A descriptive picture of radar simultaneity

[PAllen]

I'm sure someone else must have come up with this (the description in the third paragraph), but I haven't seen it. Briefly, for those not familiar with it, radar simultaneity generalizes Einstein's simultaneity convention directly to arbitrarility accelerating observer. Dalsepam has posted a paper describing it.

A key issue with accelerating observers is that there is no natural way to construct a global coordinate patch. Different approaches map different parts of spacetime, but none can be global. For Fermi-normal, your patch is bounded by areas where extended hypersurfaces intersect, but it is able to cover extensive regions that are not yet causally connected to the observer. Radar makes a different compromise (as I view it): its coordinate patch only includes the prior lightcone of an event on an observer's world line. However, the bad areas for Fermi-normal are readily mapped once they are in an observer's prior lightcone. Radar simultaneity also is inapplicable in cosmology (Observer's world line must extend to prior lightcone of distant event. This is not generally possible for cosmological distances).

Ok, now my descriptive picture. Imagine attached to any object you can see a clock and a mirror (you, the observer, have a clock too). What you see on the object's clock shows proper time progress for the object. The image of your clock in the objec's mirror tells you how to map the event you are now seeing to your own history - back from now halfway to the time you see in the reflection. You choose to follow this information from clocks and treat doppler and aberration effects (leading to rapid changes in perceived distances, etc.) as purely optical artifacts of your changes in direction motion (which of course you can feel).

 

[pervect]

How do you handle those rare cases where you might have two (or more) visual images / radar returns. Take the one with the least propagation delay?

 

[PAllen]

How do you handle those rare cases where you might have two (or more) visual images / radar returns. Take the one with the least propagation delay?

The paper posted by Dalespam didn't cover this (it only dealt with accelerated observers in SR). I would consider it equally reasonable to do as you suggest or to declare that this represents a limit on the extent of the coordinate patch. Multiply labeling points is the problem with the Fermi-normal 'bad regions'. So here you have multiple labeling, so declare it outside the bounds of the patch. Each scheme can cover different cases, but it seems obvious no scheme can be globally extended.

Trying to itemize cases where this anomaly could happen:

- gravitational lensing
- non simply connected geometry
- closed universe (for this, might be reasonable to pick one by rule)

any others?

 

[Dale]

Different approaches map different parts of spacetime, but none can be global.

I know that the D&G radar coordinates have the problem you mentioned and similarly the Fermi normal coordinates, but do you really think that means that there is no possible global coordinate system? I don't know of one, but I am not certain that they don't exist.

 

[PAllen]

I know that the D&G radar coordinates have the problem you mentioned and similarly the Fermi normal coordinates, but do you really think that means that there is no possible global coordinate system? I don't know of one, but I am not certain that they don't exist.

Maybe I should say 'natural' coordinate system. No I don't know for sure. Just never come across any proposal for useful coordinates for general accelerating observers that don't have some problem of this type. I just figured if no one ever mentions one, it is believed not to exist. Of course, it is known that there are GR solutions that admit no global coordinates at all, but that has nothing to do with accelerating observers, per se.

 

[Dale]

Yeah, I can see that, but then it could be somewhat tautological depending on what you allow as "natural" coordinates. In any case, I always like new "natural" coordinate systems for non-inertial observers.