Schwarzschild metricgreenspun.com : LUSENET : quaternions : One Thread
Is it possible to do the Conformal tranformation for a Schwarzschild metric expressed in spherical polar cordinates?
-- Minu Joy (email@example.com), April 08, 2002
I am not trained well enough to answer the question, but here are some of my thoughts anyway.
The Riemann curvature tensor has 4^4=256 components. Due to symmetries, there are only 20 that are independent. This can be broken into the Ricci tensor and the Weyl conformal tensor, each of which has 10 components. (Got these facts from MTW, p.325-7)
A conformal transformation in math means that the angles are preserved. The Weyl conformal tensor is traceless. The starting coordinate system is not important. What would matter is setting up the Cauchy-Riemann differential equations. Although someone has probably already studied this question, I do not read enough of the literature to know.
In my own work - which is both speculative and only partially complete - I have a metric that comes out of a unified field equation (EM and gravity). That metric is equivalent to the Schwarzschild metric to post-Newtonian accuracy (5/6 of the terms of the Taylor series expansion), but diverges for higher terms. One could do a conformal transformation if quaternion analysis was in good shape. Right now it is not, but that is another area I may (or may not) have made some progress on. I'll put conformal transformations of my unified metric proposal in my "try to play with this file."
Good luck in your studies, doug
-- Douglas Sweetser (Sweetser@theworld.com), April 08, 2002.