Penrose's work on singularities: Part 2

In Part 1 of this article, we saw that by 1965 there was theoretical evidence that a sufficiently massive star that is spherically symmetrical would collapse to a black hole after it has spent all its nuclear fuel. Since General Relativity postulates that spacetime is curved by the presence of matter and energy, it is expected that there would be a severe distortion of spacetime when the matter is compressed to a high density. The Oppenheimer-Snyder model of gravitational collapse showed that for a spherically symmetric, homogenous, and static distribution of dust (with no internal pressure) of a sufficiently large mass, there is no known mechanism to prevent the compression of the matter to infinite density. This would result in the formation of a black hole with an event horizon at a radius of 2GM/c^2 (Schwarzschild radius) and a singularity behind the event horizon. It is customary to rescale the dimensions so that G/c^2=1, so the Schwarzschild radius is simply 2M. The singularity represents a point of infinite curvature and also a point where all future-directed paths and light rays come to an end. In 1965 Roger Penrose proved that even if we did not make any assumptions of spherical symmetry, the geometrical constraints imposed by a very strong gravitational field will inevitably result in spacetime singularities,

As discussed in Part 1, Roger Penrose formulated his groundbreaking theorem in terms of five assumptions, which he showed were collectively inconsistent with each other. Four of those assumptions are based on assumptions of reasonable "niceness" of spacetime, namely a) The "Past" and "Future" condition, which states that there exists a consistent definition of past and future everywhere, b) The "Null Completeness" condition, which states that all paths built out of light rays can be extended indefinitely at all points and in all directions, c) The "Cauchy Hypersurface" condition, which assumes the existence of a non-compact connected Cauchy hypersurface (to be defined below), d) The "Null Energy" condition, which states that the local energy at any point is always non-negative. The fifth condition is the only one that pertains to the conditions expected near a black hole where gravity is so strong that all light rays are bent towards each other. Penrose's fifth assumption is e) The "Trapped Surface" condition, which states that there exists a 2-dimensional compact surface (like a sphere) for which all light rays emanating from the surface are bent towards each other.

Understanding the statement of the theorem and its proof entails absorbing a fair amount of geometry, topology, and terminology associated with the causal theory of spacetime and general relativity. In this article, I will attempt to explain the concepts in the simplest possible way without appealing to all the jargon that one would typically encounter in a rigorous exposition of the topic. We start with the causal theory of spacetime, which is fundamental to the entire subject.

Local Causal Structure of Spacetime

The origin of the causal theory of spacetime lies in Minkowski's reformulation of Einstein's special theory of relativity in terms of a four-dimensional spacetime. In special relativity, the speed of light occupies a very special place. Nothing can travel faster than the speed of light, a universal constant independent of any inertial observer's frame of reference. Points in the Minkowski model are events whose separation is measured by the Lorentz metric, a quantity that is invariant under Lorentz transformations. The causal theory of what events can influence other events and the domain of influence is a feature unique to the special theory of relativity and is not present in classical Newtonian mechanics.

A fundamental geometrical object when studying causal theory is the "light cone". Imagine lighting a candle at a point on Earth. If you ignore the effects of gravity, light from the candle will spread out radially in all directions in straight lines. These straight lines will sweep out a sphere in 3-dimensional space. Now if we suppress one of the spatial dimensions (say the Z-axis), then we can visualize this as an expanding circle. If we choose the vertical axis to represent time and suppress one of the space dimensions (because we cannot really visualize a 4-dimensional object!), then we will see that as time progresses vertically, the wavefront of light will spread farther and farther, on the surface of a cone. A similar cone can be envisioned going back in time.

The points in the interior and on the surface of the upper cone represent the causal future of a point (event) at the origin (at time t=0). Every event in the interior of the cone can be influenced by an object or signal traveling at a speed strictly less than the speed of light. Therefore, the interior of the cone is called the chronological future of the point at the origin. Points on the surface of the cone represent the boundary of the casual future. They are events that can only be influenced by signals traveling at the speed of light. The curves traced by such signals are called null geodesics the surface of the cone which is swept out by the null geodesics (light signals) is called a null hypersurface.

Minkowski spacetime is a "flat" manifold, meaning it has no curvature. It represents an idealized condition where one is not subject to any force or influence of any kind and all entities are moving at a uniform speed relative to each other (inertial frames of reference). For example, the spacetime very far away from any star could be approximated by Minkowski spacetime. In General Relativity, spacetime is curved by the presence of matter and energy. To quote John Wheeler, "matter tells spacetime how to curve, and spacetime tells matter how to move". The fundamental postulate of General Relativity known as the "Principle of Equivalence" states that near a point in spacetime the gravitational forces can be canceled ("transformed away by a coordinate change") by moving to a freely falling frame of reference. For example, if I am in a space station above the Earth I will experience a gravitational pull from the Earth. But if I step into a spacecraft and start freely falling in Earth's gravitational field, I will experience weightlessness. In geometric terms, this amounts to carrying out a coordinate transformation to a coordinate system wherein a small neighborhood spacetime looks and feels like Minkowski spacetime. Thus the Principle of Equivalence implies that as a manifold, spacetime is locally Minkowski at every point. So at every point, the local causal structure can be represented by a light cone. Depending on how spacetime curves due to the presence of matter, the light cones at different points might be pointing in different directions.

Timelike Curves

As a particle moves in spacetime, its trajectory in spacetime is represented by a worldline (also known as a timelike curve). At each point of that curve is a lightcone which represents the boundary of all the different spacetime directions in which a signal can travel. If the particle is a photon (quantum of light), then the curve is built out of light rays, and in that case, the lightcone is tangential to the curve at every point in spacetime. Such a curve is called a lightlike or null curve.

Lightlike (null) curves

So the concepts of timelike geodesics, null geodesics, chronological future, causal future, and the null hypersurface built out of the boundary carry over verbatim to curved spacetimes (also known as Lorentz manifolds). An essential ingredient for this causal analysis is assumption a) of Penrose's theorem namely the "Past" and "Future" assumption. It is essential that there be a consistent way to define past and future across the spacetime manifold to avoid pathologies.

For Penrose's theorem, it is important to consider not just the causal future of a point in spacetime, but the causal future of an entire "spacelike" surface in spacetime. Spacelike simply represents a slice of spacetime at a particular choice of time chosen uniformly across all points (the fact that you can do it is an assumption known as time orientability). In other words, a spacelike surface is just a region of space at a particular time. The spatial slice could not be curved.

The chronological, causal, and null future of a set satisfies some easily provable topological properties. The chronological future is an open set, meaning that for every event in the chronological future, you can find a "ball" of neighboring events in spacetime that reside in the causal future. Similarly, for any point that is on the boundary, every ball of neighboring events will have an event that is in the chronological future (interior of the causal future). Moreover, the boundary if nonempty is a closed 3-dimensional achronal C^0 submanifold of the 4-dimensional spacetime. Achronal means that no two points of the boundary can be joined by a timelike curve (worldline of an object traveling at a speed smaller than lightspeed). C^0 submanifold means that for each point on the boundary of the causal future, there is a 3-dimensional neighborhood of a point on the boundary that is topologically equivalent to an open ball in R^3 (Euclidean 3-space). In general, the boundary of the causal future will not be a smooth manifold as is evident from the lightcone and the boundary of a disconnected set.

The C^0 (topological) manifold structure by taking the so-called Riemann Normal coordinates of 4-dimensional spacetime around any point p on the boundary of a causal future. For a sufficiently small neighborhood, one can choose one of the coordinates to be timelike (since the neighborhood can be chosen to be Minkowski). The integral curves of the tangent vector of this coordinate will intersect the boundary in exactly one point because the boundary is achronal (no two points are joined by timelike curves). So the remaining 3 coordinates can be used to define a homeomorphism to R^3.

Such considerations will become important when we discuss Cauchy surfaces, trapped surfaces, and the proof of the Penrose theorem.

Global Causal Structure of Spacetime

Penrose theorem relies heavily on certain global assumptions about spacetime. Locally it is fairly clear what is happening topologically in spacetime given that it is a Lorentz manifold by the General Theory of Relativity. But when you stitch together these locally Minkowski spacetime neighborhoods, the resulting spacetime could have all sorts of pathological conditions. But when we look around with our telescopes we don't see any pathologies in spacetime. It is important to make an assumption that the spacetime starts out being nice and smooth and then determine what happens when gravity becomes too strong. For example, an "asymptotically flat" spacetime is consists of a 3-dimensional space that extends out to infinity where the gravitational field (aka the curvature of spacetime) becomes negligible far away from the source of the field (typically a massive object such as a star). Near the source spacetime is curved but far away from the source spacetime is almost "flat" (hence the name "asymptotically flat").

Asymptotically flat spacetime

The Cauchy Hypersurface condition satisfies a global niceness condition. It states, that there is an initial connected (not broken up) smooth 3-dimensional space that is spread out infinitely from which all of the spacetime can be developed in a well-defined fashion. In fact, the assumption is that the entire spacetime can be "built" out of slices of spacetime at each instance of time. Asymptotically flat spacetime is a perfect example, of a spacetime satisfying the Cauchy Hypersurface condition.

Cauchy Surfaces

A spacetime that satisfies the Cauchy surface condition has some nice properties. In fact, the technical definition of a Cauchy surface is a surface having the property that every timelike curve (a curve that is pointing in the chronological future of an event) will intersect it. It turns out (from the work of Choquet-Bruhat and Geroch) that spacetimes can be built smoothly from Cauchy surfaces. In addition, such spaces are also known to be "Globally Hyperbolic". Without getting too technical, it simply means that in such a spacetime you cannot go back in time (no closed timelike geodesics) and that there are no "holes" or gaps in the spacetime (the intersection of the causal future of an event p and the causal past of another event q that lies in the causal future of p is compact). It turns out that all Cauchy surfaces are topologically equivalent (homeomorphic to each other).

An intuitively obvious but crucial consequence of the Cauchy surface condition is that every point on the trajectory of a light curve in spacetime can be traced back to a point on the Cauchy surface using a timelike curve.

In this picture, you have two Cauchy hypersurfaces and a light signal that goes from event P1 in one Cauchy surface to event P2 in the other. But the point P2 is also the evolution of a point that is the intersection of the perpendicular timelike curve with the first Cauchy surface. The timelike curves that are perpendicular to each Cauchy surface define a homeomorphism (1-1 topological equivalence) between the two Cauchy surfaces. We saw earlier, that the boundary of the causal future of a spacelike surface is an achronal C^0 manifold generates by null geodesics (light rays as above). The timelike curves coming down from a point P2 on the boundary to the Cauchy surface Sigma_1 will map open sets to open sets, so it is a homeomorphism onto its image. This fact will become important in the proof of Penrose's theorem.

Raychaudhuri's focusing equation

Until now we have not really discussed the effect of gravity on light rays and the curvature of spacetime. The first and most famous verification of the General Theory of relativity was the observation of the bending of light during a solar eclipse by Arthur Eddington in May 1919. The phenomenon of gravitational lensing is well known today. Light rays emanating from stars behind a massive object (like a black hole) will be bent when they pass near the massive object.

Gravitational lensing (Credit: ESA/Hubble & NASA)

Raychaudhary was the first to study the implications of Einstein's equations for the collective behavior of families of geodesics in spacetime (such as families of light rays or families of trajectories of particles). He showed that since gravity is an attractive force, neighboring geodesics are bent towards each other and will eventually intersect. The intersection of infinitesimally close neighboring geodesics has a very important consequence. They are known as focal points or conjugate points. They have been studied extensively in the context of Riemannian differential geometry and they usually have important consequences for the global differential geometry of surfaces. Penrose and Hawking were the first to study them in the context of relativity and spacetime.

A familiar situation in ordinary Riemannian geometry where geodesics intersect is the example of a sphere.

Credit: Hawking-Penrose

If a geodesic in Riemannian geometry has a conjugate point, then it cannot be a length minimizing geodesic as seen on the sphere. If a great circle from a point p to q encounters a conjugate point r before reaching q, then it will not be the shortest path between p and q, because there will be another shorter great circle that directly joins p and q.

There are analogous implications of the existence of conjugate points for timelike and null geodesics in spacetime. The Raychaudhuri equation helps determine the conditions under which geodesics will encounter conjugate points. This brings us to assumption d) of Penrose's theorem - the "Null Energy" condition. The null energy condition (also known as the Weak Energy Condition) states that the local energy at any point is always non-negative. From Einstein's field equations it turns out that the Null energy condition is equivalent to the last term in the right-hand side of the Raychaudhuri equation being positive. This means that the entire right-hand side of the equation is bounded below by the square of the convergence. By solving the Raychaudhuri inequality (ignoring the shear terms and the energy term both of which are positive), one can show that the convergence factor is bounded below by a function of the affine parameter that depends on the initial convergence factor and the initial parameter value.

Now by itself, the Raychaudhari equation does not imply that all null geodesics will have conjugate points. After all light rays tend to spread out, so the initial convergence factor is usually negative. If you light a candle or if a star explodes in a large flash of light, the light rays will expand out spherically. If the initial convergence factor is negative, then even if gravity tries to focus the light rays back, it may not be enough to make them meet. That is unless gravity is so strong that the convergence factor starts out being positive. It seems counter-intuitive to imagine, but that is exactly what happens with a trapped surface. If the initial convergence factor is positive, then in a finite period of time (measured by the affine parameter), the convergence factor will blow up to infinity, meaning one will encounter a focal point.

Closed Trapped Surfaces

The final assumption of Penrose's theorem, which is based on the fact that you are in a situation of very strong gravity is e) The "Trapped Surface" condition. The following two diagrams will illustrate the contrast between a "normal" 2-dimensional surface in spacetime and a "trapped" 2-dimensional surface.

Credit: Hawking-Penrose

With a close trapped surface, you have a compact 2-dimensional surface (with no boundary) such as a sphere from where the light cones at the surface will always be tipped inwards, which is the same thing as saying that the initial convergence factor is positive (or equivalently the null expansion factor is negative). Since the Raychaudhuri equation shows that the convergence factor for light rays emanating from the trapped surface is bounded below by the initial convergence factor, the light rays will always have a positive convergence factor. If light rays (or null geodesics) can be extended indefinitely, which is assumption b) of Penrose's theorem, then eventually geodesics will have to encounter a focal point after a finite period of time. The terms caustics, conjugate points, and focal points are all used in the literature to describe the same concept.

There has been a tremendous amount of research on the conditions under which trapped surfaces will form. The most obvious example is in the case of a spherically symmetric black hole when the event horizon is formed. Compact spacelike surfaces behind the event horizon are trapped surfaces. Even if you deform the spacetime so that it is not spherically symmetric, the trapped surfaces will continue to form.

However, there are theorems that show that even in the absence of spherical symmetry, trapped surfaces can form during gravitational collapse or conditions of strong gravity.

Putting it all together - the punchline

To summarize, we know from the local causal theory of spacetime that the boundary of the chronological future of a spacelike surface is generated by null geodesics (think of the light cone in Minkowski space whose surface consists of light rays). Think of the surface of a star that is undergoing a supernova explosion. The particles emanating from the surface are moving into the chronological future of the star and the light rays from the boundary of the chronological future in spacetime. In particular, the boundary of the chronological future of a trapped surface is generated by null geodesics. What is different about a trapped surface as opposed to any other surface (such as the surface of a star) is that the light rays will all start focusing on each other. If you let the light rays travel indefinitely, they will have to intersect their infinitesimal neighbors at some point, so there will be a focal point on each geodesic. Penrose proves that this contradicts the Cauchy surface condition. The easiest way to visualize the proof is to examine the following diagrams. They are due to Penrose and are taken from his 2020 Nobel lecture.

Penrose uses normal null geodesics that are emanating from the trapped surfaces and shows them converging. A key fact is that any null geodesic that encounters a focal point stops being null and becomes timelike after it crosses the focal point. Hence it must enter the interior of the chronological future (see the image below).

The proof of this fact is a bit subtle (see Hawking-Ellis Proposition 4.5.12 or Witten Section 5.2). The heuristic argument is that if you take a geodesic \gamma from p to r containing a focal point at q will allow an infinitesimally nearby geodesic that will also join p and q. Then the neighboring geodesic plus the segment qr will have a "kink" at q, which means that this new curve is not a geodesic. But this curve has the same length as \gamma. By smoothening out the kink we can create a path that reaches its destination to the past of q. This implies that the original curve \gamma is timelike.

The fact that the null generators of the boundary leave the boundary and enter the interior after a finite amount of time implies that the boundary itself has to have a finite extent. In other words, the null boundary of the chronological future of the trapped surface is compact as can be seen in Penrose's diagrams above. Now Penrose claims that this contradicts the Cauchy hypersurface (global hyperbolicity) condition. Compare Penrose's conical diagram above with the Cauchy hypersurface diagram below.

Credit: Wald GR

The light ray from P1 to P2 is one of the generators of the cones shown above. It turns out that the null boundary of the causal future of any surface can be mapped down to the Cauchy surface using timelike geodesics that are orthogonal to the Cauchy surface. If you can extend the null geodesics indefinitely, then they would form a hypersurface that is topologically equivalent to the Cauchy surface. So the null boundary of the chronological future must be a Cauchy hypersurface. But that is not possible if the null boundary "closes up" onto itself. The technical way to state this is that the focusing of null geodesics results in the null boundary of the causal future of the trapped surface being compact. But the original assumption was that the Cauchy surface was non-compact (meaning extending out to infinity).

Mapping of future null boundary to Cauchy surface

(Credit: Sayan Kar, IIT KGP)

So the boundary curling up into itself to form a compact hypersurface cannot happen if the initial Cauchy hypersurface is non-compact. This means that the null geodesics generating the boundary of the causal future of a trapped surface must not be extendible beyond a certain point. This is incompleteness. The light rays are moving in direction of a singularity, but will never reach it. For example, the squiggly line below represents r=0 which cannot be reached by the light rays. The incompleteness (presence of singularity) means you can imagine slicing the surface along the singularity and spreading the null boundary out to map to the initial Cauchy hypersurface.

Credit: Wald GR

The technical proof arrives at a contradiction by showing that the timelike mapping to the initial Cauchy hypersurface is compact and open. Being open and closed the image of the mapping is the entire Cauchy hypersurface since the latter is connected. But that is a contradiction because the Cauchy hypersurface is non-compact. Again the perfect example of a non-compact Cauchy hypersurface is the asymptotically flat spacetime surrounding an object such as a star.

Closing Remarks

Penrose's result was a turning point in the study of collapsed objects and the subsequent work by him and Hawking started a revival of interest in General Relativity. The Hawking-Penrose singularity theorems represent a landmark in the history of General Relativity. The developments spurred by their work would have possibly shocked Albert Einstein, the discoverer of relativity who always believed that singularities were a mathematical anomaly. But the exciting aspect of singularities is that they provide a hint of new physics that is yet to be developed. The study of black holes and singularities is an active area of research in theoretical physics, astrophysics, and mathematics.

Any discussion of black holes and singularities would be incomplete without a discussion of Penrose's cosmic censorship conjecture. Since Penrose's (and Hawking's) results show that singularities are inevitable when gravity is very strong, why is it that we don't encounter or observe singularities in the universe? Remember all the evidence for black holes is about dark supermassive objects that exercise enormous gravitational influence on their neighborhood. All known models of black holes have an event horizon. The images taken of the region around black holes show either gravitational lensing of light from stars behind the black hole or light spinning around the black hole near the event horizon (photosphere) or the accretion disk, which is a region near the event horizon where the matter is sucked into the black hole.

It would be rather disconcerting if there were singularities just lying about in spacetime, but their invisibility led Penrose to make this conjecture.

Weak Cosmic Censorship Hypothesis: Nature abhors a naked singularity.

In other words, even though singularities are inevitable in general relativity, they are always hidden behind event horizons. No observer from outside (at "Null Infinity") can see a singularity. It turns out that this allows one to develop a nice theory of black holes since a lot of physics can be done without worrying about the singularities. Proving or disproving the cosmic censorship hypothesis is one of the central problems of mathematical general relativity.