Black Holes in General Relativity
S. W. HAWKING Institute of Theoretical Astronomy, University of Cambridge, Cambridge, England Received October 15, 1971 Abstract. It is assumed that the singularities which occur in gravitational collapse are not visible from outside but are hidden behind an event horizon. This means that onecan still predict the future outside the event horizon. A black hole on a spacelike surface is defined to be a connected component of the region of the surface bounded by the event horizon. As time increase, black holes may merge together but can never bifurcate. A black hole would be expected to settle down to a stationary state. It is shown that a stationary black hole must have topologicallyspherical boundary and must be axisymmetric if it is rotating. These results together with those of Israel and Carter go most of the way towards establishing the conjecture that any stationary black hole is a Kerr solution. Using this conjecture and the result that the surface area of black holes can never decrease, one can place certain limits on the amount of energy that can be extracted from blackholes.
It has been known for some time that a non-rotating star of more than about two solar masses has no low temperature equilibrium configuration. This means that such a star must undergo catastrophic collapse when it has exhausted its nuclear fuel unless it has managed to eject sufficient matter to reduce its mass to less than twice that of the sun. If the collapse isexactly spherically symmetric, the metric is that of the Schwarzschild solution outside the star and has the following properties (see Fig. 1): 1. The surface of the star will pass inside the Schwarzschild radius r = 2Gc~2M. After this has happened there will be closed trapped surfaces [1,2] around the star. A closed trapped surface is a spacelike 2-surface such that both the future directed familiesof null geodesies orthogonal to it are converging. In other words, it is in such a strong gravitational field that even the outgoing light from it is dragged inwards. 2. There is a space-time singularity. 3. The singularity is not visible to observers who remain outside the Schwarzschild radius. This means that the breakdown of our present physical theory which one expects to occur at asingularity cannot affect
Black Holes in General Relativity
what happens outside the Schwarzschild radius and one can still predict the future in the exterior region from Cauchy data on a spacelike surface. One can ask whether these three properties of spherical collapse are stable, i.e. whether they would still hold if the initial data for the collapse were perturbed slightly. This is vitalbecause no real collapse situation will ever be exactly spherical. From the stability of the Cauchy problem in general relativity  one can show that a sufficiently small perturbation of the initial data on a spacelike surface will produce a perturbation of the solution which will remain small on a compact region in the Cauchy development of the surface. This shows that property (1) is stable,since there is a compact region in the Cauchy development of the initial surface which contains closed trapped surfaces. It then follows that property (2) is stable provided one makes certain reasonable assumptions such as that the energy density of matter is always positive. This is because the existence of a closed trapped surface implies the occurrence of a singularity under these conditions. There remains the problem of the stability of property (3). Since the question of whether singularities are visible from outside depends on the solution at arbitrarily large times, one cannot appeal to the result on the stability of the Cauchy problem referred to above. Nevertheless it seems a reasonable conjecture that property (3) is indeed stable. If this is the case, we can still predict...