Is Einstein Still Right?

by Clifford M. Will

  Figure 6.3 Orbits of selected S-stars around the Galactic Center black hole Sgr A*. The black hole is located at the exact center of the graph, with a diameter about 10,000 times smaller than the distance between tick marks at 0 and 0.2 on the graph.

  Figure 6.3 shows the orbits of twenty S-stars that have been inferred from the data (over a hundred stars are currently being monitored). The location of Sgr A* is at the very center of the graph, where the lines corresponding to zero angle cross (you may be able to spot the dot just inside the very elongated orbit of S14). A black hole of its mass would be roughly 25 million kilometers in diameter, about 18 times the size of the Sun. The Galactic Center is approximately 26,000 light years away from us, which translates to about one and a half billion times the distance from the Earth to the Sun. At that distance, Sgr A* is about 20 microarcseconds, or 0.00002 arcseconds, in angular diameter. For comparison, this is the size of two American quarters on the Moon as seen from Earth. On Figure 6.3, Sgr A* would be a dot with a width about 10,000 times smaller than the distance between the tick marks at zero and 0.2 arcseconds at the bottom of the figure, much, much smaller than the dot shown.

  Sgr A* is a hot playground for testing general relativity. Many of the tests we described earlier in this book can be repeated with careful observations of the stars in its vicinity. A prime example is the recent measurements of the gravitational redshift with S2, which we mentioned at the end of Chapter 2. Because it revolves around the black hole on a highly elliptical orbit, it reaches a mere 120 astronomical units at its pericenter, with a velocity of roughly 7,650 kilometers per second. Therefore, the light that S2 emits there is redshifted relative to the light it emits later in the orbit, because of both the special relativistic time dilation and the gravitational redshift effects. This redshift was measured in 2018 by Genzel’s team and confirmed by Ghez’s team in 2019, thus verifying general relativity’s prediction one more time, but this time close to a black hole.

  If this test of Einstein’s theory sounds familiar, it is because we already encountered a similar test back on page 17, when we described the experiments conducted by Pound and Rebka at Harvard University in 1960. In those experiments, Pound and Rebka had control over the light source emitted (gamma rays with a narrow wavelength produced in the decay of an unstable isotope of iron), as well as over the height (74 feet for the Jefferson tower) between the emitter and receiver. What they did not have control over was gravity, which for Earth is so weak that the predicted shift in the frequency of light was a mere two parts in a thousand trillion. For S2, on the other hand, the predicted shift is about six parts in ten thousand, a much larger effect because of the much greater warpage of time near Sgr A*.

  We began this book with a test of general relativity using Sgr A*, one of the events of “that very good summer” of 2017. Using orbital data from S2 and S38, a star with an orbital period of 19 years, Ghez’s team searched for a specific deviation from the normal inverse square law that Newtonian gravity predicts for the gravitational force between bodies. General relativity also predicts the same law as a good approximation when the bodies are not too close to each other, which is the case here. But in some alternative theories to general relativity there could be an additional force, which could be attractive or repulsive, depending on the theory, and which falls off more rapidly with increasing distance than the inverse of the square of the distance. Since both S2 and S38 are on very elliptical orbits, they sample gravity over a wide range of distances, from the distance at the pericenter to almost ten times that distance. Thus their orbits were especially sensitive to any change in the force with distance. No such anomaly was found. This was the first test of general relativity involving orbits around a black hole.

  The elliptical nature of S2’s orbit also allows for another classical test of general relativity with Sgr A*. As we saw in Chapter 5 when discussing the orbits of binary pulsars, curved spacetime induces a precession of elliptical orbits, leading to the famous perihelion advance of Mercury and the periastron advances of binary pulsars. The close proximity of S2 to Sgr A* at pericenter and the black hole’s large mass lead to a precession of S2’s orbit at a rate of 0.2 degrees per orbit, or about three-quarters of a minute of arc per year. On 19 May 2018, S2 passed through another pericenter, and many measurements of the orbit were made during that critical period when the effects of general relativity are the strongest. These observations should soon resolve this effect, and we expect a test of general relativity by roughly 2020.

  But other tests of Einstein’s theory with Sgr A* are possible if we find stars inside the orbit of S2. So far, no other stars have been detected closer to Sgr A*, both because of limitations of the telescopes and also because S2 is so bright and so close to the black hole that it makes it difficult to detect fainter companions between the two. If a companion were observed with the same orbital eccentricity as S2 but about twenty times closer to Sgr A*, it would be possible to measure the precession of the orbit due to the dragging of inertial frames. This frame dragging or Lense–Thirring precession is the same effect that we discussed in Chapter 4, when describing the measurements of the precessions of the orbits of the LAGEOS satellites. This would make it possible to measure the rate of rotation, or “spin,” of the black hole.

  Such a measurement is very important, for two reasons. The first is that the spin of a black hole of a given mass can range from zero for a non-rotating black hole, corresponding to the Schwarzschild solution, to a maximum value, corresponding to an extreme limit of the Kerr solution. The spin cannot exceed that maximum value, for if it did, the body would not be a black hole, but instead would be something called a “naked singularity,” a bizarre object that physicists find so horrifying that they are sure that nature would never let them exist. The second reason is that a spin measurement would give hints as to how the black hole formed and grew to its large mass. If it was by the merger of two pre-existing smaller black holes, it would likely have a rather large spin, just as two ice dancers who pull together in a hug at the end of a dance are rotating quite fast. But if it was by the steady accretion of stars and gas falling across the event horizon from random directions, then its final spin might be rather low, since the matter absorbed would spin the hole up as many times as it would spin it down.

  If stars even closer to Sgr A* were to be detected and tracked, then it might be possible to perform a test of the underlying assumptions of what is called the “no-hair” theorems of black holes. As theorists began to understand the full implications of the Kerr solution during the 1960s and 70s, they came to a startling realization. This solution was the only possible solution in Einstein’s theory for a quiescent black hole sitting in otherwise empty space, with the Schwarzschild solution being the limiting case of no rotation. And all the details of the external gravitational field of a black hole depend on only two quantities: its mass and spin. If you have two black holes of the same mass and rotation rate, and one was formed from the collapse of gas while the other was formed from the collapse of a huge cloud of Toyota pickup trucks, the external gravity will be identical. This is very different from, say, the Earth, whose external gravity field depends on the rotation of its molten core, the rigidity of its crust, the peaks of mountains and the depths of valleys. Recall from Chapter 4 that the Earth’s field has been measured in exquisite detail by orbiting satellites like GRACE.

  Pondering this remarkable property of black holes, John Wheeler coined the phrase “black holes have no hair.” He imagined that if you found yourself in a room full of completely bald men, it might be hard to tell one man from another, in contrast to being in a room of men with full heads of hair. Wheeler’s aphorism has been encoded into mathematical statements about the precise nature of the field around any rotating black hole. Therefore we can contemplate using a number of stars orbiting close to the black hole (they need to be close so that the relativistic effects on their orbits are detectable) to map out the gravitational field of the hole, the way GRA
CE satellites map the field of the Earth. But if these maps don’t agree with the prediction of general relativity, then either the theory fails in the strong gravity regime of black holes, or Sgr A* is a heretofore unknown object, nothing like a black hole. Are there stars orbiting close enough? We don’t know, but the teams of astronomers peering at the Galactic Center are on the hunt. The next few years should be very exciting, as S2 moves toward apocenter, the farthest point in its orbit from Sgr A*, perhaps revealing the presence of stars even closer to the black hole.

  Seeing stars move around Sgr A* is not the same thing as seeing the black hole directly. Needless to say, we cannot see any signal that originates inside the black hole. Fortunately, we now know that the black hole is surrounded by a disk of gas that is accreting into the hole, and is radiating light. Some of this radiation is in the radio band, producing the waves detected by Balick and Brown. Follow-up observations have confirmed the presence of an accretion disk. But unlike the strong X-ray emitting accretion disks associated with black holes like Cygnus X-1, and unlike the incredibly luminous disks associated with quasars, the accretion disk around Sgr A* is a total wimp. Apparently, there just isn’t enough gas migrating into the Galactic Center, either from ambient gas or from disrupted or exploded stars, to feed a luminous disk. Is it possible that Sgr A* was once a bright quasar, now reduced to a faint ember from a lack of fuel? This is an open question at the moment. But as faint as this emission is, the Max Planck team reported in 2018 that they had detected variations in the emission from the accretion disk that are consistent with motions around the black hole of hot spots within the gas. These hot blobs of gas are moving so fast, about 40 percent of the speed of light, that they must be orbiting right at the innermost edge of the disk, close to plunging into the black hole (see Figure 6.2). These blobs are moving in an extremely warped region of spacetime!

  But what if we could take an actual “picture” of Sgr A*? What would we expect to see? The answer is complicated because the black hole can bend light in dramatic ways. Taking a picture of somebody with your smartphone is straightforward because the light rays move on straight lines from the subject to the lens of the phone’s camera. Photographing things near a black hole is more like snapping a photo of yourself in front of the warped mirrors that you find in carnivals or fun houses. Depending on where you stand, you could have a fat head (or even two heads) and a slim waist, or you could have a pea-sized head and an enormously fat waist.

  Recalling our discussion of gravitational lensing (Chapter 3), a black hole acts like an extremely strong lens, warping and distending what you see, in the same manner as do fun-house mirrors. So if a black hole suddenly appeared in the night sky you would see a number of strange things. First, stars would appear to be pushed away from the black hole, as we illustrated in Figure 3.2. Because of gravitational lensing you might see multiple images of the same constellations, for example two Orion’s belts, or two Big Dippers, as illustrated in Figure 3.10. You might also see stars or constellations that are not normally seen in that part of the sky, but that are actually behind you. In this case, some light rays from the star can pass by you from behind, swing around the black hole and then enter your camera, producing an image of a star that is not actually there (Figure 6.4). There will be a circular black disk in the center, which we would observe to be about 2.6 times the diameter of the black hole itself. This is not the actual black hole, but is about as close to it as you can see.

  Figure 6.4 Strong lensing by a black hole. Not only can the black hole displace the image of a source (A), but it can also bend light from a source (B) behind the observer, producing an image (B) of a star that is normally not in that part of the sky.

  You may recall that earlier in the chapter we mentioned that light can actually orbit the black hole in a circular path at 1.5 times the Schwarzschild radius. This is called the “light ring” (Figure 6.5). The collection of all these rings is a sphere one and a half times the radius of the black hole horizon. Any light ray that enters this “light sphere” will spiral toward the event horizon and be lost (the light ring is not a horizon because the light ray could scatter off a passing atom and be deflected back out of the sphere). Any light ray that passes the black hole just outside the light sphere (path “c” in Figure 6.5) will reach your camera. But even after leaving the light sphere, the ray’s path will still be bent quite a bit until it gets far from the hole. As a result, the black hole appears to a far-away observer as if it were casting a “shadow” about 2.6 times its diameter. In addition, you will see the black shadow outlined by a thin ring of faint light, coming from all the rays of light from all the stars and galaxies anywhere in the sky that just skim the light sphere, orbiting the black hole a number of times and then leaving the hole heading for your smartphone.

  Figure 6.5 Black hole shadow. The black disk represents the black hole itself, and the solid circle is the “light ring,” with a diameter about 1.5 times that of the black hole, where light rays can revolve around the hole on circular orbits. An observer detects a sequence of rays from the vicinity of a black hole, each ray passing closer than the previous. Ray “a” is deflected mildly; ray “b” passes closer and is deflected by around 90 degrees; ray “c” comes from just outside the light ring and is also strongly deflected before reaching the observer; ray “d” originates just inside the light ring, and has to cross the event horizon. Thus the observer cannot see any light rays from inside the dashed circle. This is the black hole’s shadow, with an apparent diameter about 2.6 times that of the black hole’s event horizon.

  Similar distortions will occur if you photograph an accretion disk near a black hole. If you are looking at the disk from directly above (see the left panel of Figure 6.6), the image will look more or less like a disk, with an inner edge, where the gas plunges quickly toward the black hole with very little emission of light. The black center shown in the figure is the shadow of the black hole. If you are looking at the disk from an angle of, say, 45 degrees, then you will see the disk in front of the black hole’s shadow, pretty much as expected, and part of the disk heading behind the shadow. But you will also see what looks like a bulge of the disk on the far side of the shadow (middle panel of Figure 6.6). This is not a physical bulge of the disk, but comes from light emitted upward from the top face of the disk that is bent by the black hole’s gravity in a direction toward us. If you are looking at the disk almost perfectly edge on, you see a truly “fun-house” image (right panel of Figure 6.6). You will see the disk in front of the black hole shadow, but you will also see what looks like another disk circling the black hole. In the top part you are seeing the upper face of the disk behind the black hole, whose light starts out going upwards, but then is deflected by 90 degrees, heading toward you. In the bottom part you are seeing the underside of the disk behind the black hole. So in one picture you can simultaneously see both faces of the disk behind the black hole.

  Figure 6.6 Schematic image of an accretion disk around a non-rotating black hole. Left: The disk seen face-on with the black hole shadow in the center. Middle: The disk is tilted relative to the line of sight. Some light rays from the top surface of the disk on the far side of the black hole are emitted vertically, then deflected by a large angle toward us in the warped spacetime of the black hole. The back side of the disk appears to be bent upward. Right: The disk is seen edge on. Some rays can be bent by very large angles by the black hole, so that we can simultaneously see both the top surface and the bottom surface of the disk behind the black hole.

  By discussing the non-rotating, or Schwarzschild, black hole, we have grossly oversimplified things to bring out the main points. In real life, we expect most black holes to be rotating, possibly at a substantial rate. If that is the case, then the dragging of inertial frames changes the shadow and distorts the images dramatically. The size of the light ring depends on the rotation rate of the black hole and on whether the light is going around the hole in the same direction as its rotation or in the o
pposite direction. And if the light is on an orbit that is tilted relative to the black hole’s equator, then its motion can be very complicated indeed. For example, in the right panel of Figure 6.6, if the black hole is spinning with its rotation axis perpendicular to the accretion disk, then the bulges will be much more pronounced on the left side than on the right side of the hole. And the black hole’s shadow will no longer be circular. Many of the calculations that lead to these pictures were done in the 1970s by James Bardeen, then at Yale University, and Jean-Pierre Luminet of the Observatory of Paris, during the period of intensive theoretical research into the properties of black holes.

  In the movie Interstellar the image of Gargantua’s accretion disk is a more detailed and correct representation of what we have shown very crudely in the right panel of Figure 6.6. The calculations done by Kip Thorne to generate that image took into account all of the gory details. However, some details had to be left on the cutting room floor. An example is various relativistic effects that make the light coming from the part of the disk that is approaching the observer much more intense than that coming from the receding part of the disk. Had these effects been included, the intense light would have completely saturated the image, leaving nothing but a bright spot. While astronomers and physicists love to observe and analyze such details, the film’s director Christopher Nolan wanted a more pleasing image for his audience, so he asked Thorne to suppress those effects.