What are valid coordinate transforms (diffeomorphisms)?

[PAllen]

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.

 

[Dale]

Well, the standard definition is that it must be a one-to-one map between points in the manifold and points in R(n), and that it must be smooth and its inverse must be smooth. So I think that your coordinate system above satisfies all of that. There is not a requirement that the coordinate basis vectors be orthonormal, so I think that the absence of that requirement implies that you are not restricted to coordinate systems with 3 spacelike and 1 timelike coordinate basis vectors.

To get a physical feeling it sometimes helps to write the metric. In this case you get two spacelike terms (y² and z²) two timelike terms (t*² and x*²) and one off-diagonal term which can be either timelike or spacelike (t* x*).

 

[bcrowell]

I agree with DaleSpam that there is nothing wrong with the coordinates defined in #1. A similar example in Minkowski space is u=x+t, v=x-t, which I think is fairly common, and gives you two null basis vectors. The basic idea is that coordinates in GR don't have built-in meanings.

 

[Dale]

A similar example in Minkowski space is u=x+t, v=x-t, which I think is fairly common, and gives you two null basis vectors. The basic idea is that coordinates in GR don't have built-in meanings.

Yes. And you can see that they are null vectors by looking at the metric again. In this case the metric is -uv-y²-z². So u and v are indeed null vectors which do not contribute to the spacetime interval by themselves at all, and again an off-diagonal term uv appears which can be either timelike or spacelike.