The black hole Pōwehi and the Mathematics

Reading time: 19 minutes

Translation by AB – December 19, 2021


Black Hole

First image of a black hole revealed by the EHT collaboration team on April 10, 2019.
Source: Event Horizon Telescope Collaboration

The image unveiled on April 10, 2019, by the international Event Horizon Telescope (EHT) collaboration team at six simultaneous press conferences in Brussels, Santiago de Chile, Shanghai, Taipei, Tokyo, and Washington, D.C., has moved scientists around the world. In the columns of Quanta Magazine, physicist Janna Levin declared1:

I am moved not just by the image; overwhelmingly I am moved by the significance of sharing this experience with strangers around the globe. I am moved by the image of a species looking at an image of a curious empty hole looming in space.

We admired with her the very first image of a “black hole”, this extraordinary celestial object foreseen since the 18th century but never directly observed before this April 10, 2019. This scientific feat alone therefore justified the exaltation of astrophysicists, but Janna Levin also considered it as an anthropological event. “We are all under the same sky” she wrote further, and beyond our cultural differences we thus contemplate a universal truth. Sheperd Doeleman, director of the EHT collaboration, said himself “we saw something so true”, and Janna Levin concluded: “it’s true for all of us”.

However, “truth” remains an equivocal concept and the image raises many questions: How was it made? What is it really given to us to see? Is it really a photograph? Reviewing these questions, we will notice again the fabulous power of mathematics.

Language games – from Laplace to Schwarzschild

Black hole: region of spacetime where gravity is so strong that nothing — no particles or even electromagnetic radiation such as light — can escape from it.2

Contrary to celestial phenomena that can be directly observed and then subjected to scientific investigation, a black hole is first a scientific hypothesis which signs were then sought in the sky. The physical sciences progress in this way: once mathematical laws consistent with observations have been established, such as Newton’s laws of motion, they acquire their own autonomy in the realm of mathematics. These writings, once spared of the necessity to correspond to the phenomena, must obey only the mathematical laws, i.e., rules of language, and can thus be pushed out of the domain of validity of the observations. The black hole hypothesis is thus the result of a temporary postponement of the judgment of existence or “truth” in the sense of Doeleman and Levin. This theoretical maneuver has enabled many hypotheses and discoveries.

Loi de NewtonThus, the mathematical law of gravitation proposed by Newton (see opposite) makes it possible to deduce the existence of a minimal speed to be reached to escape the gravitational attraction of a mass like a celestial body. This “escape velocity” is 11 km/s at the surface of the earth and 617 km/s at the surface of the sun, both values derived from a formula giving this speed vl as a function of the mass M of the body from which one seeks to escape and its radius R:

Vitesse de libération

This formula indicates that the larger the mass M and the smaller the radius R, i.e., the more massive and concentrated the star is, the higher the velocity v_l (it becomes more difficult to get away from it). So much so that by arbitrarily setting the values of M (large enough) and R (small enough), the escape velocity vl can theoretically reach or even exceed the speed of light (300 000 km/s). One would obtain under these conditions what the French mathematician, astronomer and physicist Pierre-Simon de Laplace named already in the 18th century a “dark star” from which even the light cannot escape. Such a star, if it had the mass of the earth, would be contained in an unthinkable diameter of only 2 centimeters! Let us add that the light should also have a mass and weigh to be thus trapped, what science of the time was not convinced. Finally, if the dark star were to retain its own light and thus escape observation, its existence could neither be proved nor disproved. This was too many obstacles to raise this mathematical possibility to the level of a scientific hypothesis and Laplace’s idea was put back in the curiosity cabinet.

Equation d'EinsteinBut two hundred years later, Albert Einstein proposed a more general equation than that of Newton (see opposite) and which will gradually revive the hypothesis of the dark star. It is to the German physicist Karl Schwarzschild, while he was engaged in 1915 on the Russian front, that we owe the discovery of one of the exact solutions of this very delicate equation of Einstein, a solution that he it is enough here to contemplate such an illumination:

Solution de Schwarzschild

We can see the mass M from which the light tries to escape. But apart from the complexity of this formula, what is new? The big difference with Newton’s theory is that here light, like any form of energy, really “weighs”. Its trajectory can thus really be curved by a celestial mass, like the trajectory of a satellite, a shell or a basketball. The light can really be trapped by a sufficiently massive star concentrated in a sufficiently small radius, which is called the “Schwarzschild radius”, consistent with the one deduced from the Newtonian theory3:
Rayon de Schwarzschild

Einstein’s geometrical theory of relativity received such confirmation that Schwarzschild’s hypothesis of the existence of “black holes”, as the American physicist John Wheeler would later call them, was taken seriously. Astrophysicists then began to look for these monsters in the sky.

Science and magic

Magic is only a science that has not yet been put into equations.4

If nothing escapes from black holes, not even light, how is it possible to detect them? This problem is similar to the one posed by the “Higgs boson”, an elementary particle predicted by quantum physics, that other great theory of the 20th century to which Einstein also contributed. This boson is indeed too short-lived to be observed directly. However, like the black hole, it leaves indirect traces of its presence – a wake of sorts – and it is these traces, which can only be made by these hypothetical objects, that scientists are looking for.

Let’s take the place of a layperson in science for a few moments and ask ourselves this good question: how can we believe a field of study which predicts the existence of “invisible” phenomena (Higgs bosons, black holes…) and which, moreover, tries to indirectly prove their existence by the observation of other “invisible” phenomena because they are supposed to be too distant, too microscopic, too quick…? How does faith in science differ from faith in magic or astrology, if phenomena and their causes can only be described metaphorically?

We could answer that science has observation instruments that amplify, magnify and bring to our senses these undetectable phenomena: telescopes, particle accelerators, gravitational wave detectors… But their very functioning, sometimes extremely complex, also depends on invisible reasons. What proof do we have that these instruments do not “bias” the alleged phenomena during their amplification, or even “create” the signs we wish to observe? In principle, therefore, there is no reason for anyone to believe in science. But we should try to convince our layperson that science differs from magic or astrology by a powerful safeguard: the mathematical language, this calculus ratiocinator which tunes, so as to speak, our spirits and the phenomena, thus preserving us from an absolute relativism5. Four hundred years ago, Galileo entrusted this famous formula6:

Philosophy [nature] is written in that great book which ever is before our eyes – I mean the universe – but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

Whether it is about black holes or Higgs bosons, we will admit that the “mathematical language” of the Universe expresses them as invisible causes of observable phenomena either with the naked eye or with the help of instruments themselves mathematically governed: Higgs bosons cause the mass of things; black holes cause these “rings of fire” of which we are going to speak now7.

To observe a black hole

We have seen what we thought was unseeable.

This euphoric statement by Shepherd Doeleman, if it has had the desired effect on us, “strangers around the globe”, must be properly understood. Indeed, since nothing can escape, not even the slightest information, black holes are forever unseeable. To claim that we have “seen” them is therefore inaccurate. However, black holes have what scientists nicely call an “event horizon”, a theoretical boundary that separates the “inside” of the black hole, that space confined within the Schwarzschild radius from which nothing can ever leave, from the “outside” where we are. And this horizon is observable…

The black hole “observed” here, named “Pōwehi”, is located at the center of the galaxy M87, 55 million light-years away. Its unimaginable mass is equivalent to 6.5 billion times that of the sun; its diameter, that of its horizon, is similar to that of the solar system. When approaching such a monster, the theory predicts that the matter in the vicinity acquires relativistic velocities, heats up by friction, and then radiates electromagnetic waves, in particular light. These waves are the indirect observables that the scientists and engineers of the EHT collaboration have detected and that appear in the image as a ring of fire. Thus, says Shepherd Doeleman (emphasis added)8 :

What you are seeing here is the last photon orbit, what you are seeing is evidence of an event horizon, by laying a ruler across this black hole, we now have visual evidence for a black hole.

What does this key statement mean? What Doeleman does not point out here is that the mass M of Pōwehi is otherwise known and is 6.5 billion times that of the sun or 1.3 x 1040 kilograms. So that nothing can escape it, not even light, this mass must be compacted in a sphere having at most its Schwarzschild radius. Using the formula recalled above, we obtain Rs = 19.2 billion kilometers, or a diameter of 38.4 billion kilometers. What we see, says Doeleman, is the “last photon orbit”, the one which skims the horizon of the events and thus draws its contours. By “laying a ruler” on the image to measure its diameter, we find about 38 billion kilometers, or 1.5 light days. We have superimposed on the image below, where the sun and the orbit of Pluto were represented, the Schwarzschild radius inside which the mass must be enclosed to make a black hole (the Voyager 1 probe is currently at 23 billion kilometers from the sun)9. This fits perfectly!

This is why Shepherd Doeleman uses the term “visual evidence”. What happens inside this horizon (if it is indeed an event horizon) is forever unseeable. We only know for sure that there is a gigantic mass enclosed within it (although, if it were distributed uniformly throughout the volume bounded by the Schwarzschild radius, it would have the same low density as the air at the top of Mount Everest). The mathematical theory of relativity from which Schwarzschild’s solution is derived commands this mass to gather in an infinitely small central singularity, a simple point. This is obviously even more unthinkable than an earth shrinked in a 2 centimeters ball because, as Einstein instinctively believed, “matter cannot be concentrated arbitrarily”. Scientists follow him on this point but observations now proving the existence of event horizons, it remains to understand the state of matter inside these horizons, that is to say “what” is made of a black hole. Even today, science does not have an answer …

Evenement Horizon Telescope

If the principle of the visual proof is clear (undeniable mass + undeniable radius = black hole), its realization is far from obvious as the black hole is so tiny on the astronomical scale: it is indeed necessary to observe a phenomenon of the size of the solar system at a distance of 55 million light-years, the equivalent of the size of an orange on the moon. To make matters worse, before reaching any instrument, the light of the ring of fire crosses cosmic immensities filled with matter and radiation, so many obstacles that can shatter the most delicate signals. Fortunately, while visible light passes through these obstacles poorly, radio waves are relatively insensitive to them (we cannot see through walls, but radio passes through them unhindered). The image of Pōwehi was therefore reconstructed by artificially coloring the range of radio waves emitted by the last photon orbit10:

“The yellow is the most intense emission, the red is less intense, and then black is little or no emission at all”, Fox said11. In the optical range, the ring around the black hole would likely appear white, perhaps tinged with blue or red, according to Fox.

The image thus has the same scientific value as a map like this one representing the wind speed:

Ouragan

To make visible the invisible, to bring any cause to reveal itself as a sentient phenomenon, here is the essence of scientific proof. The question of the observables of the photon orbit being thus settled, there remains the problem of distinguishing signals emitted by an object as large as an orange on the moon. The telescope capable of this feat should have a mirror as large as the earth. Such an instrument obviously does not exist but it can be simulated thanks to a remarkable technique, the Very Long Baseline Interferometry (VLBI)12, which consists in combining the observations of a network of a few radio telescopes spread over the whole surface of the Earth and aimed towards the same target at the same time. Here is the EHT array, which was initially composed of 8 telescopes (in blue):

EHT Map

However, there is still a problem: if the virtual telescope does present the dimensions of the earth, it only presents a few tiny dots, a few scattered fragments of its main mirror. It will therefore provide a very incomplete image of Pōwehi, a barely begun “puzzle” that can be pictured as follows:

Fortunately, the earth rotates, driving the telescopes, and, provided that the observed scene remains more or less stable for a few days, the “puzzle” is completed with a few trails:

The general principle of the “proof” and its demonstration having been set, it is now the digital techniques that come into play.

An image obviously does not appear in this way before the eyes of researchers. The radio waves caught by the telescopes were first converted into data and then stored on ultrafast hard drives. After 7 days of collection, 5 petabytes of data were collected and half a ton of hard drives were flown to MIT:

Katie Bouman

The image is “somewhere” in these drives and it will take almost two years to rebuild it13. Researcher Katie Bouman, seen in this cheerful photograph, is the author, with her team, of the CHIRP (“Continuous High-resolution Image Reconstruction using Patch priors”) algorithm that enabled the reconstruction of Pōwehi’s image from these scattered pieces14. We leave here temporarily the world of astrophysics for that of information processing and digital technology which philosophy is quite different. No other “matter” or “form of energy” exists in Mundus Numericus than the data which are, as we said, fictions we have agreed to believe in (in French: Données et traces numériques (sous rature)). From then on, algorithms are essentially language games, admittedly controlled by mathematics, but freed from the requirement to account for reality. In these hard drives, there is less a reality to explain than a problem to solve.

Katie Bouman’s problem was to design an algorithmic “game” to rebuild, with as little bias as possible, the image behind this digital Mashrabiya. The CHIRP algorithm that she and her team developed is a possible solution inspired by cognitive science.

Inferences

There are, logically, a multitude of possible images compatible with the data collected, but we all instinctively see more or less the same: an orange, even “hot” ring on a black background. This inference proceeds from a cognitive performance that we perform incessantly as a “supplement” to our perceptual flow, whether visual, aural, olfactory or even linguistic (e.g., “the stewardess asks him to fasten his seatbelt” instinctively allows the preverbal inference “he is on the plane”). The German physiologist and physicist Hermann von Helmholtz was the first to propose a semiological theory of perception according to which the brain unconsciously infers the cause of its perceptions15 (“top down”) rather than constructing its perceptions from the information that it receives (“bottom up”)16. The CHIRP algorithm seeks to mimic this performance to reconstruct the complete image as a “cause” of the digital fragments produced by the telescopes (technical principle also called “forward modeling”).

To infer an orange ring (or “he is on the plane”), you must first have a body of knowledge accumulated through experience and learning. It is precisely this a priori corpus that the letter “P” of CHIRP refers to: “Patch priors”, i.e., literally “previous pieces”. These are small pieces of images from previous observations but also from simulated images17. Physicists know, for example, thanks to Einstein’s equations, what a plasma orbiting a black hole might look like “seen up close”:

Black Hole Simulation

A second simulation predicts the aspect of this plasma seen by the EHT under the same observation conditions as the real collection and by applying the same rules of colorization:

Black Hole Simulation 2

The resemblance with the “real” image is amazing. The CHIRP algorithm is in a way inspired by pieces of these images to complete and smoothen the partial image coming from the EHT and stored in the hard drives. It is as if, having already observed photon orbits several times, CHIRP recognized another one behind the Mashrabiya. This process is quite smart but as the final image depends on a corpus that is a priori fixed at discretion, how can we be sure that CHIRP is not ensuring to observe a black hole? What is the scientific worthiness of this method? Let us recall that the visual evidence of the black hole relies on the presence of a glowing ring of a particular size. We cannot therefore allow the slightest bias and the process must capture the “real”18:

The dangers of false confidence and collective confirmation bias are magnified for the EHT because the array has fewer sites than typical VLBI arrays, there are no previous VLBI images of any source at 1.3 mm wavelength, and there are no comparable black hole images on event-horizon scales at any wavelength.

Therefore:

To minimize the risk of collective bias influencing our final images, in our first stage of analysis we reconstructed images of M87 in four independent imaging teams.

Ces équipes ont travaillé sans échanger leurs informations et ont chaque fois dévoilé à peu près la même image19 :

Black hole (4 teams)

Thus, the EHT collaboration concluded with reasonable certainty that the digital Mashrabiya does indeed conceal the image of a photon orbit of dimensions consistent with the presence of a black hole. This is not magic, but from Einstein’s theory to the final image, the mathematics supports a very tenuous and almost miraculous “truth”.

Photography

The process developed by the EHT team is analogous to that of digital photography: a small fraction of the photons emitted by the matter in the vicinity of the black hole reached the “lenses” of the terrestrial telescopes; their electromagnetic signals were recorded on a digital convergence plane made up of 500 kilograms of hard drives; finally, the “development” of this picture was carried out by algorithms. In principle, a digital camera does not do anything else, but the EHT is a gigantic device, designed to realize this single photograph in about ten years.

A “black process” of transformation of the real (input) into reality (output) is indeed at the origin of this image contemplated today by the human species (see The “progress” unveiled by Photography). This worldwide process involved physics, mathematics, algorithms, cognitive sciences… but also international collaboration, air transport, public and private financing, politics… We noticed that a “technical black box [like this one always] hides violent and powerful power games”. This black process has not escaped it, as scientific collaboration is rarely peaceful, especially in the extremely competitive environment of modern cosmology20. Thus, the international assembly of trolls did not fail to make Katie Bouman a target by denouncing a media narrative guided by a feminist agenda. Katie was cyber-harassed on the grounds that she was singled out as a woman at the expense of the other 200 contributors to the project21.

Beyond the scientific prowess, this photograph thus reveals many characteristics of our technological civilization, which produced it and then raised it to the rank of “truth”. And it is then, if we pay attention to the process that we have just described, ourselves that we contemplate.

Pōwehi

In front of Pōwehi’s photograph, everyone can experience a particular feeling: for the scientists, pride in the achievement or, like Janna Levin, bliss in front of a “truth” revealed to the human species; for the conspiracists, acrimony and the unhealthy joy of being able to denounce a new fake; for many, simple indifference… For us finally, astonishment and admiration for human genius. We are indeed able to foresee the rather improbable appearance of a cosmic phenomenon forever invisible to the naked eye, to predict where we should find it, and finally to engineer the camera obscura which will reveal it at the foreseen location! This feat is based on an extraordinary internal coherence of the physical sciences and an unheard-of mathematical agreement between the phenomena.

However, mathematics is in itself only a language game as the inventor Nikola Tesla pointed out:

Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.

But in this case, Tesla’s reasonable statement seems belied. What, then, is the basis of this “magical power” of mathematics to always attune us to the world, and of which Pōwehi’s discovery is a striking manifestation? We continue to think that we must look for the choreography of the body and the language games (Body and Language Games). The fact remains that the literally cosmic scope of this agreement, i.e., well beyond the sentient body, remains extraordinarily mysterious. The physicist Eugene Wigner himself described, in a famous 1960 article22, the effectiveness of mathematics as “unreasonable” and confessed a formula of underestimated significance: “fundamentally, we do not know why our theories work so well”.


1. Janna Levin / Quanta Magazine – April 10, 2019 – What the Sight of a Black Hole Means to a Black Hole Physicist
2. Wikipedia – Black hole
3. We can indeed verify that by replacing R by Rs in the formula giving the espace velocity vl, we get vl = c, that is to say the speed of light, at the surface of the star (d = 0).
4. La magie n’est qu’une science qui n’a pas encore été mise en équations” – Stefan Wul / Fleuve Noir – 1958 – Piège sur Zarkass
5. Others might also observe that the technology constitutes a “proof” of scientific truth, but this would hardly be sufficient for the layperson because the technology reigns over the realm of the sentient.
6. Galileo – 1623 – The Assayer
7. Before this image obtained by the Event Horizon Telescope there were other rather convincing indirect observables, in particular the gravitational traces of the merger of two black holes observed for the first time in 2016 (LIGO Discovers the Merger of Two Black Holes).)
8. Sheperd Doeleman / National Science Foundation, Washington – April 10, 2019 – Press conference
9. This black hole is not seen from the front but at an inclination of about sixty degrees. But that does not change anything to the measurement of its diameter.
10. Mindy Weisberger / Live Science – April 10, 2019 – Why Is the First-Ever Black Hole Picture an Orange Ring?
11. Derek Fox is an Associate Professor of Astronomy and Astrophysics at Pennsylvania State University – He was not involved in the EHT collaboration.
12. Wikipedia – Very-long-baseline interferometry
13. Larry Hardesty / MIT News – June 6, 2016 – A method to image black holes
14. Katherine L. Bouman, Michael D. Johnson, Daniel Zoran, Vincent L. Fish, Sheperd S. Doeleman, William T. Freeman / IEEE Conference on Computer Vision and Pattern Recognition (CVPR) – June 2016 – Computational Imaging for VLBI Image Reconstruction
15. Wikipedia – Unconscious inference
16. Anil Ananthaswamy / Quanta Magazine – November 15, 2021 – To Be Energy-Efficient, Brains Predict Their Perceptions
17. We are simplifying here for consistency of text. Strictly speaking, CHIRP does not reconstruct the images but rather the signals from which the images can be drawn.
18. The full paper: The Astrophysical Journal Letters, 875:L4 (52pp) – April 10, 2019 – First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole
19. Ibid. 16
20. In this regard, this “merciless” environment is well described in an edifying and humorous article by American physicist Brian Keating. This is another unfortunate astrophysical adventure that could have won him the Nobel Prize. Brian Keating / Nautilus – April 19, 2018 – How My Nobel Dream Bit the Dust
21. Jill Filipovic / The Guardian – April 17, 2019 – The misogynist trolls attacking Katie Bouman are the tip of the trashpile
22. Eugene Wigner / Communications on Pure and Applied Mathematics, vol. 13, no 1,‎ 1960, p. 1–14. – 1960 – The unreasonable effectiveness of mathematics in the natural sciences

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.