Assignment 4

Consider the metric given in problem three of Quiz II.

(1)\begin{align} ds^2 = A(r)\ dt^2 - B(r)\ dr^2 + r^2\ \left(d\theta^2 - \sin^2\theta\ d\varphi^2\right) \end{align}

Write a mathematica program that does the following:

- Computes all the components of the Christoffel connection.
- Computes all the components of the Riemann tensor using the results of part 1.
- Computes the Ricci and Einstein tensor using the results of part 2.
- Finally, imposes the condition that the Einstein tensor vanishes to obtain the independent equations for undetermined functions $A(r$) and $B(r)$.
- "Solve" for the functions $A(r)$ and $B(r)$.

I *strongly* recommend that you write your own code. Also, try to make it modular in the sense that it should be able to handle a situation where the metric is changed or "matter" is present and so on. Your solution should be presented to me in the form a mathematica notebook that should work when I run it on my machine.