Black holes: Classical properties, thermodynamics, and heuristic quantization, in

I start with a discussion of the no-hair principle. The hairy black hole solutions of recent vintage do not deprive it of value because they are often unstable. Generic properties of spherical static black holes with nonvacuum exteriors are derived. These form the basis for the discussion of the new no scalar hair theorems. I discuss the generic phenomenon of superradiance for its own sake, as well as background for black hole superradiance. First I go into uniform linear motion superradiance with some examples. I then discuss Kerr black hole superradiance in connection with a general rotational superradiance theory with possible applications in the laboratory. Adiabatic invariants have played a weighty role in theoretical physics. I explain why the horizon area of a nearly stationary black hole can be regarded as an adiabatic invariant, and support this by examples as well as a general discussion of perturbations of the horizon. The horizon area’s adiabatic invariance suggests that its quantum counterpart is quantized in multiples of a basic unit. Consideration of the quantum analog of the Christodoulou reversible processes provides support for this idea. Area quantization provides a definite discrete black hole mass spectrum. Black hole spectroscopy follows: the Hawking semiclassical spectrum is replaced by a spectrum of nearly uniformly spaced lines whose envelope may be roughly Planckian. I estimate the lines’ natural broadening. To check on the possibility of line splitting, I present a simple algebra involving, among other operators, the black hole observables. Under simple assumptions it also leads to the uniformly spaced area spectrum. In these lectures I take units for which c = 1. Occasionally, where mentioned explicitly, I also set G = 1, but always display h̄. 1 No scalar hair theorems Almost thirty years ago Wheeler enunciated the Israel-Carter conjecture, today colloquially known as “black holes have no hair” [88]. This influential conjecture has long been regarded as a theorem by large sectors of the gravity-particle physics community. But by the early 1990’s solutions for stationary black holes with exterior nonabelian gauge or skyrmion fields [105, 27, 39, 62, 44] had led many workers to regard the conjecture as having fallen by the wayside. By now things have settled down to a new paradigm not very different from Wheeler’s original one. 1.1 Early days of ‘no-hair’ By 1965 the charged Kerr-Newman black hole metric was known. Inspired by Israel’s uniqueness theorems for the Schwarzschild and Reissner-Nordström black holes [52], and by Carter’s [33] and Wald’s [106] uniqueness theorems for the Kerr black hole, Wheeler anticipated that “collapse leads to a black hole endowed with mass and charge and angular momentum, but, so far as we can now judge, no other free parameters” by which he meant that collapse ends with a Kerr-Newman black hole. Wheeler stressed that other ‘quantum numbers’ such as baryon number or strangeness can have no place in the external observer’s description of a black hole. What is so special about mass, electric charge and angular momentum ? They are all conserved quantities subject to a Gauss type law. One can thus determine these properties of a black hole by measurements from afar. Obviously this reasoning has to be completed by including magnetic (monopole) charge as a fourth parameter because it also is conserved in Einstein-Maxwell theory, it also submits to a Gauss type law, and duality of the theory permits Kerr-Newman like solutions with magnetic charge alongside (or instead of) electric charge. In the updated version of Wheeler’s conjecture, the forbidden “hair” is any field not of gravitational or electromagnetic nature associated with a black hole. But why is the issue of hair interesting ? Black holes are in a real sense gravitational solitons; they play in gravity theory the role atoms played in the nescent quantum theory of matter and chemistry. Black hole mass and charge are analogous to atomic mass and atomic number. Thus if black holes could have other parameters, such ‘hairy’ black holes would be analogous to excited atoms and radicals, the stuff of exotic chemistry. By contrast, the absence of a large number of hair parameters would support the conception of simple black hole exteriors, a situation which is natural for the formulation of black hole entropy as the measure of the vast number of hidden degrees of freedom of a black hole. Indeed, historically, the no-hair conjecture inspired the formulation of black hole thermodynamics (for the early history see review [17]), which has in the interim become a pillar of gravity theory. Originally “no-hair theorems” meant theorems like Israel’s or Carter’s [52, 33] on the uniqueness of the Kerr-Newman family within the Einstein-Maxwell theory or like Chase’s [34] on its uniqueness within the Einstein-massless scalar field theory. Wheeler’s conjecture that baryon and like numbers cannot be specified for a black hole set off a longstanding trend in the search for new no-hair theorems. Thus Hartle [45] as well as Teitelboim [98] proved that the nonelectromagnetic force between two “baryons” or “leptons” resulting from exchange of various force carriers would vanish if one of the particles was allowed to approach a black hole horizon. I developed an alternative and very simple approach [11] to show that classical massive scalar or vector fields cannot be supported at all by a stationary black hole exterior, making it impossible to infer any information about their sources in the black hole interior. In modernized form this goes as follows. Start with the action