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SR derived solely from one postulate

by grav-universe
Tags: derived, postulate, solely
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JesseM
#73
Feb16-10, 04:43 PM
Sci Advisor
P: 8,470
Quote Quote by grav-universe View Post
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'.
Quote Quote by grav-universe
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.
Quote Quote by grav-universe
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".
Quote Quote by grav-universe
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.
Quote Quote by grav-universe
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 occuring 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
#74
Feb16-10, 05:12 PM
P: 424
Quote Quote by JesseM View Post
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
#75
Feb16-10, 05:37 PM
Sci Advisor
P: 8,470
Quote Quote by grav-universe View Post
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.
Quote Quote by grav-universe
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).


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