Determining Geometry of Line Element: A General Method?

[steve1763]

TL;DR

Given an arbitrary line element, how does one calculate the geometry of the space that it describes?

Is there a general method to determine what geometry some line element is describing? I realize that you can tell whether a space is flat or not (by diagonalising the matrix, rescaling etc), but given some arbitrary line element, how does one determine the shape of the space?

Thanks

 

[Dale]

I am not sure what you mean. The line element is what describes the shape. What information are you looking for that is not already captured in the metric?

 

[PeroK]

Summary:: Given an arbitrary line element, how does one calculate the geometry of the space that it describes?

Is there a general method to determine what geometry some line element is describing? I realize that you can tell whether a space is flat or not (by diagonalising the matrix, rescaling etc), but given some arbitrary line element, how does one determine the shape of the space?

Thanks

You can also compute the Riemann curvature tensor.

 

[Ibix]

Is there a general method to determine what geometry some line element is describing?

If you mean that, given the Schwarzschild line element and the Eddington-Finkelstein line element, how do you know that they describe the same spacetime, the answer is "with difficulty". You really need to find a transform from one set of coordinates to the other and I don't think there's a recipe for that.

 

[cianfa72]

If you mean that, given the Schwarzschild line element and the Eddington-Finkelstein line element, how do you know that they describe the same spacetime, the answer is "with difficulty". You really need to find a transform from one set of coordinates to the other and I don't think there's a recipe for that.

Yes, I believe the main point is the following: if you can find (actually if it does exist !) a continuous differentiable transformation such that the metric tensor components transform into the others, then the two are actually different coordinate systems for the same underlying geometry.

 

[Ibix]

if you can find (actually if it does exist !) a continuous differentiable transformation such that the metric tensor components transform into the others

The transformation must be invertible in the regions both coordinate systems cover, as well.