SR derived solely from one postulate

[grav-universe]

So even in a Lorentz ether theory, which imagines that things "really contract" due to their velocity relative to the ether, you'd say it's just a coordinate effect since if the objects are brought to rest relative to each other (and thus have identical velocities relative to the ether) they will be the same length again?

That would be the assumption, yes. Of course, with SR, what is directly observed to take place between two observers is considered the actual physics that occurs between them since there is nothing else to relate it to such as an ether, so objects are considered to "really contract". The thing is, my assumption was that whatever contraction is observed to takes place with relative motion, being a coordinate effect, will reverse itself upon coming to rest again in the same way.

I wasn't exactly talking about SR, I was talking about applying the Lorentz transformation in a Newtonian universe where the laws of physics are not Lorentz-invariant (physicists would usually take the Lorentz-invariance of the laws of physics to be the definition of SR). You're saying that even in this case, you don't see why rigid bodies could be measured to have equal lengths in a frame where they have equal and opposite velocities, but not have equal lengths when brought to rest relative to one another?

Only if it is a coordinate effect, but I'm starting to see what you're saying in regard to the real physics.

Imagine that in a Newtonian universe, we define a single unprimed frame so that it's a standard Newtonian inertial frame where Newton's laws apply. Then we define a family of other frames using the Lorentz transformation on the unprimed coordinates:

x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)

Now suppose that in the unprimed frame, we have a rigid measuring-rod that's at rest and 10 light-seconds long, and another rigid measuring rod that's moving in the +x direction at 0.8c and is 6 light-seconds long. Now consider a coordinate system, given by the transformation above, that is moving at 0.5c in the +x direction. In this coordinate system, the first rod is moving in the -x' direction at 0.5c, while the second rod is moving in the +x' direction at 0.5c (I can prove this if you like, but consider the relativistic velocity addition formula, which says that if the unprimed frame observes the primed frame to be moving at 0.5c and the primed frame observes the second measuring-rod to be moving at 0.5c, then the unprimed frame will observe it to be moving at (0.5c + 0.5c)/(1 + 0.5*0.5) = 0.8c).

Let's say that in the unprimed frame, both measuring-rods start with their left end at x=0 at t=0. Since rod #1 is at rest in the unprimed frame, rod #1's left end will have position as a function of time given by:
x(t) = 0 light-seconds
And rod #1's right end will have position as a function of time given by
x(t) = 10 l.s.

Meanwhile since rod #2 is moving at 0.8c, rod #2's left end will have:
x(t) = 0.8c*t
And since rod #2 is 6 light-seconds long in the unprimed frame, rod #2's right end will have:
x(t) = 0.8c*t + 6

Now consider two events in the unprimed frame: (x=0, t=0) and (x=10, t=5). Obviously the first event lies on the worldline of both the left end of rod #1 and the left end of rod #2 (i.e. it's the event of the left ends of both rods lining up), since we established that both their left ends started at x=0 at t=0. But the second event happens to lie on the worldline of both the right end of rod #1 and the right end of rod #2 (so it's the event of the right ends of both rods lining up), since the right end of rod #1 remains fixed at x=10, and since plugging in t=5 into the function x(t) = 0.8c*t + 6 gives x = 0.8*5 + 6 = 4 + 6 = 10.

Finally, consider what happens when you use the coordinate transformation to find the coordinates of these two events in the primed frame. The first event will become (x'=0, t'=0) while the second event will become (x'=8.66, t'=0). So the key here is that these two events are simultaneous in the primed frame--the left ends of both rods line up at x'=0 at t'=0, while the right ends of both rods line up at x'=8.66 at t'=0. Since "length" in a given frame is just the distance between two ends of an object at a single moment in that frame, both rods must have equal lengths of 8.66 light-seconds in the primed frame. And as I said before, they also have equal and opposite velocities of 0.5c in the primed frame.

Right, looks good.

But remember that in the unprimed frame, rod #1 is 10 light-seconds long while rod #2 is 6 light-seconds long. And the unprimed frame is just an ordinary Newtonian inertial frame, where according to Newtonian laws objects will not change length when they change velocities. So if rod #2 is brought to rest next to rod #1, they will still be different lengths.

But even in Lorentz ether theory, Newton's laws do not strictly apply. Objects will still contract in the line of motion, so will "uncontract" when coming to rest again in the same way. If no contraction took place in a Newtonian universe, then all rods would remain the same lengths to all observers in the first place because no contraction took place to begin with. But once again, you do have me thinking, though, about what the physical processes are that produce contraction. If B quickly accelerates to v relative to A, then it is usually said that B will contract to A, but that isn't necessarily true, but depends upon how the acceleration took place. If B and C have some distance between them and they quickly accelerate to v simultaneously, then A will measure the same distance between them as before. Likewise, if all parts of a B's ship quickly accelerate to v simultaneously, then A will measure the ship to have the same length as before, whereas B will now measure his ship as elongated. But if all parts of B's ship quickly decelerate back to A's frame simultaneously, then since the clocks on B's ship from front to back are still synchronized to A but unsynchronized to B, then the ship will still have the same length to A, but B will say the front of his ship decelerated first and so contracted in the process. Also, if a train enters a tunnel of the same proper length, then if the tunnel observers threw spikes up all at once along the length of the tunnel, the train will quickly stop and be contracted to the tunnel, whereas the train observers say the spikes were thrown up at the front of the tunnel first and the train crunched up as it was stopped. If the train observers threw down spike simultaneously in their frame, then the train would stop all at once to them and remain longer than the tunnel, while the tunnel observers would say that the train threw down spikes at the back of the train first and stretched out as it stopped.

Still doesn't make any sense to me. In each frame you can compare the tick rates of the two clocks--for example, if the two clocks are moving at 0.8c relative to one another, then in the rest frame of clock #1 it'll be true that clock #2 is ticking at 0.6 the rate of clock #1, while in the rest frame of clock #2 it'll be true that clock #1 is ticking at 0.6 the rate of clock #2. So here each frame is comparing the tick rates of the two clocks in terms of their own coordinates. I don't know what you mean when you say talk about two clocks comparing tick rates, as opposed to using a single frame to compare the tick rates of two clocks.

The third frame would just give a third answer about the relative tick rates of the two clocks in terms of its own coordinates, no better or no worse than the answers found in either of the two clock rest frames. So like I said, I really don't understand what point you are trying to make here.

Okay now, clocks would surely dilate and "undilate" in the same way when changing frames regardless of the process involved. That is, according to SR they would, but in regard to my presentation, there would be no reason to just assume that unless I can also assume it for the lengths regardless of the process also, so it looks like I'm losing ground again.

 

[JesseM]

But remember that in the unprimed frame, rod #1 is 10 light-seconds long while rod #2 is 6 light-seconds long. And the unprimed frame is just an ordinary Newtonian inertial frame, where according to Newtonian laws objects will not change length when they change velocities. So if rod #2 is brought to rest next to rod #1, they will still be different lengths.

But even in Lorentz ether theory, Newton's laws do not strictly apply.

I wasn't talking about a Lorentz ether theory in my discussion of the numerical problem with the primed and unprimed frames. I was just talking about ordinary Newtonian physics, no shrinkage of moving rigid objects as measured in any normal Newtonian inertial frame. If you read my derivation you'll see there was no assumption that objects change lengths in the Newtonian unprimed frame, and any apparent shift in lengths in other frames was just due to the fact that we used the Lorentz transformation to generate these other frames, it's just a coordinate effect. If we instead generated other frames using the Galilei transformation on the unprimed frames, then all frames would agree that rigid objects maintain a constant length regardless of changes in velocity. This would not be true in a Lorentz ether theory!

If no contraction took place in a Newtonian universe, then all rods would remain the same lengths to all observers in the first place because no contraction took place to begin with.

This would only be true in the standard Newtonian inertial coordinate systems--the ones you would get by doing a Galilei transformation on the first (unprimed) Newtonian inertial coordinate system. If you allow arbitrary coordinate systems not constructed in the usual Newtonian way, then the coordinate length of objects need not be constant in these other coordinate systems, even though your assumptions about the laws of physics haven't changed. A coordinate system is basically just an arbitrary way of assigning position and time labels to different events, so it shouldn't be a surprise that you can assign these labels in such a way that the coordinate distance between ends of an object changes in any way you like from one time coordinate to another.

If B quickly accelerates to v relative to A, then it is usually said that B will contract to A, but that isn't necessarily true, but depends upon how the acceleration took place. If B and C have some distance between them and they quickly accelerate to v simultaneously, then A will measure the same distance between them as before.

Sure, in relativity there are no perfectly rigid objects--if there were, then pushing on one end of an object would instantaneously cause the other end to move, allowing information to be transmitted FTL. But you can have semi-rigid objects which have an equilibrium length when moving inertially, so although different parts of them may be accelerating differently when you apply a force, if you stop applying the external force then the system will return to equilibrium (similar to how if you have a relaxed spring and then stretch it and let go of it, eventually it will return to its original relaxed length). Once it reaches this equilibrium all parts of the system will once again be moving inertially and will be at rest with each other, with the same rest length in the new frame as the rest length in the old frame before you applied the force.

There is also a way of accelerating an object such that the rest length of the object in the instantaneous inertial rest frame of any part of it will be constant--this is known as Born rigid acceleration. See also the [PLAIN and the Rindler horizon

Anyway, none of this is really relevant to the example I was talking about, since I was assuming Newtonian laws of physics where objects can remain perfectly rigid even during accelerations, meaning their length will remain constant as seen in all Newtonian inertial frames. Again, even in such a universe you can use a different type of coordinate system where the length does change, that's what my example was all about.

Also, if a train enters a tunnel of the same proper length, then if the tunnel observers threw spikes up all at once along the length of the tunnel, the train will quickly stop and be contracted to the tunnel, whereas the train observers say the spikes were thrown up at the front of the tunnel first and the train crunched up as it was stopped.

Why would the train crunch up? If the train was semi-rigid in the sense I discussed above, then the train's length in its new rest frame after the acceleration (after it had returned to equilibrium) would be the same as its length in its previous rest frame before it entered the tunnel. It would be the tunnel's length that had expanded when you compare its length in the train's new rest frame with tunnel's length in the train's prior rest frame before it entered the tunnel (though in each of these frames taken on its own, the tunnel's length remains constant). Of course it's ambiguous if you meant to compare these two frames when you asked what the train observers would say, since they do not remain in the same inertial rest frame throughout this process, so there is no one inertial frame that represents their own "point of view".

Okay now, clocks would surely dilate and "undilate" in the same way when changing frames regardless of the process involved.

In SR any inertial frame says that a clock's rate of ticking at any given moment depends only on its velocity at that moment, regardless of the process that got it to that velocity--not sure if that's what you meant though.

 

[grav-universe]

I wasn't talking about a Lorentz ether theory in my discussion of the numerical problem with the primed and unprimed frames. I was just talking about ordinary Newtonian physics, no shrinkage of moving rigid objects as measured in any normal Newtonian inertial frame. If you read my derivation you'll see there was no assumption that objects change lengths in the Newtonian unprimed frame, and any apparent shift in lengths in other frames was just due to the fact that we used the Lorentz transformation to generate these other frames, it's just a coordinate effect. If we instead generated other frames using the Galilei transformation on the unprimed frames, then all frames would agree that rigid objects maintain a constant length regardless of changes in velocity. This would not be true in a Lorentz ether theory!

This would only be true in the standard Newtonian inertial coordinate systems--the ones you would get by doing a Galilei transformation on the first (unprimed) Newtonian inertial coordinate system. If you allow arbitrary coordinate systems not constructed in the usual Newtonian way, then the coordinate length of objects need not be constant in these other coordinate systems, even though your assumptions about the laws of physics haven't changed. A coordinate system is basically just an arbitrary way of assigning position and time labels to different events, so it shouldn't be a surprise that you can assign these labels in such a way that the coordinate distance between ends of an object changes in any way you like from one time coordinate to another.

Sure, in relativity there are no perfectly rigid objects--if there were, then pushing on one end of an object would instantaneously cause the other end to move, allowing information to be transmitted FTL. But you can have semi-rigid objects which have an equilibrium length when moving inertially, so although different parts of them may be accelerating differently when you apply a force, if you stop applying the external force then the system will return to equilibrium (similar to how if you have a relaxed spring and then stretch it and let go of it, eventually it will return to its original relaxed length). Once it reaches this equilibrium all parts of the system will once again be moving inertially and will be at rest with each other, with the same rest length in the new frame as the rest length in the old frame before you applied the force.

There is also a way of accelerating an object such that the rest length of the object in the instantaneous inertial rest frame of any part of it will be constant--this is known as Born rigid acceleration. See also the [PLAIN and the Rindler horizon

Looks like we are talking past each other here. I'm basically agreeing that unless the processes of acceleration from a frame are the same as the deceleration, a difference in lengths will be observed.

Anyway, none of this is really relevant to the example I was talking about, since I was assuming Newtonian laws of physics where objects can remain perfectly rigid even during accelerations, meaning their length will remain constant as seen in all Newtonian inertial frames. Again, even in such a universe you can use a different type of coordinate system where the length does change, that's what my example was all about.

Right, but if your using the Lorentz transforms, then the object cannot remain rigid to all frames. If it remains the same length in the frame of the object, then it must be seen to elongate in the other frame upon coming to rest.

Why would the train crunch up? If the train was semi-rigid in the sense I discussed above, then the train's length in its new rest frame after the acceleration (after it had returned to equilibrium) would be the same as its length in its previous rest frame before it entered the tunnel. It would be the tunnel's length that had expanded in the train's new rest frame, as compared to the tunnel's length in the train's prior rest frame before it entered the tunnel. Of course it's ambiguous if you meant to compare these two frames when you asked what the train observers would say, since they do not remain in the same inertial rest frame throughout this process, so there is no one inertial frame that represents their own "point of view".

Wait a minute, you're right. From the perspective of the passengers on the train, the tunnel throws up a spike at the front of the train, then the rest of the tunnel keeps moving, stretching out away from the spiked part, then another spike pops up, etc. The train does not change except for whatever physical occurances take place at the places the spikes pop up while the train and tunnel are moving in respect to each other.

In SR any inertial frame says that a clock's rate of ticking at any given moment depends only on its velocity at that moment, regardless of the process that got it to that velocity--not sure if that's what you meant though.

Right, so that would be the only real homogeneous observation that can be made, but it's not enough to run on, so the hypothesis falls apart.

 

[JesseM]

Right, but if your using the Lorentz transforms, then the object cannot remain rigid to all frames. If it remains the same length in the frame of the object, then it must be seen to elongate in the other frame upon coming to rest.

Yes, in all the frames except the unprimed frame, lengths will change when the object changes velocity. But since I'm assuming Newtonian physics, and the unprimed frame is a standard Newtonian inertial frame, in that frame the lengths of all these rigid objects will be constant under changes of velocity. That was the basis for my derivation showing that even though the primed frame sees the two rigid measuring-rods having equal lengths and traveling at equal and opposite velocities, the two rods have different lengths in the unprimed frame, which automatically means that when they are brought to rest relative to each other (regardless of what frame they are brought to rest in), their ends will not line up. You seemed to have misunderstood earlier when you suggested I was talking about a Lorentz ether theory, but do you follow what I'm saying now?

In SR any inertial frame says that a clock's rate of ticking at any given moment depends only on its velocity at that moment, regardless of the process that got it to that velocity--not sure if that's what you meant though.

Right, so that would be the only real homogeneous observation that can be made, but it's not enough to run on, so the hypothesis falls apart.

Well, I don't really understand what you mean by "homogenous observation", but if you don't have a definite hypothesis you're putting forward perhaps it's not that important.

 

[grav-universe]

Yes, in all the frames except the unprimed frame, lengths will change when the object changes velocity. But since I'm assuming Newtonian physics, and the unprimed frame is a standard Newtonian inertial frame, in that frame the lengths of all these rigid objects will be constant under changes of velocity. That was the basis for my derivation showing that even though the primed frame sees the two rigid measuring-rods having equal lengths and traveling at equal and opposite velocities, the two rods have different lengths in the unprimed frame, which automatically means that when they are brought to rest relative to each other (regardless of what frame they are brought to rest in), their ends will not line up. You seemed to have misunderstood earlier when you suggested I was talking about a Lorentz ether theory, but do you follow what I'm saying now?

Actually, since I've thinking about how length contractions take place, I was just thinking about something which is similar to what I think you are saying, that one could take exception to the unprimed frame, where no contractions might actually be seen to take place, so basically Newtonian in nature, but only to all other frames where the clocks dilate and things are measured differently.

Well, I don't really understand what you mean by "homogenous observation", but if you don't have a definite hypothesis you're putting forward perhaps it's not that important.

Yes, that is the assumption that was made to derive what I did, but the homogeneous observations argument seems to be falling apart, so I suppose it was just coincidence it worked out to begin with, only when applied in a particular way.

 

[phyti]

Are you disagreeing with my posts #13 and #15 which attempt to show that a coordinate transformation with an arbitrary constant A in place of gamma will still result in a constant speed of light in all frames, satisfying the second postulate?

No, I'm ignoring those posts. As usual, you're too
argumentative, like a lawyer whose going to settle
a case with a dictionary! Where did you get the
idea, everyone is going to agree with you?

Idealized math theories appeal to the
mathematician, but maybe the universe was designed
by a poet.

 

[JesseM]

No, I'm ignoring those posts. As usual, you're too
argumentative, like a lawyer whose going to settle
a case with a dictionary! Where did you get the
idea, everyone is going to agree with you?

I don't expect people to automatically agree with me, but since this board is meant to discuss mainstream claims about physics, I'd expect that people be willing to explain the reasoning behind claims that appear non-mainstream, like your claim that the Lorentz transformation can be derived from the second postulate alone.

Idealized math theories appeal to the
mathematician, but maybe the universe was designed
by a poet.

But when we discuss what assumptions are needed to derive the Lorentz transformation, this is a purely theoretical discussion about "idealized math theories", it's not a question that has anything to do with observations about the real universe.

 

[JesseM]

Actually, since I've thinking about how length contractions take place, I was just thinking about something which is similar to what I think you are saying, that one could take exception to the unprimed frame, where no contractions might actually be seen to take place, so basically Newtonian in nature, but only to all other frames where the clocks dilate and things are measured differently.

What do you mean by "take exception to"?

Yes, that is the assumption that was made to derive what I did, but the homogeneous observations argument seems to be falling apart, so I suppose it was just coincidence it worked out to begin with, only when applied in a particular way.

Right, but again I don't understand what you mean by "homogeneous observations", so I don't understand what "that" is when you say "that is the assumption that was made to derive what I did". Again, if you're no longer making an argument based on this phrase then perhaps it's not important that I understand what you meant.

 

[grav-universe]

What do you mean by "take exception to"?

Since the way an object contracts depends upon how it accelerates, objects can be made to decelerate into the unprimed frame such that the unprimed frame still measures the same length for the object as the unprimed frame did when the object was in motion.

Right, but again I don't understand what you mean by "homogeneous observations", so I don't understand what "that" is when you say "that is the assumption that was made to derive what I did". Again, if you're no longer making an argument based on this phrase then perhaps it's not important that I understand what you meant.

By homogeneous observations, I meant that if two objects that are traveling in different directions at the same realtive speed are observed to have the same time dilations and lengths, then they are identical, and will remain identical when brought to rest with each other and compared directly, but that was only if the time dilation and length contractions are coordinate effects only.

 

[JesseM]

Since the way an object contracts depends upon how it accelerates, objects can be made to decelerate into the unprimed frame such that the unprimed frame still measures the same length for the object as the unprimed frame did when the object was in motion.

Sure, even in a relativistic universe you could intentionally cause objects to accelerate such that they stayed the same length in some frame. But in a Newtonian universe, rigid objects are guaranteed to accelerate in a way that preserves their length in any Newtonian inertial frame--that's just the definition of "rigid" in Newtonian physics.

By homogeneous observations, I meant that if two objects that are traveling in different directions at the same realtive speed are observed to have the same time dilations and lengths, then they are identical, and will remain identical when brought to rest with each other and compared directly, but that was only if the time dilation and length contractions are coordinate effects only.

I still don't understand what you mean by "coordinate effects only"--can you give an example of a type of length contraction or time dilation that is not coordinate-dependent? Like I said, the only thing I can think of with time dilation is if you are comparing total time elapsed between two meetings of a pair of clocks rather than their instantaneous rates of ticking at any moment...

 

[grav-universe]

I still don't understand what you mean by "coordinate effects only"--can you give an example of a type of length contraction or time dilation that is not coordinate-dependent? Like I said, the only thing I can think of with time dilation is if you are comparing total time elapsed between two meetings of a pair of clocks rather than their instantaneous rates of ticking at any moment...

Well, maybe I should refer to it in another way or something, then, but I mean that if an object changes frames, it will always be seen to contract and time dilate. Then if changing back to the original frame, it will regain its original length and clock rate. That is true of the clock rates but the length of the object can be changed depending upon how it accelerates and decelerates, so my argument there has failed.

 

[JesseM]

Well, maybe I should refer to it in another way or something, then, but I mean that if an object changes frames, it will always be seen to contract and time dilate.

Doesn't that depend on your choice of coordinate system though? No matter what the laws of physics--relativistic or Newtonian--if you allow arbitrary coordinate systems (as opposed to the specific ones identified as 'inertial' in each case), then it's possible to find coordinate systems where a given ruler shrinks and a given clock slows down as it accelerates, and other coordinate systems where the same ruler maintains a constant length and the same clock maintains a constant rate of ticking.

 

[grav-universe]

Sure, in relativity there are no perfectly rigid objects--if there were, then pushing on one end of an object would instantaneously cause the other end to move, allowing information to be transmitted FTL. But you can have semi-rigid objects which have an equilibrium length when moving inertially, so although different parts of them may be accelerating differently when you apply a force, if you stop applying the external force then the system will return to equilibrium (similar to how if you have a relaxed spring and then stretch it and let go of it, eventually it will return to its original relaxed length). Once it reaches this equilibrium all parts of the system will once again be moving inertially and will be at rest with each other, with the same rest length in the new frame as the rest length in the old frame before you applied the force.

Yes, that's true too, isn't it? So once again it seems my homogeneous coordination deal has not failed after all, again :) . If forces were applied in the same way at points all along a ship from the rest frame, then the ship would remain the same length as viewed from the rest frame but elongated in the moving frame. And either way a ruler on the ship would still contract unless forces were applied all along its length as well, since rulers are what we are really comparing. If the forces on the ship, however, were to continue in this way, they would eventually just tear the ship apart quicker than they would actually elongate it in the moving frame, and the rest frame would see breaks occurring along its length while each piece of the ship that breaks off then contracts if there is no further acceleration at the other end of the piece, but only the distance between the pieces remains the same from the perspective of the rest frame. If the forces only acted for a short while without tearing the ship apart, then the ship would either become noticably deformed or pull back to its original proper length which would be then contracted to the rest frame in that case also.

Why would the train crunch up? If the train was semi-rigid in the sense I discussed above, then the train's length in its new rest frame after the acceleration (after it had returned to equilibrium) would be the same as its length in its previous rest frame before it entered the tunnel. It would be the tunnel's length that had expanded when you compare its length in the train's new rest frame with tunnel's length in the train's prior rest frame before it entered the tunnel (though in each of these frames taken on its own, the tunnel's length remains constant). Of course it's ambiguous if you meant to compare these two frames when you asked what the train observers would say, since they do not remain in the same inertial rest frame throughout this process, so there is no one inertial frame that represents their own "point of view".

Okay, I've been thinking about this, and if the spikes all spring up simultaneously from the tunnel according to the tunnel frame, then since they all spring up simultaneously with equal distances between them in the same way, then still from the perspective of the tunnel, whatever the train does to one, bending them or pushing them along the tunnel somewhat or whatever, it will do to all in the same way, so the same distance still remains between the spikes overall and the train is contracted to the tunnel when it comes to a stop. From the perspective of passengers on the train, however, the spikes did not spring up simultaneously, but from the back of the tunnel first, catching the front of the train, then as the tunnel contines to move in respect to them while dragging the front of the train along with the spike, the next spike spring up a little further along the train, and so on. The only way that this can occur to gain the same end result as from the perspective of the tunnel is if the train were being crunched up as the spikes spring up to catch it as the tunnel and spikes continue to move according to the passengers of the train.

 

[grav-universe]

Doesn't that depend on your choice of coordinate system though? No matter what the laws of physics--relativistic or Newtonian--if you allow arbitrary coordinate systems (as opposed to the specific ones identified as 'inertial' in each case), then it's possible to find coordinate systems where a given ruler shrinks and a given clock slows down as it accelerates, and other coordinate systems where the same ruler maintains a constant length and the same clock maintains a constant rate of ticking.

How could clocks maintain a constant rate of ticking from the perspective of another frame if time dilation occurs between the frames?

 

[JesseM]

How could clocks maintain a constant rate of ticking from the perspective of another frame if time dilation occurs between the frames?

The time dilation equation only relates the time of inertial frames in SR (the ones that are related to one another by the Lorentz transformation), but that's why I specified that I wanted to talk about what happens "if you allow arbitrary coordinate systems (as opposed to the specific ones identified as 'inertial' in each case)". It's certainly possible to construct a non-inertial coordinate system where a clock that has nonzero proper acceleration (so it is accelerating, and its rate of ticking is changing, from the perspective of every inertial frame) is ticking at a constant rate relative to coordinate time.

 

[grav-universe]

The time dilation equation only relates the time of inertial frames in SR (the ones that are related to one another by the Lorentz transformation), but that's why I specified that I wanted to talk about what happens "if you allow arbitrary coordinate systems (as opposed to the specific ones identified as 'inertial' in each case)". It's certainly possible to construct a non-inertial coordinate system where a clock that has nonzero proper acceleration (so it is accelerating, and its rate of ticking is changing, from the perspective of every inertial frame) is ticking at a constant rate relative to coordinate time.

If you mean we could increase the ticking in one frame to match the ticking as viewed from another frame, then sure. Is that what you mean? You are not saying that a clock that changes reference frames can still tick at the same rate as in the original frame without tampering with the clock, right? By the way, I made a second post before my last one in case you missed it.

 

[JesseM]

If you mean we could increase the ticking in one frame to match the ticking as viewed from another frame, then sure. Is that what you mean?

I'm not talking about physically messing with the ticking of the clock relative to a normal clock traveling alongside it, if that's what you mean. I'm just saying that since non-inertial coordinate systems are totally arbitrary ways of labeling events with position and time coordinates (see the last animated diagram in http://www.aei.mpg.de/einsteinOnline/en/spotlights/background_independence/index.html would be an example of a non-inertial coordinate system where the clock is ticking at a constant rate relative to coordinate time.

You are not saying that a clock that changes reference frames can still tick at the same rate as in the original frame without tampering with the clock, right?

If the original frame is an inertial frame, then it won't tick at the same rate in the inertial frame. But whatever rate it was ticking relative to coordinate time in the original frame before it accelerated, you can construct a non-inertial coordinate system where the clock ticks at that same rate relative to the coordinate time of this separate coordinate system throughout the acceleration.

By the way, I made a second post before my last one in case you missed it.

I did see it, I'll get back to it soon but I thought this issue could be addressed with a shorter reply so I did that first...

 

[grav-universe]

Okay, so are you applying this to a non-inertial observer watching the clock, rather than the clock itself, then? If the observer is inertial, then all clocks that were originally at rest with an inertial observer will now be seen to have time dilation regardless of how the motion of a clock occurs by integrating over the path of acceleration, as I'm sure you know. As for non-inertial observers, though, there would be one place, with constant acceleration applied at least, that the clock would be seen to tick at the same rate and remain at a constant distance from the observer, thereby having zero relative speed, and that would indeed be at the Rindler horizon to which you referred.

 

[JesseM]

Okay, so are you applying this to a non-inertial observer watching the clock, rather than the clock itself, then?

I'm just talking about coordinate systems. As a convenient shorthand we can talk about inertial coordinate systems and inertial observers pretty much interchangeably, since each inertial observer has a unique rest frame (well, not completely unique since you still have the freedom to place the origin wherever, but this won't affect things like velocity and energy and time dilation in that observer's frame). Of course a real human observer is perfectly free to use a frame other than their own rest frame when making calculations, but like I said it's just a widely-recognized shorthand. With non-inertial coordinate systems this doesn't work so well, since the complete freedom you have in defining a non-inertial coordinate system (aside from basic rules like making it smooth) means that for any non-inertial observer, you could define an infinite number of distinct non-inertial coordinate systems where that observer would be at rest and yet the coordinate systems would disagree about things like the velocity of other objects, or how a given clock's ticking rate relates to coordinate time.

If the observer is inertial, then all clocks that were originally at rest with an inertial observer will now be seen to have time dilation regardless

You mean, if all these clocks accelerate? If so, yes, in the observer's inertial rest frame the clocks will change their rate of ticking when they accelerate.

As for non-inertial observers, though, there would be one place, with constant acceleration applied at least, that the clock would be seen to tick at the same rate and remain at a constant distance from the observer, thereby having zero relative speed, and that would indeed be at the Rindler horizon to which you referred.

Huh? It has nothing to do with any location in space like the Rindler horizon, it just has to do with what coordinate system you're using. If all the accelerating observers use the same Rindler coordinate system, they'll all agree that an accelerating clock at rest in these coordinates is ticking at a constant rate relative to coordinate time. So I don't understand what you mean when you say "there would be one place...and that would indeed be the Rindler horizon", especially since it's impossible for any sublight observer to remain on the Rindler horizon for more than an instant, since the Rindler horizon is expanding outward at the speed of light as seen in any inertial frame (see the first two diagrams on this page).

 

[grav-universe]

I'm just talking about coordinate systems. As a convenient shorthand we can talk about inertial coordinate systems and inertial observers pretty much interchangeably, since each inertial observer has a unique rest frame (well, not completely unique since you still have the freedom to place the origin wherever, but this won't affect things like velocity and energy and time dilation in that observer's frame). Of course a real human observer is perfectly free to use a frame other than their own rest frame when making calculations, but like I said it's just a widely-recognized shorthand. With non-inertial coordinate systems this doesn't work so well, since the complete freedom you have in defining a non-inertial coordinate system (aside from basic rules like making it smooth) means that for any non-inertial observer, you could define an infinite number of distinct non-inertial coordinate systems where that observer would be at rest and yet the coordinate systems would disagree about things like the velocity of other objects, or how a given clock's ticking rate relates to coordinate time.

Okay, right, so I am considering just inertial observers in the postulates and all of the mathematics is found from the perspectives of inertial observers only.

Huh? It has nothing to do with any location in space like the Rindler horizon, it just has to do with what coordinate system you're using. If all the accelerating observers use the same Rindler coordinate system, they'll all agree that an accelerating clock at rest in these coordinates is ticking at a constant rate relative to coordinate time.

Right, a constant rate I suppose, but not the same rate as a clock in the observing frame.

So I don't understand what you mean when you say "there would be one place...and that would indeed be the Rindler horizon", especially since it's impossible for any sublight observer to remain on the Rindler horizon for more than an instant, since the Rindler horizon is expanding outward at the speed of light as seen in any inertial frame (see the first two diagrams on this page).

Yes, I believe you're right. The Rindler horizon would probably be more like an event horizon where another clock's time would be seen to slow to zero. It's been a while since I've attempted to study Rindler, so that would be another reason I am steering away from non-inertial observers, although I still don't see offhand how an arbitrary choice of coordinates could make the clocks tick any differently than whatever rate they are observed to tick with some time dilation applied.

 

[JesseM]

Okay, right, so I am considering just inertial observers in the postulates. All of the mathematics is found from the perspectives of inertial observers in order to derive SR only, not GR or any form of it.

But if you're supposed to be deriving the coordinates, you can't assume from the start that they will end up being the coordinates given by the Lorentz transformation, can you? Especially if you don't start from the same postulates that are used to derive the Lorentz transformation. See my example of assuming only the second postulate and getting a more general set of coordinate transformations where gamma is replaced by an arbitrary constant A.

Right, a constant rate, but not the same rate as a clock in the observing frame.

"same rate" relative to whose coordinates? If you have an inertial coordinate system A and a clock that starts out moving inertially but then accelerates, you can design the non-inertial coordinate system B such that the ratio between clock ticks and increments of B's time coordinate is constant, and it's the same as the what the ratio between clock ticks and A's time coordinate was before the clock started accelerating (but not after obviously, since the ratio of clock ticks to A's time coordinate will change as soon as the clock starts accelerating). That's what I thought you might be saying was impossible when you said "You are not saying that a clock that changes reference frames can still tick at the same rate as in the original frame".

I don't agree. Two observers that simultaneously attain a constant and equal proper acceleration from a rest frame and are separated by the distance of the Rindler horizon according to the leading observer

First of all, observers at different positions in Rindler coordinates don't have equal proper acceleration, the ones closer to the Rindler horizon must have greater proper acceleration (the observers are undergoing what is known as Born rigid acceleration, which means that the distance between them stays constant in their instantaneous inertial rest frame at each moment, and the math works out so that this can only happen if the trailing observer has a greater proper acceleration). Second, if the leading observer saw the trailing observer at the Rindler horizon at some moment, the trailing observer couldn't stay on the Rindler horizon since that horizon is expanding outward at the speed of light as defined by any inertial frame. Again, look at the second diagram from the Rindler horizon page I linked to:

This diagram is drawn from the perspective of an inertial frame. The curved lines represent observers with constant position in Rindler coordinates, and the dotted line is the Rindler horizon. You can see that the Rindler horizon is moving outward at c, and that observers closer to the horizon increase their speed more quickly (the tangent to each curve is vertical at t=0 in the inertial frame, so they are all instantaneously at rest at that moment, and you can see that the slopes of worldlines closer to the dotted line become closer to diagonal more quickly)

 

[grav-universe]

But if you're supposed to be deriving the coordinates, you can't assume from the start that they will end up being the coordinates given by the Lorentz transformation, can you? Especially if you don't start from the same postulates that are used to derive the Lorentz transformation. See my example of assuming only the second postulate and getting a more general set of coordinate transformations where gamma is replaced by an arbitrary constant A.

I can only use inertial observers as stated in the postulates in order to measure the speed of light the same in all inertial reference frames and then to perform the mathematics accordingly. This does not mean that non-inertial observers won't measure the speed of light differently, but I have not included them, just inertial observers only measuring c for the speed of light. What non-inertial observers will measure for the speed of light can then be worked out from there.

"same rate" relative to whose coordinates? If you have an inertial coordinate system A and a clock that starts out moving inertially but then accelerates, you can design the non-inertial coordinate system B such that the ratio between clock ticks and increments of B's time coordinate is constant, and it's the same as the what the ratio between clock ticks and A's time coordinate was before the clock started accelerating (but not after obviously, since the ratio of clock ticks to A's time coordinate will change as soon as the clock starts accelerating). That's what I thought you might be saying was impossible when you said "You are not saying that a clock that changes reference frames can still tick at the same rate as in the original frame".

Okay, well if one messes with the distance coordinization in order to make the ticking working out the same in the reality of the non-inertial observer, then sure, but then one would have to design a different coordinate system for accelerating away from the rest frame as accelerating back, otherwise how would something like the twin paradox work out? In any case, I am only considering inertial observers with each having the same coordinate system to keep things simple.

First of all, observers at different positions in Rindler coordinates don't have equal proper acceleration, the ones closer to the Rindler horizon must have greater proper acceleration (the observers are undergoing what is known as Born rigid acceleration, which means that the distance between them stays constant in their instantaneous inertial rest frame at each moment, and the math works out so that this can only happen if the trailing observer has a greater proper acceleration). Second, if the leading observer saw the trailing observer at the Rindler horizon at some moment, the trailing observer couldn't stay on the Rindler horizon since that horizon is expanding outward at the speed of light as defined by any inertial frame. Again, look at the second diagram from the Rindler horizon page I linked to:

This diagram is drawn from the perspective of an inertial frame. The curved lines represent observers with constant position in Rindler coordinates, and the dotted line is the Rindler horizon. You can see that the Rindler horizon is moving outward at c, and that observers closer to the horizon increase their speed more quickly (the tangent to each curve is vertical at t=0 in the inertial frame, so they are all instantaneously at rest at that moment, and you can see that the slopes of worldlines closer to the dotted line become closer to diagonal more quickly)

Right, I editted my post since my last reply, sorry. The trailing observer having a greater acceleration must be what I was thinking. Anyway, I can now see that what is occurring with the light catching up to the accelerating observer can all be worked out from the frame of an inertial observer as you said.

 

[JesseM]

I can only use inertial observers as stated in the postulates in order to measure the speed of light the same in all inertial reference frames and then to perform the mathematics accordingly.

But "inertial observers" doesn't necessarily imply the Lorentz transformation unless you assume both postulates of SR. Inertial observers as defined in Newtonian physics all observe the same laws of physics (first postulate satisfied), and all see each other traveling at constant velocity, but there's no invariant speed postulate and their coordinates transform according to the Galilei transformation. Likewise, I showed that if you just wanted to satisfy the second postulate but not the first, you could have a family of coordinate systems that all see light moving at c, and that all see each other traveling at constant velocity, but where the coordinates transform according to a different transformation. If you're going to go mucking about with the postulates, you can't start out assuming that the phrase "inertial observer" will mean exactly the same thing as it does in SR, with different frames related by the Lorentz transformation, that'd just be circular reasoning rather than an actual "derivation'.

Okay, well if one messes with the distance coordinization in order to make the ticking working out the same

It's the time coordinate that determines the rate of ticking, not the distance coordinate.

in the reality of the non-inertial observer

Again, "the reality of the non-inertial observer" is meaningless since there is no single way to construct a coordinate system where a non-inertial observer is at rest. You have to talk about coordinate systems, not "observers".

but then one would have to design a different coordinate system for accelerating away from the rest frame as accelerating back, otherwise how would something like the twin paradox work out?

No, you'd have a single non-inertial coordinate system, not two different ones for different parts of the trip. Since time dilation doesn't work the same way in non-inertial coordinate systems as it does in inertial ones, there's no problem getting the twin paradox to work out, at some point the inertial twin would just have his clock ticking faster relative to coordinate time than the non-inertial one. This section of the twin paradox FAQ features a diagram showing what lines of simultaneity could look like in a single non-inertial coordinate system (drawn relative to the space and time axis of the inertial frame where the inertial twin is at rest):

You can see that during the phase where the non-inertial twin "Stella" is accelerating, the clock of the inertial twin "Terence" will elapse much more time than hers. Lines of constant position in this non-inertial system aren't drawn in, you could draw them any way you like (including curved lines so that Stella could be at a constant position throughout her trip) and have a valid non-inertial system.

In any case, I am only considering inertial observers with each having the same coordinate system to keep things simple.

Right, I editted my post since my last reply, sorry. The trailing observer having a greater acceleration must be what I was thinking. Anyway, I can now see that what is occurring with the light catching up to the accelerating observer can all be worked out from the frame of an inertial observer as you said.

And you understand how in Rindler coordinates, any given clock in that family of accelerating clocks can be ticking at a constant rate relative to coordinate time, and occupying a fixed coordinate position?

 

[grav-universe]

But "inertial observers" doesn't necessarily imply the Lorentz transformation unless you assume both postulates of SR. Inertial observers as defined in Newtonian physics all observe the same laws of physics (first postulate satisfied), and all see each other traveling at constant velocity, but there's no invariant speed postulate and their coordinates transform according to the Galilei transformation. Likewise, I showed that if you just wanted to satisfy the second postulate but not the first, you could have a family of coordinate systems that all see light moving at c, and that all see each other traveling at constant velocity, but where the coordinates transform according to a different transformation. If you're going to go mucking about with the postulates, you can't start out assuming that the phrase "inertial observer" will mean exactly the same thing as it does in SR, with different frames related by the Lorentz transformation, that'd just be circular reasoning rather than an actual "derivation'.

I am only applying the observations from non-accelerating observers as stated in the second postulate, but you're right that I do have to make an additional assumption about the homogeneity of space where if clocks and lengths with the same relative speed are observed the same regardless of direction, then they are considered identical, of course, as we've discussed, although not necessarily including the first postulate in that case.

No, you'd have a single non-inertial coordinate system, not two different ones for different parts of the trip. Since time dilation doesn't work the same way in non-inertial coordinate systems as it does in inertial ones, there's no problem getting the twin paradox to work out, at some point the inertial twin would just have his clock ticking faster relative to coordinate time than the non-inertial one. This section of the twin paradox FAQ features a diagram showing what lines of simultaneity could look like in a single non-inertial coordinate system (drawn relative to the space and time axis of the inertial frame where the inertial twin is at rest):

You can see that during the phase where the non-inertial twin "Stella" is accelerating, the clock of the inertial twin "Terence" will elapse much more time than hers. Lines of constant position in this non-inertial system aren't drawn in, you could draw them any way you like (including curved lines so that Stella could be at a constant position throughout her trip) and have a valid non-inertial system.

But if according to the non-inertial observer, Stella, both observer's clocks tick at the same rate away and back, then they will read the same time when they meet back up, so I'm still not sure what you're saying there about having an arbitrary choice of coordinate systems.

And you understand how in Rindler coordinates, any given clock in that family of accelerating clocks can be ticking at a constant rate relative to coordinate time, and occupying a fixed coordinate position?

Right. We can have the trailing observer with a greater acceleration that remains a constant distance behind according to the leading accelerating observer so with zero relative speed and the clocks tick at the same rate according to the leading observer also.

 

[JesseM]

But if according to the non-inertial observer, Stella, both observer's clocks tick at the same rate away and back, then they will read the same time when they meet back up, so I'm still not sure what you're saying there about having an arbitrary choice of coordinate systems.

My point was that you can design a non-inertial coordinate system where a non-inertial clock ticks at a constant rate relative to coordinate time--in this case, Stella's clock. I didn't say all clocks would tick at a constant rate in such a coordinate system, and in the type where simultaneity is defined as in the diagram, the clock of the inertial twin Terence would necessarily speed up and tick faster than Stella's during the middle part of the journey.

Right. We can have the trailing observer with a greater acceleration that remains a constant distance behind according to the leading accelerating observer so with zero relative speed and the clocks tick at the same rate according to the leading observer also.

My only quibble is that it's not really "according to the leading observer", it's according to Rindler coordinates (which the leading observer doesn't necessarily have to use if he doesn't want to, even if he's restricting his attention to coordinate systems where he's at rest).