The Schwarzschild solution

Abstract This chapter deals with the Schwarzschild metric. To find the gravitational potential U produced by a spherically symmetric object in the Newtonian theory, it is necessary to solve the Poisson equation ΔU = 4πGρ. Here, the matter density ρ and U depend only on the radial coordinate r and possibly on the time t. Outside the source the solution is U = –GM/r, where M = 4π ∫ ρr2dr is the source mass. In general relativity the problem is to find the ‘spherically symmetric’ spacetime solutions of the Einstein equations, and the analog of the vacuum solution U = –GM/r is the Schwarzschild metric.