Differences between different simultaneity conventions

[analyst5]

Hi guys, I'm reading a text regarding different possible ways of synchronizing clocks in an inertial frame, so maybe somebody could help me with some details. The standard ways, the Einstein synchronisation, is characterized by the synchronisation parameter which is 1/2.

Since I've red that basically we can put any number between 0 and 1 instead of 1/2 as a paremeter, can somebody explain to me what kinds of differences arise in simultaneity while using different synchronisation methods, compared to the Einstein method, maybe with a space time diagram so I can figure out what is really meant by the oftenly used sentence 'we may synchronise the clocks the way we like'.

Thanks in advance.

 

[WannabeNewton]

Since I've red that basically we can put any number between 0 and 1 instead of 1/2 as a paremeter, can somebody explain to me what kinds of differences arise in simultaneity while using different synchronisation methods, compared to the Einstein method, maybe with a space time diagram so I can figure out what is really meant by the oftenly used sentence 'we may synchronise the clocks the way we like'.

The differences lie in the geometry and topology of the simultaneity hypersurfaces defined by the synchronization or simultaneity convention. This results, in particular, in the Lorentz transformations having different forms, and in general much more complicated when compared to $\epsilon = \frac{1}{2}$ synchrony, relative to different synchronization conventions.

See here for more details and see the references therein: http://www.mcps.umn.edu/assets/pdf/8.13_friedman.pdf

I would also recommend checking out: https://www.amazon.com/dp/0691020396/?tag=pfamazon01-20
as it goes into much more explicit, calculational detail on non-standard simultaneity conventions for inertial frames.

 

[bcrowell]

See here for more details and see the references therein: http://www.mcps.umn.edu/assets/pdf/8.13_friedman.pdf

Friedman makes a couple of points:

- Only the $\epsilon=1/2$ definition of simultaneity is consistent with defining simultaneity according to the slow transport of clocks.

- Picking $\epsilon\ne 1/2$ is equivalent to defining simultaneity in a noninertial frame.

 

[analyst5]

@ WBN, thanks for the link, I quickly looked it up and the description of what happens seems a little complicated, but I guess I'll invest more time. I found this on the net: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/significance_3/index.html#Conventionality.

In one of the chapters, there is a good example what happens when the parameter is 1/4 and not 1/2, but unfortunately I don't understand what do surfaces of simultaneity look like from the perspective of the other clock (the right one) and how would the surfaces differ if we used a smaller, or bigger number than 1/2 as our parameter. I hope somebody could give me an explanation.

 

[WannabeNewton]

In one of the chapters, there is a good example what happens when the parameter is 1/4 and not 1/2, but unfortunately I don't understand what do surfaces of simultaneity look like from the perspective of the other clock (the right one)...

The same exact thing that happens when regarding $\epsilon = \frac{1}{2}$ synchrony will happen for the $\epsilon = \frac{1}{4}$ synchrony: the simultaneity surface at an angle $\theta$ to the 4-velocity of one inertial time-like curve at some event will be rotated through some angle $\Delta \theta$ to the 4-velocity of a different, time-like curve passing through the same event. The only difference in the case of $\epsilon = \frac{1}{4}$ is $\theta \neq \frac{\pi}{2}$ but is rather some oblique angle. $\Delta \theta$ is determined by the Lorentz transformations as usual, with the latter being expressed in terms of the chosen synchronization convention. See the Friedman paper for more details on how to calculate this angle explicitly.

and how would the surfaces differ if we used a smaller, or bigger number than 1/2 as our parameter. I hope somebody could give me an explanation.

If $\epsilon \in (0,1)$ and $\epsilon = \text{const}$ then different values of $\epsilon$ simply yield space-like hyperplanes of different angles $\theta$ to the 4-velocity of a given inertial time-like curve.

If we allow $\epsilon = \epsilon(x)$ then the simultaneity surfaces will look a lot more interesting i.e. they can be arbitrarily curved embeddings into Minkowski space-time, so long as $\epsilon(x)\in (0,1)$. Friedman gives an example of such a simultaneity surface.

 

[analyst5]

Does this angle dramatically differ between different simultaneity parameters from 0 to 1, and what about trying to synchronize other way round, I mean from the perspective of the other clock?

 

[WannabeNewton]

Does this angle dramatically differ between different simultaneity parameters from 0 to 1...

No it varies smoothly. Again see the Friedman article.

...and what about trying to synchronize other way round, I mean from the perspective of the other clock?

Huh? You can't synchronize ideal inertial clocks that are in relative motion. Whichever $\epsilon$-synchronization convention you choose relative to a given inertial world-line, the clock synchronization is between all spatially separated clocks at rest with respect to the clock described by said world-line.

 

[Dale]

Does this angle dramatically differ between different simultaneity parameters from 0 to 1

The angle is probably not the thing to think about. What you probably should focus on is the speed of light. With the value of 1/2 the speed of light is isotropic, c in both directions. At the extremes the speed of light is infinite in one direction and c/2 in the other direction. The difference between 0 and 1 is just which direction is infinite.