Coordinate Transformations in GR

[Mad Dog]

As I try to understand GR, I find coordinate transformations just about everywhere. My question is simply: What is the reason coordinate transformations play such an important role in GR? Thanks.

 

[GDogg]

I'd say the reason is that one of the most important aspects of GR is the principle of relativity and the general covariance of its equations. Basically, that the laws of physics should be the same in any reference frame so naturally you'll want to perform coordinate trasnformations.

 

[Crazy Tosser]

That and also coordinate transformation is the simplest (read: best) way to visualize reference frames.

 

[atyy]

In special relativity, you can can use 3 spatially orthogonal plane wave solutions of Maxwell's "free space" equations as rulers for 3 orthogonal spatial axes. The 3 rulers won't interact with each other and Maxwell's equations are linear. In general relativity, each plane wave has energy or "gravitational mass" and should attract the other plane waves, so Maxwell's "free space" equations should become nonlinear indicating that you cannot get independent plane wave solutions. This will be true of any rulers you set up, so you will have no orthogonal coordinates, except very locally. The bending of your rulers and clocks or "metric" appears as spacetime curvature.

 

[cesiumfrog]

Another reason is that GR permits solutions in which it is mathematically impossible for a single choice of coordinates to fully suffice at every event.

 

[Mad Dog]

I'm replying here to my own question, as a contribution to the Forum:

Even after having received the answers to my question, I continued to be concerned, and finally realized the answer is simple and obvious:

The name of the field is "Relativity" and the key is right there - we're talking about how different observers will measure the same events. Different observer's points of view are expressed as observations from different reference frames, and "reference frames" is simply another way of saying "coordinate systems," so of course coordinate transformations are found everywhere in General Relativity.

The second reason is that:
-For generations whe have dealt with coordinate translations, rotations, and moving coordinate systems without thinking much about it.
-In Special Relativity we add the Lorentz Transformation (and have to think a good bit about that).
-One of the basic lessons of General Relativity is that there are no preferred coordinate systems, so now we must deal with arbitrary coordinate systems and their transformations, and learning to do this is an essential part of GR.