# Kerr Geometry

## Initialization

Run the following code to initialize $\LaTeX$ output and load some macros, 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, enter the components $s^i{}_j$ of an orthonormal basis $\{\sigma^i\}$ of 1-forms with respect to a coordinate basis $dx^j$ (so that $\sigma^i=s^i{}_j\,dx^j$), and run the MakeS command (whose first argument is the signature). 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. (You can convert the answer to a coordinate basis with the command nab.display(f).)
(This computation can take several minutes.)

Compute and display the components of the Ricci tensor $R_{ij}$. The Kerr geometry is a vacuum solution of Einstein's equation!
(This computation can take several minutes.)

Compute and display the components of the Einstein tensor $G_{ij}$.

Enter any further code you wish below.