The Mexican School on Nuclear Astrophysics was held in the Hotel Castillo Santa Cecilia, Guanajuato, México, from August 13 to August 20, 1997. The goal of the school was to gather together researchers and graduate students working on related problems in astrophysics – to present areas of current research, to discuss some important open problems, and to establish and strengthen links between researchers. The school consisted of eight courses and material presented in these forms the basis of this book.

Non–stop interaction between the participants, through both formal and informal discussions, gave the school a relaxed and productive atmosphere. It provided the opportunity for researchers from a wide range of backgrounds to share their interests in and different perspectives of the latest developments in astrophysics.

The productivity of the meeting reflected the strong interest of the Mexican and Latin American scientific communities in the subjects covered, Indeed, a second school is planned for 1999.

Professor David Schramm very sadly died not long after the conference, in December 1997. His lectures at the School were fascinating. He will be sorely missed by us and the rest of the astrophysics community.

A brief review of key concepts in multifrequency observational astronomy is presented. The basic physical scales in astronomy as well as the concept of stellar evolution are also introduced. As examples of the application of multifrequency astronomy, recent results related to the observational search for black holes in binary systems in our Galaxy and in the centers of other galaxies is described. Finally, the recently discovered microquasars are discussed. These are galactic sources that mimic in a smaller scale the remarkable relativistic phenomena observed in distant quasars.

Introduction

There have been many outstanding observational and theoretical discoveries made in astronomy during the twentieth century. However, in the future this ending century will most probably be remembered not by these achievements, but by being the time when astronomers started observing the Cosmos with a variety of techniques and in particular when we started to use all the “windows” in the electromagnetic spectrum.

During our century we started to investigate systematically the Universe using:

• The whole electromagnetic spectrum. At the beginning of the century, practically all the data was coming from the visible photons (that is, those detected by the human eye) only.

• Cosmic rays. These charged particles hit the Earth's atmosphere and can be detected by the air showers they produce. The origin of the most energetic cosmic rays (1019 ergs or more) remains a mystery.

• […]

Introduction, formalism, and cosmological bound

In these four lectures I will present a brief and rather elementary description of the physics of massive neutrinos as it emerges from studies involving nuclear physics, particle physics, astrophysics and cosmology. The lectures are meant for physicists who are not experts in this field, which I believe covers most of the participants in this School, and many potential readers elsewhere. I hope that such readers can find here enough information that they will be able to understand and appreciate the connection between the hunt for neutrino mass and mixing described here, and their own field of expertize.

Throughout I will use original references sparingly. Instead I refer to several monographs, written and published during the last decade [Boehm & Vogel (1992), Kayser, Gibrat–Debu k Perrier (1989), Winter (1991), Mohapatra & Pal (1991), Kim & Pevsner (1993), Klapdor–Kleingrothaus & Staudt (1995)] where an interested reader can find references to the original papers. When appropriate I will also refer to review papers on various aspects of the neutrino mass or related topics. For the experimental data, including the list of the most recent original experimental papers, the best source is the Review of Particle Physics, periodically updated, with the latest printed version in PDG (1996). The update of this very useful publication is available even between printed editions on the World–Wide Web at http://pdg.lbl.gov/.

The mechanism for a core–collapse or type II supernova is a fundamental unresolved problem in astrophysics. Although there is general agreement on the outlines of the mechanism, a detailed model that includes microphysics self–consistently and leads to robust explosions having the observational characteristics of type II supernovae does not exist. Within the past five years supernova modeling has moved from earlier one–dimensional hydrodynamical simulations with approximate microphysics to multi–dimensional hydrodynamics on the one hand, and to much more detailed microphysics on the other. These simulations suggest that large–scale and rapid convective effects are common in the core during the first hundreds of milliseconds after core collapse, and may play a role in the mechanism. However, the most recent simulations indicate that the proper treatment of neutrinos is probably even more important than convective effects in producing successful explosions. In this series of lectures I will give a general overview of the core–collapse problem, and will discuss the role of convection and neutrino transport in the resolution of this problem.

Introduction

A type II supernova is one of the most spectacular events in nature, and is a likely source of the heavy elements that are produced in the rapid neutron capture or r–process. Considerable progress has been made over the past two decades in understanding the mechanisms responsible for such events.

If we wish to quantize (2+1)-dimensional general relativity, it is important to first understand the classical solutions of the Einstein field equations. Indeed, many of the best-understood approaches to quantization start with particular representations of the space of solutions. The next three chapters of this book will therefore focus on classical aspects of (2+1)-dimensional gravity. Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions.

In this chapter, I will introduce two fundamental approaches to classical general relativity in 2+1 dimensions. The first of these, based on the Arnowitt–Deser–Misner (ADM) decomposition of the metric, is familiar from (3+1)-dimensional gravity; the main new feature is that for certain topologies, we will be able to find the general solution of the constraints. The second approach, which starts from the first-order form of the field equations, is also similar to a (3+1)-dimensional formalism, but the first-order field equations become substantially simpler in 2+1 dimensions.

In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions. The next chapters will describe the resulting spaces of solutions in more detail. I will also derive the algebra of constraints in each formalism – a vital ingredient for quantization – and I will discuss the (2+1)-dimensional analogs of total mass and angular momentum.

The focus of the past few chapters has been on three-dimensional quantum cosmology, the quantum mechanics of spatially closed (2+1)-dimensional universes. Such cosmologies, although certainly physically unrealistic, have served us well as models with which to explore some of the ramifications of quantum gravity. But there is another (2+1)-dimensional setting that is equally useful for trying out ideas about quantum gravity: the (2+1)-dimensional black hole of Bañados, Teitelboim, and Zanelli introduced in chapter 2. As we saw in that chapter, the BTZ black hole is remarkably similar in its qualitative features to the realistic Schwarzschild and Kerr black holes: it contains genuine inner and outer horizons, is characterized uniquely by an ADM-like mass and angular momentum, and has a Penrose diagram (figure 3.2) very similar to that of a Kerr–anti-de Sitter black hole in 3+1 dimensions.

In the few years since the discovery of this metric, a great deal has been learned about its properties. We now have a number of exact solutions describing black hole formation from the collapse of matter or radiation, and we know that this collapse exhibits some of the critical behavior previously discovered numerically in 3+1 dimensions. We understand a good deal about the interiors of rotating BTZ black holes, which exhibit the phenomenon of ‘mass inflation’ known from 3+1 dimensions. Black holes in 2+1 dimensions can carry electric or magnetic charge, and can be found in theories of dilaton gravity. Exact multi-black hole solutions have also been discovered.

In this chapter, we shall concentrate on the quantum mechanical and thermodynamic properties of the BTZ black hole.

The universe in which we live is not (2+1)-dimensional, and the quantum theories described in this book are not realistic models of physics. Nor is (2+1)-dimensional quantum gravity fully understood; as I have tried to emphasize, many deep questions remain open. Nevertheless, the models developed in the preceding chapters can offer us some useful insights into realistic quantum gravity.

Perhaps the most important role of (2+1)-dimensional quantum gravity is as an ‘existence theorem’, a demonstration that general relativity can be quantized without any new ingredients. This is by no means trivial: there has long been a suspicion that quantum gravity would require a radical change in general relativity or quantum mechanics. While this may yet be true in 3+1 dimensions, the (2+1)-dimensional models suggest that no such revolutionary overhaul of known physics is needed. This does not mean that our existing frameworks are correct, of course, but it makes it less likely that major changes will come merely from the need to quantize gravity.

At the same time, (2+1)-dimensional quantum gravity serves as a sort of ‘nonuniqueness theorem’. We have seen that there are many ways to quantize general relativity in 2+1 dimensions, and that not all of them lead to equivalent theories. This is perhaps not surprising, but it is a bit disappointing: in the absence of clear experimental tests of quantum gravity, there has been a widely held (although often unspoken) hope that the requirement of self-consistency might be enough to guide us to the correct formulation.

Interest in (2+1)-dimensional gravity – general relativity in two spatial dimensions plus time – dates back at least to 1963, when Staruszkiewicz first showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. Over the next 20 years occasional papers on classical and quantum mechanical aspects appeared, but until recently the subject remained largely a curiosity.

Two discoveries changed this. In 1984, Deser, Jackiw, and 't Hooft began a systematic investigation of the behavior of classical and quantum mechanical point sources in (2+1)-dimensional gravity, showing that such systems exhibit interesting behavior both as toy models for (3+1)-dimensional quantum gravity and as realistic models of cosmic strings. Interest in this work was heightened when Gott showed that spacetimes containing a pair of cosmic strings could admit closed timelike curves; (2+1)-dimensional gravity quickly became a testing ground for issues of causality violation. Then in 1988, Witten showed that (2+1)-dimensional general relativity could be rewritten as a Chern–Simons theory, permitting exact computations of topology-changing amplitudes. The Chern–Simons formulation had been recognized a few years earlier by Achúcarro and Townsend, but Witten's rediscovery came at a time that the quantum mechanical treatment of Chern–Simons theory was advancing rapidly, and connections were quickly made to topological field theories, three-manifold topology, quantum groups, and other areas under active investigation.

Together, the work on point particle scattering and the Chern–Simons formulation ignited an explosion of new research.

The first-order path integral formalism of the preceding chapter allows us to compute a large number of interesting topology-changing amplitudes, in which the universe tunnels from one spatial topology to another. It does not, however, help much with one of the principle issues of quantum cosmology, the problem of describing the birth of a universe from ‘nothing’.

In the Hartle–Hawking approach to cosmology, the universe as a whole is conjectured to have appeared as a quantum fluctuation, and the relevant ‘no (initial) boundary’ wave function is described by a path integral for a compact manifold M with a single spatial boundary ∑ (figure 10.1). In 2+1 dimensions, it follows from the Lorentz cobordism theorem of appendix B and the selection rules of page 157 that M admits a Lorentzian metric only if the Euler characteristic χ(∑) vanishes, that is, if ∑ is a torus. If M is a handlebody (a ‘solid torus’), it is not hard to see that any resulting spacelike metric on ∑ must be degenerate, essentially because the holonomy around one circumference must vanish. The case of a more complicated three-manifold with a torus boundary has not been studied, and might prove rather interesting. It is, however, atypical.

To obtain more general results, we can imitate the common procedure in 3+1 dimensions and look at ‘Euclidean’ path integrals, path integrals over manifolds M with positive definite metrics. Since path integrals cannot be exactly computed in 3+1 dimensions, research has largely focused on the saddle point approximation, in which path integrals are dominated by some collection of classical solutions of the Euclidean Einstein field equations.

The approaches to quantization described in chapters 5–7, although quite different, share one common feature. They are all ‘reduced phase space’ quantizations, quantum theories based on the true physical degrees of freedom of the classical theory.

As we saw in chapter 2, not all of the degrees of freedom that determine the metric in general relativity have physical significance; many are ‘pure gauge’, describing coordinate choices rather than dynamics, and can be eliminated by solving the constraints and factoring out the diffeomorphisms. Indeed, we have seen that in 2+1 dimensions only a finite number of the ‘6 × ∞3’ metric degrees of freedom are physical. In each of the preceding approaches to quantization, our first step was to eliminate the nonphysical degrees of freedom, sometimes explicitly and sometimes indirectly through a clever choice of variables; only then were the remaining degrees of freedom quantized.

An alternative approach, originally developed by Dirac, is to quantize the entire space of degrees of freedom of classical theory, and only then to impose the constraints. In Dirac quantization, states are initially determined from the full classical phase space; in quantum gravity, for instance, they are functionals ψ[gij] of the full spatial metric. The constraints act as operators on this auxiliary Hilbert space, and the physical Hilbert space consists of those states that are annihilated by the constraints, acted on by physical operators that commute with the constraints.

The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The Weinberg–Salam model has successfully unified electromagnetism and the weak interactions, and quantum chromodynamics (QCD) has proven to be an extraordinarily accurate model for the strong interactions. While we do not yet have a viable grand unified theory uniting the strong and electroweak interactions, such a unification no longer seems impossibly distant. At the phenomenological level, the combination of the Weinberg–Salam model and QCD – the Standard Model of elementary particle physics – has been spectacularly successful, explaining experimental results ranging from particle decay rates to high energy scattering cross-sections and even predicting the properties of new elementary particles.

These successes have a common starting point, perturbative quantum field theory. Alone among our theories of fundamental physics, general relativity stands outside this framework. Attempts to reconcile quantum theory and general relativity date back to the 1930s, but despite decades of hard work, no one has yet succeeded in formulating a complete, self-consistent quantum theory of gravity. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics.

The obstacles to quantizing gravity are in part technical. General relativity is a complicated nonlinear theory, and one should expect it to be more difficult than, say, electrodynamics. Moreover, viewed as an ordinary field theory, general relativity has a coupling constant G1/2 with dimensions of an inverse mass, and standard power-counting arguments – confirmed by explicit computations – indicate that the theory is nonrenormalizable, that is, that the perturbative quantum theory involves an infinite number of undetermined coupling constants.

This appendix provides a quick summary of the topology needed to understand some of the more complicated constructions in (2+1)-dimensional gravity. Readers familiar with manifold topology at the level of reference or will not learn much here, although this appendix may serve as a useful reference. The approaches I present here are not rigorous: this is ‘physicists’ topology', not ‘mathematicians’ topology', and the reader who wishes to pursue these topics further would be well advised to consult more specialized sources. A good intuitive introduction to basic concepts can be found in reference, and a very nice source for the visualization of two- and three-manifolds is reference.

Mathematically inclined readers may be somewhat surprised by my choice of topics. I discuss mapping class groups, for example, but I largely ignore homology. In addition, I introduce many concepts in rather narrow settings – for instance, I define the fundamental group only for manifolds. These choices represent limits of both space and purpose: rather than giving a comprehensive overview, I have tried merely to highlight the tools that have already proven valuable in (2+1)-dimensional gravity.

Homeomorphisms and diffeomorphisms

Let us begin by recalling the meaning of ‘topology’ in our context. Two spaces M and N are homeomorphic – written as M ≈ N − if there is an invertible mapping f : M → N such that

1. f is bijective, that is, both f and f−1 are one-to-one and onto; and

2. both f and f−1 are continuous.

In the two preceding chapters, we derived solutions of the vacuum field equations of (2+1)-dimensional gravity by using rather standard general relativistic methods. But as we have seen, the field equations in 2+1 dimensions actually imply that the spacetime metric is flat – the curvature tensor vanishes everywhere. This suggests that there might be a more directly geometric approach to the search for solutions.

At first sight, the requirement of flatness seems too strong: we usually think of the vanishing of the curvature tensor as implying that spacetime is simply Minkowski space. We have seen that this is not quite true, however. The torus universes of the last chapter, for example, are genuinely dynamical and have nontrivial – and inequivalent – global geometries. The situation is analogous to that of electromagnetism in a topologically nontrivial spacetime, where Aharanov–Bohm phases can be present even when the field strength Fµν vanishes.

It is true, however, that locally we can always choose coordinates in which the metric is that of ordinary Minkowski space. That is, every point in a flat spacetime M is contained in a coordinate patch that is isometric to Minkowski space with the standard metric ηµν. The only place nontrivial geometry can arise is in the way these coordinate patches are glued together. This is precisely what we saw in chapter 3 for the spacetime surrounding a point source: locally, the geometry was flat, but a conical structure arose from the identification of the edges of a flat coordinate patch.

In the last chapter, we investigated two formulations of the vacuum Einstein field equation in 2+1 dimensions. In this chapter, we will solve these field equations in several fairly simple settings, finding spacetimes that represent a collection of point particles, a rotating black hole, and a variety of closed universes with topologies of the form [0, 1] ×∑. In contrast to (3+1)-dimensional general relativity, where it is almost always necessary to impose strong symmetry requirements in order to find solutions, we shall see that for simple enough topologies, it is actually possible to find the general solution of the (2+1)-dimensional field equations.

The reader should be warned that this chapter is not a comprehensive survey of solutions of the (2+1)-dimensional field equations. In particular, I will spend a limited amount of time on the widely studied point particle solutions, and I will say little about solutions with extended (‘string’) sources and solutions in the presence of a nonvanishing matter stress–energy tensor. The latter are of particular interest for quantum theory – they offer models for studying the interaction of quantum gravity and quantum field theory – but systematic investigation of such solutions has only begun recently, and they are not yet very well understood.

Point sources

As a warm-up exercise, let us use the ADM formalism of chapter 2 to find the general stationary, axisymmetric solutions of the vacuum field equations with vanishing cosmological constant. Such spacetimes are the (2+1)-dimensional analogs of the exterior Schwarzschild and Kerr metrics, representing the region outside a circularly symmetric gravitating source.

Having examined the classical dynamics of (2+1)-dimensional gravity, we are now ready to turn to the problem of quantization. As we shall see in the next few chapters, there are a number of inequivalent approaches to quantum gravity in 2+1 dimensions. In particular, each of the the classical formalisms of the preceding chapters – the ADM representation, the Chern–Simons formulation, the method of geometric structures – suggests a corresponding quantum theory.

The world is not (2+1)-dimensional, of course, and the quantum theories developed here cannot be taken too literally. Our goal is rather to learn what we can about general features of quantum gravity, in the hope that these lessons may carry over to 3+1 dimensions. Fortunately, many of the basic conceptual issues of quantum gravity do not depend on the number of dimensions, so we might reasonably hope that even a relatively simple model could provide useful insights.

After a brief introduction to some of the conceptual issues we will face, I will devote this chapter to a quantum theory based on the ADM representation of chapter 2. As we saw in that chapter, the ADM decomposition and the York time-slicing make it possible to reduce (2+1)-dimensional gravity to a system of finitely many degrees of freedom. Quantum gravity thus becomes quantum mechanics, a subject we believe we understand fairly well. This approach has important limitations, which are discussed at the end of this chapter, but it is a good starting place.