The Drawing that Earned Sir Roger Penrose a Nobel Prize
This hand drawn figure represents a mathematical masterpiece
On October 14th, I will have the pleasure and honor to join Sir Roger Penrose in conversation at his home institution in Oxford. Please join us too if you happen to kick around that town. I’ll post details here soon. This week, with Sir Roger on my mind, here is a piece on the drawing that earned him a Nobel prize.
In 1965, in an inspired and disarmingly clear article—so brief that the text does not quite fill three journal pages—Roger Penrose draws a figure to summarize his ingenious mathematical proof that would earn him the 2020 Nobel Prize in Physics “for the discovery that black hole formation is a robust prediction of the general theory of relativity.” The now legendary Penrose distilled his mathematical calculation into a single paragraph and then distilled the proof even further into a single image that can be read by the initiated. The powerful conclusion: black holes are an inevitable consequence of unhindered gravitational collapse.
The term “black hole” had not yet been coined, and wouldn’t be until the influential American relativist John Archibald Wheeler concluded a lecture in 1967 with the image, “[The star] like the Cheshire cat fades from view. One leaves behind only its grin, the other, only its gravitational attraction…light and particles go down the black hole…” Quasi-stellar radio sources were a newly observed mystery, so named since the signals appeared to come from a single location like a star. Some astronomers hypothesized that the emission originated in the formation of a supermassive black hole (using the moniker “black hole” anachronistically) millions or hundreds of millions of times the mass of our Sun.
Although intriguing hypothetical candidates for unexplained sources, black holes were not assured to exist even theoretically. The descriptions of the spacetime structure around collapsed stars leaned on extremely simplified symmetries for their construction. The first mathematical model of a black hole was presented to Einstein in a letter within six months of the publication of the General Theory of Relativity. The letter arrived from the Russian front during WWI from his friend Karl Schwarzschild, an astronomer and enlisted German soldier, and includes words of such bewildering optimism they are often quoted with some bemusement: “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.” Schwarzschild imagined a perfect sphere with all the mass of a star crushed to a point. He didn’t ask how nature would create such a thing. He simply chose to imagine it. Nearly a half century later, in 1963, Roy Kerr discovered the mathematical solution for an idealized, perfect, spinning black hole. Despite these formal descriptions, the skepticism towards the existence of black holes in nature was reasonable. The mathematical models were clever but so idealized as to be impossible to achieve in reality.
In that one beautiful paragraph in that admirably concise paper, Roger Penrose defied the prevailing skepticism and proved that in fact black holes were inevitable, thereby throwing open the gates of possibility to real black holes in the universe.
The figure itself is a pleasure to read. Due to our inability to visualize four spacetime dimensions (three space and one time), Penrose drew only two dimensions of space and one dimension of time. So, if a star is a sphere in spacetime, on the drawing it would be represented by a circle. Time elapses from the bottom of the figure to the top. At the bottom of the figure, a star begins to collapse and he draws the lines of the outer atmosphere shrinking as we move up the figure and into the future. At a certain time, the star has shrunk so much that a “trapped surface” forms—what we now refer to as the event horizon. Here it gets interesting. Penrose illustrates that the star is forced to continue to collapse. The proof hinges on the argument that once the event horizon forms, even light is driven mercilessly inward as represented by the cones of light pointing toward the future. Since nothing can travel faster than the speed of light, everything is driven inevitably away from the event horizon deeper into the hole.
The implications are already tremendous: the star is gone and the event horizon left in its wake is an empty shadow that not even light can escape; To an outside observer—the eyes Penrose draws on the left—time appears to stop on the horizon since no more light reaches them; What the observer on the outside presumed to be the center of a sphere is, to the falling star, a point in the future and not a point in space at all. The star can no more halt its descent inside the black hole than we can halt our flow through time.
We read the diagram further and realize that the star collapses catastrophically to a singularity, a spacetime point of infinite curvature, and all light from the star goes with it. The singularity is anathema to physicists. Matter and light hit the singularity and effectively cease to be, cease to exist. Singularities are in violation of the entire paradigm of physics as the future beyond the singularity is not only unknowable, it is meaningless. In the face of his own proof that singularities are inevitable within event horizons because they lie in the collapsed star’s future, he then posits that they must be a false prophecy. He suggests that singularities are signaling the failure of general relativity at that point. He goes further to suggest that the quantum paradigm may rewrite the story of the deep interior of black holes without them.
Sir Roger proved that black holes are inevitable. Slowly, steadily, they were then found in nature. We now understand that quasi-stellar radio sources, or quasars, are supermassive black holes inducing magnetic storms in the debris around them. In the decade following Penrose’s article, smaller black holes that formed as the death states of stars were observed cannibalizing companion stars. In 2017, the Nobel Prize was awarded to the architects of LIGO, the extravagant antennae that record the sounds of spacetime ringing around colliding black holes. Just a few weeks ago we lost a recipient of that Nobel prize, Rai Weiss. In 2019, the Event Horizon Telescope gave humanity its first look at an actual resolved image of an event horizon’s shadow, a 6.5 billion solar mass behemoth in the galaxy M87, 55 million light years away.
Penrose is now Sir Roger, knighted by Queen Elizabeth II to the honor Order of Merit, one of only twenty-two living recipients of the highest order. His scientific legacy is staggering in range, profundity, and originality. His truly exceptional mathematical mind is complemented by the rare ability to invoke complex mathematics in imagery. He invented a formal technique to map an infinite spacetime into a finite drawing known as a Penrose diagram. He discovered tilings, now called Penrose tilings, which are coverings of a plane in a set of geometric figures to generate non-repeating patterns. He would doodle and create impossible triangles and other impossible figures that inspired M.C. Escher and Pablo Picasso.
He is the ultimate visual mathematician. And the proof this handmade drawing represents is a masterpiece.
This piece was first published in Pioneer Works Broadcast on the occasion of Sir Roger's winning of the Nobel Prize. Find the Broadcast substack here.
Can you shortly explain why the singularity is not a point in space? Clearly it is a point in time, but from the figure it seems like it clearly is a point in space also, the point r=0.
Looks like an orange juicer.