The Trouble With Black Holes Part 1: A Brief History of Singularities

In this brief series of articles I aim outline the current state of knowledge in the theory of black holes. My particular focus is in the area of singularities and the ongoing efforts to remove them and the consequent breakdown in predictability they imply from models of physical black holes. This first article is written from a historical point of view and aims to set the scene for subsequent articles which will present the scope of the problem and the proposed solutions to it.

Introduction 

Since the early work of Karl Schwarzschild on spherical solutions to Einstein's gravitational field equations, the notion of a black hole has presented a unique set of challenges to modern physics. They often form in the highly collapsed cores of massive dying stars, where gravity is so extreme that even light, moving at its phenomenal speed, cannot escape to reach the eyes of some distant onlooker. This simple fact means that the conventional tools of science, observation and experimentation, appear to have little chance of finding a foothold on the interior of a black hole, and a precise description of their exact nature is, by necessity, entirely theoretical.

For this reason, the question of the very existence of black holes divided scientific opinion for decades. Many, including Einstein and his Soviet couterpart Lev Landau, believed that technicalities in the collapse of stars would prevent them from forming at all. Later theorists, including Stephen Hawking, conjectured that they not only exist, but are remarkably common within observable Universe. Their mysterious nature, coupled with their reputation for absolute destructive power, ensured that black holes captured the public imagination from the 1970s onward. Black hole physics became an endeavour of high technology and equally high expense, with many international scientific collaborations, among them LIGO, the Event Horizon Telescope, and the Large Hadron Collider, including the investigation of these strange objects in their research programs. 

The study of mathematical models of black holes continues to be a favoured occupation of theoreticians, as well, on account of how they blur the lines between the sub-microscopic physics of the quantum and the macroscopic physics of classical gravity and thermodynamics. A successful description of gravity within a quantum mechanical framework (a so-called quantum theory of gravity) is suspected to be the resolution to a number of puzzles that exist within the classical description of black holes, most pertinently the so-called information paradox and the problem of the central singularity that has plagued the black hole field equations since their discovery.

Einstein, Schwarzschild, and the General Theory of Relativity

The prehistory of black hole physics dates back as far as the eighteenth century. John Mitchell, an English astronomer, proposed, in a letter dated 1784, the concept of a star with such immense gravity that its escape velocity, the speed an orbiting body would have to exceed to leave the gravitating influence altogether, would be greater than the speed of light. Mitchell’s reasoning was flawed only in that it was based on Newtonian ideas, ill equiped for high speeds and strong gravity, but the notion later resurfaced in the early twentieth century on a sounder foundation: Einstein’s General Theory of Relativity; a theory of classical (non-quantum) motion containing the most robust description of gravity currently known to science.

The general theory of relativity represented a radical reinvisioning of gravity as geometry, specifically the geometry of spacetime. Spacetime is a four-dimensional mathematical construct consisting of the three dimensions of space unified with time by the speed of light, the fundamental upper limit to the speed of relative motion and thus of the propogation of information through the Universe. All events can be uniquely described as a point in spacetime and, in the absence of gravity and other forces, all processess move between spacetime events along smooth, straight paths, as defined by Einstein's 1905 special theory of relativity. The general theory of relativity conjectures that gravity is intrinsic curvature of spacetime. Just as parallel lines drawn perpendicularly from the the equator of the Earth will be drawn closer to together by the curvature of the planets surface, eventually converging at the poles, the curvature of spacetime draws distributions of matter and energy together in the manner that appereared to Newton to be a force of mutual attraction. The presence of matter or energy, in turn, generates this curvature about itself in space and time. Objects in curved spacetime deviate from their usual linear trajectories into the elliptical orbits of planets about the sun or the freefalling paths of tragically ill-placed cups of tea. As American theoritician John Wheeler put it, decades later, "spacetime tells matter how to move, matter tells spacetime how to curve". Curved spacetime acts, in popular language, as the gravitational field, but General Relativity is unique among field theories in that it speaks directly of the causal structure of the Universe; the "fabric of reality" as it were, rather than just the events that take place against that background.

The general theory of relativity is succinctly summarised in a series of ten equations that can be written, through the use of suitably compact notation (and a metric convention that sets the speed of light to one), in a single line:

Rab − ½ Rgab = 8𝝅GTab 

Einstein first presented these field equations of gravitation to the German Academy of Sciences in a series of lectures, culminating in December 1915. At that time, complicated and non-linear as they are, he did not believe a simple analytic solution to them would be found. 

Several months later however, a solution was found by his fellow countryman Karl Schwarzschild (and independently by Johannes Droust). Employing simplifying assumptions, including spherical symmetry (that a field-generating object like a planet or a star is exacly a sphere, and therefore that the field is the smooth and regular) and that the space a field occupies is mostly vacuum leading the stress-energy tensor (the right-hand side of the field equations) to vanish, Schwarzschild was able to construct the simplest working model of gravitational field surrounding an object like the Sun.

However, while it worked well to describe the behaviour of gravitating bodies, such as the motion of the planets, Schwarzschild’s solution predicted singularities at two distinct radii: the centre of the spherically symmetric coordinate system (at the heart of the star), and at the so-called Schwarzschild radius.

Singularities

Singularities can be thought of as regions of spacetime for which certain quantities avalanche to infinity. The standard mathematical tools of physics are ill-equiped to handle infinitely large parameters, and as things in the real world tend to be measurably finite, singularities are invariably things to be avoided in a theory that means to act as a reasonable description of nature. They may indicate that the theory has broken down and can no longer predict the behaviour of physical systems. They appear in relativity in two distinct flavours: coordinate singularities and curvature singularities. 

Coordinate singularities are artefacts of an unsuitable choice of coordinate system, the arbitrary artificial background against which the action of physics occurs, used to quantitively define intervals in distance and time (for example, the longitude-latitude-altitude system coupled with Greenwich Mean Time). A simple change of coordinate basis will remove such a singularity from a system of equations. This is similar to the way a flat map of the Earth cannot represent the north and south poles or, if it does, will leave at least one other point similarly ill-defined. Of course this doesn't mean that the cartographer was unaware of the existence of these points, only that they have chosen a basis ill-suited to representing them. 

Curvature singularities are a far more sinister matter, however. These are singularities in the mathematical objects used to represent the curvature of spacetime and, thus, the strength of gravitational field - something with, in principle, directly measurable physical significance. As an infinitely large measurable quantity isn't something that any reasonable description of nature can be thought to entertain, a curvature singularity is clear evidence of a defect in a theory. 

Most scientists ignored the apparent flaws in Schwarzschild's spacetime. For a star with a mass comparable to that of the sun, all of the offending points would be buried deep inside the object’s structure, where the vacuum solution would cease to be valid anyway and some new solution, with a non-zero stress energy tensor, would be required to accomodate the distribution of matter and energy within the solar interior. Some continued to speculate on the nature of the surface at the Schwarzschild radius, however. In 1924, British astronomer Arthur Eddington was able to show that a change of coordinates (to what are now known as Eddington-Finklestein coordinates) would lead to the disappearance of the singularity. Georges Lemaitre would be the first to comment, in 1933, that this suggested that this singularity was merely a coordinate singularity rather than a physically meaningful result. The suspicion that the same might be true of the central singularity, on the other hand, would soon prove false. 

Collapsing Stars

Research into the collapsed Schwarzschild solution continued to be something of a speculative backwater for most of the early twentieth century. The question seemed to many to be too abstract to be considered a legitimate avenue of research. This attitude was based in the widespread belief that no object with the density required to trap light could form in a physically realistic scenario. At this stage, the mechanisms of star collapse were too poorly understood. This began to change in 1931 when the Indian physicist Subrahmanyan Chandrasekhar applied quantum statistics to the problem, demonstrating that a static distribution of so-called electron-degenerate matter (matter for which no two particles could share a quantum state) of mass exceeding 1.4 times the mass of the sun would not form a stable object, but rather undergo runaway collapse under self-gravity. This mass upper-bound mass became known as the Chandrasekhar limit.

A number of respected physicists, including Einstein, Eddington, and Landau, remained unconvinced, however, believing that angular momentum in a more realistic rotating collapse model, or some unknown short-range repulsive force could be invoked to stabilise the star. Indeed, many white dwarf stars with masses slightly higher than the Chandrasekhar limit can collapse into stable neutron star configurations. The question was definitively settled by, among others, the American J.R. Oppenheimer, who was able to show that even neutron stars with masses in the range 1.5 to 3 times the mass of the sun (now known as the Tolman-Oppenheimer-Volkoff limit) would collapse in the manner described by Chandrasekhar. Runaway collapse seemed to be the inevitable fate of a sufficiently massive star - no new repulsive forces could be invoked to save them.

The Event Horizon

With the possibility of highly-collapsed gravitating objects now seeming more realistic, the apparent pathology in Schwarzschild spacetime metric could no longer be ignored. Oppenheimer and his colleagues were the first to tackle the question of the nature of the surface at the Schwarzschild radius. They suggested that it was the set of points at which time stopped for any approaching object.

It is well known that the gravitational field, described in terms of geometrical spacetime, has interesting implications for the experience of time and space for different observers. This is becuase observers taking different paths through curved spacetime are exposed to different amounts of 'time' and 'space' while preserving certain relationships among themselves in spacetime. This means, for example, that from the point of view of an astronaut on the International Space Station life on Earth appears to be moving more slowly than in their own cabin. In this case the difference is negligible (the astronaut gains about a hudredth of a second per year) as the Earth's gravity is relatively weak. In the presence of much stronger gravity, such as that produced by the objects described by the collapsed Schwarzschild solution, however, the difference in the rates of passage of time between a distant observer and one falling towards the object becomes more significant until the latter reaches the Schwarzschild surface where, Oppenheimer and his colleagues argue, time appears to stop altogether. An impression of the object becomes frozen on the  surface: a ghostly after-image, gradually reddening and fading from view as the waves of light from which the image is assembled, streteched by the extreme gravity at the horizon, escape this point of no return in diminishing frequency. 

True to the spirit of relativity, Oppenheimer's point of view is valid for distant observers, but not those crossing the horizon. A simple change over to Eddington-Finklestein coordinates would show that an in-falling observer could reach and even cross the surface in a finite time. Such observers would ascribe no particular significance to the horizon at all. 

 Oppenheimer’s results were extended by David Finkelstein and Martin Kruskal, the former of whom was the first to identify the Schwarzschild radius with the term "event horizon". In Finkelstein’s own words, the Schwarzschild surface is "a perfect, unidirectional membrane. Causal influences can pass through it in one direction only". Nothing, not even rays of light, could cross the horizon from the inside. 

 The Golden Age of General Relativity and the Birth of Black Hole Astrophysics

Many new developments in astronomy, including the 1967 discovery of a "pulsar" (a then enigmatic pulsing source of radio waves in deep space) by Oxford’s Jocelyn Bell-Burnell,  identified two years later as the first observed example of a neutron star, spurred interest in the study of highly-collapsed objects and legitimised General Relativity as an empirical science. This period became known as "the Golden Age of General Relativity". Around about this time the term "black hole" entered scientific vocabulary in reference to the objects described by the collapsed Schwarzschild solution. The term, widely believed to have been coined by Wheeler, aimed to create an impression of how these objects would appear (or rather fail to appear) in the night sky. It was also possibly a macabre allusion to an infamous Calcuttan prison from which there was said to be no escape.

Other solutions to Einstein’s Field Equations that could describe black holes were soon found including a rotating solution, discovered by Roy Kerr in 1963, and a charged, rotating solution, discovered by Ezra Newman in 1965. Studying the Kerr-Newman solution suggested that a black hole appeared to be completely specified by just three parameters: mass, charge, and angular momentum. The proposal, by Werner Israel, Brandon Carter, and David Robinson, that this was universally true of physical black hole systems became known as the "No Hair" Theorem. It implied both a remarkable simplicity in the nature of black holes, from the point of view of General Relativity, as well as troubling loss of the individual character of the star from which the black hole was formed. As a tool of expedience, the no-hair theorem was unparalled. A great deal of black hole simulation could be boiled down to manipulating these three numbers to model different kinds of parent star. However, taken at face value, the no-hair theorem implies that a large amount of information, the immutable currency of quantum physics, was destroyed by the process of stellar collapse. They may not have then realised it at the time, but these theorticians were begining to reach the limits of what classical gravity could say about black holes.
 

The Singularity Theorems

Despite the pace of development of black hole physics in the latter half of the twentieth century, the problem of the central singularity never disappeared as some initially believed it would. In fact, an important piece of work by two eminent British physicist served to further crystallise the issue.

 Unlike the apparent singularity at the event horizon hyper-surface, the central singularity is not the result of an inappropriate choice of coordinate system. The divergence in question arises when attempting to evaluate curvature scalars, such as the so-called Kretschmann scalar (a contraction of the Riemann curvature tensor with itself in technical jargon), at the centre of the black hole. These quantities directly relate to the curvature of spacetime - a measurable physical quantity that should have a finite value. It was possible to imagine that some flaw existed in the assumptions used in constructing the Schwarzschild model; an inappropriate symmetry condition, for example, and that the singularity would not occur in a generic situation. This view was discredited in the 1960s by the Penrose-Hawking Singularity Theorems.

The singularity theorems are based on the concept of "geodesic incompleteness". A spacetime is said to be geodesically incomplete if worldlines, the spacetime trajectories of physical objects, terminate at some proper time (i.e. the time measurably experienced by the object itself in its passage through spacetime for a "time-like" path such as that of a massive particle moving at sub-light speeds) or "affine parameter" (for a "null" path such as that of a photon of light in vacuum, which have no direct experience of time in the ordinary sense and must therefore have some different label for their passage through spacetime) beyond which the history of an object cannot be further extrapolated. These points are the singularities. Spacetime becomes ill-defined at a singularity, appearing as geometrically impossible object in a self-consistent four-dimensional space, similar to the manner in which the Penrose Staircase, for example, is an impossible object in three-dimensions (note that this definiton is equivalent to that given earlier in terms of curvature tensors. This new definiton stresses the logical self-consistency of relativity theory in the language of spacetime geometry).

 
Oxford’s Roger Penrose was able to use this definition to show that a singularity always forms whenever an event horizon does, provided a certain reasonable (albeit weak) energy condition was satisfied: that gravity always draws rays of light together, and never causes them to diverge - something that is valid for any matter with non-negative energy density (all matter currently known to science). Stephen Hawking, then a PhD student at Cambridge, later applied Penrose’s ideas to the global (cosmological) structure of spacetime, arguing that the begining of time at the Big Bang must have been a singularity according to classical theory. Hawking assumed a "dominant energy condition" that was stronger than Penrose’s: that energy of a field is always larger than pressure it exerts - something that is true for nearly all matter (except the vacuum expectation value of a scalar field such as the Higgs field). It doesn't seem that this "initial singularity" can be avoided even in inflationary cosmologies.

The work of Penrose and Hawking showed that singularities are an inevitable feature of certain classical spacetimes. These singularities will lie either in the absolute past of all worldlines (as is the case of the initial Big Bang singularity) or in the absolute future of any worldline that crosses an event horizon (as is true of black hole solutions). This is a manifestation of what later became known as the 'cosmic censorship principle' sometimes whimisically paraphrased "God abhors a naked singularity". The only saving grace for classical theory is that its catastrophic breakdown seems to be decently hidden from the view of a distant observer in almost every case. For black holes, however, an in-falling observer will always reach the singularity in finite proper time - an encounter with a physical singularity is unavoidable, even if a living observer will not survive to reach it.

 The singularity theorems provide the clearest known demonstration of the incompleteness of General Relativity as a theory of nature. Clearly, Einstein's conception of gravity must be extended to account for new phenomena at the heart of a black hole; what we currently call a singularity is very probably a new kind of physics. Given the scale on which it exists, large amounts of matter and energy compressed into unimaginably small dimensions by extreme gravity, that new physics will very probably be quantum mechanical in nature.  

 End of Part One.