What are valid coordinate transforms (diffeomorphisms)?

In summary, the conversation discusses the use of different coordinate systems in general relativity and the lack of inherent meaning in these coordinates. Examples are given of coordinate systems with two timelike or two null basis vectors, which do not violate the standard definition of a coordinate system. The importance of looking at the metric to understand the properties of a coordinate system is also emphasized.
  • #1
PAllen
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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.
 
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  • #2
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*).
 
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  • #3
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.
 
  • #4
bcrowell said:
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.
 

What are valid coordinate transforms (diffeomorphisms)?

Valid coordinate transforms, also known as diffeomorphisms, are mathematical functions that map points from one coordinate system to another in a smooth and invertible manner. They are used to transform geometrical objects, such as curves and surfaces, from one coordinate system to another while preserving their properties.

How are diffeomorphisms different from other coordinate transforms?

Diffeomorphisms are different from other coordinate transforms, such as rigid transformations or scaling, because they allow for more complex deformations and can map points from one coordinate system to another in a non-linear manner. This makes them more versatile and useful for a wider range of applications.

What are the benefits of using diffeomorphisms?

Diffeomorphisms have many benefits, including their ability to accurately map points from one coordinate system to another without distorting the underlying geometry. They also allow for more flexibility in transforming objects and can be used to solve complex mathematical problems.

How are diffeomorphisms used in scientific research?

Diffeomorphisms are used extensively in scientific research, particularly in fields such as physics, engineering, and computer science. They are used to model and analyze complex systems, simulate physical phenomena, and solve differential equations. They are also used in image processing and computer graphics to transform and manipulate digital images.

Are there any limitations or drawbacks to using diffeomorphisms?

While diffeomorphisms have many benefits, they also have some limitations. They can be computationally expensive to compute and may not always provide an accurate representation of the underlying geometry. Additionally, they may not be applicable in certain situations where the underlying data is too complex or noisy.

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