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Hitting the Books: The 'symphony' of Fourier transformations that first imaged a black hole

·Senior Editor
·8 min read

Our understanding of the universe changed a bit in April, 2019 when an international team of astronomers coordinated an octet of telescopes spread across the globe to capture the first photographic evidence of black holes. Renowned astrophysicists and Professor of astroparticle physics and radio astronomy at Radboud University, Dr. Heino Falcke wasn't only on hand for the momentous imaging session, he was leading the way. 

Light in the Darkness is a meditation on those efforts, exploring the nature of black holes through the technologic, scientific, philosophic and spiritual. In the excerpt below, Falcke examines the inner workings of Fourier transformations and how they helped reveal the darkest spots of the universe. 

Light in the Darkness cover
Light in the Darkness cover

Excerpted from Light in the Darkness by Heino Falcke, reprinted with permission by HarperOne, an imprint of HarperCollins Publishers. Copyright 2021.

Images of outer space don’t just fall from the sky. Quite the contrary, every astronomer knows how much effort and patience are necessary to capture an image of the cosmos — especially when the light waves are stored on hard drives. After gathering the data, we essentially have to assemble a globe-spanning telescope on the computer and figure out what the dish or mirror of this giant telescope would do with real waves.

The mathematical operation that a mirror carries out when it focuses light from space is called a Fourier transform. The operation is named after the French mathematician Jean-Baptiste Joseph Fourier, who introduced the idea in 1822, and today it is used in every conceivable area of our day-to-day lives. Anyone who stores compressed JPEG images or MP3 music files on their computer uses aspects of Fourier transforms. Our ears do as well, transforming oscillations into notes. It turns out our ears are mathematical geniuses, as is a simple concave mirror: they can handle complex mathematical operations automatically, in their sleep — as anyone who has been startled awake in the middle of the night by the beeping of a malfunctioning alarm clock has experienced. On the computer, though, we first have to complete the arduous task of programming these transforms ourselves, which means teaching the computer to carry out the operation step by step.

A unique quality of the Fourier transform is the ability to leave out information without losing the total impression of an image or a piece of music. Electronic compression processes take advantage of this every day: you make a Fourier transform of the image or the piece of music, deleting unimportant parts of the data and saving the data left over, and at any time you can use these to transform the image or the sound file back to its previous state. The differences are hardly visible or audible, but the data quantities have become substantially smaller, which means, for example, that more images can be stored on a single memory card.

The same thing happens when there’s dust on a camera lens, or when we look into the night sky with a telescope with a scratched-up mirror. We lose information as a result, and the mirror can’t completely carry out the Fourier transform. Still, we don’t get an image with holes or perforations in which certain stars are missing, but rather one in which every star looks a little less clear. Without our noticing, the disruptions caused by missing information are distributed across the entire image. Every flaw in the mirror influences all the stars in the image equally. With a computer algorithm, however, you can for the most part calculate and remove these flaws, and in so doing polish up the image.

For this reason, a global radio interferometer, which rather than being made of a big reflecting mirror is made up of many small telescopes linked together, doesn’t have to be complete. It works even if its telescopes don’t cover every inch of the world’s surface. This is the equivalent of a scratched-up mirror with lots of holes—in fact, a lot more scratches and holes than mirror. Nevertheless, with a little skill and some valuable mathematical knowledge, a precise image can be reconstructed. This saves a lot of antennae and even more money. Totally covering the Earth’s surface with radio telescopes would also be something of an imposition.

A good comparison for the Fourier transform of an image is a symphony. The image that you see is like the music you hear. The Fourier transform of the image, then, is like the score of the symphony, and a radio interferometer is a measuring device that records the music and divides it back up into the individual notes of the score.

At any given time within our VLBI network, every combination of two telescopes is measuring exactly one note of the image, which the correlator calculates. The distances between the telescope pairs are the baselines; you can imagine them as being like the different-size strings of a harp that are responsible for different notes. Only here it’s the other way around: the strings don’t produce notes; rather, they listen for them, and the longer they are, the higher the “image-note” is that they pick up. Returning to the symphony analogy, the short baselines would mainly hear the timpani and the double bass, and the long baselines just the piccolos and the triangle.

If you were to carry out a Fourier transform of the image of a person’s head, for example, the low image-notes would just capture the shape of the head, but not the facial details. The high image-notes, on the other hand, would make the distinct contours of the mouth and nose visible, but not the head around them. What’s important is how long the virtual strings between the telescopes are from the perspective of the radio source. If you look at the string from an angle, it appears shorter than if you look at it from directly above. As a result of the Earth’s rotation, the projected string length and the direction change, and over the course of a night of several hours of observation, the telescope gets tuned.

To get a good image from a VLBI network, the measuring sensitivity of each individual telescope must be precisely calibrated relative to every other telescope, and relative delays between the telescopes must be corrected. This is equivalent to adjusting and evenly polishing a mirror that consists of several segments, or precisely tuning a piano. Our calibration group1 takes on this task in spring 2018. It makes sure we have the right mix, adjusting the different volume levels in a big piece of music with a lot of instruments; it conducts the “sound check” before our concert can begin. Only then can a harmonious, symphonic image of a black hole emerge from the cacophony of data.

One day in mid-May I’m leaving my office as usual when Sara Issaoun comes up to speak to me: “Have you seen our first calibration plots from Sgr A* and M87? I think you’ll find them pretty interesting,” she says in a suspiciously calm manner. Sara is always in a cheery mood, but today her eyes have a particular roguish gleam to them. I’m curious and take a look at her screen. Then I do a double take. Stunned, I ask, “Do you all believe what you’re seeing there?” “Well, sure, it’s preliminary data; of course we’ll have to examine it more carefully . . . ,” she answers.

What the calibration team is looking at is a curve of faint points. It’s the musical scale for M87 and shows the volume of every note we measured from the object, arranged by frequency. The volume decreases steadily as it moves in the direction of the image’s high notes, and eventually hits zero. If the image of the black hole were a portrait, we would now know exactly how big the head was. The fewer the high notes, the larger the head. But then the curve suddenly starts to go up again. We also measured a lot of loud high notes. The head has a face, too, and we’ve captured it! The highest—and most crucial—notes reached us in the very last minutes, when we were observing simultaneously in Spain and Hawaii—truly astounding!

I take a deep breath, relieved and yet nervous at the same time. “This is too good to be true!” Every one of us knows the shape of this curve; it’s in every radio astronomy textbook.2 “I don’t want to say it, but this corresponds rather exactly to the Fourier transform of a ring. If that’s true, then M87 really is as big as some people say it is, and we can see the shadow,” I say, almost in awe. “Yes, six to seven billion solar masses,” Sara adds, smiling.

“Well, okay, first we’ll just have to wait and see,” I reply, determined to sound casual, and try to put my poker face on. Nevertheless I spend the rest of the day pacing nervously around my office. It’s as if you’d received news that a very special guest, whose visit you’d spent decades waiting for, was about to arrive—that’s what the situation feels like for me. Soon we would be able to see this guest for the first time. A prayer of thanks fills the otherwise so sober room.