Is c invariant in Accelerating frames?

[Austin0]

Simple question.
I would like to know if there is a definitive answer , consensus in the field, on the question of the measurement of light in an accelerating system.
Whether one way measurements from the front to the back and vice versa would result in (c +v) = (c-v) = c as usual??

Thanks,,, any answers appreciated.

 

[Ich]

No .

 

[Dale]

I agree with Ich as long as you are talking about the coordinate speed of light. If you make a local experimental measurement of the speed of light you will always obtain c, even in an accelerating frame.

 

[Austin0]

I agree with Ich as long as you are talking about the coordinate speed of light. If you make a local experimental measurement of the speed of light you will always obtain c, even in an accelerating frame.

Hi DaleSpam I am confused
In this context ,,what do you mean by the coordinate speed? I was asking about measuring the speed in a single coordinate system. Frame. Ship.

From the front to the back [one way] and from the back to the front [one way]

Do you mean that these measurements would not be isotropic and return c but that very local measurements taken at either the front or back would ,,or what??



Thanks for your reply

 

[Dale]

In this context ,,what do you mean by the coordinate speed?

Most coordinate systems have three space coordinates and one time coordinate. The coordinate velocity is then the derivative of the three spatial coordinates wrt the time coordinate. This value is dependent on the coordinate system used, particularly the synchronization convention defined by the time coordinate.

Contrast this with the four-velocity which is the unit tangent to the worldline and is thus defined in a coordinate-independent geometric fashion.

I was asking about measuring the speed in a single coordinate system. Frame. Ship.

From the front to the back [one way] and from the back to the front [one way]

AFAIK a one-way measurement of the speed of light is always coordinate dependent since it relies on the synchronization convention chosen.

Do you mean that these measurements would not be isotropic and return c but that very local measurements taken at either the front or back would ,,or what??

A two way measurement (eg a mirror at a known distance) will always return c as long as the curvature of spacetime is negligible over the measurement.

 

[Dale]

In this context ,,what do you mean by the coordinate speed?

Most coordinate systems have three space coordinates and one time coordinate. The coordinate velocity is then the derivative of the three spatial coordinates wrt the time coordinate. This value is dependent on the coordinate system used, particularly the synchronization convention defined by the time coordinate.

Contrast this with the four-velocity which is the unit tangent to the worldline and is thus defined in a coordinate-independent geometric fashion.

I was asking about measuring the speed in a single coordinate system. Frame. Ship.

From the front to the back [one way] and from the back to the front [one way]

AFAIK a one-way measurement of the speed of light is always coordinate dependent since it relies on the synchronization convention chosen.

Do you mean that these measurements would not be isotropic and return c but that very local measurements taken at either the front or back would ,,or what??

A two way measurement (eg a mirror at a known distance) will always return c as long as the curvature of spacetime is negligible over the measurement.

 

[Ich]

A two way measurement (eg a mirror at a known distance) will always return c as long as the curvature of spacetime is negligible over the measurement.

Not for accelerating observers, even in flat spacetime.

 

[sylas]

Hi DaleSpam I am confused
In this context ,,what do you mean by the coordinate speed? I was asking about measuring the speed in a single coordinate system. Frame. Ship.

In an accelerating space ship, any local measurement taken over a sufficiently small region will figure the speed of light as c.

However, the distance from the front of the ship to the back of the ship is ambiguous; it can be defined in different ways. In brief; remote distances are a co-ordinate issue, and hence so too are speeds at a remote location within the ship.

Here are two ways to define distances in the ship. One is to place a clock at some fixed location, and then put mirrors everywhere else in the ship, and see how long it takes light to get there and back, by your clock. Using this distance co-ordinate, the speed of light is everywhere constant. (By definition; you use this as an assumption to derive the distance!)

The other way is to move your clock all over the ship, and use it as a "light ruler" at every point, and mark off distances that way. This will give you different distances from the "radar" method. It will also mean that the speed of light varies in different parts of the ship, from the perspective of an observer with a clock seeing how long it takes light to get from one point to another remote point.

If you are the front of the ship, clocks at the back of the ship are running slow, and distances are also compressed. The speed of light is reduced, because by YOUR clock, it takes longer for light to get to a point at the rear and then back to your clock than distance/c.

Conversely, if you are at the rear then the speed of light at the front of the ship is more than c, in this co-ordinate system.

Cheers -- sylas

 

[Dale]

Not for accelerating observers, even in flat spacetime.

Yes, even for accelerating observers. Of course, the clock and the mirror need to move in a Born-rigid fashion.

 

[atyy]

For free falling guys, "locality" is determined by negligible curvature.

Accelerating guys can also set up locally Minkowski coordinates, but for them "local" is determined by negligible curvature and their acceleration.

Curvature produces second order deviations, acceleration produces first order deviations from Minkowski coordinates.

http://www.csun.edu/~vcmth00m/fermi.pdf

 

[Ich]

Yes, even for accelerating observers. Of course, the clock and the mirror need to move in a Born-rigid fashion.

I think sylas explained why this would not work: You have to pick a definition of distance. The most sensible would be IMHO the "light ruler at every position", but it does not matter which one you choose. As long as we agree that forward distance is the same as backward distance, the measured two way speed of light will be different in both directions.

 

[Dale]

As long as we agree that forward distance is the same as backward distance, the measured two way speed of light will be different in both directions.

I am not sure what you mean by this, especially how "different in both directions" applies to a two-way measurement of the speed of light.

 

[Ich]

I am not sure what you mean by this, especially how "different in both directions" applies to a two-way measurement of light.

If you pick one specific line (from point A to point B), and assign to it one and only one length, by whatever procedure you like,
then you will determine different two way light speeds: A-B-A differs from B-A-B.
Both trips take the same amount of coordinate time, but in the first experiment you measure A's proper time (I think that's how a two-way measurement is defined), in the second B's. Both are different due to time dilation.

 

[Dale]

If you pick one specific line (from point A to point B), and assign it one and only one length, by whatever procedure you like,
then you will determine different two way light speeds: A-B-A differs from B-A-B.
Both trips take the same amount of coordinate time, but in the first experiment you measure A's proper time (I think that's how a two-way measurement is defined), in the second B's. Both are different due to time dilation.

The two measurements may not take the same amount of coordinate time, but now I have doubts about my assertion. I will have to work out the math for a couple of different coordinate systems.

 

[Ich]

The two measurements may not take the same amount of coordinate time

In static coordinates they have to; you can simply rearrange
A-B-A -> A-B + B-A
B-A-B -> B-A + A-B = A-B + B-A,
because time translation changes nothing.

 

[Dale]

because time translation changes nothing.

That is not true in general for non-inertial coordinate systems.

 

[Ich]

That is not true in general for non-inertial coordinate systems.

That's why I said "In static coordinates...", meaning A and B being at constant coordinate positions in a time-independent metric like Rindler or Schwarzschild coordinates.

 

[Dale]

Sorry, I missed that. Anyway, I will do the math and see. I am not as confident as I initially was.

 

[Austin0]

=sylas;2299032] However, the distance from the front of the ship to the back of the ship is ambiguous; it can be defined in different ways. In brief; remote distances are a co-ordinate issue, and hence so too are speeds at a remote location within the ship.

Hi Sylas

I am not sure how to interpret this. If there is a ruler running the length of the ship it would seem that the observed measurements must be the same at both ends. So wherein is the possibiity of ambiguity? Clocks at disparate locations can disagree on measurements but it is hard to picture what it might mean wrt to rulers and distance.
Are you suggesting that even though the observed measurements might be the same at both ends ,somehow the "actual" distance might be different? Or are you assuming that the front and the back are two different frames with different metrics with the ship as a whole being some hybrid interpolation of the two ?

Here are two ways to define distances in the ship. One is to place a clock at some fixed location, and then put mirrors everywhere else in the ship, and see how long it takes light to get there and back, by your clock. Using this distance co-ordinate, the speed of light is everywhere constant. (By definition; you use this as an assumption to derive the distance!)

Then I would ask , what is the real point of this procedure. It is only relevant to a single clock at a single location and, as you pointed out,,, "by definition" it can only return c under any conditions and therefore not only is that information without real meaning, so are any distance measurements derived from the process.

The other way is to move your clock all over the ship, and use it as a "light ruler" at every point, and mark off distances that way. This will give you different distances from the "radar" method. It will also mean that the speed of light varies in different parts of the ship, from the perspective of an observer with a clock seeing how long it takes light to get from one point to another remote point.

Excuse me but isn't there an inherant problem with this method of moving a clock to determine distance or degree of desynchronization?
Doesn't the basic foundation of SR include the recognition that the act of moving a clock must result in time dilation in the clock and consequently loss of synchronization.
I had always assumed that the light method of synchronization for clocks that are spatially separated was, in part, a responce to this inherant problem.?

If you are the front of the ship, clocks at the back of the ship are running slow, and distances are also compressed. The speed of light is reduced, because by YOUR clock, it takes longer for light to get to a point at the rear and then back to your clock than distance/c.

Conversely, if you are at the rear then the speed of light at the front of the ship is more than c, in this co-ordinate system.

I understand completely this description and concept. But isn't it simply a projection onto the accelerating frame of assumptions derived from calculations made in an inertial frame.
Is it possible to arrive at these conclusions based on our knowledge of physics ,completely within the reference frame of the accelerating system on first priciples alone?
Thanks

 

[Hurkyl]

I am not sure how to interpret this. If there is a ruler running the length of the ship it would seem that the observed measurements must be the same at both ends. So wherein is the possibiity of ambiguity?

The ambiguity is that this solution is by no means the only reasonable one.

What your ship-long ruler does for you is that it defines a system of spatial coordinates. You can then define lengths inside of your one-dimensional rocket in terms of some function defined upon those coordinates.

But this notion of length isn't particularly closely related to the notion of distance defined by the geometry of space-time -- and a priori, we have no good reason to think it consistent with any other reasonable scheme for measuring distances.

 

[Austin0]

Originally Posted by Austin0
In this context ,,what do you mean by the coordinate speed?

=DaleSpam;2298089]Most coordinate systems have three space coordinates and one time coordinate. The coordinate velocity is then the derivative of the three spatial coordinates wrt the time coordinate. This value is dependent on the coordinate system used, particularly the synchronization convention defined by the time coordinate.

Hi Would I be mistaken to say that basically physics boils down to clocks, rulers, and gauges. They embody the metric and that "coordinate systems" are simply a way of providing a global context, specifc locations (space +time) and directions (angles,vectors) for the clocks and rulers and the measurements they provide. They are invaluable for comparisons of disparate measurements and events and for comparisons between different systems, but in the end, all calculations are simply predictions of measurements of clocks and rulers,et al.

So in this context ie: on board this hypothetical ship. The assumption is the clocks were synchronized by SR convention while inertial, prior to initiating acceleration.
There are two clocks and one long ruler between them. They are the whole coordinate system as far as this question is concerned.
So I am still unsure what you mean by coordinate speed wrt this situation. Are you also implying that there is more than one coordinate system to be considered here?

A couple of more questions:

1) Would you agree that as far as establishing synchronization between separated, unsynched clocks is concerned ; the One way method is absolutely equivalent to the two way, reflected method . That it in all cases it would establish the same actual readings and synchronization (simultaneity) as the standard convention ?

2) Would you agree, or not, that regarding testing synchronization of clocks already in place ; Two, one way, measurements between clocks A--> B B-->A is a sure test.
That two reflected tests with mirrors , from two locations, cannot itself provide information regarding the synchronization of the two clocks??

thanks

 

[Dale]

Hi Austin0, I will come back and answer this question, but it may be a while. I need to work through the math to become confident in my response, and I am not sure when I will have the time to do that.

 

[Austin0]

=Hurkyl;2301318]The ambiguity is that this solution is by no means the only reasonable one.

What your ship-long ruler does for you is that it defines a system of spatial coordinates. You can then define lengths inside of your one-dimensional rocket in terms of some function defined upon those coordinates.

You seem to be saying that space time itself doesn't define lengths .
In an inertial context; are rulers in other frames contracted because of how we choose definitions or because we assume that will be the observed reality, spacetime geometry in action??

Thanks Stephen

 

[sylas]

I am not sure how to interpret this. If there is a ruler running the length of the ship it would seem that the observed measurements must be the same at both ends. So wherein is the possibiity of ambiguity?

It comes about in how you decided to mark off distances along the ruler. I've given two methods for doing that. Both make sense; and both give the same result in an inertial frame. In an accelerating spaceship, the two methods mark off distances on your long ruler in two different ways.

Clocks at disparate locations can disagree on measurements but it is hard to picture what it might mean wrt to rulers and distance.

How might you propose to figure the lengths of rulers which are not right adjacent to you?

Suppose you measure the distance taken by a beam of light, over 1 nanosecond. That will be 30 centimeters; assuming that the speed of light is constant everywhere. Suppose you measure the time using the clock at that remote location; but that you also know the clock is running slow, by a certain factor. The 1 nanosecond by that clock will be longer than a nanosecond as you measure it locally; and the distance light has moved will be longer as well, by the same factor; longer than would be measured had you gone over to that part of the ruler and measured it yourself.

That is: if the speed of light is constant; then time dilation and length contraction go hand in hand. Or perhaps the speed of light ISN'T constant, and it varies at different locations. Pick which you like; it's basically the same as deciding what co-ordinate system you like best.

Is it possible to arrive at these conclusions based on our knowledge of physics ,completely within the reference frame of the accelerating system on first priciples alone?

If special relativity is one your first principles; yes.

Cheers -- sylas

 

[Hurkyl]

You seem to be saying that space time itself doesn't define lengths .

That is certainly not what I meant.

Between any two events in space-time, geometry defines a distance between them (or a duration between them, if they are time-like separated rather than space-like separated).

However, the marks on your ruler are not points. The marks define world-lines: entire trajectories through space-time.

 

[Ich]

Is it possible to arrive at these conclusions based on our knowledge of physics ,completely within the reference frame of the accelerating system on first priciples alone?

You need the principle of relativity and the constancy of c in any inertial frame. From that, you can deduce time dilation in accelerating frames to first order.
Of all distances in accelerating frames, there is a best choice, IMO. That's distance in the comoving frame = distance in Rindler coordinates = radar chain distance. All other have some weird properties that you don't expect when speaking about "distance".

 

[Austin0]

That is certainly not what I meant.

Between any two events in space-time, geometry defines a distance between them (or a duration between them, if they are time-like separated rather than space-like separated).

However, the marks on your ruler are not points. The marks define world-lines: entire trajectories through space-time.

Hi I tried to make the conditions very explicit and unambiguous in my simplistic, two -dimensional spaceship.

SO if two clocks ,synchronized by convention while inertial, are then used to test light speed and synchronicity after they are in an accelerating state , aren't the events of emission at one location [ ie:end of the ruler ] at time T(e) and the reception event at the other location [end] at T'(r) , "points"?
Then either L /T'(r) -T(e) = c = L / T(r)-T'(e) or it doesn't.

This question was framed as a hypothetical empirical situation. Two observers at opposite ends of a ship test and synch their clocks and then the system accelerates and they simply repeat the procedure.

This question was related to another question I asked in another thread.

After a period of acceleration would the clocks have to be resynchronized to correctly measure light and achieve simultaneity??

In that thread I encountered the, apparently, mutually exclusive ideas that 1) the clocks would be out of synch ... but 2) all clock systems ,even accelerating ones would invariantly measure c in both directions relative to motion.

I posed the question of this thread to
1) attempt to get more information as to the consensus regarding the idea that all systems ,even accelerating ones would measure light at c
2) find out if there was a definite general answer to the question as defined simply in the above. So far the single unequivical answer has been Ich's " NO" [ Hardly a comprehensive census.]
3) find out the thought process required to encompass both desynchronization and invariance in the same two clocks.

Thanks for your help in this quest

 

[Austin0]

=sylas;2301417]It comes about in how you decided to mark off distances along the ruler.

I've given two methods for doing that. Both make sense

; and both give the same result in an inertial frame. In an accelerating spaceship, the two methods mark off distances on your long ruler in two different ways.

The first method, the "radar method", is obviously valid in inertial frames for deriving distance.
The second ,the "light ruler" ,moving a clock around as a measure, is not valid in either inertial or accelerating frames for determining distance or ,testing and establishing synchronicity unless the fundamental principle that moving a clock [ ie:at a relative velocity]
neccessarily causes time dilation and loss of synchronicity with the system is not valid.
Would you agree with this or not?

How might you propose to figure the lengths of rulers which are not right adjacent to you?

Within the conditions of the original question as I framed it, this is not necessary.
The assumption was that the ruler was to be taken as singular , the distance singular, and the tests would proceed from there. I said assumption as a condition of the hypothetical scenario without any implication that that assumption was correct.
As far as a proposal to figure the lengths of distant rulers; how about cutting off unit lenghts from each end and tranporting them to the opposite sites for a direct comparison??
Of course this would only apply in an actual real world situation, if we could achieve significant accelerations. So scratch that idea.

Suppose you measure the distance taken by a beam of light, over 1 nanosecond. That will be 30 centimeters; assuming that the speed of light is constant everywhere. Suppose you measure the time using the clock at that remote location; but that you also know the clock is running slow, by a certain factor. The 1 nanosecond by that clock will be longer than a nanosecond as you measure it locally; and the distance light has moved will be longer as well, by the same factor; longer than would be measured had you gone over to that part of the ruler and measured it yourself.

How do we know?? Because we are taking coordinate measurements from another frame [equivalent to coordinate determination of clock desynch between frames] and then assuming they would apply in this one.

Thanks Sylas

 

[sylas]

The first method, the "radar method", is obviously valid in inertial frames for deriving distance.
The second ,the "light ruler" ,moving a clock around as a measure, is not valid in either inertial or accelerating frames for determining distance or ,testing and establishing synchronicity unless the fundamental principle that moving a clock [ ie:at a relative velocity]
neccessarily causes time dilation and loss of synchronicity with the system is not valid.
Would you agree with this or not?

No, I do not agree; but mainly because we are talking at cross purposes.

The rulers are not moving with respect to each other. Rather, you have each point equipped with a clock and a ruler, and all the rulers are stationary with respect to each other... determined by the fact that the distances obtained between two rulers remains unchanged.

You seem to have taken me as saying that the rulers are actually moving with respect to each other when they make the measurement. That's not the definition used. The rulers remain at unchanging distances from each other to make the measurement.

As far as a proposal to figure the lengths of distant rulers; how about cutting off unit lenghts from each end and tranporting them to the opposite sites for a direct comparison??

That is precisely the method I proposed; except rather than cut lengths, I use a light ruler as the common reference. It's the same thing. You move the rulers (transport them) all over the ship, and then when they are all at rest with respect to each other (fixed distances) you have your co-ordinates defined by those rulers.

Note that in your method also, having the rulers moving past any point brings in a Lorentz contraction. That's why we require the rulers to be at rest -- at fixed separation.

Now... how do you propose to define time? I have been supposing that we have a single reference clock, and give the time at any other point as given by that clock. You can define "simultaneous" with the reference clock as being the midpoint of the interval for light to get from the reference clock to an event and then back.

By this method of defining co-ordinates, the speed of light throughout the ship is not constant, as explained previously. It is c at the reference clock.

Cheers -- sylas

 

[atyy]

In that thread I encountered the, apparently, mutually exclusive ideas that 1) the clocks would be out of synch ... but 2) all clock systems ,even accelerating ones would invariantly measure c in both directions relative to motion.

I posed the question of this thread to
1) attempt to get more information as to the consensus regarding the idea that all systems ,even accelerating ones would measure light at c
2) find out if there was a definite general answer to the question as defined simply in the above. So far the single unequivical answer has been Ich's " NO" [ Hardly a comprehensive census.]
3) find out the thought process required to encompass both desynchronization and invariance in the same two clocks.

I think the difference between Ich's and DaleSpam's answers is that Ich is giving it for any finite region of spacetime, while DaleSpam is giving it for an infinitesimal region of spacetime.