The Schwarzschild Solution
A derivation of the Schwarzschild metric and a discussion of its main properties (including a detailed computation of the precession of the planetary orbits). The Schwarzschild solution is also used as a simple example of “black hole” geometry, in order t
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The Schwarzschild Solution
So far we have only used the linearized Einstein equations, and considered geometric configurations typical of the weak-field approximation. In this chapter we will apply for the first time the full Einstein equations without approximations, and we will obtain a particular exact solution for a static, spherically symmetric gravitational field. Such a solution—the well known Schwarzschild solution—will be used to illustrate one of the most celebrated predictions of general relativity: the perihelion precession of the planetary orbits. Such an effect, experimentally known since the nineteenth century for the planets of our solar system, has provided one of the most convincing observational test of the Einstein theory. It should be immediately stressed, however, that the Schwarzschild solution plays a very important role in general relativity not only for its phenomenological applications, but also for its formal properties. It gives indeed a simple and basic example of how gravity can modify the (classical) causal structure of space–time, by introducing “event horizons” which limit our possibility to get information from a given portion of space (the interior of the so-called “black hole”). Also, extrapolated to the limit r → 0, it represents one of the simplest models of geometric singularity, i.e. of “geodesically incomplete” space–time manifold.
10.1 Spherically Symmetric Einstein Equations in Vacuum Let us look for solutions of the Einstein equations (7.29) describing the geometry associated to a spherically symmetric gravitational field, generated by a central source. We are interested, in particular, in the vacuum geometry (“externally” to the matter sources): we can thus set Tμν = 0, and in that case the equations simply reduce to Rμν = 0. All we need, therefore, are the components of the Ricci tensor for a metric gμν which describes a spherically symmetric three-dimensional space. This means, more precisely, that the spatial part (gij ) of our metric has to be rotationally invariant, i.e. it must admit the rotation group SO(3) as its isometry M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_10, © Springer-Verlag Italia 2013
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The Schwarzschild Solution
group. We can also say, using the terminology of Sect. 6.3, that it must be possible to choose an “adapted” foliation of the space–time manifold based on threedimensional space-like sections containing maximally symmetric subspaces with n = 2 dimensions, and then characterized by n(n + 1)/2 = 3 Killing vectors (corresponding, in this case, to the three generators of the spatial rotations). Using polar coordinates, x μ = (ct, r, θ, ϕ), the above condition can be easily satisfied by imposing that the space–time sections specified by fixed values of t and r correspond to two-dimensional spherical surfaces. The most general line-element satisfying this property is then the following: ds 2 = A1 (r, t)c2 dt 2 − A2 (r, t) dr 2 − A3 (r, t) dr dt − A4
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