Bondi k and everyday clock synchronization

[bernhard.rothenstein]

Dr.Greg has presented on the Forum the Lorentz-Einstein transformations (LET) expressed as a function of k=[(1+v/c)/(1-v/c)]^1/2. The important fact is that the LET are clock synchronization independent.
Because special relativity is a very flexible chapter of physics, I have tried to express the LET as a function of the time at which the radar signal is emitted. Let e' be that time and t' the time at which it arrives at the location of the detected event, the detection taking place in the I' inertial reference frame. Let x' be the space coordinate of that event. We have
x'=c(t'-e') (1)
t'=(x'/c)+e' (2)
Equations (1) and (2) enable us to express the space-time coordinates of the detected event in I as a function of e'
x=kx'+$$\gamma$$Ve' (3)
t=$$\gamma$$[t'+(v/c)(t'-e')=kt'-$$\gamma$$(V/c)e' (4)
In the case of the "everyday clock synchronization" which is equivalent with the synchronization performed with a signal that propagates with infinite speed
t'=e'
(3) remains unchanged.
(4) becomes
t=$$\gammat'$$
Do you find some flow in the derivations above.
What is the way to Selleri?

 

[DrGreg]

Bernhard, if you want anyone other than me to respond to your questions, you really should help other readers to understand what you are talking about. It would help to give links to other threads where the topic(s) has been discussed before. Even I can't remember the details of everything I've said in the past.

I believe you are referring to this post.

It would also make your posts more readable if you could use LaTeX. If you use the $\Sigma$ button in the "Go Advanced" edit window, it's not much harder that using Microsoft's Equation Editor, with which I know you are competent. You can also get help by clicking on anybody else's equation and following the link to "LaTeX code reference".

In this case I don't see where (3) has come from, you need to explain what other result you are quoting to obtain it.


If you follow my original proof in the post I linked to above, the equation

$$t = (r + e)/2$$​

(just before equation (4)) is the point where I assume Einstein synchronisation. That line can be replaced by

$$t = \epsilon r + (1 - \epsilon) e$$​

following Reichenbach. Use the appropriate values of $\epsilon$ to get either Selleri or Leubner synchronisation.
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P.S. you meant "flaw", not "flow".
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P.P.S. I am guilty of not practising wbat I preach. I haven't explained what "Selleri", "Leubner" or "Reichenbach" refer to. Interested readers should Google these words along with "DrGreg" and "site:physicsforums.com".

 

[bernhard.rothenstein]

Thanks
My last question here going to all the participants on the forum is:
Conider that I have expressed in the standard Lorentz-Einstein transformations the time displayed by the standard synchronized clocks as a function of the time displayed by a clock which is not standard synchronized, the readings of the two clocks being related by a correct relationship e.g. as proposed by Reichenbach.
Obtaining the nonstandar transformation equations should I check if causality is not violated, if the principle is respected, if transitivity takes place...?

 

[DrGreg]

Thanks
My last question here going to all the participants on the forum is:
Conider that I have expressed in the standard Lorentz-Einstein transformations the time displayed by the standard synchronized clocks as a function of the time displayed by a clock which is not standard synchronized, the readings of the two clocks being related by a correct relationship e.g. as proposed by Reichenbach.
Obtaining the nonstandar transformation equations should I check if causality is not violated, if the principle is respected, if transitivity takes place...?

If you had a coordinate system that said that a photon was received before it was emitted, I think that would not be a very useful coordinate system. (Although such coordinate systems are possible, e.g. Rindler coordinates extended beyond the singularity at the "centre of hyperbolic curvature", see this thread[/color] for example.)

To be honest I'm not entirely sure what, precisely, various authors mean by "transitive" in this subject. Mathematically a relationship is transitive if whenever {A is related to B} and {B is related to C} then {A is related to C}. But I'm not sure what "is related to" means in this context. If "is related to" means "has the same time coordinate as", that must always be true by the laws of mathematics. I'm guessing what they really mean is "B is simultaneous with A according to A's definition of simultaneity". (And I assume A, B and C are all stationary in the frame of reference.) That definition is a condition that could be false; for example the omnidirectional Leubner definition of "everyday simultaneity", where the sync-signal always goes from observer to object, fails to be transitive in that sense. (But the unidirectional 1D version where the sync-signal always travels from left to right, regardless of whether the object is to the left or right of the observer, is transitive in that sense.)

That's my guess at what "transitive" means here, but I could be wrong.

It's not immediately obvious to me whether transitivity is related to causality.

____________________

(Bernhard, was this the thread you meant when you referred to an "unanswered thread" in a private email to me?)

 

[bernhard.rothenstein]

If you had a coordinate system that said that a photon was received before it was emitted, I think that would not be a very useful coordinate system. (Although such coordinate systems are possible, e.g. Rindler coordinates extended beyond the singularity at the "centre of hyperbolic curvature", see this thread[/color] for example.)

To be honest I'm not entirely sure what, precisely, various authors mean by "transitive" in this subject. Mathematically a relationship is transitive if whenever {A is related to B} and {B is related to C} then {A is related to C}. But I'm not sure what "is related to" means in this context. If "is related to" means "has the same time coordinate as", that must always be true by the laws of mathematics. I'm guessing what they really mean is "B is simultaneous with A according to A's definition of simultaneity". (And I assume A, B and C are all stationary in the frame of reference.) That definition is a condition that could be false; for example the omnidirectional Leubner definition of "everyday simultaneity", where the sync-signal always goes from observer to object, fails to be transitive in that sense. (But the unidirectional 1D version where the sync-signal always travels from left to right, regardless of whether the object is to the left or right of the observer, is transitive in that sense.)

That's my guess at what "transitive" means here, but I could be wrong.

It's not immediately obvious to me whether transitivity is related to causality.

____________________

(Bernhard, was this the thread you meant when you referred to an "unanswered thread" in a private email to me?)

Thanks. I will simplify my question. Consider the Lorentz-Einstein transformation. It works with times displayed by standard synchronized clocks t_E and t'_E
satisfying a certain number of physical conditions. I establish a relationship between the readings of two clocks of the same inertial reference frame, one standard synchronized the other nonstandard synchronized related as
t_E=t_n+(x/c)(1-n)
where n is a synchronization parameter. I express the Lorentz transformation as a function of t_n obtaining a nonstandard Lorentz transformation.
The question is: Does the nonstandard transformation satisfy the same conditions as the standard transformation does?

 

[DrGreg]

The question is: Does the nonstandard transformation satisfy the same conditions as the standard transformation does?

I'm sorry, which "conditions of the standard transformation" do you mean? That is too vague a phrase for me to interpret.