Speed of light in NON-inertial frames

[Dickfore]

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.

 

[Passionflower]

Optical coordinates with a projected distance measure was developed by G. Temple in 1938 (http://www.jstor.org/pss/97172).

 

[yuiop]

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 textbook 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.

 

[Passionflower]

That's how to measure the average speed of an object between two points, not necessarily its instantaneous speed (dx/dt) at any point.

Yes, and your point is?
Why is for some people only the instantaneous speed of any interest?

 

[Mentz114]

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 textbook 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 ?

https://www.physicsforums.com/showthread.php?p=1786556

 

[Fredrik]

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]

Yes, and your point is?
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]

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.

[Edit] 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.

 

[bcrowell]

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]

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]

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.

 

[bcrowell]

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.

Agreed.

 

[Passionflower]

[Edit] 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.

You do not have to invoke Born rigidity for the Sagnac effect? How so?