Principle of relativity for proper accelerating frame of reference

[DrGreg]

Any feedback about this? Thank you.

Yes, post #89. (Which is a way to determine you are using "stationary" coordinates in a stationary spacetime.)

 

[DrGreg]

Sorry, I do not see post #89. I don't know why it goes directly from #88 to #90 skipping #89.

Odd. Do you need to refresh the page?

This is post #89 :

At rest relative each other: is actually involved a physical procedure that each of them (i.e. let me say each of those objects) has to use to check it is actually at rest w.r.t. each of the others ? Thanks.

yes, constant round trip signal times.

 

[cianfa72]

This is post #89 :

okay now I see it, thank you.
That is the unique test (let me say 'The test') to check other bodies are at rest w.r.t. you ? In other words: is 'body at rest relative w.r.t. you' defined that way from an operational point of view?

 

[PAllen]

okay now I see it, thank you.
That is the unique test (let me say 'The test') to check other bodies are at rest w.r.t. you ? In other words: is 'body at rest relative w.r.t. you' defined that way from an operational point of view?

SR case:
There are several operational definitions of at mutual rest. They all agree in inertial frames. They do not agree in noninertial frames. Consider:

A. No mutual Doppler shift.
B. Mutual round trip signal times constant over time.
C. An object considered rigid in the every day sense continues to span two bodies over time. While in SR, there are fundamental limitations on rigidity, a case where it is plausible to talk about a stable mutual rest is one case where it still makes sense.

So, in inertial frame, all of these agree. However, in a Rindler frame, e.g. an accelerating rocket, B and C agree, while A does not. Further, A does not make physical sense in this case. Consider a given uniformly accelerating Rindler observer. Then another world line at Doppler rest from it, extended to the past and future, will necessarily touch the reference observer. So two different bodies at alleged mutual rest come to touch each other.

GR case

In general, all of these definitions disagree. However, in a stationary spacetime, B and C will agree. This includes the region around an isolated planet or star, whether rotating or not.

Also, in an FLRW cosmology, all three agree (C must be replaced by a mathematical analog, since real light year long massless rods are too hard to come by) to very high precision out to substantial distances. Further, they don‘t agree at all with mutual rest defined as those observers seeing isotropy. Instead, such comoving observers are all found to be moving apart per these notions of mutual rest.

[edit: mixed up letters fixed, pointed out by @PeterDonis ]

 

[cianfa72]

SR case:
There are several operational definitions of at mutual rest. They all agree in inertial frames. They do not agree in noninertial frames.

Sorry, why frames are actually involved ? The definition of relative rest between bodies should involve only them, I believe.

Are you thinking of those bodies at rest w.r.t. a given reference frame (inertial vs non-inertial) ?

 

[PAllen]

Sorry, why frames are actually involved ? The definition of relative rest between bodies should involve only them, I believe.

Are you thinking of those bodies at rest w.r.t. a given reference frame (inertial vs non-inertial) ?

You can replace frames with statements about the bodies, or just about one of the bodies considered as a 'reference'. Then, instead of inertial frames, substitute inertial motion (zero proper acceleration) for the reference body. Instead of Rindler frame, substitute uniform acceleration read by an accelerometer for the reference body. Similar statements can be made for the GR cases.

 

[cianfa72]

You can replace frames with statements about the bodies, or just about one of the bodies considered as a 'reference'. Then, instead of inertial frames, substitute inertial motion (zero proper acceleration) for the reference body. Instead of Rindler frame, substitute uniform acceleration read by an accelerometer for the reference body. Similar statements can be made for the GR cases.

Sorry, not sure to grasp what you mean with 'reference body'. Do you refer to the body w.r.t. define if every other body is at rest ?

 

[PeterDonis]

SR case:
There are several operational definitions of at mutual rest. They all agree in inertial frames. They do not agree in noninertial frames. Consider:

A. No mutual Doppler shift.
B. Mutual round trip signal times constant over time.
C. An object considered rigid in the every day sense continues to span two bodies over time. While in SR, there are fundamental limitations on rigidity, a case where it is plausible to talk about a stable mutual rest is one case where it still makes sense.

So, in inertial frame, all of these agree. However, in a Rindler frame, e.g. an accelerating rocket, A and C agree, while B does not. Further, B does not make physical sense in this case.

I think you mean A does not agree and does not make physical sense, correct? A is Doppler rest.

 

[PAllen]

I think you mean A does not agree and does not make physical sense, correct? A is Doppler rest.

Yes, mixed up my letters. I’ll fix it.

 

[PAllen]

Sorry, not sure to grasp what you mean with 'reference body'. Do you refer to the body w.r.t. define if every other body is at rest ?

Yes.

 

[valenumr]

My point is the following: Galileo principle of relativity applies not only to inertial frames but even to not-inertial constant (proper) accelerated frames having constant relative velocity (it definitely makes sense in the context of Newtonian mechanics).

Then what about in the context of SR ? I was trying to single out two (proper) accelerated frames (spaceships) having constant relative velocity to ask if we can continue to apply the principle of relativity even to them.

Even in classical physics you can go through some really simple 2d math. Define some inertial observers (i.e constant velocity), and then define a subject that undergoes acceleration for a period of time. All energy and momentum of every observer will be conserved for inertial observers for the starting and ending time of the accelerated subject, but the same will not hold true when using calculating total energy and momentum of observers from the accelerated subjects reference frame before and and after the acceleration.

sorry if that's wordy. I can formalize it with math if you want, and its just basic algebra.

 

[valenumr]

This formulation is different from the principle of relativity as stated by Galileo. AFAIK the Galileo formulation makes no assumption about the state of motion of the frames involved (proper accelerated or not).

In other words Galileo principle of relativity applies as well even if the first frame is proper accelerated (i.e. bodies at rest in it have got the same proper acceleration as measured by accelerometers attached to them) and the second frame is moving with constant relative velocity w.r.t the first frame.

The laws of physics in both frames will be the same even if, of course, they are not in the 'simplest' form as in any inertial frame.

Maybe this difference with SR is 'summarized' by the keyword 'special', I guess...

I don't think that's correct if I understand what you are claiming. Even in classical physics, you can't make energy and momentum conserved from an accelerating frame.

 

[valenumr]

Special relativity uses the definition you gave earlier: no force is acting on the body if an accelerometer attached to the body reads zero.

The part that requires general relativity is how the presence of gravitating masses affects things. Special relativity assumes that you can construct global inertial frames using bodies on which no forces are acting. But that turns out not to be possible in the presence of gravitating masses. General relativity gives you a way to handle that case.

My humble observation: if an observer can compute energy and momentum and see that those values are conserved, then they are inertial. If these values are not conserved, the subject is non-inertial.

 

[valenumr]

okay now I see it, thank you.
That is the unique test (let me say 'The test') to check other bodies are at rest w.r.t. you ? In other words: is 'body at rest relative w.r.t. you' defined that way from an operational point of view?

I think you mean inertial, not accelerating. Or am I missing a subtle point?

 

[PeterDonis]

if an observer can compute energy and momentum and see that those values are conserved, then they are inertial.

This is obviously false for curved spacetimes; in curved spacetimes, either there is no valid global energy and momentum at all (in non-stationary spacetimes), or, in stationary spacetimes, observers who have constant energy and momentum, viewed globally, are not inertial (for example, "hovering" observers in Schwarzschild spacetime, with constant kinetic and potential energy, have nonzero proper acceleration).

 

[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.