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:

  1. Computes all the components of the Christoffel connection.
  2. Computes all the components of the Riemann tensor using the results of part 1.
  3. Computes the Ricci and Einstein tensor using the results of part 2.
  4. Finally, imposes the condition that the Einstein tensor vanishes to obtain the independent equations for undetermined functions $A(r$) and $B(r)$.
  5. "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.

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