Is there any generalized Schwarzschild solution for an arbitrary number of dimensions? Is it necessary to calculate each individually, or is there a relationship between them?
Answer
In $d>4$ dimensions, the analogue of the Schwarzschild solution is $$ ds^2 = - \left( 1 - \frac{2 M}{r^{d-3} } \right) dt^2 + \left( 1 - \frac{2M}{r^{d-3}} \right)^{-1} dr^2 + r^2 d \Omega_{d-2}^2 $$
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