Kerr Geometry

Initialization

Run the following code to initialize $\LaTeX$ output and load (Tevian's frontend to) the differential forms package, either by clicking on "Evaluate" or by typing Shift+Enter.

Code

Now enter a line element in the box below, adapting the given code as needed. First declare any parameters or functions, then provide a list of coordinates using the MakeC command as shown below (note the double parentheses). Finally, set the (covariant) metric to the (inverse of the) matrix relating the coordinate and orthonormal 1-form bases. The result should be the line element in tensor notation.

List the nonzero components of the (covariant) metric tensor (in a coordinate basis!).

List the nonzero Christoffel symbols (in an orthonormal frame).

Compute and display the components of the Ricci tensor $R_{ij}$. The Kerr geometry is a vacuum solution of Einstein's equation!

Enter any further code you wish below.

The remaining computations do not (yet) complete in a reasonable amount of time using an orthonormal basis. You can try them in a coordinate basis instead by going here.

Compute the components $R^i{}_{jkl}$ of the (Riemann) curvature tensor (in an orthonormal frame), but only print a single component.