Fastest way to find the curvature terms from a given metric I want to find the spherically symmetric, static solutions to Einstein's equations
$$
R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0
$$
in four dimensions using the metric
$$
g_{\mu \nu}dx^{\mu}dx^{\nu} = -A(r)dt^2 + B(r)\left[ dr^2 + r^2 \left( d\theta^2 + \sin^2\theta d\phi^2\right)\right]
$$
My question is: What is the fastest way of doing that? I am eliminating the terms by using the obvious simplifications such as "only $r$ and $\theta$ derivatives may survive" or "off-diagonal metric elements give zero" but it is still so lengthy and complicated. It took more than an hour for me to find the $tt$ equation. So, I want to know if there is a faster way to handle this kind of equations. I will be grateful if you can help.
 A: I would recommend using Mathematica to calculate curvatures, unless there's a good reason to do it by hand (for example, perhaps you want to calculate the curvatures for a metric while keeping the dimension general). It's not hard to write your own code to do this, and I think it's a nice idea actually. I also have found this code to be very useful: http://www.inp.demokritos.gr/~sbonano/RGTC/. It's good enough to handle differential forms also.
The solution you'll find for your ansatz above is the Schwarzschild solution, but you've written it in non-standard coordinates known as isotropic coordinates. The second term in parentheses is just flat space in spherical coordinates. 
If you are calculating the curvatures by hand for a simple warped product metric like this one, there's a slick trick you can use. If you preform a Weyl transformation, 
$ds'^2 = \Omega^2 ds^2$, with $\Omega^2 = B^{-1},$ 
then the resulting metric is very simple:
$ds'^2 = -\frac{A}{B} dt^2 + \sum_{i=1}^3 dy_i^2, $
and the curvature of this new metric is very easily to calculate since it's a direct product (and one of the products is flat space!). Then, using the Weyl transformation formula for the curvature tensor allows you to find the curvature for the original metric $ds^2$. This formula can be found in any GR textbook.
