Coordinate transformation and choice of a suitable coordinate system.

In summary, the conversation discusses the use of a line element in a particular coordinate system and the possibility of transforming it into a different coordinate system. The speaker expresses doubt about whether the results obtained will be the same in both systems and mentions the use of a tensor for transformation. They also question the order in which the transformation and metric derivation should be done.
  • #1
arroy_0205
129
0
Consider the line element:
[tex]
ds^2=-f(x)dt^2+g(x)dx^2
[/tex]
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
[tex]
g(x)dx^2=dy^2
[/tex]
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?
 
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  • #2
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

[tex]A_n^{n'} = \frac{\partial x^n'}{\partial x^n}[/tex]

The new metric is given by

[tex]g_{m'n'} = g_{nm}A^n_{n'}A^m_{m'}[/tex]

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
 
  • #3
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.
 

1. What is coordinate transformation?

Coordinate transformation is the process of changing the coordinates of a point or object from one coordinate system to another. This is often necessary when working with different coordinate systems in mathematics, physics, or engineering.

2. Why is coordinate transformation important?

Coordinate transformation allows us to describe the same point or object in different ways, making it easier to solve problems and analyze data. It also allows us to compare and combine information from different coordinate systems.

3. How do you choose a suitable coordinate system?

The choice of a suitable coordinate system depends on the problem or application at hand. It is important to consider factors such as the type of data being analyzed, the symmetry of the system, and the ease of mathematical operations in a particular coordinate system.

4. What are some commonly used coordinate systems?

Some commonly used coordinate systems include Cartesian coordinates, polar coordinates, cylindrical coordinates, and spherical coordinates. Each of these systems has its own advantages and is suitable for different types of problems.

5. Can multiple coordinate systems be used at the same time?

Yes, it is possible to use multiple coordinate systems simultaneously. This is often done in more complex problems where different coordinate systems are used to describe different aspects of the problem. However, it is important to carefully keep track of which coordinate system is being used for each part of the problem.

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