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The thread bcrowell had on time reversal in GR got me thinking about this. Some limitations are obvious: mapping two events onto one, discontinuity,...
I will use x*, t*, etc. to refere to tansformed coordinates (primes always confuse me with derivatives).
Similarly, the transform x* = t, t* =x, is really just relabeling the time axis with the letter x instead of t; it isn't really doing anything physical. You would have to identify x* as the coordinate with all the properties of time after this transform. Physics doesn't care what letter I use.
But what about something like the following, based on a starting coordinate system that is maximally inertial minkowski in some local region (I assume c=1):
x*=t-x/2
t*=t+x/2
I now have two 'equally timelike' coordinates. Two of the basis vectors would be inside light cones instead of one. I can't see anything that prohibits this, yet I don't quite understand how to understand physics in such a coordinate system (for example, it doesn't identify any spacelike hypersurface).
Any insights welcomed.
I will use x*, t*, etc. to refere to tansformed coordinates (primes always confuse me with derivatives).
Similarly, the transform x* = t, t* =x, is really just relabeling the time axis with the letter x instead of t; it isn't really doing anything physical. You would have to identify x* as the coordinate with all the properties of time after this transform. Physics doesn't care what letter I use.
But what about something like the following, based on a starting coordinate system that is maximally inertial minkowski in some local region (I assume c=1):
x*=t-x/2
t*=t+x/2
I now have two 'equally timelike' coordinates. Two of the basis vectors would be inside light cones instead of one. I can't see anything that prohibits this, yet I don't quite understand how to understand physics in such a coordinate system (for example, it doesn't identify any spacelike hypersurface).
Any insights welcomed.
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