The theory we have constructed can be compared with observational results. But for the consideration of concrete processes, the theory in some cases should be detailed. In this chapter, we therefore make both a theoretical analysis, which allows us to obtain concrete calculational formulas, and a direct comparison of the theory with experiment. We analyse the structure of the active region (Section 7.1), the generation of electron–positron plasma and of high-frequency radiation (Section 7.2), the dynamics of a neutron star due to its current-induced deceleration (Section 7.3), the statistical distribution of pulsars (Section 7.4), the generation of radio emission (Sections 7.5 and 7.6) and, finally, nonstationary processes (Section 7.7).

The structure of the active region

The model of a partially filled magnetosphere

As shown in Chapters 3–5, the theory of plasma generation in the region of the double layer associates the pulsar rotation energy losses Wtot with the magnitude of the longitudinal electric current j∥ flowing in the magnetosphere (see (4.215)). This current is specified by the ‘compatibility relation’ (4.174) and by the pulsar ‘ignition’ condition (5.66). Owing to this, we can reconstruct the structure of the active region in the polar cap and determine important parameters of a neutron star as the magnitude of the magnetic field B0 from the observed quantities P and dP/dt.

It should be noted, however, that to be compared with observational data, the theory developed in Chapter 4 should be extended.

INTRODUCTION

The presumed breakdown of the general theory of relativity (GTR) at the Planck scale without, as yet, a complete quantum theory of gravity (QG) to replace it is used to motivate consideration of a simpler problem—the quantum mechanics of cosmological models. In a pioneering paper, Misner (1969a) coined the term “quantum cosmology” (QC) for the quantization of the dynamical system whose degrees of freedom describe a spatially homogeneous universe. He also introduced the term “minisuperspace” (MSS) (Misner 1972) for the finite-dimensional configuration space of the dynamics of homogeneous cosmologies—a finite subspace of Wheeler's “superspace” (Wheeler 1968), the space of all three-geometries. The first invocation of quantized MSS models was that of DeWitt (1967) to apply Dirac (1958, 1959) quantization of gravity to a tractable system. He considered the closed Friedmann-Robertson-Walker (FRW) model (see Misner et al 1973). Misner (1969a, 1970, 1972) considered the application of Arnowitt, Deser, and Misner (1962) (ADM) quantization methods to the Bianchi Type cosmologies (e.g. Ryan and Shepley 1975, MacCallum 1975). In both treatments, it became clear that issues of time, factor ordering, and interpretation which plague the canonical quantization of gravity survive in the truncated models. It was, of course, recognized that, while perfectly valid as classical solutions to Einstein's equations, quantized cosmologies where degrees of freedom have been zeroed by hand need have no relation to a true QG theory. [A systematic attack on the validity of the MSS “approximation” has only recently begun (Kuchař and Ryan 1986, 1989).] Misner argued (1969a) that one might reasonably expect the quantum universe to be dominated by the dynamics of its spatially homogeneous mode.

INTRODUCTION

The internal workings of a black hole constitutes a relatively new field, still in its first faltering steps. These pages are intended as a brief introduction and progress report on what I think has been gleaned so far.

As far as external appearances go, black holes are matchlessly simple objects. I have described elsewhere (Israel 1987; Thorne 1993) how Charles Misner was probably the first to fully appreciate this. In the wake of a gravitational collapse, quadrupole and other deformations originally anchored in the star get swallowed by the hole or carried off by gravitational radiation. The external field and event horizon settle, like a newly-formed soap bubble, into the simplest configuration that is compatible with the external constraints—mass, charge and angular momentum. (The soap-bubble analogy, I believe, stems also from Misner.)

The immaculate exterior of a black hole hides an egregious inner disorder. A solarmass black hole holds 10 times as much entropy as a ball of radiation of the same mass-energy and volume.

The key to this dichotomy is the peculiar causal structure of the hole. The event horizon not only encloses but precedes the inner core where curvatures approach Planck levels. The surface of a black hole, unlike the surface of a star, is not influenced by processes in the core.

INTRODUCTION

In China, I have learned, there is a valuable tradition: to appreciate and show great respect for one's teachers. Undoubtedly, the most successful method of teaching is by example. Certainly this was true in my case. Charles W. Misner was a pioneer in applying modern differential geometry (Misner 1964) especially differential forms (Misner and Wheeler 1957) to gravitational theory, in investigating the canonical Hamiltonian formulation of gravity, and in formulating suitable expressions for conserved quantities, in particular mass-energy (Arnowitt, Deser and Misner (ADM) 1962). I have, as this work indicates, at least to some extent, followed my teacher's directions.

The outline of this work is as follows. First, differential form methods are used to obtain a covariant Hamiltonian formulation for any gravitational theory. The Hamiltonian includes a covariant expression for the conserved quantities of an asymptotically flat or constant curvature space. Next the positive total energy test, a promising and appropriate theoretical test for alternate gravitational theories, is described. The final topic concerns application to Einstein gravity of new rotational gauge conditions and their associated special orthonormal frames. These frames provide a good localization of energy and parameterization of solutions.

COVARIANT HAMILTONIAN FORMALISM

There are significant benefits in the canonical Hamiltonian formulation of a theory, in particular the identification of constraints, gauges, degrees of freedom and conserved quantities (Isenberg and Nester 1980), however the usual approach (e.g., in Misner, Thorne and Wheeler (MTW) 1973 and ADM 1962) exacts a heavy price: the loss of manifest 4-covariance. Differential form techniques can be used to largely avoid this cost (Nester 1984).

In the Introduction to this volume we began with an appreciation of Charles Misner as a scholar and an educator, and we felt that it would be fitting to end it with a selection of reminiscences that would allow us to show something of Charlie Misner the man and the teacher. Few of the readers of this book in the year that it is published will not know the dry facts of his life, but since all books are at least a reach for immortality, we should consider the reader who may see it long after all of us are dust and give him or her the framework of a life on which so many memories rest.

Charles was born on June 13, 1932, attended Notre Dame University from 1948 to 1952, and received his Ph.D. from Princeton, where his advisor was John Wheeler, in 1957. He married Susanne Kemp in 1959 and has four children. From 1956 to 1963 he was an Instructor and then an Assistant Professor at Princeton. Since 1963 he has been on the faculty of the University of Maryland. He has been a visiting faculty member in universities and institutes throughout the United States and Europe. He is a fellow of the American Physical Society, the Royal Astronomical Society and the American Association for the Advancement of Science, and a member of the International Society on General Relativity and Gravitation, the Association of Mathematical Physicists, the American Mathematical Society, the International Astronomical Union, the Philosophy of Science Association and the Federation of American Scientists.

ABSTRACT

A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In “Copenhagen” quantum mechanics, probabilities can be assigned to histories of a subsystem that have been “measured”. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system's initial condition and dynamics. Such sets of histories are said to decohere. We define decoherence for closed systems in the simplified case when quantum gravity can be neglected and the initial state is pure. Typical mechanisms of decoherence that are widespread in our universe are illustrated.

Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact.

Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained.

I first met Charles Misner at the genial general relativity meetings that James Anderson and Ralph Schiller organized at Stevens Institute of Technology in the mid-‘50s. His birthday reminds me of our collaboration in topological physics in 1959, when we found the topological spin-statistics connection and gravitational kinks. Misner's contributions to the discovery of the relativistic theory of the black hole are not adequately appreciated. Let me say a little about these matters here.

They all hang on the thread of anomalous spin. The first anomalous spin was the spin ½ of the electron. It was anomalous in that the very possibility of spin ½ was initially overlooked by quantum theorists. Then experiment and Uhlenbeck & Goudsmit forced it to our attention. Wigner explained this spin by examining how the electron wavefunction behaved under the Wigner waltz W: a path in the rotation group describing a continuous rotation of the physical system through 2π. In three dimensions W cannot shrink continuously to a point (W is nontrivial), but W2, a 4π rotation, can (W2 is trivial). It followed that W can change quantum phase by π. This happens for spin ½.

In 1954, fresh out of graduate school, I set out to find a unified theory of the known particles and forces, as I imagined one was supposed to do. As an undergraduate my ambition was to carry out Von Neuman's quantum set theory program. What I describe next began a decade detour from that program, to which I later returned.

INTRODUCTION

In this article we want to describe a long-term research project that the authors plan to carry out over a period in the immediate future. We would like to outline the basic ideas of the project and give a few preliminary calculations the bear on the validity of our ideas as well as some speculations on where the research will lead. We chose this volume to do this because it occured to us that each of the components of our plan lies in a field that Charlie has touched on at some point in his varied career. These components are the Hamiltonian formulation of the gravitational field equations, path integral quantization, and quantum cosmology. Such a wide-ranging list shows how much we all owe to Charlie as a scientist and we hope that our efforts will demonstrate our debt to his lead in these fields.

The genesis of our paper lies in a long-term concern that both of us have had about the structure of quantum mechanics that has been the subject of years of blackboard discussions between us (the speculations of one of us [M.R.] go all the way back to fondly-remembered discussions with Charlie and other Charlie-students in Maryland). The focal point of our ideas has always been the cookbook nature of quantum mechanics as based on a Hamiltonian action functional for the classical equations of motion and canonical commutation relations derived from the Poisson bracket relations of the variables in the action.

Abstract

Misner [1] suggested kinetic theory ideas could explain the smoothing of the universe at early times. Recently [2], [3] the inverse has been investigated: the generation of Bianchi I anisotropy in cosmologies with an exact FRW geometry, again for kinetic theory reasons. At the time it was stated that more general anisotropy and inhomogeneity could be generated by the same methods. In this paper the more general case is addressed to show how the model can be produced and to identify some open questions connected with it.

Introduction

One of the major contributions Charles Misner has made to cosmology was the first serious investigation of the chaotic cosmology idea [1], namely that the universe started off in a very chaotic state and then developed (by physical processes) towards a smooth state. This idea has since been taken up with a vengeance by the Inflationary Universe school of thought, utilising Guth's insight [4] that the vast expansion associated with the false vacuum (a scalar field) could provide the required smoothing mechanism.

The resulting theory is very interesting but perhaps over-stated [5]. In particular it does not overcome the Stewart remark [6] that no matter what smoothing mechanisms one might find there are always initial data that will lead to universes more lumpy than the observed universe (simply run the equations backwards from any considered present state to find the needed initial conditions). Furthermore entropy arguments have been adduced by Penrose [7] to suggest that the universe must have started off in a smooth state rather than the very chaotic state suggested by many inflationists.