Is Einstein Still Right?

by Clifford M. Will

  But because Dyson’s paper appeared in a book on extraterrestrial life, nobody working in general relativity noticed it or followed up on it. It was the discovery of the binary pulsar in 1974 that made the idea of binary inspiral respectable (see Chapter 5). The binary pulsar was proof that binary neutron stars exist, and the measurement of its decreasing orbital period proved that such systems spiral inward because of the emission of gravitational waves. As Taylor and his collaborators found, the rate of inspiral is absolutely tiny, only 76 microseconds per year in its orbital period, or 4 meters per year in separation. But the measured rate agreed with the prediction of general relativity. The theory goes on to predict that as the orbit shrinks, the stars speed up, thereby emitting stronger gravitational waves, which thus accelerates the inspiral, leading to even stronger waves, and so on, culminating in a runaway rush toward a final merger. For the Hulse–Taylor system, the formula predicted that the merger would occur in a few hundred million years, while for the double pulsar, it would occur in 85 million years. These are ridiculously long times—don’t bother to mark your calendars—but they are only a few percent of the thirteen-billion-year age of the Milky Way galaxy. So one could easily imagine counterparts to these binary pulsars starting their evolutions a few hundred million years ago and merging today, thus fulfilling Dyson’s prophesy.

  As planning for large-scale laser interferometers moved forward, the idea that neutron star inspirals might be a leading potential source of gravitational waves began to take hold. The discovery of a few more binary pulsars in our galaxy similar to the Hulse–Taylor system made it possible to estimate, albeit very crudely, that the rate of final inspirals of such systems in our galaxy could be of the order of one every 100,000 years. Obviously this is not a rate of occurrence on which to build a thriving career in gravitational wave detection. And indeed this fact was the Achilles heel for Joe Weber’s attempts to detect gravitational waves. His detectors might have been capable of sensing waves from a nearby binary neutron star inspiral in our galaxy, but the rate of such events was far too small.

  However, the strength of the gravitational wave signal received on Earth is inversely proportional to the distance of the source. If the waves from a source at a given distance have a certain strength, then waves from an equivalent source twice as far away are half as strong; waves from such a source ten times farther way are one tenth as strong, and so on. So if one could build an interferometer sensitive enough to detect waves from such a distance that a sphere surrounding the Earth of that radius contained about a million galaxies, then all of a sudden the rate of detectable neutron star inspirals could be a few (maybe up to ten) per year. A graduate student could even get a Ph.D. out of detections that frequent.

  It is important to remember that a laser interferometer differs from a normal telescope in a crucial respect. A telescope can see light coming from only a very narrow range of directions and therefore has to be pointed toward a potential source. It is blind to all other directions. By contrast, a laser interferometer can hear gravitational waves coming from essentially every direction. There are only four source directions to which the interferometer is “deaf.” If the wave approaches the interferometer traveling parallel to the ground at the site, but making an angle of 45 degrees with respect to the arms, the interference pattern when the laser beams recombine will not change at all (see Figure 8.5). But if the wave is approaching at just 7 degrees off from those special directions, the interferometer will hear the signal with 25 percent efficiency. At 15 degrees, the efficiency is up to 50 percent. Only a tiny number of signals unlucky enough to arrive from the vicinity of those four special directions would be missed. So if the designers and builders of LIGO and Virgo could reach the required sensitivity to hear inspirals from the nearest million galaxies (roughly a million times more sensitive than Weber’s resonant bars), then one could imagine a decent rate of detections.

  Figure 8.5 An interferometer is deaf to any gravitational waves approaching it parallel to the ground at an angle of 45 degrees relative to the arms.

  Another reason that the neutron star inspiral took hold as the most popular potential source is that the gravitational wave signal could be calculated very accurately from Einstein’s theory. General relativity is a notoriously difficult and complicated mathematical theory, but fortunately the problem of two very small bodies orbiting each other was one that could be attacked by a number of methods. In one method, the equations of the theory could be replaced by an approximate set of equations which could be solved for the orbital motion and the gravitational waves emitted. This approximation could then be systematically improved in a step-by-step procedure. The results of these calculations were enormously long formulae that would typically span two or three pages of paper. But using these formulae and a simple laptop computer or tablet, one could generate a very accurate prediction for thousands of cycles of the inspiral part of the gravitational signal in a tiny fraction of a second. The last few cycles of a characteristic inspiral signal are displayed in Figure 8.4. In another method, the exact equations of the theory were formulated in a way that they could be solved numerically using large computers. This method was particularly important for accurately predicting the merger part of the signal, also shown in Figure 8.4. By the early 1990s, as inspirals were being recognized as the leading potential source for detection by the interferometers, Kip Thorne realized the importance of having accurate predictions of the waves in advance, and he urged theorists to begin the arduous task of developing these methods for predicting the wave shapes. Progress was slow at first, but by the time LIGO began its advanced run in September 2015, the tools for making accurate predictions were in place.

  Why did this matter? The answer is a version of the child’s puzzle that requires spotting a character named Waldo in the middle of a page of hundreds of cartoon characters; this version is called “Where’s Nico?”. If you were asked to spot a certain Nico in a teeming crowd of random people, knowing that Nico is a man would not be very helpful. But knowing that he is wearing a purple shirt with a black vest and a white gaucho hat would be very useful in finding him. The more you know about a signal, the easier it is to find it, even if it is buried in a lot of noise. In the same way, one could compare a library of predicted wave shapes (similar to the wave shown in Figure 8.4) to the output of the detector to see if there was a match. This would be done, not by eye, which is notoriously unreliable, but using sophisticated and extremely fast data analysis algorithms. Also, since the wave shape depends on things like the masses of the two bodies, that wave from the library that gives the best match to the data provides information about the system that produced the gravitational wave in the first place.

  The neutron star inspiral idea was so popular that if you had asked anybody in the field just prior to the start of the Advanced LIGO run in 2015 what would be the most likely first detection, a large majority would have predicted a binary neutron star merger. The authors of this book certainly would have predicted that, though neither of us felt strongly enough to put serious money or seriously expensive wine on the table. But according to a Danish proverb (also attributed to Danish physicist Niels Bohr), prediction is difficult, especially about the future. Our predictions were wrong.

  Not only was the first signal detected that of a binary black hole inspiral and merger, but the masses of the two black holes were completely unexpected. Until then, all the observational evidence and theoretical modeling on black holes pointed to two basic classes of black holes. The first was the class of stellar mass black holes, with masses between 6 and 15 solar masses, the classic example being the 10 solar mass black hole in Cygnus X-1. The second was the class of massive black holes, with masses between a hundred thousand and several billion solar masses, residing in the centers of galaxies, such as Sgr A* or the black hole in M87. We encountered these black holes in Chapter 6. A third class of intermediate mass black holes, between 100 and 100,000 solar masses, has been proposed, but the eviden
ce is still not very solid.

  Suffice it to say that black holes with masses of 36 and 30 solar masses were not expected. This was the set of values that gave the best fit between the theoretical wave shapes and the observed wave shapes during the inspiral phase. From that portion of the signal (the left-hand part of Figure 8.2), we can infer that the black holes were separated by roughly 700 kilometers, each revolving around the other at about one fifth of the speed of light, emitting waves at a frequency of about 50 hertz. By the time they collided and merged, each was moving at roughly a quarter of the speed of light. Using those masses and the equations of general relativity, and recalling that the overall strength of the waves decreases inversely with distance, one can then calculate how far the source would have to be so that the overall size of the wave agreed with the measured size shown in Figure 8.2. The answer turned out to be 1.4 billion light years. The final waves were emitted 1.4 billion years ago, around the time when the first green algae were forming in the oceans of the primitive Earth. Even though the two black holes were extremely far away, their larger than expected masses made the waves still “loud” enough at Earth for LIGO to detect them in 2015.

  When two black holes merge they form a single black hole in a process that pictorially is similar to the way two soap bubbles merge into one big soap bubble. The main difference is that the “surface” of each black hole, known as the event horizon (see Chapter 6), is not made of any material such as soap, but is a surface where the warpage of spacetime has specific characteristics (such as allowing you to go in but never come back out). But just as the newly formed soap bubble may initially have a convoluted shape, the black hole remnant is a highly distorted beast, unlike the simple pictures you might see that represent black holes as dark spherical objects. The spacetime surrounding it also has bumps and distortions that vibrate and generate additional gravitational waves as the distorted black hole rotates. The frequency of the waves is related to the mass of the black hole, just as a bell whacked by a hammer emits sound of a specific tone. Eventually, friction within the metal of the bell reduces the vibrations and the bell goes quiet. But for the black hole “bell,” these waves, called “ringdown” waves, carry the energy of the vibrations away so effectively (some waves go into the hole as well) that the black hole goes quiet after only a few cycles.

  Unfortunately, the gravitational waves detected in 2015 were not loud enough during this ringdown phase to extract the ringdown frequency and decay time directly from the data, but something else could be done. The LIGO collaboration was able to extract the masses of the black holes from the earlier part of the signal, as we described earlier. Using this, together with numerical simulations of the Einstein equations, scientists were then able to predict that the mass of the final black hole was 63 solar masses, which is roughly consistent with the frequency of the ringdown part of the signal shown in Figure 8.2. Incidentally, those final cycles of radiation also provided evidence that the final object is a black hole. Other stellar objects, such as neutron stars, white dwarfs or ordinary stars, also vibrate at specific frequencies (the Sun has its own set of modes of oscillation), but their frequencies and decay rates are nowhere near the values inferred for the remnant of GW150914.

  If you noticed a discrepancy between the initial total mass (66 solar masses) and the final mass (63 solar masses), you are correct. Three solar masses were converted into the energy of the outward-flowing gravitational waves, and most of this happened during the two tenths of a second of the signal detected by LIGO (see Figure 8.2). At its peak this represented a rate of energy output that is larger than that of all the stars in the observable universe combined! The energy emitted was the equivalent of a decillion (i.e. a million billion billion billion, or 1033) 1 megaton hydrogen bombs. Interestingly, while stars and hydrogen bombs convert one form of matter into another (mainly hydrogen into helium), with the mass difference being converted into energy, here there is no matter. Whatever matter was involved in forming the black holes is inside their event horizons and can no longer affect the outside world. Instead, the conversion is from the mass imprinted on the spacetime curvature surrounding each hole into the energy of the spacetime ripples propagating outward.

  In fact, using the characteristics of the detected wave and the distance to the source, one can estimate the total energy carried by the waves in all directions (some of which might conceivably have been detected by a LIGO in a galaxy on the far side of the universe). That energy turns out to agree reasonably well with Einstein’s famous formula E = mc2, where m is the 3 solar mass difference between the initial and final masses of the source, and c is the speed of light. That iconic formula, so well verified in the laboratory, also holds on cosmic scales!

  You can think about this loss of mass–energy in another way. We saw in Chapter 5 that the orbital period of binary pulsars decreased as gravitational waves took energy away from the system. In that case, the rate of change of orbital period was minuscule, roughly 70 microseconds per year. But for the LIGO observations, the period change was so fast that one could see it in the 0.2 second data stream by eye (Figure 8.2)! Using the same formula that relates energy loss to period change for the binary pulsar, one could show that the the amount of energy lost during the inspiral in order to cause the observed period change was perfectly consistent with the 3 solar mass difference.

  Everything about the event GW150914 supported key predictions of general relativity: gravitational waves exist, and carry energy away from the source. We knew this indirectly already from binary pulsars, but a direct detection and confirmation was essential. Binary black holes exist. There was already abundant observational evidence for black holes using electromagnetic radiation from material swirling or stars orbiting around them (see Chapter 6), but this was the first evidence using gravitational signals alone. Until then there was no observational evidence for binary black holes. This was the first, but it would not be the last. Newly formed, distorted black holes settle down by emitting “ringdown” gravitational radiation. This was confirmed.

  GW150914 provided another important test of general relativity. The theory predicts that gravitational waves travel at exactly the speed of light, and, as with light traveling in vacuum, the speed is independent of the frequency of the waves. But during the inspiral, the frequency of the emitted waves changes dramatically, yet Einstein’s theory predicts the waves always travel at the speed of light. In some modified theories of gravity, however, this is not the case.

  One example is a class of theories developed in an effort to explain the fact that the universe appears to be expanding faster with time rather than slowing down with time, as standard general relativity predicts (see page 8). In these theories the “particle” associated with gravitational waves is given a small mass. Just as light can be thought of both as a manifestation of Maxwell’s electromagnetic waves and as a quantum mechanical particle called the photon, vibrations in gravity can also be thought of as a manifestation of Einstein’s gravitational waves or as a quantum mechanical particle called the graviton. But while the particle aspect of light has been abundantly demonstrated and even finds practical applications in things like solar panels, gravity is so weak that we will never be able to measure the particle aspect of gravity directly. If gravitons had a mass, then they would travel more slowly than light, with their velocity dependent on the wavelength of the waves associated with them. In particular, waves of long wavelength would travel slower than waves of short wavelength. Figure 8.6 illustrates what happens: waves emitted in the final part of the inspiral and merger travel a bit faster than waves emitted in the early inspiral. This then allows the merger waves to “catch up” with the inspiral waves in their long, long journey from the source to the Earth. The waves received by a detector would therefore be compressed or squashed a bit in time, compared to the wave that was emitted. But the signature of such a distortion was not seen at all in the LIGO observations, once again confirming Einstein’s predictions in a spectacul
ar way. The LIGO observation places a bound on how large the mass of the graviton could be, as otherwise, if it were any larger, they would have been able to detect its effects. The bound says that the mass of the graviton must be smaller than one part in ten octodecillion kilograms, in other words one divided by ten followed by fifty-seven zeroes! General relativity predicts that its mass is exactly zero, in agreement with the measured bound.

  Figure 8.6 If short-wavelength gravitational waves travel more quickly than long-wavelength waves, then the later “merger” waves will catch up with the earlier inspiral waves, so that the wave received by the interferometer is slightly squashed in time compared to the emitted wave. No such effect was detected in the data, placing a very tight upper limit on the mass of a hypothetical “graviton.”

  Despite the string of experimental successes for general relativity that we have been describing in the first part of this book, experimental verifications like these matter. The reason is that Einstein’s theory had not yet been seriously tested in extreme gravity situations, where gravity is very strong and rapidly changing. At the same time, observations of the universe as a whole have revealed apparent anomalies such as the accelerating expansion of the universe. The combination of these two facts has led theorists to propose that some of these anomalies could be resolved by modifying general relativity. And the potential outcome of these tests could be dramatic: any measurement that demonstrates and confirms a deviation from Einstein’s predictions could direct us toward a resolution of observational anomalies, potentially leading to a theory of gravity that is “beyond Einstein.”