SR derived solely from one postulate

[JesseM]

Using only the 2nd postulate, the measured speed of
light is constant in space for all inertial frames,
(and independent of its source), you can derive
time dilation, addition of velocities, and
variation of length measurements.

Mapping constant linear motion in 2 or 3 dimensions
to the perception space of the observer,
demonstrates the hyperbolic form (gamma).
Time is linear only in radial directions for the observer,
as noted in the popular and simplistic one dimensional expositions.
It's been done already!

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?

 

[grav-universe]

What do you mean by "identical", though? Are you concluding that because they have equal length in the third observer's own rest frame, they must have identical rest lengths in their own respective rest frames? If so that would not be a valid conclusion, one can find coordinate transformations where this is not true. If that's not what you're concluding, can you explain what you meant by "then the rulers are identical"? Presumably you don't just mean they have the same length in the frame of the third observer, because in that case "If the lengths are equal, then the rulers are identical" would just be repeating the same thing twice using different phrasing.

edit: Maybe I answered my own question below? When you say the rulers are identical, do you mean that if brought to rest relative to one another, they would line up?

Right, the rulers have identical lengths if when compared side by side in the same frame, they line up end to end. The physical properties don't matter, as they may be composed of different materials.

But does the second postulate justify the assumption that "observations are homogenous in any direction"? How so?

No, that space is homogeneous is an assumption made, or extra postulate required.

The fact that the third observer measures the objects going in opposite directions to have identical lengths in his frame is a statement about coordinate length, while the idea that the two rulers would line up if brought next to each other is a statement about physical length which depends on the form of the laws of physics. For example, if we assumed the laws of physics were Newtonian but used the Lorentz transformation to define different coordinate systems, I don't think it would in fact be true that just because two objects going at equal and opposite velocities in some frame had the same length in that frame, they would necessarily have the same physical length when brought next to each other in the same frame.

If space is homogeneous, then there shouldn't be any difference in the observation of Bob traveling to the right of Alice at v and Bob traveling to the left of Alice at v, so would be the equivalent of Alice just turning around 180 degrees to face the other way in that case, the observations being the same regardless of direction.

Anyway, even if your statement were true, how is this supposed to prove that the Lorentz transformation follows from the second postulate? You never addressed my request for a violation of the second postulate in the non-Lorentzian coordinate transformation I mentioned in post #13. Do you or do you not think that it is possible to find a numerical example where a signal is moving at c in one of the frames given by that transformation, but not moving at c in another?

I addressed it in post #24. You are right about that of course, but working through the exercise helped me to realize what extra assumption I was making in order to derive the values precisely, which was the assumption that space is homogeneous.

 

[grav-universe]

It is clear that it is totally obvious to derive SR from the first postulate alone.
You can easily come to the general transformation with one unkown parameter, that you can label 'c' if you want.
The Galilean Relativity appears then as a very special case, almost unlikely.
If Galileo had been able of such an approach, he would have wondered why the special case would prevail and he would have tried to find out an experimental value for 'c'.

One more evidence that mankind (on an historic time scale) is still unable to free itself from empiricism and still has hard times with abstraction!

see: http://adsabs.harvard.edu/abs/1994AmJPh..62..157S

That is not necessarily true, since that would be assuming that the speed of something must still be measured istropically. If nothing is measured isotropically but light had traveled ballistically with the source instead, for instance, then the physics would still be the same in every inertial frame, although we would not have derived SR from the first postulate, but should have instead derived ballistic theory in this case.

 

[JesseM]

No, that space is homogeneous is an assumption made, or extra postulate required.

But what do you mean by that phrase, exactly? When Einstein talked about the homogeneity of space, I assumed he was talking about one or more spacetime symmetries like translation invariance or rotation invariance. Newtonian physics certainly respects all these symmetries, but as I pointed out above, if you assume Newtonian laws but use the Lorentz transformation to define your coordinate systems, you can have a situation where in one coordinate system two rigid objects have equal and opposite velocities and equal lengths, but if the two objects are brought to rest relative to one another, their lengths are not equal.

 

[bcrowell]

The way I've often presented it in the past is that there is only one postulate: the laws of physics are the same in every inertial frame. Maxwell's equations are a law of physics, so #2 follows from #1.

I'm not a big fan of this view. What you're describing isn't a derivation of SR from the first postulate. It can at best be described as a derivation of SR from a mathematical reformulation of the first postulate and Maxwell's equations (which of course are already mathematical statements). But Maxwell's equations are absurdly complicated compared to the axiom that the invariant speed is finite, so what you're suggesting isn't a very elegant solution. It also seems to be wildly inconsistent with your own point (b):

(b) ...and there's clearly no reason to describe c as the speed of light. c is a geometrical property of spacetime, the maximum speed of causality.

There is certainly an inconsistency there. That's essentially why the first quote begins with "The way I've often presented it in the past is..." That was the way I used to present it until I understood it more deeply.

Einstein presented the structure of relativity as being closely tied to the theory of electromagnetic waves. With the benefit of another century's worth of hindsight, we can see that that's not really the right way to look at the foundations of relativity. It's just an accident of history that the only fundamental field known in 1905 was the EM field. The more modern point of view is that c is a property of spacetime, and massless particles just happen to propagate at that speed. Before I understood that, I used to teach SR from a point of view that followed Einstein's 1905 presentation, except that I telescoped the two postulates into one.

 

[grav-universe]

But what do you mean by that phrase, exactly? When Einstein talked about the homogeneity of space, I assumed he was talking about one or more spacetime symmetries like translation invariance or rotation invariance. Newtonian physics certainly respects all these symmetries, but as I pointed out above, if you assume Newtonian laws but use the Lorentz transformation to define your coordinate systems, you can have a situation where in one coordinate system two rigid objects have equal and opposite velocities and equal lengths, but if the two objects are brought to rest relative to one another, their lengths are not equal.

Rotational invariance sounds about right for what I am describing. In other words, the physics is observed the same in a particular frame in every direction. That is not the same as the physics being the same in every frame, since potentially the physics could still be observed differently in differrent frames, but still the same in every direction in the same frame if space is homogeneous.

 

[JesseM]

Rotational invariance sounds about right for what I am describing. In other words, the physics is observed the same in a particular frame in every direction. That is not the same as the physics being the same in every frame, since potentially the physics could still be observed differently in differrent frames, but still the same in every direction in the same frame if space is homogeneous.

But without the first postulate, I don't see why rotational invariance in the laws of physics themselves should imply rotational invariance in the set of coordinate systems you happen to be using. As I said, the laws of Newtonian physics themselves exhibit rotational invariance, but if you use the coordinate systems given by the Lorentz transformation to describe a Newtonian universe, you will find that in some frames physically identical objects are shrunk more when going in one direction than the other. Likewise, in our own relativistic universe, if you choose to use some non-inertial coordinate system you may find that in that coordinate system objects behave differently in different directions, that doesn't mean that the laws of physics aren't rotationally invariant.

 

[grav-universe]

But without the first postulate, I don't see why rotational invariance in the laws of physics themselves should imply rotational invariance in the set of coordinate systems you happen to be using. As I said, the laws of Newtonian physics themselves exhibit rotational invariance, but if you use the coordinate systems given by the Lorentz transformation to describe a Newtonian universe, you will find that in some frames physically identical objects are shrunk more when going in one direction than the other. Likewise, in our own relativistic universe, if you choose to use some non-inertial coordinate system you may find that in that coordinate system objects behave differently in different directions, that doesn't mean that the laws of physics aren't rotationally invariant.

It is not required with the first postulate. For instance, if an object is moved from a first frame to a second, the physics might say that the object is physically contracted to 1/2 the length in the process. Then, if the physics is the same in all frames, if one were to move the object back from the second frame to the first in the same way, then the object will again contract by 1/2, making it 1/4 the length when coming back to rest in the first frame, at least potentially that could happen although it still doesn't with SR of course because the other frame is always measured as contracted anyway. However, if space is homogeneous, then if one moves the object to a second frame and back again without the actual physics changing the length at all but only in reference to the coordinization of what is observed in space, the coordinization in the first frame is still the same as it was before, and if the physical length of the object hasn't changed as part of our definition of identical, it will be measured at the same original length.

 

[aery]

It is clear that it is totally obvious to derive SR from the first postulate alone.

If only the first postulate describes SR, what is preventing us from deriving SR as "observers in all inertial frames measure the speed of light as a multiple of the number of coins they have in their left pocket".

(I'm not joking, I'm new to SR and having trouble imagining SR without a second postulate, be it the constancy of c, Maxwell's equations or some other way to put it)

 

[grav-universe]

I thought the second postulate actually were: "the speed of light is independent on the speed of the source of light", which is not exactly the same

I've always seen the second postulate stated in terms of light moving at c in every inertial frame--saying only that it's independent of the source's speed doesn't rule out the possibility that the speed of light rays could vary with time or spatial region, for example. Einstein did state the postulate in terms of light moving at c in section 2 of his original 1905 paper: 2. Any ray of light moves in the "stationary'' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. (technically it seems he is saying here only that all light moves at c in one inertial frame, the one he has labeled the "stationary" system, but if you combine that with the first postulate it of course implies that light must move at c in every inertial frame)

It is clear that it is totally obvious to derive SR from the first postulate alone.

If only the first postulate describes SR, what is preventing us from deriving SR as "observers in all inertial frames measure the speed of light as a multiple of the number of coins they have in their left pocket".

(I'm not joking, I'm new to SR and having trouble imagining SR without a second postulate, be it the constancy of c, Maxwell's equations or some other way to put it)

Actually, what lightarrow and JesseM stated hits the nail on the head right there, and I should have extended the second postulate accordingly. If we were to try to derive SR from the first postulate alone, then since ballistic theory also includes the same physics in all frames, we would still need some generalized form of the second postulate, something on the order of "the speed of some (massless) particle exists that will always be measured to travel isotropically and not ballistically with the source, so at the same speed whether the source is stationary or moving", but of course light has been determined to be such a massless particle that travels at this isotropic speed. That postulate makes all the difference for what is derived.

 

[JesseM]

It is not required with the first postulate. For instance, if an object is moved from a first frame to a second, the physics might say that the object is physically contracted to 1/2 the length in the process.

What does "physically contracted" mean? You can't really compare the lengths of objects moving at different velocities in a coordinate-independent way.

However, if space is homogeneous,

You haven't explained what you mean by that phrase. Is it a physical statement, or a coordinate-dependent one? As I said, with Newtonian physics we normally say that the laws of physics have translation and rotation symmetries (which I guess just means that it's possible to come up with a coordinate system such that moving the origin or rotating the axes doesn't change the equations of the laws of physics, not that this would be true in all coordinate systems), yet at the same time it's also true that if you use the Lorentz transformation to define your family of coordinate systems in a universe with Newtonian laws, there will be some frames where objects which have the same length when moving at equal and opposite velocities would not have the same length if brought to rest relative to one another. Do you disagree about that, or not fully understand it? If you do understand and agree, then would you say "space is homogeneous" is true of the Newtonian universe in general, or is it a statement that's meant to be specific to your choice of coordinate system so that you might say that in a Newtonian universe, space is homogeneous in some coordinate systems but not others?

then if one moves the object to a second frame and back again without the actual physics changing the length at all but only in reference to the coordinization of what is observed in space

Again, I don't see what it means to talk about "actual physics changing the length" as opposed to just a coordinate-dependent change in length, since there doesn't seem to be any coordinate-independent way to compare length for objects moving at different velocities (contrast with time dilation, where even though there is no coordinate-independent way to say which of two clocks is ticking slower at a given instant, if two clocks cross paths twice we can say which elapsed less total time between the crossings, since proper time is coordinate-independent).

 

[grav-universe]

What does "physically contracted" mean? You can't really compare the lengths of objects moving at different velocities in a coordinate-independent way.

By "physically contracted", I am assuming I mean the same thing that you are referring to for a Newtonian universe, where objects physically contract rather than being a coordinatization effect. If it were Newtonian, then lengths could physically change when brought back to the initial frame, but not if the contraction is a coordinate effect.

You haven't explained what you mean by that phrase. Is it a physical statement, or a coordinate-dependent one? As I said, with Newtonian physics we normally say that the laws of physics have translation and rotation symmetries (which I guess just means that it's possible to come up with a coordinate system such that moving the origin or rotating the axes doesn't change the equations of the laws of physics, not that this would be true in all coordinate systems), yet at the same time it's also true that if you use the Lorentz transformation to define your family of coordinate systems in a universe with Newtonian laws, there will be some frames where objects which have the same length when moving at equal and opposite velocities would not have the same length if brought to rest relative to one another. Do you disagree about that, or not fully understand it? If you do understand and agree, then would you say "space is homogeneous" is true of the Newtonian universe in general, or is it a statement that's meant to be specific to your choice of coordinate system so that you might say that in a Newtonian universe, space is homogeneous in some coordinate systems but not others?

Right, coordinate only. I suppose that has to be included with the homogeneity of space, that the observations are coordinate effects, not physical.

Again, I don't see what it means to talk about "actual physics changing the length" as opposed to just a coordinate-dependent change in length, since there doesn't seem to be any coordinate-independent way to compare length for objects moving at different velocities (contrast with time dilation, where even though there is no coordinate-independent way to say which of two clocks is ticking slower at a given instant, if two clocks cross paths twice we can say which elapsed less total time between the crossings, since proper time is coordinate-independent).

Right, two clocks cannot compare rates when passing each other, but a frame set between the two frames of the clocks can compare rates for the clocks traveling at the same relative speed in opposite directions when viewing the time dilation as a coordinate effect within the homogeneity of space.

 

[JesseM]

By "physically contracted", I am assuming I mean the same thing that you are referring to for a Newtonian universe, where objects physically contract rather than being a coordinatization effect.

What does "physically contract" mean? Consider that when we say rigid objects don't contract in a Newtonian universe, we just mean that in any of the inertial reference frames given by the Galilei transformation, the object's coordinate length remains constant regardless of change in velocity. So, for an object to contract in Newtonian physics would presumably just mean that all Galilean frames would agree the length contracted. It's still a "coordinatization effect" in the sense that all statements comparing the length of objects in relative motion depend on your choice of coordinate system, as far as I can tell.

If it were Newtonian, then lengths could physically change when brought back to the initial frame, but not if the contraction is a coordinate effect.

I was talking about rigid objects in a Newtonian universe, though. In the set of inertial frames given by the Galilei transformation, these objects never change length regardless of velocity. I take it you agree that even for such rigid objects in a Newtonian universe, if we instead use the frames given by the Lorentz transformation, then we can find cases where one frame measures two objects traveling at equal and opposite velocities to have equal lengths, yet when these rigid objects are brought to rest next to each other they are found to have unequal lengths?

Right, coordinate only. I suppose that has to be included with the homogeneity of space, that the observations are coordinate effects, not physical.

But if you're imposing the requirement that laws of physics exhibit rotational invariance in the particular coordinate systems you're using--as opposed to just saying there has to be some coordinate system where they exhibit rotational invariance--then this is basically a special case of the first postulate. Instead of requiring that the laws of physics are the same in all the coordinate systems allowed by the transformation, you're requiring that the laws of physics are the same in each subset of coordinate systems which can be related to one another by a spatial rotation transformation.

Again, I don't see what it means to talk about "actual physics changing the length" as opposed to just a coordinate-dependent change in length, since there doesn't seem to be any coordinate-independent way to compare length for objects moving at different velocities (contrast with time dilation, where even though there is no coordinate-independent way to say which of two clocks is ticking slower at a given instant, if two clocks cross paths twice we can say which elapsed less total time between the crossings, since proper time is coordinate-independent).

Right, two clocks cannot compare rates when passing each other, but a frame set between the two frames of the clocks can compare rates for the clocks traveling at the same relative speed in opposite directions when viewing the time dilation as a coordinate effect within the homogeneity of space.

Huh? Of course each clock can compare their own rate to the rate of the other clock, in terms of their own frame. Why shouldn't they be able to? The answer will be coordinate-dependent (and thus the two frames will have different answers about which clock is ticking at a slower rate), but then so is the answer found by the frame "set between the two frames of the clocks" (I don't understand the phrase 'when viewing the time dilation as a coordinate effect within the homogeneity of space'--instantaneous time dilation is always coordinate-dependent, and the phrase 'within the homogeneity of space' is unclear). My point was that unlike these comparisons of instantaneous rates of ticking which are always frame-dependent, if you compare total elapsed time on two clocks between two local meetings of these clocks then there will be an objective answer that does not depend on your choice of frame (since this is just the proper time between two events on a given clock's worldline, and proper time along a worldline is frame-invariant)

 

[grav-universe]

What does "physically contract" mean? Consider that when we say rigid objects don't contract in a Newtonian universe, we just mean that in any of the inertial reference frames given by the Galilei transformation, the object's coordinate length remains constant regardless of change in velocity. So, for an object to contract in Newtonian physics would presumably just mean that all Galilean frames would agree the length contracted. It's still a "coordinatization effect" in the sense that all statements comparing the length of objects in relative motion depend on your choice of coordinate system, as far as I can tell.

That was the definition of what I referring to with "physically contract", but maybe it would be easier to define what I mean by a coordinate effect. With a coordinate effect, if two objects are at rest with identical lengths and then one accelerated to some v, then turned around and came back to rest again, the lengths would still be identical.

I was talking about rigid objects in a Newtonian universe, though. In the set of inertial frames given by the Galilei transformation, these objects never change length regardless of velocity. I take it you agree that even for such rigid objects in a Newtonian universe, if we instead use the frames given by the Lorentz transformation, then we can find cases where one frame measures two objects traveling at equal and opposite velocities to have equal lengths, yet when these rigid objects are brought to rest next to each other they are found to have unequal lengths?

I don't see that with the coordinate effects of SR, no. Times, sure, since different time dilations will give different readings on clocks brought back together, but if an observer reads identical time dilations or identical lengths for two clocks or rulers with the same relative speed and observations of each, regardless of their directions, then if brought to rest in the same frame next to each other, they should continue to have an equal tick rate and the same lengths.

But if you're imposing the requirement that laws of physics exhibit rotational invariance in the particular coordinate systems you're using--as opposed to just saying there has to be some coordinate system where they exhibit rotational invariance--then this is basically a special case of the first postulate. Instead of requiring that the laws of physics are the same in all the coordinate systems allowed by the transformation, you're requiring that the laws of physics are the same in each subset of coordinate systems which can be related to one another by a spatial rotation transformation.

I agree with that to a point, but the observations in different inertial frames for the same relative speed to those frames can still potentially be different, and the physics can also be different overall. However, since applying homogeneous space as I did still worked out to having the same physics in each frame, then that must be the natural result.

Huh? Of course each clock can compare their own rate to the rate of the other clock, in terms of their own frame. Why shouldn't they be able to? The answer will be coordinate-dependent (and thus the two frames will have different answers about which clock is ticking at a slower rate), but then so is the answer found by the frame "set between the two frames of the clocks" (I don't understand the phrase 'when viewing the time dilation as a coordinate effect within the homogeneity of space'--instantaneous time dilation is always coordinate-dependent, and the phrase 'within the homogeneity of space' is unclear). My point was that unlike these comparisons of instantaneous rates of ticking which are always frame-dependent, if you compare total elapsed time on two clocks between two local meetings of these clocks then there will be an objective answer that does not depend on your choice of frame (since this is just the proper time between two events on a given clock's worldline, and proper time along a worldline is frame-invariant)

I mean that two clocks with a relative speed between them cannot directly compare tick rates any more than two frames can directly compare rulers in order to determine that they are identical. It would require a third frame set between these two speeds in homogeneous space that can directly compare them this way.

 

[JesseM]

That was the definition of what I referring to with "physically contract", but maybe it would be easier to define what I mean by a coordinate effect. With a coordinate effect, if two objects are at rest with identical lengths and then one accelerated to some v, then turned around and came back to rest again, the lengths would still be identical.

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?

I don't see that with the coordinate effects of SR, no.

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?

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.

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.

I mean that two clocks with a relative speed between them cannot directly compare tick rates any more than two frames can directly compare rulers in order to determine that they are identical.

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.

It would require a third frame set between these two speeds in homogeneous space that can directly compare them this way.

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.

 

[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.