The method of the renormalization group was originally introduced by Gell-Mann and Low as a means of dealing with the failure of perturbation theory at very high energies in quantum electrodynamics. An n-loop contribution to an amplitude involving momenta of order q, such as the vacuum polarization Πμν(q), is found to contain up to n factors of In() as well as a factor αn, so perturbation theory will break down when is large, even though the fine structure constant a is small. Even in a massless theory like a non-Abelian gauge theory we must introduce some scale μ to specify a renormalization point at which the renormalized coupling constants are to be defined, and in this case we encounter logarithms In(E/μ), so that perturbation theory may break down if E ≫ μ or E ≪ μ, even if the coupling constant is small.

Fortunately, there is a modified version of perturbation theory that can often be used in such cases. The key idea of this approach consists in the introduction of coupling constants gμ defined at a sliding renormalization scale μ — that is, a scale that is not related to particle masses in any fixed way. By then choosing μ to be of the same order of magnitude as the energy E that is typical of the process in question, the factors In(E/μ) are rendered harmless. We can then do perturbation theory as long as gμ remains small.

There are subtleties in the implications of symmetries in quantum field theory that have no counterpart in classical theories. Even in renormalizable theories, the infinities in quantum field theory require that some sort of regulator or cut-off be used in actual calculations. The regulator may violate symmetries of the theory, and even when this regulator is removed at the end of the calculation it may leave traces of this symmetry violation. This problem first emerged in trying to understand the decay rate of the neutral pion, in the form of an anomaly that violates a global symmetry of the strong interactions. Anomalies can also violate gauge symmetries, but in this case the theory becomes inconsistent, so that the condition of anomaly cancellation may be used as a constraint on physical gauge theories. The importance of anomalies will become even more apparent in the next chapter, where we shall study the non-perturbative effects of anomalies in the presence of topologically non-trivial field configurations.

The π° Decay Problem

By the mid-1960s the picture of the pion as a Goldstone boson associated with a spontaneously broken SU(2)⊗ SU(2) symmetry of the strong interactions had scored a number of successes, outlined here in Chapter 19. However, this picture also had a few outstanding failures. The most disturbing had to do with the rate of the dominant decay mode of the neutral pion, π0→ 2γ.

Most of this book has been devoted to applications of quantum field theory that can at least be described in perturbation theory, whether or not the perturbation series actually works well numerically. In using perturbation theory, we expand the action around the usual spacetime-independent vacuum values of the fields, keeping the leading quadratic term in the exponential exp(iI), and treating all terms of higher order in the fields as small corrections. Starting in the mid-1970s, there has been a growing interest in effects that arise because there are extended spacetime-dependent field configurations, such as those known as instantons, that are also stationary ‘points’ of the action. In principle, we must include these configurations in path integrals and sum over fluctuations around them. (In Section 20.7 we have already seen an example of an instanton configuration, applied in a different context.) Although such non-perturbative contributions are often highly suppressed, they are large in quantum chromodynamics, and produce interesting exotic effects in the standard electroweak theory.

There are also extended field configurations that occur, not only as correction terms in path integrals for processes involving ordinary particles, but also as possible components of actual physical states. These configurations include some that are particle-like, such as magnetic monopoles and skyrmions, which are concentrated around a point in space or, equivalently, around a world line in spacetime. There are also string-like configurations, similar to the vortex lines in superconductors discussed in Section 21.6, which are concentrated around a line in space or, equivalently, around a world sheet in spacetime.

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.