances to their immediate neighbors (approximately) the same in all its comoving inertial frames. That approximation is only exact in the limit where the distance between neighbors go to zero.
It's trivially true that if you apply a small enough force, the object doesn't deform very much, because the acceleration is small. If that's a good enough approximation to rigidity, then you don't need to talk about Born rigidity at all. The only reason for talking about Born rigidity is if you want rigidity that's greater than the fundamental limits on the properties of materials imposed by relativity, e.g., you want disturbances to propagate through the material at >c.

JesseM
It's trivially true that if you apply a small enough force, the object doesn't deform very much, because the acceleration is small. If that's a good enough approximation to rigidity, then you don't need to talk about Born rigidity at all. The only reason for talking about Born rigidity is if you want rigidity that's greater than the fundamental limits on the properties of materials imposed by relativity, e.g., you want disturbances to propagate through the material at >c.
If an object is being accelerated from one end with constant proper acceleration, then even though the acceleration right after the initial "push" won't be Born rigid since it takes some time for the push to have any effect on the other end, eventually (if the acceleration isn't large enough to break up the object) the object should reach some equilibrium where it is accelerating in a Born rigid way, with the atoms closer together near the side being pushed, so the intra-atomic forces and thus the accelerations are greater on that side. This is just like how solid objects at rest in a gravitational field reach an equilibrium where the spacing of atoms near the bottom is compressed more under the object's weight than that of atoms near the top and thus the object maintains a constant shape.

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bcrowell
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