- #1

Freixas

- 307

- 42

We place an observer at location A. This observer releases a photon at time ##t_1 = 0## in the direction of a mirror at B, located at distance ##d## from A. The mirror is at rest with respect to the observer and aligned to reflect the photon back to A. The photon returns to A at time ##t_2##. The two-way speed of light is then ##2d / t_2##. This is an invariant and is represented by ##c##.

The mirror at B has a clock that is set when the photon reaches the mirror. How might we choose the clock’s setting, ##t##?

- We could allow
*any*function ##t = f(d)##. - We could allow only functions where the time values are monotonically nondecreasing (for every event that forms the worldline of light, the time value of the event must be greater than or equal to that of any event occurring earlier).
- We could further limit the functions to ones where the velocity in one direction is a constant. I believe this limits the possible formulas to ##t = 2{\epsilon}d/c##, where ##\epsilon## ranges from 0 to 1.
- Finally, we could limit the function to only ##t = d/c##, which is Einsteinian synchronization, where the one-way speed in every direction is the same as the two-way speed (##\epsilon## = ½).

- If we allow using any function, some functions may make some calculations difficult or impossible. If a choice is dysfunctional, that is a good reason not to choose it, but not clearly a reason for forbidding it. Is that correct?
- This limitation would still allow functions such as ##t = floor(d)/c## (I believe this satisfies the causal requirement).
- I believe math currently exists to handle physics in which the possible simultaneity conventions match the formula ##t = 2{\epsilon}d/c##, for any valid value of ##\epsilon##.
- David Malament proposed a theorem that he used to argue that the only valid simultaneity convention was the one Einstein chose. Others think that Malament’s Theorem is invalid. I don’t believe he has anything testable.

Simultaneity is an odd concept. Simultaneity is clear when we are talking about events that are co-located, but not when events are separated by space. If simultaneity is somewhat (or totally) arbitrary, then things that are commonly discussed, such as the length or velocity of a moving object, become equally arbitrary.

I don’t have any preferred answer—I am just trying to understand the meaning and any issues relating to the term “simultaneity convention”. I did enough research to realize there is a philosophical war around the conventionality of simultaneity (summarized here: https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/significance_conv_sim/index.html). Physics is not philosophy, though—I’d like the viewpoint of physicists.

As a reminder, I am discussing this strictly with respect to S.R. I know little about G.R., but I’ve heard it messes with the concept of simultaneity even more.