Is Einstein Still Right?

by Clifford M. Will

  We are going to need a really good camera to take a picture of the disk around Sgr A*. After all, we already explained how Sgr A* subtends a tiny angle in the sky, a mere 20 microarcseconds, as viewed from Earth. Enter Very Long Baseline Interferometry (VLBI). In Chapter 3 we saw how connecting radio telescopes can lead to very precise measurements of directions of sources such as quasars (page 39). Combining the data from such telescopes in a specific way can also produce images of the source, given enough resolution.

  The interferometer used by Balick and Brown only allowed them to limit the size of the region where Sgr A* is located, but not to resolve the source itself. To do better, you need longer baselines. In the late 1990s, astrophysicists Heino Falcke, Fulvio Melia and Eric Agol pointed out that if the baseline between telescopes is about the diameter of the Earth and the observing wavelength is about a millimeter, then the angular resolution at the Galactic Center would be around 20 microarcseconds, smaller than the diameter of the shadow of Sgr A* as seen from Earth. It was later realized that, with the same resolution, it would be possible to detect the shadow of the supermassive black hole in the center of the galaxy, Messier 87, generally called M87. Although, at a distance of 53 million light years, the galaxy is two thousand times farther away than Sgr A*, the black hole is 1,500 times more massive than Sgr A*, and therefore, as seen from Earth, its shadow is about the same size. Sgr A* and M87 seem to be the only two massive black holes so far that have just the right “Cinderella” combination of mass and distance to make this possible.

  This inspired Sheperd Doeleman of MIT’s Haystack Radio Observatory and Harvard’s Center for Astrophysics to try to forge a collaboration of radio astronomers working at telescopes around the world to create an array with Earth-sized baselines. This would not be a simple task. The telescopes were operated by different agencies in different countries and had competing scientific priorities. Since each telescope would independently but simultaneously observe the Galactic Center, it was important that the instrumentation at each telescope be either identical or sufficiently similar that the data quality was the same. Each location had to have excellent atomic clocks so that the data could be time stamped accurately enough to permit merging the various data sets properly. The data from each observing session (thousands of terabytes worth of data) would be shipped to a central location for processing into images. It was also essential that the weather be excellent simultaneously at many different locations around the globe. Good luck with that!

  As a proof of principle, Doeleman and colleagues managed in 2007 to use a triangle of short-wavelength radio telescopes in Hawaii, California and Arizona to detect something at the Galactic Center at a scale of the order of Sgr A*’s event horizon. This was the breakthrough needed to push ahead.

  In 2012, Doeleman and colleagues formally kicked off the project, now called the Event Horizon Telescope. Currently, the array of telescopes in the collaboration numbers ten: two in the mainland USA, three in Chile, two in Hawaii, one in Spain, one in Mexico and one at the South Pole (see Figure 6.7). There are plans to add telescopes in Greenland, France and the USA. The initial full observation run occurred over ten days in April 2017, with eight observatories taking part. One obstacle to analyzing all the data was logistical: by the end of the run in April, winter had set in at the South Pole, and so the data files had to be placed in (literal) cold storage until December 2017, when they could be flown out. The data were copied and sent to four teams who analyzed them independently of each other, and with strict secrecy to guard against mistakes and to enable later checks and counter checks. Finally, on 10 April 2019, they announced that they had obtained an image of the shadow of M87, and that its size was consistent with the prediction of Einstein’s theory. The data on Sgr A* is still being analysed; obtaining an image is more difficult, in part because the accretion rate of gas is on the low side, so it is not very bright, but also because the accretion is highly variable. More observations may be needed before they will be able to see Sgr A*’s shadow.

  Figure 6.7 The Event Horizon Telescope. The radio telescopes of the project and the baselines joining them. Credit: Event Horizon Telescope Collaboration.

  We all delighted in the breathtaking beauty of the pictures they produced; you would be hard pressed to find a newspaper anywhere in the world that did not have the image on its front page. But this is not the only (or even the main) reason for this major undertaking. The accretion disk around both black holes is not static, but changes with time as hot blobs of matter within the disk circle the black hole, thus changing the light emitted. As we already mentioned, such hot spots have already been detected around Sgr A* by the Galactic Center group at the Max Planck Institute. By combining a sequence of joint pictures, EHT hopes to produce a movie of how the accretion disk behaves as its inner edge is swallowed by the black hole. This, in turn, will provide a detailed window into the physics and the dynamics of accretion disks, allowing astrophysicists to compare their models and predictions to actual data.

  We may also be able to test general relativity with EHT. The idea is to test the black hole no-hair theorems much closer to the black hole than we could do using stars. In general relativity, the shadow cast by the black hole is a circle for the non-rotating hole, but is off center and somewhat flattened on one side if the black hole is rotating. Once you know the mass and angular momentum of the black hole, this shape is predicted precisely by general relativity, given a particular gastrophysical accretion disk model. But if rotating black holes are not described by the Kerr solution of general relativity, then this shadow will be different, perhaps more flattened or less flattened, perhaps shifted in a different way relative to the Kerr expectation. A precise enough observation of the shadow could therefore test general relativity.

  Similarly, because the external gravitational field of the black hole is completely fixed by its mass and angular momentum, the response of an accretion disk and the behavior of the light it emits are predictable and fixed. The black hole no-hair theorems of general relativity leave no wiggle room. Of course, the light that EHT will observe will depend on the nitty-gritty details of the disk itself, such as its density, temperature and composition. But because, unlike its cousins in X-ray binaries and quasars, this disk around Sgr A* appears to be a relative weakling, there is reason to hope that there won’t be too much dirty gastrophysics to complicate the interpretation of the observations. The same tests can be done with follow-up observations of M87.

  We have come a long way from a time when Einstein and his contemporaries were sure that Schwarzschild’s Massenpunkt solution would never happen in nature because of its strange singularity. Today we know that black holes exist, and they may soon provide remarkable new tests of Einstein’s theory. But the black holes we have described in this chapter pretty much sit there being, well, black holes. What if we could detect two black holes colliding with each other?

  1 Pericenter is the generic term denoting the closest approach in an eccentric orbit. For specific systems there are suitable variants, such as perihelion for orbits about the Sun, perigee for the Earth, perijove for Jupiter, periastron for binary stars and so on. The relativity community has not managed to come up with a good term for orbits around black holes. “Periholion” doesn’t thrill the community. Our colleague Scott Hughes at MIT has proposed “peribothros,” using the ancient Greek word βóθρoσ for hole, but Greek colleagues have pointed out that in modern Greek this word has a different (curse-word) meaning. We invite readers to send suggestions.

  CHAPTER 7

  Gravitational Waves Detected At Last!

  The room is small and windowless. At the front of the room stand five chairs, two large video panels and a podium displaying the logo of the NSF, the US National Science Foundation. The audience comes from around the world and includes scientists, government officials and reporters. They whisper in anticipation of a major announcement that the scientific rumor mill has been mongering for a couple of months. At 10:30 a.m
. on Thursday 11 February 2016, NSF Director France Cordova welcomes the audience to the National Press Club in Washington DC.

  Two thousand miles away, in the small town of Bozeman, Montana, Nico and twenty people sit at a table in a small room of the eXtreme Gravity Institute at Montana State University. This room has windows with a beautiful view of the mountains, but nobody is paying attention to the scenery. All eyes are focused on the television in the front of the room. The screen has a live internet stream from the National Press Club. Nobody pays attention to the celebratory cake waiting on the center of the table.

  Five hundred miles to the south, in Aspen, Colorado, Cliff and eighty physicists and astronomers watch the same feed in an auditorium of the Aspen Center for Physics. They are participating in a workshop on stars and gas at the centers of galaxies, but the day’s schedule has been pushed back by two hours so that everybody can watch this event.

  After briefly extolling the NSF’s commitment to funding cutting-edge research in fundamental science, Cordova sits down and the man sitting next to her approaches the podium. He is tall, middle aged, with graying hair, wearing a blue suit, with a blue shirt and a paisley tie. His tired eyes reveal that the past few months have seen very, very long hours. He places some notes on the podium.

  “Ladies and gentlemen,” he says. “We have detected gravitational waves. We did it!” he exclaims, and the audience bursts into applause. The twenty people in the room in Bozeman applaud; the audience in Aspen applauds. At institutes and universities around the world, scientists of all stripes break out in applause. David Reitze, the Director of the Laser Interferometer Gravitational-Wave Observatory, or LIGO for short, has just announced the most important scientific discovery of the twenty-first century (at least so far).

  Around 1.2 billion years ago, in a very distant galaxy, two black holes crashed against each other. Each black hole was roughly thirty times more massive than our Sun, but in actual size was only about as big as Albania or Haiti. They were circling around each other at roughly half the speed of light, locked by gravity in a fatal dance, when they merged to form a single black hole. The event created ripples in the fabric of space and time that traveled outward in all directions at the speed of light. On 14 September 2015, those same spacetime waves finally arrived at the Earth, passed through the LIGO instruments and produced an unmistakable gravitational wave reading. This was the event that David Reitze had just announced to the world.

  Within hours, congratulations poured in, from Stephen Hawking, from President Barack Obama, from leaders of the CERN accelerator center in Geneva. Twenty months later, a remarkably short time for the normally glacial Swedish Academies, the 2017 Nobel Prize in Physics was awarded to three of the founders of LIGO: Rainer Weiss, Kip Thorne and Barry Barish. “Gravitational wave astronomy” became an “official” field, hailed even by some of the astronomers who once lobbied against LIGO.

  Gravitational waves were not always so in vogue.

  At one point, Einstein himself thought he had proven that gravitational waves were not real! The definitive proof by theorists that they are real would not be achieved until the late 1950s, and the first experimental evidence that they exist would come in 1979, as we saw in Chapter 5. A 1969 claim by physicist Joseph Weber to have actually detected the elusive waves would soon be undone by the failure of other scientists to replicate his results. The story of gravitational waves is rich in science, of course, but is also a story of human personalities and foibles, of debates and controversies, and of big science, politics and money. It is a hundred-year-long saga that starts with a botched paper by Einstein himself.

  In May 1916, Einstein published a major review article on his general theory of relativity that pulled together all the bits and pieces of the short papers that he had presented at the Prussian Academy of Sciences the previous November into a coherent exposition. He then immediately began to work on gravitational waves.

  Einstein was a devotee of James Clerk Maxwell (1831–1879), the Scottish physicist who in 1867 united the seemingly disparate phenomena of electricity and magnetism into a single framework, known as electromagnetism. Maxwell’s equations are still a central ingredient of modern physical science, from electrical engineering to high-energy particle physics. A deep understanding of Maxwell’s theory underlies the technology in our most beloved devices, such as televisions, cellphones and laptops. Maxwell’s equations are at the core of physics and engineering education today, and that was also the case in the late nineteenth century when Einstein was a student.

  Maxwell’s key insight was that electricity and magnetism could be understood through the idea of an electromagnetic field, a physical quantity that encodes information about the force exerted on a charged object anywhere in space. Even if you don’t realize it, you have probably been exposed to the concept of a field before. You may have observed the way iron filings on a sheet of paper array themselves to display the field of the magnet under the paper. You know that Earth’s magnetic field helps to protect us from the harmful energetic particles streaming from the Sun and creates the aurora borealis and the aurora australis. And you have heard of the gravitational field of the Earth, which is responsible for the force that allegedly caused the famous apple to fall on Newton’s head, and that also holds the Moon in its orbit.

  In addition, Maxwell showed that his equations had solutions in which the electric and magnetic fields oscillate, feeding off each other to produce a wave that travels at the same speed as light. He suggested that these waves were light, an idea that was confirmed experimentally by Heinrich Hertz in Germany in 1887.

  Several scientists, including Hendrik Lorentz in the Netherlands and Henri Poincaré in France, began to wonder well before Einstein whether there could also be waves of gravity itself, simply by analogy with Maxwell’s electromagnetic waves. Furthermore, because Einstein’s special theory of relativity said that nothing could travel faster than light, it seemed logical that gravity should not be instantaneous. The effects of gravity should travel with a finite speed, and this speed should not be greater than that of light. But speculate was the best they could do, because they didn’t have an actual theory of gravity to work with.

  But in 1916 Einstein had an actual theory, and he set out, in the spirit of his hero Maxwell, to see if his equations had solutions that would resemble waves. He completed the calculations and published the result in June 1916. Unfortunately, the paper was full of what could charitably be called “bone-headed” mathematical and conceptual errors. Einstein’s colleague, Norwegian physicist Gunnar Nordström, helped Einstein find and correct the mistakes, and Einstein published a second gravitational wave paper in 1918.

  Einstein showed that a varying system, such as a dumbbell spinning about an axis perpendicular to its handle (Figure 7.1), will emit gravitational waves that travel at the speed of light. He also found that the waves carry energy away from the rotating dumbbell, just as light waves carry energy away from a light source. As we saw in Chapter 5, this loss of energy in a binary pulsar system is what Hulse and Taylor measured, thereby verifying, albeit indirectly, that gravitational waves exist. It’s the same loss of energy that brought the two black holes to their final embrace, emitting the burst of gravitational waves that LIGO detected.

  Figure 7.1 Einstein’s spinning dumbbell generating gravitational waves.

  But there were aspects of his gravitational wave solution that Einstein didn’t fully understand. In Maxwell’s theory there are two solutions for electromagnetic waves. For example, if you have a wave propagating horizontally in the laboratory, one solution could have the oscillating electric field pointing vertically (it is always perpendicular to the direction of propagation); the other solution would then have it pointing horizontally but still perpendicular to the direction of propagation. These two cases are called the “modes of polarization” of the electromagnetic wave, and a general light wave consists of a combination of the two modes. A charged particle encountering the electric
field of the first mode would move up and down, just as a ball moves up and down on a water wave at the beach. This is in contrast to sound waves, where the motion of the molecules of the medium which carries the sound is always along the direction of propagation of the waves. The concept of polarization is exploited for example in polarized sunglasses, which are designed to block one mode of polarization preferentially. They are most useful at protecting your eyes when the mode that is blocked is the one that is dominant when light scatters off the pavement in front of your car or off the water at the beach.

  In the case of gravity, Einstein also found two modes of polarization of the gravitational waves (we’ll get to what those modes look like shortly), and those modes traveled with the same speed as light. But there were additional solutions to the equations whose meaning was not so clear, and to make matters worse, the speed of these modes was not fixed by the equations. In a 1922 paper, Eddington analyzed Einstein’s gravitational waves carefully. He pointed out that Einstein had made a small calculational error in his 1918 paper, making his formula for the energy lost off by a factor of two. He also observed that the additional modes of Einstein were probably not physically real, but instead might be waves in the coordinates used to describe the problem. He made the dismissive remark that “the only speed of propagation relevant to them is the speed of thought.”