This question is very simple, but it is driving me mad.
Show that the Schwarzschild metric in Kruskal coordinates takes the form
ds2 = (32M3/r)e-r/2M(-dv2+du2) +r2(d(theta)2 + sin2(theta)*d(phi)2)
The equations are just those defining the Kruskal coordinates
(1) u2 - v2 = (r/2M -1)er/2M
(2) v/u = tanh(t/4M)
The Attempt at a Solution
I've tried this numerous times without luck. Usually I start by solving the second eqt'n for t and then take its total time derivative. This gets me dt2 to plug into the regular Schwarzschild metric. Next, I take total derivative of LHS and RHS of first eqt'n and then solve for dr. This leaves me with a hopelessly complex expression for the metric that I have not been successful in reducing to the form given by the problem.
I can't find derivations of this anywhere on the internet (Wikipedia just states it). This makes me think that either the problem is exceedingly difficult, or exceedingly trivial. If it is trivial, I'm missing something. Any ideas?