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The Schwarzschild Solution and Classical Tests of General Relativity

Imagine my joy at the feasibility of general covariance and the result that the equations give the perihelion motion of Mercury correctly. For a few days I was beside myself with joyous excitement. —A. Einstein (To P. Ehrenfest, Jan. 17, 1916)

The solution of the field equations, which describes the field outside of a spherically symmetric mass distribution, was found by Karl Schwarzschild only two months after Einstein published his field equations. Schwarzschild performed this work under rather unusual conditions. In the spring and summer of 1915 he was assigned to the eastern front. There he came down with an infectious disease and in the fall of 1915 he returned seriously ill to Germany. He died only a few months later, on May 11, 1916. In this short time, he wrote two significant papers, in spite of his illness. One of these dealt with the Stark effect in the Bohr–Sommerfeld theory, and the other solved the Einstein field equations for a static, spherically symmetric field. From this solution he derived the precession of the perihelion of Mercury and the bending of light rays near the sun. Einstein had calculated these effects previously by solving the field equations in post-Newtonian approximation.

4.1 Derivation of the Schwarzschild Solution The Schwarzschild solution is the unique static, spherically symmetric vacuum spacetime and describes the field outside a spherically symmetric body. It is the most important exact solution of Einstein’s field equations. We know that a static spacetime (M, g) has locally the form (see Sect. 2.9) M = R × Σ,

g = −ϕ 2 dt 2 + h,

(4.1)

where h is a Riemannian metric on Σ and ϕ is a smooth function on Σ. The timelike Killing field K with respect to which (M, g) is static is ∂t . We assume that this is the only static Killing field and thus that there is a distinguished time. Note also that ϕ 2 = −K, K. N. Straumann, General Relativity, Graduate Texts in Physics, DOI 10.1007/978-94-007-5410-2_4, © Springer Science+Business Media Dordrecht 2013

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4 The Schwarzschild Solution and Classical Tests of General Relativity

Fig. 4.1 Construction of Schwarzschild coordinates

We say that a Lorentz manifold is spherically symmetric, provided it admits the group SO(3) as an isometry group, in such a way that the group orbits, i.e., the subsets on which the group acts transitively, are two-dimensional spacelike surfaces. If (4.1) is also spherically symmetric, the action of SO(3) is only non-trivial on Σ, i.e., the time t is preserved. This follows from the fact that K = ∂t is invariant under SO(3) (because of the uniqueness assumption). Indeed, this implies that K is perpendicular to the orbits of SO(3). (A non-vanishing orthogonal projection of K on an orbit would also be invariant, but such an invariant vector field can not exist.) Another way to see that t remains invariant is to note that the 1-form ω = K, K dt belonging to K (see Eq. (2.114)) is also invariant. Since K, K is obviously invarian