December 1997
Lecture Notes on General Relativity
Sean M. Carroll
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The Schwarzschild Solution and Black Holes
We now move from the domain of the weak-field limit to solutions of the full nonlinearEinstein’s equations.With the possible exception of Minkowski space, by far the mostimportant such solution is that discovered by Schwarzschild, which describes sphericallysymmetric vacuum spacetimes. Since we are in vacuum, Einstein’s equations becomeRμν=0. Of course, if we have a proposed solution to a set of differential equations such as this,it would suffice to plug in the proposed solution in order to verify it; we would like to dobetter, however. In fact, we will sketch a proof of Birkhoff’s theorem, which states that theSchwarzschild solution is theuniquespherically symmetric solution to Einstein’s equationsin vacuum.The procedure will be to first present some non-rigorous arguments that anyspherically symmetric metric (whether or not it solves Einstein’s equations) must take on acertain form, and then work from there to more carefully derive the actual solution in sucha case.“Spherically symmetric” means “having the same symmetries as a sphere.”(In thissection the word “sphere” meansS2, not spheres of higher dimension.) Since the object ofinterest to us is the metric on a differentiable manifold, we are concerned with those metricsthat have such symmetries. We know how to characterize symmetries of the metric — theyare given by the existence of Killing vectors. Furthermore, we know what the Killing vectorsofS2are, and that there are three of them.Therefore, a spherically symmetric manifoldis one that has three Killing vector fields which are just like those onS2.By “just like”we mean that the commutator of the Killing vectors is the same in either case — in fancierlanguage, that the algebra generated by the vectors is the same. Something that we didn’tshow, but is true, is that we can choose our three Killing vectors onS2to be (V(1), V(2), V(3)),such that[V(1), V(2)]=V(3)[V(2), V(3)]=V(1)[V(3), V(1)]=V(2).(7.1)The commutation relations are exactly those of SO(3), the group of rotations in three di-mensions. This is no coincidence, of course, but we won’t pursue this here. All we need isthat a spherically symmetric manifold is one which possesses three Killing vector fields withthe above commutation relations.Back in section three we mentioned Frobenius’s Theorem, which states that if you havea set of commuting vector fields then there exists a set of coordinate functions such that thevector fields are the partial derivatives with respect to these functions. In fact the theorem164

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