Principle of relativity for proper accelerating frame of reference

[guptasuneet]

I don't know where you are getting this from. It is perfectly possible to have an accelerating frame in which all objects with constant spatial coordinates are at rest relative to each other. The canonical example is Rindler coordinates in Minkowski spacetime.

An accelerating frame might have objects in constant spatial coordinates relative to each other (within its frame of reference). However, All objects in any such uniformly accelerating frame shall feel a constant force and proper acceleration. Therefore it is not an inertial frame.

In Special Relativity, two frames can be said to be at rest with respect to each other if and only if both of them are in non-accelerated frames of reference. For accelerated frames of reference we would need General Relativity.

 

[PeterDonis]

All objects in any such uniformly accelerating frame shall feel a constant force and proper acceleration. Therefore it is not an inertial frame.

This is true, but it's not what you said in your previous post.

In Special Relativity, two frames can be said to be at rest with respect to each other

It doesn't even make sense to say that frames are "at rest with respect to each other". Frames aren't the kinds of things to which the concept of "being at rest" even applies. Objects can be at rest or not at rest relative to each other, but not frames.

For accelerated frames of reference we would need General Relativity.

This is false; accelerated frames in flat spacetime can be handled by SR perfectly well. I already gave you one example of such a frame: Rindler coordinates.

 

[cianfa72]

I don't know where you are getting this from. It is perfectly possible to have an accelerating frame in which all objects with constant spatial coordinates are at rest relative to each other. The canonical example is Rindler coordinates in Minkowski spacetime.

In Rindler coordinates in Minkowski spacetime objects at rest in it have - by definition - constant values of their spatial coordinates (btw it applies to any coordinate system as well).

You are claiming that they are at rest relative to each other: this is not actually the same as saying that each of them is at rest in the chosen coordinate system. Are you thinking of a specific "way" (or procedure) w.r.t. an object has to be considered at rest with respect to another?

 

[Ibix]

Are you thinking of a specific "way" (or procedure) w.r.t an object has to be considered at rest with respect to another?

Any obvious method will do (we hold opposite ends of a thread or a rod, or keep track of each other on radar). Assuming we're both at rest in the same Rindler frame we'll both see the other at a constant distance by these methods.

That isn't the case for objects at constant coordinates in some arbitrary non-inertial frame.

 

[cianfa72]

we hold opposite ends of a thread or a rod

You are really saying 'hold opposite hands of a thread (or rod) without it breaks' I believe.

 

[Ibix]

You are really saying 'hold opposite hands of a thread (or rod) without it breaks' I believe.

Yes.

 

[cianfa72]

That isn't the case for objects at constant coordinates in some arbitrary non-inertial frame.

This is the general arbitrary case.

Rindler coordinates in flat spacetime is not an inertial coordinate system (aka coordinate chart or frame) since accelerometers at rest in it measure non-zero proper acceleration. Yet each body at rest in it is also at rest w.rt. any other body at rest in that frame.

 

[Ibix]

This is the general arbitrary case.

Rindler coordinates in flat spacetime is not an inertial coordinate system (aka coordinate chart or frame) since accelerometers at rest in it measure non-zero proper acceleration. Yet each body at rest in it is also at rest w.rt. any other body at rest in that frame.

Yes.

 

[PeterDonis]

You are claiming that they are at rest relative to each other: this is not actually the same as saying that each of them is at rest in the chosen coordinate system.

Yes, those are distinct concepts.

Are you thinking of a specific "way" (or procedure) w.r.t. an object has to be considered at rest with respect to another?

The simplest invariant way to specify it is constant round-trip light travel time, as measured by clocks attached to each object.

The other way @Ibix described is workable in practical terms, but it requires you to specify that the thread or rod is also at rest with respect to the two objects, which adds an element of circularity to the definition. No such specification is required for light signals since their worldlines are already specified by the light cone structure of the spacetime geometry, independently of any other objects.

 

[cianfa72]

A though about what we said in last posts. A family of Rindler observers (i.e. observers at rest in Rindler coordinates -- namely observers described by worldlines with constant values of spatial Rindler coordinates) are actually able to build a grid of rigid rods (rigid in the sense of last post). On the other hand, what is not feasible is the persistence of the synchronization between clocks at rest at different Rindler spatial coordinates when they were initially synchronized (i.e. clocks at rest initially synchronized lost the synchronization).

 

[Ibix]

what is not feasible is the persistence of the synchronization between clocks at rest at different Rindler spatial coordinates when they were initially synchronized

Well, you can do as the GPS does and adjust the clock rates to match some standard - it's just a constant multiplier.

 

[cianfa72]

As discussed recently in another thread: what if we employed for example sound signals to synchronize standard clocks at rest in Rindler coordinates ? Would they lose the synchronization as well ?

AFAIK sound signals share the property of light that the propagation speed does not depend on the state of motion of the source.

 

[PAllen]

As discussed recently in another thread: what if we employed for example sound signals to synchronize standard clocks at rest in Rindler coordinates ? Would they lose the synchronization as well ?

AFAIK sound signals share the property of light that the propagation speed does not depend on the state of motion of the source.

Yes, in precisely the same way. The point is, once synchronized, what happens over 'time' is that any periodic physical process (thus any form of clock tick) goes slower toward the back of the rocket, compared to the front. In an inertial frame, this is just the Doppler effect because the front is continuously moving faster than the back at time of emission from the back - because it is accelerating during the time it takes for any signal to get from back to front.

 

[cianfa72]

Yes, in precisely the same way. The point is, once synchronized, what happens over 'time' is that any periodic physical process (thus any form of clock tick) goes slower toward the back of the rocket, compared to the front. In an inertial frame, this is just the Doppler effect because the front is continuously moving faster than the back at time of emission from the back

Sorry, in this specific case the periodic physical process you were talking about is actually the emission (with a given fixed frequency w.r.t. the back clock) of periodic 'sound signals' from back clock towards front clock ?

 

[PAllen]

Sorry, in this specific case the periodic physical process you were talking about is actually the emission (with a given fixed frequency w.r.t. the back clock) of periodic 'sound signals' from back clock towards front clock ?

That, and also any clock mechanism.

 

[cianfa72]

That, and also any clock mechanism.

The dotted lines in the following picture are the paths of light rays from event A to B and from B to C respectively (events A and B belong to the worldlines of observers at rest in Rindler coordinates -- solid lines in the picture).

The main point is that -- regardless the state of motion of light source -- its path in spacetime is always the same (here a straight line drawn at 45 degree in inertial coordinates t - x)

From what we said in last posts I believe the same is also true for sound signals (of course they are not straight line at 45 degree) since the speed of sound does not depend on the state of motion of the source, too.

At the end of the day I think it is actually the reason why the synchronization procedure also works employing sound signals.

Any thought ?

 

[anuttarasammyak]

ok, but as you said that does not imply they have a constant relative velocity, though. Nevertheless we can continue to apply the principle of relativity in terms of symmetries formulation (boost in this case).

Say bodies moving free on the xy floor of a z proper accelerating rocket. They do not represent xy IFRs as they do on the Earth ground. They will keep losing speed against the floor even without friction so the floor plays the role of absolute at rest. This is the observation of spaceship crew. Do I take it wrong?

 

[Ibix]

the speed of sound does not depend on the state of motion of the source, too.

It does depend on the state of motion of the medium, however, which is presumably accelerating with the rocket. So I would expect the worldlines of sound waves to be curved.

 

[cianfa72]

It does depend on the state of motion of the medium, however, which is presumably accelerating with the rocket. So I would expect the worldlines of sound waves to be curved.

We can assume the medium is actually at rest w.r.t. the global inertial frame, I believe.

If we assume isotropy for all physical processes I think the sound worllines should be straight in the global inertial frame.

 

[Ibix]

We can assume the medium is actually at rest w.r.t. the global inertial frame, don't you ?

That depends. I was assuming it was a sound wave in an enclosed rocket. If you want to assume your medium is not carried with the rocket then you will not get a uniform tick rate in the rocket frame because the speed of sound the rocket measures changes. Not to mention all the turbulence and drag issues you need to idealise away...

 

[cianfa72]

If you want to assume your medium is not carried with the rocket then you will not get a uniform tick rate in the rocket frame because the speed of sound the rocket measures changes. Not to mention all the turbulence and drag issues you need to idealise away...

Sorry, maybe I didn't grasp the point.

The idea was employ sound signals just in order to synchronize the front clock from the rear clock.

My understanding is that from the point of view of global inertial frame it will be just a Doppler effect (i.e. the frequency of the sound signal received changes). I think it is the same as for light signals (i.e. gravitational redshift). Regardless of synchronization process employed, we know clocks at front and rear do not tick at uniform rate.

Not to mention all the turbulence and drag issues you need to idealise away...

Sure, that was idealized of course.

 

[Ibix]

The idea was employ sound signals just in order to synchronize the front clock from the rear clock.

But in the rocket frame the distance from nose to tail remains constant while the speed of sound falls, so the time between "clock 1 emits a tick" and "clock 2 receives the tick" increases over time - so you can't use this as a synchronisation process without also measuring your speed relative to the medium and calculating sound travel times. And you can't do it at all once you go supersonic.

The point about the light pulses is that there's symmetry. Imagine a pulse bouncing backwards and forwards in the rocket for ever. If I draw a Minkowski diagram with the rocket instantaneously at rest when there's a reflection at the rear, then boost to any other frame where there is a rear reflection "at rest" then I have the same diagram. Not so with sound pulses because the medium is only at rest in one frame.

 

[cianfa72]

The point about the light pulses is that there's symmetry. Imagine a pulse bouncing backwards and forwards in the rocket for ever. If I draw a Minkowski diagram with the rocket instantaneously at rest when there's a reflection at the rear, then boost to any other frame where there is a rear reflection "at rest" then I have the same diagram. Not so with sound pulses because the medium is only at rest in one frame.

I took what you said as depicted in the picture:

The (t' , x') frame should be the global inertial frame in which the rocket is instantaneously at rest when there's a reflection at the rear (event A), right ?

Thanks for your time !

 

[PAllen]

I think we need to separate time dilation form synchronization. In a rocket, the only physically plausible model is that the air is carried by the rocket, and that at each point inside the rocket, the air may be taken to be 'stationary' i.e. not have wind, i.e. the local average momentum in the local instantaneous inertial frame of the given Rindler observer at one event is zero. This is the @Ibix was saying.

Given this, if clocks at front and back of the rocket are synchronized using sound, the synchronization will be different from using light. To make this concrete, assume you prepare two rockets. Both have front and back clocks initially synchronized in inertial frame of rocket construction. One is filled with air, the other vacuum. They both launch and accelerate uniformly for a day. Then they both synchronize their clocks, the air rocket using sound, the vacuum rocket using light. In each case, this is accomplished by the back clock sending a signal to the front clock, to which the front clock responds with a signal containing its reading at reception event. When the back clock receives this response, it sends a correction amount signal to the front clock. The amount of correction it sends is the difference between the received time reading and half way between its sent time and receipt time.

Doing this, the corrections in the two cases will be different. Two different synchronizations have been performed. It is not only because of different speed, but that in any coordinates where one is a straight path, the other will be curved.

Despite all this, pseudo-gravitation time dilation will be identical. Specifically, if a month later (of uniform acceleration), each rocket performs its synchronization procedure again, both will find that front clock is out of synch by same amount - the accumulated pseudo-gravitational time dilation.

 

[Ibix]

Doing this, the corrections in the two cases will be different. Two different synchronizations have been performed. It is not only because of different speed, but that in any coordinates where one is a straight path, the other will be curved.

I think some care is needed when stating this. For a rocket carrying its air, the offsets would be different for a light synchronised and a sound synchronised clock, but the result would be that the clocks showed the same time. Consider a concrete example with an absurdly high speed of sound of 0.5c and a rocket proper length of 1ls. The front clock hears the rear clock strike 12.00.00 and sees it read 12.00.01 - in either case they would (after an exchange of signals) deduce that it currently reads 12.00.02.

 

[PAllen]

I think some care is needed when stating this. For a rocket carrying its air, the offsets would be different for a light synchronised and a sound synchronised clock, but the result would be that the clocks showed the same time. Consider a concrete example with an absurdly high speed of sound of 0.5c and a rocket proper length of 1ls. The front clock hears the rear clock strike 12.00.00 and sees it read 12.00.01 - in either case they would (after an exchange of signals) deduce that it currently reads 12.00.02.

Let me clarify what I claim, and see if you agree or not. Let’s have one rocket, prepared in some inertial frame, with one rear clock and two front clocks. All are initially synchronized per the starting inertial frame. The rocket is filled with air, but there is a vacuum tube running the length of the rocket. Then, after a day of uniform acceleration, one front clock is synchronized with the rear using light in the vacuum tube, the other via sound in the air. These operations begin at the same moment per the rear clock. The sound procedure will take longer, but after both are complete, we find the two front clocks disagree. Further, the disagreement is not explained just by gravitational time dilation occurring during the procedures.

Note that if this were done with the rocket never having proper acceleration, the two front clocks would not be different. This set of operations distinguishes the non-inertial character of Rindler motion.

 

[PeterDonis]

in any coordinates where one is a straight path, the other will be curved.

I'm not sure what you mean by this. It is perfectly possible to model the path of the sound signal as a timelike geodesic, just as the path of the light signal is a null geodesic. In what sense is the former path "curved" while the latter is "straight"? It seems like you have some coordinate-dependent sense of those terms in mind, but coordinate-dependent properties should not affect the physics. It should be possible to describe what is going on entirely in terms of invariants.

 

[PAllen]

I'm not sure what you mean by this. It is perfectly possible to model the path of the sound signal as a timelike geodesic, just as the path of the light signal is a null geodesic. In what sense is the former path "curved" while the latter is "straight"? It seems like you have some coordinate-dependent sense of those terms in mind, but coordinate-dependent properties should not affect the physics. It should be possible to describe what is going on entirely in terms of invariants.

No. If the air is carried by the rocket and there is never wind observed by any of the contained Rindler congruence world lines, then the path of the sound pulse will not be a a geodesic. This is precisely the invariant fact I am getting at.

 

[PeterDonis]

If the air is carried by the rocket and there is never wind observed by any of the contained Rindler congruence world lines, then the path of the sound pulse will not be a a geodesic.

Can you give a brief explanation of why?

 

[PAllen]

Can you give a brief explanation of why?

My physical model is that the sound will have standard speed of sound in air in the tetrad of each event on each Rindler congruence line it passes. This I take to follow from absence of wind within the rocket as observed by the Rindler congruence. Then the path meeting this requirement has proper acceleration - in fact, changing proper acceleration over its path. Maybe this model is wrong, but it’s the one that make most sense to me.

 

[PeterDonis]

My physical model is that the sound will have standard speed of sound in air in the tetrad of each event on each Rindler congruence line it passes.

This is the initial model I came up with as well. I'm not 100% sure it's correct.

Then the path meeting this requirement has proper acceleration

I assume this is because the rocket itself has proper acceleration, so "same velocity relative to the rocket" translates to proper acceleration?

in fact, changing proper acceleration over its path.

Is the change due to the fact that the speed of sound in the air will change with height in the rocket (because the density will change)?

 

[PAllen]

This is the initial model I came up with as well. I'm not 100% sure it's correct.

Neither am I, but I haven’t come up with anything I find more convincing.

I assume this is because the rocket itself has proper acceleration, so "same velocity relative to the rocket" translates to proper acceleration?

Yes.

Is the change due to the fact that the speed of sound in the air will change with height in the rocket (because the density will change)?

No, because successive Rindler observers have different proper acceleration. But, of course, your observation is true as well. It might be interesting try to quantify all of this, but I don’t plan to do it.

 

[Ibix]

These operations begin at the same moment per the rear clock. The sound procedure will take longer, but after both are complete, we find the two front clocks disagree. Further, the disagreement is not explained just by gravitational time dilation occurring during the procedures.

I don't agree. I do agree that if it's the front clocks we are adjusting and we start the sync process at the rear clocks then the light-based one will complete first and unless it will drift due to gravitational time dilation before the sound-based sync completes (although we could measure the apparent rate of the lower clock and adjust the upper clock to match as in the GPS or repeatedly reset the clock). I don't see why you say the disagreement is not explained just by gravitational time dilation.

My synchronisation procedure is as follows: the front clocks simultaneously (they are co-located so this is unambiguous, and also simultaneous per the rear clocks) emit a pulse of their chosen waves. They also begin to watch/listen to the rear clocks to determine their apparent tick rate. Once their signal returns each clock concludes that the echo event was half way through the wait time, they can see the time shown on their rear clock, and they have the rate so they can add half the time multiplied by the tick rate ratio to get the time "now" on the rear clock.

 

[PAllen]

I don't agree. I do agree that if it's the front clocks we are adjusting and we start the sync process at the rear clocks then the light-based one will complete first and unless it will drift due to gravitational time dilation before the sound-based sync completes (although we could measure the apparent rate of the lower clock and adjust the upper clock to match as in the GPS or repeatedly reset the clock). I don't see why you say the disagreement is not explained just by gravitational time dilation.

My synchronisation procedure is as follows: the front clocks simultaneously (they are co-located so this is unambiguous, and also simultaneous per the rear clocks) emit a pulse of their chosen waves. They also begin to watch/listen to the rear clocks to determine their apparent tick rate. Once their signal returns each clock concludes that the echo event was half way through the wait time, they can see the time shown on their rear clock, and they have the rate so they can add half the time multiplied by the tick rate ratio to get the time "now" on the rear clock.

The Einstein synchronization convention has nothing to do with observing rates. It says one clock sends a signal, gets the signal reflected back, and declares that the reflection event was simultaneous to half way between the transmission and reception events. A simple way to synchronize a distant clock consistent with this simultaneity definition is to have the distant clock send its own clock reading as the 'reflected return signal'. Then, when the first clock gets the return signal, it can note the difference between this sent time and half way between transmission and reception. Then it sends an amount to correct the distant clock. To repeat, by design, this is purely based on a simultaneity convention, not any adjustment for differing clock rates (it is, of course assumed that clocks are identically constructed, so if colocated, would tick at the same rate). The adjustment to the distant clock ensures that the reflection event has the same clock reading as halfway between transmission and reception of the initial sending clock.

Then because sound wave transmission in the accelerating rocket is anisotropic in any inertial frame (due to the proper acceleration of the air), while light travel is defined to be isotropic in such a frame, different events are paired as simultaneous by sound as compared to light; and this has nothing to do with different tick rates due to gravitational time dilation.

 

[cianfa72]

To repeat, by design, this is purely based on a simultaneity convention, not any adjustment for differing clock rates (it is, of course assumed that clocks are identically constructed, so if colocated, would tick at the same rate). The adjustment to the distant clock ensures that the reflection event has the same clock reading as halfway between transmission and reception of the initial sending clock.

So, the adjustment sent from the first clock (rear clock in the case at hand) is employed by the distant clock (front clock) just to 'set' accordingly its time reading without making any rate adjustment (clock rate adjustment, if any, is really a separate matter).

Then because sound wave transmission in the accelerating rocket is anisotropic in any inertial frame (due to the proper acceleration of the air), while light travel is defined to be isotropic in such a frame, different events are paired as simultaneous by sound as compared to light; and this has nothing to do with different tick rates due to gravitational time dilation.

For example in the Minkowski global inertial frame the sound wave propagation is anisotropic while light propagation is defined to be isotropic in that frame.

Does it make sense ?

Sorry to point out all this things, but it is really important for me to get a clear understanding Thanks.