# Transformation to get a Lorentzian spherically symmetric metric to the Sylvester form

1. Jun 26, 2011

### 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.

2. Jul 5, 2011

### henry_m

Re: Transformation to get a Lorentzian spherically symmetric metric to the Sylvester

This is essentially nothing more than the orthogonal diagonalisation of a symmetric matrix that you probably did loads of times when you first learnt 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.

3. Jul 5, 2011

### George Jones

Staff Emeritus
Re: Transformation to get a Lorentzian spherically symmetric metric to the Sylvester

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,