In a cosmic sense, the collision of the ninth periodic comet discovered by the team of Carolyn and Gene Shoemaker and David Levy with the planet Jupiter was unremarkable. The history of the solar system, indeed its very genesis, has been marked by countless such events. The cratered surfaces of planetary bodies are a testament to this ubiquitous phenomenon; even the Earth's ephemeral surface records the continued action of this elemental process in impact craters and in the fossil record.

In human terms, on the other hand, the impact of Comet Shoemaker-Levy 9's 20-odd fragments into Jupiter was an unprecedented event of global significance. After a year of planning and preparation, the largest astronomical armada in history focussed on the planet Jupiter in July 1994. News of each successively more astonishing image or spectrum was broadcast with almost instantaneous speed over the world's increasingly sophisticated computer communications network. Astronomers were, for a time, to be found on daily newscasts and the front pages of newspapers. For a week in July, the world looked up from its normal preoccupations long enough to notice, and to ponder, the awesome beauty of the natural world and the surprising unpredictability of the universe.

Still one more perspective on this event remains. What has science gained from the terabytes of images, lightcurves and spectra obtained over the entire range of the electromagnetic spectrum?

This paper reviews spectroscopic measurements relevant to the chemical modifications of Jupiter's atmosphere induced by the Shoemaker-Levy 9 impacts. Such observations have been successful at all wavelength ranges from the UV to the centimeter. At the date this paper is written, newly detected or enhanced molecular species resulting from the impacts include H2O, CO, S2, CS2, CS, OCS, NH3, HCN and C2H4. There is also a tentative detection of enhanced PH3 and a controversial detection of H2S. All new and enhanced species were detected in Jupiter's stratosphere. With the exception of NH3 (and perhaps H2S and PH3), apparently present down to the 10–50 mbar level, the minor species are seen at pressures lower than 1 mbar or less, consistent with a formation during the plume splashback at 1–100 microbar. NH3 may result from upwelling associated with vertical mixing generated by the impacts. The main oxygen species is apparently CO, with a total mass of a few 1014 g for the largest impacts, consistent with that available in 400–700 m radius fragments. The observed O/S ratio is reasonably consistent with cometary abundances, but the O/N ratio (inferred from CO/HCN) is much larger, suggesting that another N species was formed but remained undetected, presumably N2. The time evolution of NH3, S2, CS2 shows evidence for photochemical activity taking place during and after the impact week.

What did the break-up of comet Shoemaker-Levy 9 (SL9) and its subsequent impact on Jupiter teach us about the nature and constitution of this comet? The break-up of the comet apparently triggered activity of the fragments. Although a dust coma was continuously present around the fragments that orbited Jupiter, spectroscopic observations did not reveal any sign of gas. The impact itself was so energetic that most molecules of the impactor were dissociated and that any chemical memory was lost. Ultraviolet and visible spectroscopy of the impact sites revealed emission lines from several atoms, giving potential information on elemental abundances. However, the fact that both neutral and ionized atoms are emitting, and that both fundamental and inter-system lines are present, suggest that the medium is out-of-equilibrium and that emitting mechanisms other than simple resonance fluorescence are at work. Ultraviolet, infrared, and radio spectroscopy revealed lines of several molecular species, in emission and/or absorption, that are not normally present in Jupiter's upper atmosphere. In the visible, dark spots due to aerosols developed at the impact sites. It is not clear at the present time which part of this material is coming from preserved impactor material, from the recombination of the dissociated impactor material, from reactions between the impactor's and Jupiter's material, or from material coming from the lower layers of Jupiter's atmosphere. Realistic modelling of the impacts and of the following chemical reactions will be necessary to address all these issues.

In this chapter we present the arguments which establish that the Schwarzschild metric describes the only static, asymptotically flat vacuum spacetime with regular (not necessarily connected) event horizon (Israel 1967, Müller zum Hagen et al. 1973, Robinson 1977, Bunting and Masood–ul–Alam 1987). We then discuss the generalization of this result to the situation with electric fields; that is, we demonstrate the uniqueness of the 2–parameter Reissner–Nordström solution amongst all asymptotically flat, static electrovac black hole configurations with nondegenerate horizon (Israel 1968, Müller zum Hagen et al. 1974, Simon 1985, Ruback 1988, Masood–ul–Alam 1992). Taking magnetic fields into account as well, we finally establish the uniqueness of the 3–parameter Reissner–Nordström metric. We conclude this chapter with a brief discussion of the Papapetrou-Majumdar metric, representing a static configuration with M = |Q| and an arbitrary number of extreme black holes (Papapetrou 1945, Majumdar 1947). This metric is not covered by the static uniqueness theorems, since the latter apply exclusively to electrovac solutions which are subject to the inequality M > |Q|.

Throughout this chapter the domain of outer communications is assumed to be static. In the vacuum or the electrovac case staticity is, as we have argued in the previous chapter, a consequence of the symmetry conditions for the matter fields.

Our main objective in this chapter concerns the “modern” approach to the static uniqueness theorem, which is based on conformal transformations and the positive energy theorem. We shall, however, start this chapter with some comments on the traditional line of reasoning, which is due to Israel, Müller zum Hagen, Robinson and others.

In this chapter we consider self–gravitating electromagnetic fields which are invariant under the action of one or more Killing fields. As an application, we present the derivation of the Kerr–Newman metric in the last section.

In the first section we introduce the electric and magnetic 1–forms, E = –iKF and B = iK * F, which can be defined in terms of the electromagnetic field tensor (2–form) F and a Killing field K. We also express the stress–energy tensor in terms of K, E and B. We then focus on the case where spacetime admits two Killing fields, k and m, say, and establish some algebraic identities between their associated electric and magnetic 1–forms.

The invariance conditions for electromagnetic fields are introduced in the second section. The homogeneous Maxwell equations imply the existence of a complex potential, Λ, which can be as-sociated with E and B if the Killing field K acts as a symmetry transformation on F. In terms of Λ, the remaining Maxwell equations reduce to one complex equation for ΔΛ, involving the twist, ω, and the norm, N, of the Killing field. In the Abelian case, to which we restrict our attention in this chapter, the presence of gauge freedom does not require a modified invariance concept for F. In contrast, the symmetry conditions must be reexamined if one deals with arbitrary gauge groups (see, e.g., Forgács and Manton 1980, Harnad et al. 1980, Jackiw and Manton 1980, Brodbeck and Straumann 1993, 1994, Heusler and Straumann 1993b).

In this chapter we present the uniqueness theorem for the Kerr–Newman metric. The latter describes the only asymptotically flat, stationary and axisymmetric electrovac black hole solution with regular event horizon. The proof of this fact involves the following steps: First, one has to establish circularity of the domain of outer communications as a consequence of the symmetry properties of the electromagnetic field. The Einstein–Maxwell equations are then reduced to a 2–dimensional elliptic boundary–value problem for the complex Ernst potentials E and Λ. One then takes advantage of the symmetries of the Ernst equations to derive a divergence identity for the difference of two solutions. Since the boundary and regularity conditions are completely parametrized in terms of the total mass, angular momentum and charge, Stokes' theorem finally yields the desired result.

The chapter is organized as follows: The first section gives a short outline of the reasoning. In the second section we parametrize the Ernst potentials in terms of the hermitian matrix Φ, describing the nonlinear sigma model on the symmetric space G/H = SU(1, 2)/S(U(1) × U(2)) (or G/H = SU(1, 1)/U(1) in the vacuum case). We then establish the variational equation for Φ and derive an identity for the difference of two solutions to this equation. Evaluating the expressions in a circular spacetime, we obtain the Mazur identity (Mazur 1982) in the third section. This identity - or a related identity found by Bunting in 1983 - must be considered the key to the uniqueness theorems for rotating black holes.

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.