Image of a black hole (2019)
Credits: Event Horizon Telescope collaboration et al.
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. - Albert Einstein
The year of black holes
For most people, 2020 will be remembered as the year that a pandemic raged across the globe killing hundreds of thousands and disrupting many lives. It was the year of social and political upheaval culminating in a contentious US election. But it was also the year that Black Hole research received the recognition that it deserved. The 2020 Nobel Prizes in Physics were awarded to Sir Roger Penrose, Andrea Ghez, and Reinhard Genzel for their pioneering work on Black Holes. In announcing the prizes the Nobel committee stated that half of the Nobel Prize was awarded to Roger Penrose "for the discovery that black hole formation is a robust prediction of the theory of relativity". This post will try to unpack this statement and provide an intuitive feeling for the uninitiated reader of Penrose's remarkable work. The discovery in question refers to a short 2-1/2 page paper entitled "Gravitational Collapse and Singularities" that Penrose published in January of 1965 in the journal "Physical Review Letters". In this paper, Penrose provided rigorous mathematical proof that under certain conditions the formation of a singularity in space-time is unavoidable. What is a singularity and what does it have to do with black holes? A singularity is a place and time where something really "bad" happens. It could be things like the curvature of space-time "blowing up" to infinity or a "tear in the very fabric of space-time". For example, a sufficiently massive object could collapse under its own gravitational force, and if there is nothing to resist the collapse it could distort space-time so badly that its curvature could end up becoming infinite. But such descriptions are somewhat misleading. What it really means is that there is a breakdown in the physical theory, and that a broader theory is needed to explain what is going on. Singularities don't have to be associated with points of infinite curvature. For example, in the case of the Big Bang, things seem to come out of nowhere, meaning particles don't have a history beyond a certain point in time in the past. Similarly, in the interior of a black hole particles or light rays could reach a point beyond which spacetime simply ceases to exist.
Penrose Lecturing on the Big Bang in Berlin, 2015
Incidentally, any mention of Penrose's work and singularities would be incomplete without mention of Stephen Hawking. As portrayed in the movie "The Theory of Everything" both were young graduate students at Cambridge when Penrose made his discovery on singularities. Hawking immediately understood the significance of Penrose's work and applied it to Cosmology and the Big Bang. By essentially reversing the time direction of Penrose's argument, Hawking was able to prove that there had to be a singularity at the time of the Big Bang (the birth of space and time!). Both Penrose and Hawking were awarded the prestigious Adams Prize in 1966 for their research. They then went on to collaborate and publish a series of singularity theorems that are now collectively known as the Hawking-Penrose theorems. Hawking became an iconic figure in Science who overcame a debilitating disease (ALS) to make groundbreaking discoveries in physics. It is unfortunate that Hawking died in 2018, else he would have surely shared the Nobel prize with Penrose.
In the next couple of sections, we will discuss the history of black holes and singularity research prior to Penrose's publication of his 1965 paper.
The Schwarzschild Singularities
In 1915 Albert Einstein made history when he presented his General Theory of Relativity to the Prussian Academy of Sciences. Newspapers around the world hailed the discovery as the most important since Newton and Einstein became a household name. It heralded the dawn of a new era in Science with a new and transformed understanding of the universe. Just one month after the publication of his results on space, time, and gravitation, Einstein was stunned to receive a postcard from a lieutenant in the German army containing the first-ever exact solutions to his field equations of gravitation. Einstein's equations are highly non-linear differential equations and notoriously difficult to solve. Einstein had only been able to supply an approximate solution in the context of the planetary motion of Mercury. But here was a postcard from someone posted at the Russian front that said "As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas." Karl Schwarzschild was the Director of the Astrophysical Observatory in Potsdam, but as a patriot had decided to join the army to fight in the war. During breaks from the fighting on the Russian front, he had managed to find the time to not only read Einstein's latest papers but also solve Einstein's equations for the space-time surrounding a spherically symmetric, non-rotating, non-charged body. Einstein was impressed and replied "I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation".
Schwarzschild's postcard to Einstein and his metric
Elegant and beautiful as Schwarzschild's solution was, it had a problem. There were two singularities in it, one at the center of the body at radius r=0 and one at the radius r=2GM/c^2, known today as the gravitational radius (or Schwarzschild radius). Here G is Newton's gravitational constant, M is the mass of the body and c is the speed of light. At these two points the solutions "blew up", meaning that they shot up to infinity. Einstein did not consider these singularities as physically meaningful. In fact in the spirit of classical electrostatic and Newtonian gravitational potentials, he made the assumption that the Schwarzschild solution applied only outside the spherical region of radius r=2GM/c^2. It is unclear what he expected to happen inside that sphere, but he considered them mathematical pathologies that had no physical meaning. In fact, Einstein was wrong to dismiss the singularity at the Schwarzschild radius. Today, the sphere at this radius is known as the "event horizon" of a black hole. Inside this spherical region, nothing can escape the gravitational attraction towards the center. The escape velocity exceeds the speed of light, so even light cannot escape, hence the term "black hole". As for the singularity, it was discovered by Arthur Eddington and David Finkelstein that it was indeed just an artifice of the choice of coordinates used by Schwarzschild. Eddington and Finkelstein showed that the singularity could be "transformed away" by simply choosing a different set of coordinates (now known as the Eddington-Finkelstein coordinates). However, r=0 was still a bonafide singularity and could not be transformed away. That did not cause any worry for Eddington or anyone else since the mass was centered at r=0, so the belief was that the field equations did not apply there.
Collapsing Stars
The first sign of trouble came in 1929 when a 19-year old Indian astrophysicist named S. Chandrashekhar performed some calculations on the final fate of stars on his sea voyage from India to England. The prevailing wisdom at that time was that a star that had spent all its nuclear fuel, would start collapsing causing outgoing shockwaves that would eject its outer shell in a "supernova explosion".
SN2018gv observed by the Hubble telescope
The inner core of such a collapsing star would settle into a stable object known as a white dwarf. It was believed that the white dwarf was prevented from further collapsing by something known as the "electron degeneracy pressure". In essence, the electron degeneracy pressure is a consequence of the Pauli exclusion principle in quantum mechanics which states that no two electrons can be in the same quantum state at the same time. So if you squeeze a collection of cold electrons in a small space, their repulsion due to Pauli exclusion principle and the electrostatic repulsion would result in an outward-facing pressure. The famous astrophysicist Ralph Fowler had shown that the electron degeneracy pressure was sufficient to resist the gravitational force and prevent the white dwarf from collapsing into itself. In doing so he ignored the relativistic motion of particles. Chandrashekhar produced a "relativistic degeneracy formula" that showed that if the star's mass was greater than 1.4 times the solar mass, then the electron degeneracy pressure was insufficient to prevent the star from collapsing beyond a white dwarf. This implied that the star would keep shrinking and collapsing ad infinitum. This was a startling and highly disconcerting discovery. While most experts including Fowler believed that Chandra's results were correct, Arthur Eddington who was highly influential at that time reacted with derision. At a conference in 1935, Eddington told his audience that Chandrasekhar's work “was almost a reductio ad absurdum of the relativistic degeneracy formula. Various accidents may intervene to save a star, but I want more protection than that. I think there should be a law of Nature to prevent a star from behaving in this absurd way!” Roger Penrose gave a nice talk on the topic, where he made the point that even though Chandrashekhar was correct in his calculations, Eddington was also right in believing that something in nature should prevent a star from collapsing indefinitely. Penrose does point out that Chandrashekhar (who was of a conservative bent of mind) was careful not to speculate about the eventual state of such an endlessly collapsing star. And it is known today that Eddington was wrong about the relativistic degeneracy formula. By dismissing Chandra's work, Eddington may have delayed much-needed research in the area of stellar collapse.
Penrose talking about Chandra
Chandrashekhar was eventually vindicated in his work and today the size of 1.4 solar mass is called the Chandrashekhar limit (Chandra also won the Nobel prize in Physics in 1983 for his work on the structure and evolution of stars). Meanwhile, Walter Bade and Fritz Zwicky proposed the existence of a neutron star just two years after the discovery of the neutron by James Chadwick in 1931. They predicted that a bigger star could squeeze the electrons and protons together to form neutrons which would generate a "neutron degeneracy pressure" that would resist runaway gravitational collapse in a manner similar to the electron degeneracy pressure. In 1939, Oppenheimer and Volkoff calculated an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrashekhar limit for white dwarf stars. This is known today as the Tolman-Oppenheimer-Volkoff limit and is estimated to be between 1.5 and 3 solar masses.
But what if the mass was greater than 3 solar masses. Was there a "law of Nature" as Eddington expected that would prevent a star from collapsing indefinitely? In a 1939 paper entitled "On Continued Gravitation Contraction", Oppenheimer and Snyder showed, that a spherically symmetric ball of gas of sufficient mass would necessarily collapse beyond the stage of a neutron star. In their words "When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely." They further showed that "the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened and can escape over a progressively narrower range of angles". In essence, they showed that for a spherically symmetric ball of gas, the gravitational collapse will result in infinite density and the creation of an event horizon in finite time.
The 1965 Paper
The reception to the Oppenheimer-Snyder paper was lukewarm due to the assumptions made about spherical symmetry. Ironically, at about the same time Einstein published a paper claiming that singularities could never form in General Relativity. His paper contained a mistake. Until Penrose's 1965 paper, there was a question as to whether objects like black holes and singularities were mathematical pathologies that could never exist in nature. The Russians Kalatnikov and Lifshitz claimed to have proved that singularities could not occur in cosmology. Their paper contained an error that was later corrected by Belinski. But it did not constitute a categorical proof that singularities could not occur and Penrose himself was skeptical of their methods. The objection towards assumptions made in prior work is best articulated by Penrose himself in his 1965 paper:
"The question has been raised as to whether this singularity is, in fact, simply a property of
the high symmetry assumed. The matter collapses radially inwards to the single point at the
center, so that a resulting space-time catastrophe there is perhaps not surprising. Could not
the presence of perturbations which destroy the spherical symmetry alter the situation
drastically? The recent rotating solution of Kerr [also possesses a physical singularity, but
since a high degree of symmetry is still present (and the solution is algebraically special), it
might again be argued that this is not representative of the general situation. Collapse
without assumptions of symmetry will be discussed here."
There are a few notable facts about Penrose's result.
- Penrose makes a very generic argument without making any assumptions of symmetry. Unlike prior results on black holes and singularities, which rely on explicit solutions of Einstein's equations, Penrose's work makes use of differential topology and global methods in geometry.
- Second, Penrose gives a very precise definition of singularity which is broader than the usual definition based on infinite curvature. Specifically, Penrose uses a concept in differential geometry called "geodesic incompleteness" as a proxy for the presence of singularities. Incompleteness means you cannot go past a certain point in space-time, which is an indication of a breakdown in the predictability of space-time.
- Third, Penrose's result is a negative statement in the sense that it simply says that under certain reasonable assumptions about space-time and gravitational collapse, space-time has to become incomplete. It says nothing about where exactly the completeness breaks down and it does not even make any claim about the nature of the singularities that would lead to such incompleteness. However, what Penrose's paper does do is provide rigorous mathematical proof that incompleteness is inevitable if certain reasonable conditions are met.
With regard to the implications of his result, Penrose makes this intriguing remark. "If, as seems justifiable, actual physical singularities in space-time are not to be permitted to occur, the conclusion would appear inescapable that inside such a collapsing object at least one of the following holds: (a) Negative local energy occurs. (b) Einstein’s equations are violated. (c) The space-time manifold is incomplete. (d) The concept of space-time loses its meaning at very high curvature – possible because of quantum phenomena. In fact (a), (b), (c), (d) are somewhat interrelated, the distinction being partly one of attitude of mind."
Geodesic Incompleteness
Geodesics refer to the "straightest possible" curves on a surface. For example, a straight line is a geodesic on a flat plane. The longitudinal lines on a sphere (great circles) are geodesics. They are curves of extremal (maximum or minimum) length between two points. Typically one thinks of the shortest path between two points. However, in space-time, it is more appropriate to consider the time it takes for an object or a signal to travel between two points (events). In General Relativity, freely falling bodies in a gravitational field follow geodesics. In addition light rays also follow geodesics. They are the analogs of "straight" lines on the plane.
Penrose's diagram showing a singularity
A surface or a manifold is said to be geodesically complete if starting at any point p you can follow a smooth path indefinitely in any direction. A plane and a sphere are both geodesically complete. But if you remove a point from the plane ("punctured plane") you get a geodesically incomplete space. If you follow a straight line going towards the missing point at a certain speed, then after a finite amount of time you will hit the puncture ad you cannot go any further. You can think of the point that is removed as the singularity. Of course, a punctured plane is an artificial example, because it resides in an ambient smooth space namely the plane and one can remedy the incompleteness by simply adding back the point. But in general, that may not be possible. In differential geometry, you can define manifolds without immersing them in a larger space. If you follow a geodesic path and the path cannot be continued after a certain point, then the manifold is said to be geodesically incomplete. For example, a light ray or a spaceship moving towards the center of a Schwarzschild black hole will not have a future after a finite amount of time because it will encounter the singularity at r=0. Time and space literally come to an end at that point. Penrose has a rather amusing footnote in relation to his reference to space-time incompleteness: "The “I’m all right, Jack” philosophy with regard to singularities would be included under
this heading!" "I'm all right, Jack" is a well known English expression indicating smug and complacent selfishness. It was also the title of a well known British movie starring Peter Sellers. Clearly, singularities are not very accomodating when it comes to letting things and signals from getting through. Technically speaking Penrose's singularity theorem should truly be called "Penrose's Incompleteness Theorem".
The Singularity Theorem
Penrose's paper is a mathematician's dream to dig into. But sadly for the uninitiated, it would be daunting to comprehend as it uses sophisticated mathematical concepts. The physicist Ed Witten said jokingly in a lecture that there are a small set of ideas in the paper that if understood would make even the uninitiated an expert. However, Witten's comment is addressed to his colleagues at the Institute for Advanced Study who could be called anything but uninitiated.
Witten's lecture at IAS on singularities
Regardless of my misgivings about Witten's assessment, I will attempt to convey the key ideas Penrose's theorem and its proof. Penrose lists five assumptions, which by themselves are reasonable but together lead to a mathematical inconsistency. The five assumptions are as follows:
- "Past and Future": Space-time is a smooth manifold with a clear definition of past and future everywhere.
- "Null Completeness": Every path built out of light rays can be extended indefinitely into the future.
- "Cauchy hypersurface condition": Initial condition of space based on the distribution of matter allows one to determine its evolution over time (in relativity space-time is a dynamical entity that evolves with time). A Cauchy surface is a "nice" initial surface in space-time that can be used to predict the future dynamic evolution of spacetime. Penrose makes a crucial assumption that there exists a "non-compact" Cauchy hypersurface. Non-compact Cauchy hypersurface means a 3-dimensional surface that extends out to spatial infinity. For example, if there is a "non-compressed" spatial distribution of matter and energy, then locally space will be curved but it will be surrounded by "flat" space that extends out to infinity.
- "Non-negativeness of local energy": The local geometry of space-time is affected by the local distribution of energy. The non-negativeness of energy implies the non-negativeness of curvature. This is essentially stating that gravity is an "attractive force".
- "Trapped surface": This is the one and only assumption that is unique to a black hole situation. Trapped surfaces are special types of spherical surfaces where all light rays that emerge from them start bending towards each other. In general, outgoing light rays will spread out from the surface of a sphere. But behind an event horizon gravity is so strong that space itself is shrinking with time. If the shrinking of space is faster than the spreading of light, then over time the light rays will start focusing towards each other. The great advantage that trapped surfaces have over previous approaches to singularities is that they are robust to small perturbations away from spherical symmetry. So even if the space-time is distorted so that it is not spherically symmetric, trapped surfaces will continue to form behind the event horizon.
As a consequence, if we assume that assumptions 1, 3, and 4 are true and if we assume the presence of a trapped surface (assumption 5), then we have to conclude that assumption 2 is false, which implies that incompleteness of space-time is inevitable. So stated more simply Penrose's theorem states that for any reasonable space-time where the matter is distributed in a concentrated region of space and space-time becomes close to "flat" as we move away from that region, the presence of trapped surfaces will inevitably result in the incompleteness of space-time (aka singularities). We will discuss these assumptions in more detail and the proof of Penrose's theorem in Part 2 of the article.
