Space-time transformations with different shape

[bernhard.rothenstein]

It's an unusual way to do things (mixing primed and unprimed coordinates) but, assuming the author makes no mistakes, there's nothing actually wrong with this and it could be of some practical use. A "moving" observer may be able to measure their distance on a "static" scale but only have access to their own "moving" clock, for example. (Of course, the words "static" and "moving" are both relative to a single inertial frame.) Dividing "static" distance by "moving" (i.e. proper) time you get something called "celerity" or "proper velocity" (see this post).



For the benefit of other readers I'll reproduce the abstract:

It was only a few years ago, soon after I started using this forum, that I realized that some of the standard relativistic effects, as described above, are coordinate-dependent and that other coordinate systems were permissible.

The point the authors are making is that there are lots of different coordinate systems to choose from apart from the standard "Einstein-synced" coordinates. Einstein coords are arguably the best coords but they are not the only coords, and it is educational to consider some of the other possibilities. The "everyday" coords that the authors define are one possibility.

The "radar coordinates" (e,r) which I defined in this post are another possibility. These are unusual because their two axes are not timelike and spacelike respectively, like most systems, but both null. (For mathematicians, radar coordinates are particularly interesting because they diagonalise the Lorentz transform; the two coord directions are eigenvectors and the k and k^-1 red- and blue-shift Doppler factors are the eigenvalues.)

Time and space coordinates (t(\epsilon), x) can be defined from (e,r) by

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

x = c(r - e)/2

(Sorry: you'll need to look closely at the above equations to see the difference between "epsilon" and "e".)

You have a choice of \epsilon:

• Standard Einstein-synced coordinates result when \epsilon = ½. (t(E) in your notation.)
  
• Leubner et al’s "everyday" coords result when \epsilon = 0. (t(r) in your notation.)
  
• Tangherlini coords (which I discussed in an earlier post in this thread) result when \epsilon = ½(1 + v/c), where v is the supposed velocity of the observer relative to a postulated aether.

I think it is useful to be aware of these different coord systems, to help understand which relativistic effects are "intrinsic" (coord independent) – e.g. the twin "paradox" -- and which are not. However I’m not sure whether it’s a good idea to present all this to someone learning relativity for the first time; it might just confuse them.

I have a particular fondness for radar coordinates because of how, with k-calculus, they can be used to derive many results with quite simple proofs, and some without even having to define "simultaneity". The concept of "relative simultaneity" seems to be what most people have most difficulty understanding when learning relativity.

It is not clear to me whether Leubner’s "everyday" coords help with the original navigation problem. Nor am I convinced that using "base vectors" (I would call them "unit basis vectors") is the easiest method. I would think you just need to write down all the relevant equations to convert from one coord system to another and then it’s just maths (algebra) to solve them.

I’m not sure I’ve answered your question. Does any of this help you?

Consider please that I have performed the synchronization of the clocks of I using a signal that propagates in the positive direction of the OX axis with speed c(f)=c/n where n>1 represents a synchrony parameter. Please let me know how does n transform i.e. what is the relationship between n and n'.

 

[DrGreg]

Consider please that I have performed the synchronization of the clocks of I using a signal that propagates in the positive direction of the OX axis with speed c(f)=c/n where n>1 represents a synchrony parameter. Please let me know how does n transform i.e. what is the relationship between n and n'.

It depends exactly what you mean by n and n'. I'm hoping that you don't mean the refractive index of some medium (as it did in some other of your threads), but instead it means n=2\epsilon in Reichenbach notation, or equivalently n=1-\lambda in the Edwards notation I used in post #25 and earlier.

And I'm assuming that n' then refers to the equivalent concept according to some other inertial observer.

If that is the case, then the two inertial observers are free to choose whatever values of n and n' they wish; they are, in general, two independent parameters reflecting the two observers' choices of sychronisation. In a particular context, e.g. Selleri or Leubner or Einstein, there may be additional information that links two observers' choices together. But without such linkage, the two values are arbitary (and leading to the Edwards transform).