This book originated as a symposium at the American Association for the Advancement of Science annual meeting in San Francisco in 1989. The topic, The Farthest Things in the Universe, suggested itself to me as the most interesting and significant topic that people could hear about. An earlier AAAS Symposium had led to a book, The Redshift Controversy, that was still in use, and we hope that this volume will prove itself of similarly lasting interest.

Two of the original speakers, Hyron Spinrad of the University of California at Berkeley, and Patrick Osmer, then of the National Optical Astronomy Observatories, revised their pieces to bring them up-to-date for inclusion in this book. Further, Ed Cheng of the COBE Science Team and NASA's Goddard Space Flight Center agreed to write a new piece for inclusion in the book. We appreciate his taking time during the period of his duties as Chief Scientist for the Hubble Space Telescope's repair mission to complete his piece. During the interval from the time of the symposium to the present, the Cosmic Background Explorer spacecraft was launched and has had its tremendous successes in showing that the Universe has a blackbody spectrum and in finding ripples in space that may be the seeds from which galaxy-formation began. Thus this book appears at an optimum time.

The technical ability of astronomers to obtain images and spectra of very faint galaxies has improved greatly over the last decade. Since galaxies are vast collections of gas and stars, they must physically evolve with time. We should be able to directly observe the time-evolution of galaxies by studying very distant systems; the look-back internal corresponding to the mostdistant galaxies known in 1992 now approaches 15 billion years (80% of the total expansion age of the Universe)!

The line spectra of these faint galaxies are invaluable for redshift determination and physical study. The realization that Ly α (121.6 nm), formed in neutral hydrogen gas, is a strong emission line in most active galaxies and perhaps normal star-forming galaxies also, has helped us measure much larger redshifts in 1987–92 than was previously possible. Recall that this wavelength is in the ultraviolet; it can be observed only by satellites. But when galaxies are very far away, their Doppler effect shifts this spectral line into the region of the spectrum that we can observe with large telescopes on Earth. The largest redshifts for radio galaxies now approach z=3.8. Differing selection effects control which galaxies can be seen/isolated that far away. At least some red galaxies must form at redshift zf>5 (where the subscript f stands for the epoch of star formation).

When we look out into space at night, we see the Moon, the planets, and the stars. The Moon is so close, only about 380000 kilometers (240000 miles) that we can send humans out to walk on it, as we did in the brief glorious period from 1969 to 1972. Even the planets are close enough that we can send spacecraft out to them, notably the Voyager spacecraft, one of which has passed Neptune. Whereas light and radio signals from spacecraft take only about a second to reach us from the Moon, the radio signals from Voyager 2 at Neptune took several hours to travel to waiting radio telescopes on Earth. We say that the distance to the Moon is 1 light-second and the distance to Neptune is several light-hours.

Aside from our Sun, the nearest star at 8 light-minutes away, the distances to the stars are measured in light-years. The nearest star system is Alpha Centauri, visible only in the southern sky, and the single nearest star is known as Proxima Centauri, about 4.2 light-years away. We know so little about the stars that new evidence in 1993 indicates that Proxima Centauri might not be a member of a triple-star system along with the other parts of alpha Centauri, as has long been thought. The speeds at which those stars are moving through space may be sufficiently different that Proxima is only temporarily near Alpha's components.

Looking up at the clear night sky, it is hard to avoid wondering about the many objects that we can see. It is simple to recognize with the naked eye that there are planets, countless stars, and the band of light from the disk of our own Galaxy, the Milky Way. With the help of binoculars or a small telescope, the complexity of the scene increases dramatically, and it becomes apparent that the glow of the Milky Way is the light from many faint stars. We also start to notice that there are numerous faint and fuzzy objects which are the nearby galaxies and the star-forming regions in our own Galaxy. Probing with more and more sophisticated instruments, the level of detail and structure that can be resolved using visible light increases until the light becomes so exceedingly faint that even the best detectors on the largest telescopes see only darkness. This is the regime of the farthest objects in the Universe.

Before discussing these objects in any detail, I would like to take a brief moment and address the question of how we can possibly know about things so remote in both distance and experience. After all, we invent and test the physical sciences here on Earth by making experiments, interacting with the world around us, and creating a system of beliefs (theories) that ties all these experiments together into a consistent and testable story.

The central aim of this book is the development of the results and techniques needed to determine when it is possible to extend a space-time through an “apparent singularity” (meaning, a boundary-point associated with some sort of incompleteness in the space-time). Having achieved this, we shall obtain a characterisation of a “genuine singularity” as a place where such an extension is not possible. Thus we are proceeding by elimination: rather than embarking on a direct study of genuine singularities, we study extensions in order to rule out all apparent singularities that are not genuine. It will turn out, roughly speaking, that the genuine singularities which then remain are associated either with some sort of topological obstruction to the construction of an extension, or with the unboundedness of the Riemann tensor when its size is measured in a suitable norm.

I had at one stage hoped that there would be a single simple criterion for when such an extension cannot be constructed, which would then lay down once and for all what a genuine singularity is. But it seems that this is not to be had: instead one has a variety of possible tools and concepts for constructing extensions, and when these fail one declares the space-time to be singular on pragmatic grounds. The main such tools are the use of Hölder and Sobolev norms of functions, used for measuring the extent to which the metric or the Riemann tensor is irregular.

Introduction

Although many of our considerations will be purely geometrical, treating space-time as a pseudo-Riemannian manifold and asking whether or not this geometrical structure is breaking down, it must always be remembered that we are really working with a physical theory, governed by particular physical equations for fields and particles, and that it is the breakdown of the physics that is primarily of interest. The breakdown of the geometry is simply one possible manifestation of the breakdown of the physics.

Unfortunately there is a conflict between the mathematical contexts appropriate to, on the one hand, geometry and, on the other hand, physically significant differential equations. In differential geometry one deals with geodesies, domains of dependence and so on. For this to be valid one requires that the connection should satisfy a Lipshitz condition, which ensures the existence of unique geodesies and normal coordinate neighbourhoods. Providing this holds, the differentiability of the metric has little geometrical significance and it is customary to require it to be C∞ for convenience. By contrast, in the study of hyperbolic differential equations (a type to which Einstein's equations belong) questions of differentiability are crucial. The differentiability chosen reflects the character of the solutions allowed: by choosing a low differentiability one admits solutions like shock-waves or impulse-waves which may be very significant; conversely, by choosing too high a level of differentiability one will brand as “singular” shock-wave solutions that from the point of view of fluid dynamics may be entirely legitimate.

In the first chapter we defined a singular space-time as one containing incomplete inextendible curves that could not be continued in any extension of the space time. We must now give the definition (at times already anticipated) of the noun “singularity”. The fundamental idea is that space-time itself (the structure (M, g)) consists entirely of regular points at which g is well behaved, while singularities belong to a set ∂M of additional points – “ideal points” – added onto M. We denote the combined set M ∪ ∂M by ClM, the closure of M, and define the topology of this set to be such that phrases like “a continuous curve in M ending at a singularity p in ∂M”, or “The limit of R as x tends to a singularity p is …” all have meanings corresponding to one's intuitive picture of what they ought to mean.

The construction can be carried out in various ways and the set of ideal points, ∂M, could contain points other than singularities. Two important classes of ideal points that are not singularities are

1. endpoints of incomplete inextendible curves that can be continued in some extension of M (such endpoints being called regular boundary points) and

2. points “at infinity” such as I+.

If the construction is carried out in such a way that ClM consists only of singularities and points of type (1) then ∂M will consist precisely of the endpoints of all incomplete curves.

In this chapter I shall describe various situations in which it is possible to extend through a boundary point; in these cases the boundary point is not a singularity. As has been explained, we are proceding by elimination, so that the remaining cases must either be regarded as genuine singularities, or be amenable to extension by more powerful means than used here. There is no absolute criterion for what sorts of extension are “legitimate”, and hence no absolute criterion for what is, and what is not, a singularity.

Spherical symmetry

In this situation (which is of considerable interest because of the ease of obtaining exact solutions) it is possible to prove the existence of extensions under weaker assumptions than is normally the case. The results are thus not only of interest in their own right, but may be an indication of the “best possible” results that might be obtainable in the general case.

Definition of the problem

We shall be dealing with a space-time in which the rotation group SO(3,ℝ) acts transitively on spacelike 2-surfaces. So through every point p of the space-time there exists in a neighbourhood of p a totally geodesic timelike 2-surface S orthogonal to the orbits of the group; the surfaces maximal with respect to these properties through a given group orbit are equivalent.