SR derived solely from one postulate

[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).