Transformation to get a metric to diagonal form

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SUMMARY

The discussion focuses on transforming a spherically symmetric spacetime metric into a diagonal form using appropriate coordinate transformations. The metric is represented in spherical coordinates as [P,Q,0,0;Q,R,0,0;0,0,S,0;0,0,0,Ssin^(theta)], where P, Q, R, and S are functions of time (t) and radial distance (r). The key takeaway is that the transformation to achieve diagonalization is analogous to matrix diagonalization, leveraging principles from linear algebra.

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tut_einstein
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Hi,

If you have a spherically symmetric spacetime metric in a set of spherical coordinates t,r,theta,phi: [P,Q,0,0;Q,R,0,0;0,0,S,0;0,0,0,Ssin^(theta)]. Here P,Q,R,S are functions of t and r.


Now, if I want to choose cooridnates to get the metric in the generic diagonal form (that is by choosing appropriate t' and r' (the theta and phi would remain the same), is there a simple way to determine the transformation that would bring the metric to this diagonal form?

My main problem is that I'm working in a general case, where P,Q,R,S are unspecified, so I don't know their explicit dependence on t and r. I really really need help with this.


Thank you!
 
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That is basic Linear Algebra. Works exactly the same way as diagonalisation of matrices.
 

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