Get Lorentzian Spherically Symmetric Metric to Sylvester Form

[tut_einstein]

Hi,

I'm trying to determine the exact transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form (that is, with just 1 or -1 on its main diagonal, with all other elements equal to zero.) Assuming that the metric has Lorentzian signature, does anyone know how to determine the exact transformation that achieves this?

Thanks.

 

[henry_m]

This is essentially nothing more than the orthogonal diagonalisation of a symmetric matrix that you probably did loads of times when you first learned about matrices. The coordinate transformation can be calculated from the matrix of eigenvectors. This gives a diagonal matrix, which should have one negative and the rest positive entries. Then you just have to rescale the coordinates to make the entries -1 and +1.

 

[George Jones]

Then you just have to rescale the coordinates to make the entries -1 and +1.

tut_einstein, this, in general, results in a non-holonomic basis. i.e., one that is not induced by any coordinate system. As a specific example, consider Schwarzschild spacetime,

https://www.physicsforums.com/showthread.php?t=102902