A question on proper time in special relativity

In summary, proper time between two events as observed in an unprimed frame is calculated along the timelike worldline between the two events. This can be achieved by setting the primed coordinates to zero, but this does not mean that only the time coordinates remain. On the spacetime diagram, the two events will not necessarily lie on the primed time axis, but there exists a frame in which they are in the same place. This is analogous to rotating a line to be parallel to an axis. The formula for calculating proper time is the integral of 1/γ in any frame, and in an unprimed spacetime diagram, the object can have any timelike motion.
  • #36
Ibix said:
A frame is an infinite family of observers, one at every place in space, all following parallel worldlines, all carrying a three number spatial coordinate, and all agreeing a mechanism for when they should zero their watches. Then the proper time since zero of one of this family is this frame's coordinate time at their location.
In this definition of frame, the agreed 'mechanism' you were talking about for zeroing their watches, is actually the Einstein synchronization procedure in the rest frame of that infinite family of observers ?

Are other mechanisms possibile in principle to do that ?
 
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  • #37
cianfa72 said:
In this definition of frame, the agreed 'mechanism' you were talking about for zeroing their watches, is actually the Einstein synchronization procedure in the rest frame of that infinite family of observers ?

Are other mechanisms possibile in principle to do that ?
Send machine gun bullets between the clocks in the grid at a velocity that is verifiably consistent in both the "forward" and "reverse" directions. (Measure momentum at launch and arrival, for instance).

Measure round trip time. Synchronize clocks by zeroing the shooter's clock when a shot is made and setting the target's clock to RTT/2 upon arrival.Or, slow clock transport. Synchronize one grid clock with the wrist watch of a traveler. Send the traveler to the next grid clock. Synchronize the next grid clock with the traveler's watch. In the limit of slower and slower traveler's, you get perfect synchronization.

For speedier convergence, send the traveler and his watch at an arbitrarily great fixed speed and synchronize in both directions, using a round trip. Measure the amount by which the original clock was adjusted after the return and only adjust it by half that much.
 
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  • #38
jbriggs444 said:
Send machine gun bullets between the clocks in the grid at a velocity that is verifiably consistent in both the "forward" and "reverse" directions. (Measure momentum at launch and arrival, for instance).

Measure round trip time. Synchronize clocks by zeroing the shooter's clock when a shot is made and setting the target's clock to RTT/2 upon arrival.
ok, assume the mechanism of machine gun bullets to synchronize clocks in the grid the way you said. It defines the coordinate time ##t## for (I would say one of) the frame (aka coordinate chart) in which clocks in the grid are at rest.

Now I think there is actually no logical reason why the one-way speed of light should result c in that frame. In other words there is no logical reason why the metric in that frame should be (two spatial dimension and speed of light normalized to 1) $$ds^2=dx^2 + dy^2 - dt^2$$
 
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  • #39
cianfa72 said:
ok, assume the mechanics of machine gun bullets to synchronize clocks in the grid the way you said. It defines the coordinate time ##t## for (I would say one of) the frame (aka coordinate chart) in which clocks in the grid are at rest.

Now I think there is actually no logical reason why the one-way speed of light should result c in that frame. In other words there is no logical reason why the metric in that frame should be (two spatial dimension and speed of light normalized to 1) $$ds^2=dx^2 + dy^2 - dt^2$$
The speed of light could be less than c. Yes. Light pulses could behave like bullets with source-relative speeds rather than always traveling at the invariant speed.

One would have to re-derive special relativity without reference to light. But that has been done anyway.
 
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  • #40
jbriggs444 said:
One would have to re-derive special relativity without reference to light. But that has been done anyway.
ok, but in that case (light did not travel at invariant speed) which procedure we would use to physically synchronize clocks in the grid in order to define the rest frame coordinate time?
 
  • #41
cianfa72 said:
ok, but in that case (light did not travel at invariant speed) which procedure we would use to physically synchronize clocks in the grid in order to define the rest frame coordinate time?
Slow clock transport would be the easiest.
 
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  • #42
Dale said:
Slow clock transport would be the easiest.
So using it we would be able to define the coordinate time ##t## for a frame (coordinate chart) in which clocks are at rest and the metric is in the standard Minkowski form, though. All that even if light nor other physical process had the property to travel at the invariant speed.
 
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  • #43
cianfa72 said:
So using it we would be able to define the coordinate time ##t## for a frame (coordinate chart) in which clocks are at rest and the metric is in the standard Minkowski form, though. All that even if light nor other physical process had the property to travel at the invariant speed.
Yes. It isn’t the only way, but the easiest
 
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  • #44
cianfa72 said:
ok, assume the mechanism of machine gun bullets to synchronize clocks in the grid the way you said. It defines the coordinate time ##t## for (I would say one of) the frame (aka coordinate chart) in which clocks in the grid are at rest.

Now I think there is actually no logical reason why the one-way speed of light should result c in that frame. In other words there is no logical reason why the metric in that frame should be (two spatial dimension and speed of light normalized to 1) $$ds^2=dx^2 + dy^2 - dt^2$$
There is a logical reason, why the bullet clock synchronization leads to the same time-coordinate definition as the Einstein synchronization with light:

Rindler said:
The basic principle of clock synchronization is to ensure that the coordinate description of physics is as symmetric as the physics itself. For example, bullets shot off by the same gun at any point and in any direction should always have the same coordinate velocity dr/dt .
Source:
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations
 
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  • #45
Sagittarius A-Star said:
There is a logical reason, why the bullet clock synchronization leads to the same time-coordinate definition as the Einstein synchronization with light.
With light-based Einstein synchronization procedure applied to a grid of clocks mutually at rest, by very definition the one-way speed of light is c (provided that its two-way speed is actually c). For the same reason if the speed of bullets on a closed path was costant (say b) then the one-way speed of bullets would be b.

However I do not understand your logic about why using bullets clock synchronization the one-way speed of light should be c in that frame.
 
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  • #46
cianfa72 said:
With light-based Einstein synchronization procedure applied to a grid of clocks mutually at rest, by very definition the one-way speed of light is c (provided that its two-way speed is actually c). For the same reason if the speed of bullets on a closed path was costant (say b) then the one-way speed of bullets would be b.

However I do not understand your logic why using bullets clock synchronization the one-way speed of light should be c in that frame.
The logical reason is assuming isotropy in your synchronization. This is the only free choice - isotopropy or a very special type of one way anisotropy that preserves two way isotropy of lightspeed. Once you adopt isotropy for any synchronization method, then constant one way speed of light is forced. For example, if you simply assume that if two identical colocated clocks are moved apart with identical acceleration profile, then they are syncchronized, that alone is sufficient for one way lightspeed isotropy to follow.
 
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  • #47
PAllen said:
The logical reason is assuming isotropy in you synchronization. This is the only free choice - isotopropy or a very special type of one way anisotropy that preserves two way isotropy of lightspeed. Once you adopt isotropy for any synchronization method, then constant one way speed of light is forced.
Do you mean isotropy of the synchronization method chosen or isotropy of lightspeed ?
PAllen said:
if you simply assume that if two identical colocated clocks are moved apart with identical acceleration profile, then they are synchronized, that alone is sufficient for one way lightspeed isotropy to follow.
Is it just an assumption or it can be checked in some way ?
 
  • #48
cianfa72 said:
Do you mean isotropy of the synchronization method chosen or isotropy of lightspeed ?Is just an assumption or it can be checked in some way ?
1) of synchorinization method

2) Yes, it can be checked. It is a perfectly feasible experiment (in principle).
 
  • #49
cianfa72 said:
With light-based Einstein synchronization procedure applied to a grid of clocks mutually at rest, by very definition the one-way speed of light is c (provided that its two-way speed is actually c). For the same reason if the speed of bullets on a closed path was costant (say b) then the one-way speed of bullets would be b.

However I do not understand your logic why using bullets clock synchronization the one-way speed of light should be c in that frame.
With Einstein-synchronized clocks we have Reichenbach's ε = 1/2. Then the one-way-bullet-speed is measured as isotropic. The same is valid backwards.
 
  • #50
PAllen said:
1) of synchronization method
But...if the synchronization method (e.g bullets synchronization method) was isotropic but light propagation process was not, it did not follow the constancy of one-way speed of light, I believe.
 
  • #51
cianfa72 said:
But...if the synchronization method (e.g bullets synchronization method) is isotropic and light propagation process was non isotropic, it did not follow the constancy of one-way speed of light, I believe.
Wrong. There is no logically consistent model in which identically launched bullets can be assumed to have isotropic one way speed while light does not. The epsilon parameter describing the only allowed form of anisotropy applies to all physics. That is, any clock synchronization technique in SR amounts to a choice of epsilon. Any that assumes full isotropy for any synchronization method is choosing epsilon of 1/2.
 
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  • #52
Sagittarius A-Star said:
Then the one-way-bullet-speed is measured as isotropic.
...only if the process of sending bullets is actually isotropic.
 
  • #53
cianfa72 said:
...only if the process of sending bullets is actually isotropic.
is assumed to be isotropic for a fully specified launch method.
 
  • #54
PAllen said:
Wrong. There is no logically consistent model in which identically launched bullets can be assumed to have isotropic one way speed while light does not. The epsilon parameter describing the only allowed form of anisotropy applies to all physics. That is, any clock synchronization technique in SR amounts to a choice of epsilon. Any that assumes full isotropy for any synchronization method is choosing epsilon of 1/2.
So in this case at hand the point is that if we assume full isotropy (epsilon of 1/2) it applies to identically launched bullets as well to one-way light speed
 
  • #55
cianfa72 said:
So in this case the point is that if we assume full isotropy (epsilon of 1/2) it applies to identically launched bullets as well to one-way light speed
Yes.
 
  • #56
cianfa72 said:
But...if the synchronization method (e.g bullets synchronization method) was isotropic but light propagation process was not, it did not follow the constancy of one-way speed of light, I believe.
It does follow, actually. That is part of what Reichenbach showed. If the identical bullets are isotropic then the identical lasers are isotropic.
 
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  • #57
Dale said:
It does follow, actually. That is part of what Reichenbach showed. If the identical bullets are isotropic then the identical lasers are isotropic.
So choosing epsilon=1/2 two spatially separated events are said simultaneus if and only if they are simultaneous accoring any specific "isotropic" synchronization method chosen.
 
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  • #58
cianfa72 said:
So choosing epsilon=1/2 two events are said simultaneus if and only if they are simultaneous accoring any specific "isotropic" synchronization method chosen.
Yes, that is correct
 
  • #59
Dale said:
Yes, that is correct
Furthermore it follows that whatever "isotropic" method used -- assuming the constancy of two-way speed of light on closed paths equals to c -- then the one-way lightspeed has to be c (or just 1 when normalized).
 
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