Coordinate transformation and choice of a suitable coordinate system.

[arroy_0205]

Consider the line element:
$$ds^2=-f(x)dt^2+g(x)dx^2$$
in a coordinate system (t,x) where f(x) and g(x) are two functions to be determined by solving Einstein equation. But I can always make a transformation
$$g(x)dx^2=dy^2$$
and then calculate everything in the (t,y) coordinate system. My doubt is whether the results obtained will be physically same in the two coordinate systems. In my opinion the results will be same, and the second approach is easier than the first one. But I notice in some of the papers the authors use the first approach. I do not understand why they do so. However they work in higher dimensions and I have formulated my problem in 1+1 dimension for simplicity. Can anybody explain if there is anything wrong in the second approach which I prefer?

 

[Mentz114]

When you make a co-ordinate transformation, the metric will also be transformed. The transformation of the metric is through a tensor A defined by

$$A_n^{n'} = \frac{\partial x^n'}{\partial x^n}$$

The new metric is given by

$$g_{m'n'} = g_{nm}A^n_{n'}A^m_{m'}$$

My point is that the new space may not be the same as the old space. I think you should do this calculation, I don't have the energy right now.

M

 

[JesseM]

I'm not that knowledgeable about GR, but my understanding is that normally you start with the coordinate transformation, and from that you derive the new metric tensor in this coordinate system, and from that the new line element. Here you seem to be going in reverse, first picking the line element you want in the new coordinate system--but would it be trivial or difficult to derive the appropriate coordinate transformation from that? The new line element is of no use unless you know how the coordinates of the new system are related to the coordinates of the original system which you were previously using to describe physical events and worldlines.