On April 10, 2019, the Event Horizon Telescope (EHT) collaboration released the first image of a black hole: a glowing orange ring surrounding a dark central shadow. The object was the supermassive black hole at the center of the galaxy Messier 87 (M87), located 55 million light-years from Earth. It had a mass of about 6.5 billion times the mass of the Sun.
The image was immediately recognized as one of the great scientific achievements of the 21st century. But it was not a photograph in any ordinary sense. No single telescope took the picture. No camera recorded the light. The image was reconstructed from radio wave data collected by eight telescopes scattered across the globe, using mathematical algorithms that assembled fragments of information into a coherent picture. The first image of a black hole was, in a very real sense, a mathematical creation.
Why Black Holes Are Invisible
A black hole is a region of space where gravity is so intense that nothing, not even light, can escape. The boundary beyond which escape is impossible is called the event horizon. Anything that crosses the event horizon is lost to the outside universe forever. Since black holes emit no light from within their event horizons, they are, by definition, invisible.
What is visible is the material surrounding the black hole. Supermassive black holes at the centers of galaxies are typically surrounded by accretion disks: swirling rings of superheated gas falling inward. This gas reaches temperatures of billions of degrees and emits intense radiation, including radio waves, X-rays, and visible light. The accretion disk is one of the brightest objects in the universe.
The black hole itself appears as a dark shadow against this bright background. The shadow is not the event horizon itself but a slightly larger region called the photon sphere, where light is bent so strongly by gravity that it orbits the black hole. Light passing just outside the photon sphere is deflected toward the observer, creating a bright ring. Light passing inside the photon sphere falls in and is lost, creating the dark center.
The shape and size of this shadow are predicted precisely by general relativity. Einstein’s equations describe how spacetime curves around a massive object, and this curvature determines the paths that light follows. The shadow of a black hole is a direct test of general relativity in its most extreme regime.
The Problem of Resolution
The M87 black hole is enormous by astronomical standards (its event horizon is larger than our entire solar system), but it is 55 million light-years away. At that distance, the angular size of the shadow on the sky is about 42 microarcseconds. This is roughly the angular size of a coin on the Moon as seen from Earth, or a grain of sand in New York viewed from Paris.
No single telescope on Earth can resolve an object that small. The resolution of a telescope is limited by its aperture (the diameter of its mirror or dish): larger apertures resolve finer details. To resolve the M87 black hole shadow, you would need a radio telescope with a dish the size of the Earth.
Building such a telescope is impossible. But simulating one is not.
The Earth-Sized Telescope
The Event Horizon Telescope is not a single instrument but a network of eight radio telescopes at six locations around the world: Hawaii, Arizona, Mexico, Chile, Spain, and the South Pole. These telescopes simultaneously observed the M87 black hole during a ten-day window in April 2017, recording the radio waves emitted by the accretion disk at a wavelength of 1.3 millimeters.
The technique used is called very long baseline interferometry (VLBI). Each pair of telescopes in the network acts as a single telescope with an effective aperture equal to the distance between them. The farther apart the telescopes, the higher the resolution. With telescopes spanning the entire planet, the effective aperture equals the diameter of the Earth, roughly 12,700 kilometers.
But VLBI does not directly produce an image. Each pair of telescopes measures a single component of the image: the amplitude and phase of a specific spatial frequency. The full image is the sum of all these components. The problem is that the EHT has only eight stations, which means it measures only a sparse subset of the spatial frequencies needed to reconstruct a complete image. The data is incomplete, and the image must be inferred from what is available.
The Mathematics of Image Reconstruction
Reconstructing an image from incomplete data is a mathematical problem, specifically a problem of inverse theory. The EHT data constrains the image but does not uniquely determine it. Many different images are consistent with the measured data. The challenge is to find the image that best fits the data while also being physically plausible.
The EHT collaboration used multiple independent algorithms to reconstruct the image, including CLEAN (a standard radio astronomy algorithm), regularized maximum likelihood methods, and Bayesian approaches. Each algorithm made different assumptions and used different mathematical techniques. The fact that all algorithms produced qualitatively similar images (a bright ring with a dark center) was strong evidence that the reconstructed image was real, not an artifact of any particular method.
The mathematical techniques involved draw on Fourier analysis (decomposing signals into frequency components), optimization theory (finding the best fit to data), and statistical inference (quantifying uncertainty). The Fourier transform, which converts between an image and its spatial frequency components, is the central mathematical tool. The EHT data are measurements in the Fourier domain, and image reconstruction is essentially the problem of inverting an incomplete Fourier transform.
What the Image Showed
The reconstructed image showed a bright, asymmetric ring of emission surrounding a dark central region. The ring diameter was approximately 42 microarcseconds, consistent with the shadow of a black hole of 6.5 billion solar masses as predicted by general relativity. The asymmetry of the ring (brighter on the south side) was consistent with relativistic effects: the accretion material is moving toward the observer on one side (Doppler boosted, appearing brighter) and away on the other (Doppler dimmed, appearing fainter).
The image was a direct confirmation of general relativity’s predictions about black holes. The size, shape, and brightness profile of the ring matched the theoretical calculations with remarkable precision. Einstein’s equations, formulated in 1915 for a universe that he did not imagine contained black holes, correctly described an object of almost incomprehensible scale and ferocity.
From Einstein to the Image
The theoretical chain that connects Einstein’s equations to the EHT image passes through several key developments. Karl Schwarzschild found the first exact solution of Einstein’s equations for a spherically symmetric mass in 1916, predicting what we now call the Schwarzschild radius (the event horizon). Roger Penrose proved in 1965 that black holes form inevitably from sufficiently massive collapsing stars. Jean-Pierre Luminet calculated the first theoretical image of a black hole’s accretion disk in 1979, using a computer to trace the paths of photons through curved spacetime.
Each of these advances required the mathematical framework of general relativity: Riemannian geometry, tensor calculus, and the Einstein field equations. These tools, in turn, descend from the tradition of mathematical physics that began with Newton’s formulation of gravity in the Principia. Newton’s inverse-square law describes gravity for weak fields and slow speeds. Einstein’s equations generalize it to all conditions. The EHT image is a test of Einstein’s generalization at the most extreme conditions in the universe.
Sagittarius A*
In May 2022, the EHT released a second black hole image: Sagittarius A* (Sgr A*), the supermassive black hole at the center of our own Milky Way galaxy. Sgr A* is much closer than M87 (about 27,000 light-years away) but also much smaller (about 4 million solar masses), so its angular size is similar. The image showed the same characteristic ring-and-shadow structure, confirming that the phenomenon is universal.
Imaging Sgr A* was actually harder than imaging M87, because the gas around Sgr A* orbits much faster (the orbital period is minutes rather than days), causing the image to change during the observation. The reconstruction algorithms had to account for this variability, adding another layer of mathematical complexity.
The Role of Computation
The EHT generated approximately five petabytes (5,000 terabytes) of data during the April 2017 observations. The data was too voluminous to transmit electronically and was physically shipped on hard drives to processing centers at MIT Haystack Observatory and the Max Planck Institute in Bonn. Correlation and calibration of the data required months of computation. Image reconstruction required additional months.
The computational infrastructure that made this possible traces back to the development of electronic computers in the mid-20th century. The stored-program architecture described in von Neumann’s 1945 EDVAC report, available in Kronecker Wallis’s edition of the EDVAC Report, is the blueprint for every computer that processed the EHT data. From vacuum tubes to petabyte arrays, the architecture remains fundamentally the same.
Mathematics Made It Visible
The first image of a black hole is often described as a triumph of engineering, and it is. But at its core, it is a triumph of mathematics. General relativity predicted the shadow. Fourier analysis made the reconstruction possible. Optimization algorithms extracted the image from sparse data. Statistical methods quantified the uncertainty.
Without mathematics, the EHT data would be meaningless: a collection of numbers from eight telescopes. Mathematics turned those numbers into a picture of something that, by its very nature, cannot be seen. The dark circle at the center of the M87 image is a region from which no information can escape. And yet we know its size, its mass, and its shape, because the mathematics that describes how spacetime curves around it is precise enough to predict what the surrounding light should look like.
The black hole is invisible. Mathematics made it visible. That is perhaps the deepest lesson of the EHT image: the universe can hide its most extreme objects behind absolute darkness, and mathematics can still reveal them.
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