SR derived solely from one postulate

[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).