In a manuscript communicated to the Royal Society by Henry Cavendish in 1783, an English scientist, Reverend John Michell, presented the idea of celestial bodies whose gravitational attraction was strong enough to prevent even light from escaping their surfaces. Both Michell and Laplace, who came up with the same concept in 1796, based their arguments on Newton's universal law of gravity and his corpuscular theory of light.

During the nineteenth century, a time when the notion of “dark stars” had fallen into oblivion, geometry experienced its fundamental revolution: Gauss and Lobachevsky had already found examples of non–Euclidean geometry, when Riemann became aware of the full consequences which arise from releasing the parallel axiom. In a famous lecture given at Göttingen University in 1854, the former student of Gauss introduced both the notion of spatial curvature and the extension of geometry to more than three dimensions.

It is these features of Riemannian geometry which, more than fifty years later, enabled Einstein to reveal the connection between the gravitational field and the metric structure of spacetime. In February 1916 - only three months after having achieved the final breakthrough in general relativity - Einstein presented, on behalf of Schwarzschild, the first exact solution of the new equations to the Prussian Academy of Sciences.

It took, however, almost half a century until the geometry of the Schwarzschild spacetime was correctly interpreted and its physical significance was fully appreciated.

In this chapter we establish the uniqueness of the Kerr metric amongst the stationary black hole solutions of self–gravitating harmonic mappings (scalar fields) with arbitrary Riemannian target manifolds. As in the vacuum and the electrovac cases, the uniqueness proof consists of three main parts: First, taking advantage of the strong rigidity theorem (see section 6.2), one establishes staticity for the nonrotating case, and circularity for the rotating case. One then separately proves the uniqueness of the Schwarzschild metric amongst all static configurations, and the uniqueness of the Kerr metric amongst all circular black hole solutions.

The three problems mentioned above are treated in the first section and the last two sections, respectively: The staticity and circularity theorems are derived from the symmetry properties of the scalar fields and the general theorems given in sections 8.1 and 8.2. The static uniqueness theorem is then proven along the same lines as in the vacuum case, that is, by means of conformal techniques and the positive energy theorem. The uniqueness theorem for rotating configurations turns out to be a consequence of the corresponding vacuum theorem (see chapter 10) and an additional integral identity for stationary and axisymmetric harmonic mappings.

Besides dealing with general harmonic mappings, we shall also pay some attention to ordinary scalar (Higgs) fields. By this, we mean harmonic mappings into linear target spaces with an additional potential term in the Lagrangian. By 1972, Bekenstein had already established the static no–hair theorem for ordinary massive scalar fields, by means of a divergence identity.

Einstein's equations simplify considerably in the presence of a second Killing field. Spacetimes with two Killing fields provide the framework for both the theory of colliding gravitational waves and the theory of rotating black holes (Chandrasekhar 1991). Although they deal with different physical objects, the theories are, in fact, closely related from a mathematical point of view. Whereas in the first case both Killing fields are spacelike, there exists an (asymptotically) timelike Killing field in the second situation, since the equilibrium configuration of an isolated system is assumed to be stationary. It should be noted that many stationary and axi-symmetric solutions which have no physical relevance give rise to interesting counterparts in the theory of colliding waves. We refer the reader to Chandrasekhar (1989) for a comparison between corresponding solutions of the Ernst equations. In this chapter we discuss the properties of circular manifolds, that is, asymptotically flat spacetimes which admit a foliation by integrable 2–surfaces orthogonal to the asymptotically timelike Killing field k and the axial Killing field m.

In the first section we argue that the integrability conditions imply that locally M = Σ × Γ and (4)g = σ + g. Here (Σ, σ) and (Γ, g) denote 2–dimensional manifolds where, in an adapted coordinate system, the metrics σ and g do not depend on the coordinates of Σ.

In the second section we discuss the properties of (Σ, σ), the pseudo–Riemannian manifold spanned by the orbits of the 2–dimensional Abelian group generated by the Killing fields.

The strong rigidity theorem implies that stationary black hole spacetimes are either axisymmetric or have a nonrotating horizon. The uniqueness theorems are, however, based on stronger assumptions: In the nonrotating case, staticity is required whereas the uniqueness of the Kerr–Newman family is established for circular spacetimes. The first purpose of this chapter is therefore to discuss the circumstances under which the integrability conditions can be established.

Our second aim is to present a systematic approach to divergence identities for spacetimes with one Killing field. In particular, we consider the stationary Einstein–Maxwell equations and derive a mass formula for nonrotating - not necessarily static - electrovac black hole spacetimes.

The chapter is organized as follows: In the first section we recall that the two Killing fields in a stationary and axisymmetric domain fulfil the integrability conditions if the Ricci–circularity conditions hold. As an application, we establish the circularity theorem for electrovac spacetimes.

The second section is devoted to the staticity theorem. As mentioned earlier, the staticity issue is considerably more involved than the circularity problem. The original proof of the staticity theorem for black hole spacetimes applied to the vacuum case (Hawking and Ellis 1973). Here we present a different proof which establishes the equivalence of staticity and Ricci–staticity for a strictly stationary domain. Since our reasoning involves no potentials, it is valid under less restrictive topological conditions. We conclude this section with some comments on the electrovac staticity theorem, which is still subject to investigations.

In the previous chapter we compiled the basic geometric identities for stationary and axisymmetric spacetimes. We shall now use these relations to derive the Kerr metric. Although we have to postpone the general definitions of black holes and event horizons to a later chapter, we feel that this is the right time to present the Kerr solution. As we shall argue when going into the details of the uniqueness theorems, the Kerr metric occupies a distinguished position amongst all stationary solutions of the vacuum Einstein equations.

The nonrotating counterpart of the Kerr solution was found by Schwarzschild (1916a, 1916b) immediately after Einstein's discovery of general relativity (Einstein 1915a, 1915b). In contrast to this, it took almost half a century until Kerr (1963) was eventually able to derive the first asymptotically flat exterior solution of a rotating source in general relativity. As is well known, both the Schwarzschild and the Kerr metric have charged generalizations, which were found by Reissner (1916) and Nordstrom (1918) in the static case, and by Newman et al. (1965) in the circular case.

The fact that it was not until 1963 that the Kerr metric was discovered reflects the difficulties of its derivation. As was pointed out by Chandrasekhar (1983), this does, however, not imply that “there is no constructive analytic derivation of the Kerr metric that is adequate in its physical ideas…” (Landau and Lifshitz 1971). In fact, the derivation of the Kerr solution appears fairly transparent when based on a discussion of the general properties of stationary and axi-symmetric spacetimes.

In 1931 Chandrasekhar established an upper bound for the mass of a cold self–gravitating star in thermal equilibrium (Chandrasekhar 1931a, 1931b). This leads one to consider the ultimate fate of a star which, having radiated all its thermo–nuclear energy, still has a mass beyond the critical limit (a few solar masses). Once the nickel and iron core has been formed, there exists no possibility for any further nuclear reactions; the core must therefore undergo gravitational collapse. The collapse may cease by the time the core has reached nuclear densities, which leads to the formation of a neutron star, provided that the mass of the collapsing part lies below the critical value. If this is not the case, then nothing can prevent total gravitational collapse (Chandrasekhar 1939, Oppenheimer and Snyder 1939, Oppenheimer and Volkoff 1939), resulting in the formation of a black hole (Wheeler 1968; see Israel 1987 for a historical review).

Birkhoff's theorem (Birkhoff 1923), which states that a spherically symmetric spacetime is locally isometric to a part of the Schwarzschild–Kruskal metric (Kruskal 1960), yields a significant simplification in the discussion of the spherically symmetric collapse scenario (Harrison et al. 1965). However, in order to treat more general situations, one has to find the generic features of gravitational collapse in general relativity. This was achieved by Geroch, Hawking, Penrose and others in the late sixties and early seventies (Hawking and Penrose 1970; see also Hawking and Ellis 1973, Clarke 1975, 1993): The singularity theorems show that - in contrast to Newtonian gravity - deviations from spherical symmetry, internal pressure or rotation do not prevent the formation of a singularity.

Einstein's field equations form a set of nonlinear, coupled partial differential equations. In spite of this, it is still sometimes possible to find exact solutions in a systematic way by considering space-times with symmetries. Since the laws of general relativity are covariant with respect to diffeomorphisms, the corresponding reduction of the field equations must be performed in a coordinate–independent way. This is achieved by using the concept of Killing vector fields. The existence of Killing fields reflects the symmetries of a spacetime in a coordinate–invariant manner.

A spacetime (M, g) admitting a Killing field gives rise to an invariantly defined 3–manifold Σ. However, Σ is only a hypersurface of (M, g) if it is orthogonal to the Killing trajectories. In general, Σ must be considered to be a quotient space M/G rather than a subspace of M. (Here G is the 1–dimensional group generated by the Killing field.) The projection formalism for M/G was developed by Geroch (1971, 1972a), based on earlier work by Ehlers (see also Kramer et al. 1980). The invariant quantities which play a leading role are the twist and the norm of the Killing field.

In the first section of this chapter we compile some basic properties of Killing fields. The twist, the norm and the Ricci 1–form assigned to a Killing field are introduced in the second section. Using these quantities, we then give the complete set of reduction formulae for the Ricci tensor.

In the third section we apply these formulae to vacuum space-times. In particular, we introduce the vacuum Ernst potential and derive the entire set of field equations from a variational principle.

The area theorem is probably one of the most important results in classical black hole physics. It asserts that (under certain conditions which we specify below) the area of the event horizon of a predictable black hole spacetime cannot decrease. This result bears a resemblance to the second law of thermodynamics. The analogy is reinforced by the similarity of the mass variation formula to the first law of ordinary thermodynamics. Within the classical framework the analogy is basically of a formal, mathematical nature. There exists, for instance, no physical relationship between the surface gravity, κ, and the classical temperature of a black hole, which must be assigned the value of absolute zero. Nevertheless, on account of the Hawking effect, the relationship between the laws of black hole physics and thermodynamics gains a deep physical significance: The temperature of the black–body spectrum of particles created by a black hole is κ/2π. This also sheds light on the analogy between the entropy and the area of a black hole.

The Killing property of a stationary event horizon implies that its surface gravity is constant. If the Killing fields are integrable (that is, in static or circular spacetimes), the zeroth law of black hole physics is a purely geometrical property of Killing horizons. Otherwise, it is a consequence of Einstein's equations and the dominant energy condition.

The Komar expression for the mass of a stationary spacetime provides a formula giving the mass in terms of the total angular momentum, the angular velocity, the surface gravity and the area of the horizon.

Quantum mechanics, as for example in the case of a non-relativistic particle, can be treated in either of two ways. One can work with the differential-equation form of the theory, by studying the Schrödinger equation. Alternatively, one can study the Feynman path integral, which gives the integral form of the Schrödinger differential approach. The Feynman path integral has the advantage of incorporating the boundary conditions on the particle, for example that the particle is at spatial position xa at an initial time ta, and at position xb at final time tb. The path integral leads naturally to a semi-classical expansion of the quantum amplitude, valid asymptotically as the action of the classical solution of the equations of motion becomes large compared to Planck's constant ħ.

One moves from quantum mechanics to quantum gravity by replacing the spatial argument x of the wave function by the three-dimensional spatial geometry hij(x). A typical quantum amplitude is then the amplitude to go from an initial three-geometry hijI to a final geometry hijF, specified (say) on identical three-surfaces ΣI, ΣF. To complete the description in the asymptotically flat case, one needs to specify asymptotic parameters such as the time T between the two surfaces, measured at spatial infinity. To make the classical boundary-value problem elliptic and (one hopes) well-posed, one rotates to imaginary time –iT. The Feynman path integral would again give a semi-classical expansion of the quantum amplitude, were it not for the infinities present in the loop amplitudes.