Coordinate transformation

[Tangchao Liu]

Hi all,

I have a question about coordinate transformation using xCoba.

My ultimate goal is to solve an exercise to go from Boyer-Lindquist coordinate to Kerr-Newmann and Kerr-Schild coordinate, especially I expect the principle null vectors first written in BL to be (0,1,0,0) in KN. As a warm-up exercise I was doing a more trivial case of transforming from spherical to Cartesian coordinate in 3D Euclidean space. Specifically I expect the radial unit vector (1,0,0) in spherical coordinate to take the form of (x/r, y/r, z/r) where r is given in terms of sqrt(x^2 + y^2 + z^2). But I couldn't do it. Attached is the notebook. I suggest you to go first right to the bottom before viewing some of my attempts before.

The key is, I can do the transformation of the metric because I can code the Jacobi matrix. But I don't know how to tell xAct the functional dependence of the two coordinates. 

Later I would like to transform differential operators between different coordinate. In this case it would be the Laplacian. In 4D, it can be the wave operator. And in Kerr, I'd like to do the transformation for Teukolsky operators. 

Thank you very much if you could help me solve this simple exercise, and if you could give some hints on the more complicated things I want to achieve later. 

Best,

Tangchao