Chapter 4 Gravitational waves

  Gravitational waves have already been mentioned, in Chapter 3. Now let’s consider them in a little more detail. You will recall from Chapter 1 that, in Einstein’s theory, gravity is due to the curvature of space-time. Massive objects like stars and planets deform the shape of the space-time in which they exist, so that other bodies that move through it appear to have their trajectories bent. It is the mistaken interpretation of the motion of these bodies as occurring in a flat space that leads us to infer that there is a force called gravity. In fact, it is just the curvature of space-time that is at work.

  The relevance of this for gravitational waves is that if a group of massive bodies are in relative motion (such as in the Solar System, or in a binary pulsar), then the curvature of the space-time in which they exist is not usually fixed in time. The curvature of the space-time is set by the massive bodies, so if the bodies are in motion, the curvature of space-time should be expected to be constantly changing. The scientific way to describe this situation is to say that, in Einstein’s theory, space-time is a dynamical entity.

  As an example of this, consider the supernovae that we discussed previously. Before their cores collapse, leading to catastrophic explosion, they are relatively stable objects, much like our own Sun. In this stage of their life they should therefore be expected to curve the space-time around them in the same way that the Sun does, and should therefore have a similar gravitational field. After they explode they settle down to a neutron star or a black hole, and once again return to a relatively stable state, with a gravitational field that doesn’t change much with time. During the explosion, however, they eject huge amounts of mass and energy. Their gravitational field changes rapidly throughout this process, and therefore so does the curvature of the space-time around them.

  Like any system that is pushed out of equilibrium and made to change rapidly, this causes disturbances in the form of waves. A more down-to-earth example of a wave is what happens when you throw a stone into a previously still pond. The water in the pond was initially in a steady state, but the stone causes a rapid change in the amount of water at one point. The water in the pond tries to return to its tranquil initial state, which results in the propagation of the disturbance, in the form of ripples that move away from the point where the stone landed. Likewise, a loud noise in a previously quiet room originates from a change in air pressure at a point (e.g. a stereo speaker). The disturbance in the air pressure propagates outwards as a pressure wave as the air tries to return to a stable state, and we perceive these pressure waves as sound.

  So it is with gravity. If the curvature of space-time is pushed out of equilibrium, by the motion of mass or energy, then this disturbance travels outwards as waves. This is exactly what occurs when a star collapses and its outer envelope is ejected by the subsequent explosion. The violent process that occurs in the star causes a disturbance in the curvature of space-time, much like the stone falling into the pond causes a disturbance in the water. And again, just as in the examples just given, the disturbance propagates outwards as waves.

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  The speed with which waves propagate usually depends on the medium through which they travel. For example, we know that sound waves travel slightly faster in warm air than they do in cold air. The medium for gravitational waves is space-time itself, and according to Einstein’s theory, they propagate at exactly the same speed as light. Just as with light, they are also expected to have a speed that is independent of the motion of the person who observes them, and that is independent of the motion of their source. They therefore propagate information at the maximum possible speed, as nothing can travel faster than light.

  The efect of a gravitational wave

  To try and make sense of what a gravitational wave is, it might be useful to consider what its efects should be on a group of objects as it passes through them. For example, if you were trying to describe the waves on a pond to an alien who wasn’t familiar with water, you might start by suggesting that the efect of the wave is to make the lilies on the surface of the pond move up and down in a smooth, repetitive way. Let us consider the corresponding situation with gravitational waves.

  Let’s start by considering a uniform cloud of gas 』oating in space. This removes the efect of the Earth’s own gravitational field, which is many times stronger than any gravitational wave that we are ever likely to see, as well as removing any disturbance the gas might feel from interacting with anything else. If the gravitational wave travels through the cloud of gas, as in Figure 8, then the principal efect of the wave is to displace the gas in the directions transverse to its direction of propagation. That is, if the wave travels from left to right, then the particles in the gas are displaced up and down, and in and out of the plane of the page.

  The efect of the gravitational wave on the cloud of gas may initially make it look a little bit like the wave is being supported by the gas itself, much like the wave in the pond is supported by the water. It is, however, quite diferent. With the gravitational wave it is space-time that carries the wave. The efect of the gravitational wave on the gas is therefore more analogous to the efect that the water wave has on a lily that sits on the surface of the pond, rather than the water in the pond itself. That is, the gravitational wave is not a wave in the gas, but rather a propagating disturbance in the space-time in which the gas exists.

  8. An illustration of a wave passing through a cloud of gas. The wave travels from left to right, and displaces the particles of gas up-and-down, and in-and-out of the plane of the page.

  In fact, what the gravitational wave is really doing in this example is changing the amount of space that exists in the directions that are transverse to its direction of propagation. This means that although the atoms in the gas might be closer together (or further apart) than they were before the wave passed through them, it is not because the atoms have moved, but because the amount of space between them has been decreased (or increased) by the wave. The gravitational wave changes the distance between objects by altering how much space there is in between them, not by moving them within a fixed space. This is only possible because of the fact that space is not fixed in Einstein’s theory but is itself dynamical.

  To consider the efects of gravitational waves in more detail, let us now consider a ring of particles, and think what would happen to this ring if a gravitational wave passed through it. This is illustrated in Figure 9, where in this case the gravitational wave is being imagined to be coming upwards, out of the plane of the page. The efect of the wave is only really in the directions transverse to its direction of propagation. A ring of particles in the plane of the page should therefore give us a pretty good idea of its consequences.

  9. The deformation that a ring of particles would experience if a gravitational wave passed through it, in an upwards direction (i.e. coming directly out of the page). The leftmost image shows the initial configuration, and the other images (from left to right) show the ring at four subsequent instances of time.

  If the particles are initially arranged in a perfect circle, and are not attached to anything or each other, then the efect of the wave will be to compress the circle in one direction and to stretch it in another. The result is that the particles start to form an ellipse. As the wave passes through, it stretches the circle smoothly until it reaches a maximum deformation, at which point it stops and then reverses the process until the direction in which it initially stretched is at a minimum. This stretching and squashing in diferent directions is illustrated in Figure 9, and continues until the wave has passed.

  The consequences of the emission of gravitational waves has already been discussed for the binary pulsar. The waves carry energy away from the system, so that the two neutron stars slowly circle in towards each other. The measured rate of in-spiral in the binary pulsar has given good evidence for the existence of gravitational waves, but it is of great scientific interest to try and see the efects of these waves directly as well. This was done for the first time in September 2015 by the LIGO experiment. The direct detection of gravitational waves is particularly exciting because it provide us with an entirely new way to view the Universe. For the first time, we don’t need to rely on light to see distant objects as we can now look at them through their gravitational fields directly. This allows us to see what happens, for example, when black holes collide.

  The direct detection of gravitational waves also allows for new and exciting tests of Einstein’s theory of gravity. In general, one could conceive of a number of diferent possible efects from a passing gravitational wave. For example, the area of the circle of particles in Figure 9 could have been changed, or there could be deformations of space along the direction of propagation of a gravitational wave (as well as in transverse directions). It is the equations that Einstein proposed for the curvature of space and time that forbids these possibilities within his theory, but if he was wrong then they could be there in nature. By using experiments like LIGO, we can look to see if gravitational waves have the specific efects that Einstein predicted. This provides yet another test of his theory. Beyond this, being able to gain a new window into what happens when black holes collide provides a number of exciting possibilities for the further study of gravitational physics.

  Gravitational wave detectore

  The existence of gravitational waves was first predicted by Einstein in 1916, and, after nearly a century of efort, they were detected for the first time in 2015. The reason it took so long to do this is largely a consequence of the extremely low amplitude of the signal. In the diagram shown in Figure 9, the efect of the gravitational waves was enormously exaggerated in order to try and make it easier to visualize their efect. In reality, the 』uctuations in the shape of the ring should only be expected to be at the level of about one part in a hundred million trillion. That is, if we made a ring of particles that was a 1,000km across, we shouldn’t expect its shape to change by more than a trillionth of a centimetre. This is obviously very dicult to detect.

  Despite the diculty, or perhaps because of it, a huge amount of efort has been expended in trying to detect gravitational waves directly. Much of the early history of this work involved what are now called Weber bars. These instruments, named after Joseph Weber of the University of Maryland, consisted of large cylinders of metal. They were about a metre wide, and a couple of metres long. The idea was that if a gravitational wave passed through the Earth, and hence also through the detector, then it might cause the bar to start ringing, like a bell hit by a hammer. For this to happen, the gravitational wave needs to be of just the right frequency. If such a wave were to exist, it should be expected to cause small vibrations in the bar.

  The detectors that were used to see if the bar was vibrating were sensitive enough to detect changes in length of 1 part in 1,000 trillion. Remarkably sensitive as this is, it’s not enough to measure the gravitational waves that we now know to exist. This didn’t stop a few false alarms being sounded though. In 1968, Weber claimed he had evidence for the existence of gravitational waves, which would have made him a prime contender for a Nobel Prize. Unfortunately, the claimed detection couldn’t be reproduced, and it is now widely believed that it was erroneous.

  Some modern versions of the Weber bar are still being used today. An example is the MiniGRAIL experiment in Leiden. It consists of a 1,150kg metal sphere, and is about 1,000 times more sensitive than the ones used by Weber himself. However, most modern attempts at detecting gravitational waves use a diferent technology, known as interferometry. Detectors based on this technology are similar, in principle, to the Michelson—Morley device described in Chapter 2. The modern devices are, however, much, much larger. The largest of all, at the time of writing, is the LIGO detector, in the USA.

  LIGO, or the Laser Interferometer Gravitational-Wave Observatory, has two bases of operation; one is in Livingston, Louisiana, and the other is in Richland, Washington. Each site contains a giant interferometer. The interferometers consist of two 『arms』, which are built to be at right angles to each other, as illustrated in Figure 6. Each arm is several kilometres long, and contains a tube of near vacuum about a metre wide. Lasers are fired down each of these tubes, and are re』ected from hanging masses at the end of each. When the laser light makes its way back to the point where the arms meet it can be studied, and the length of each of the arms inferred from the pattern it makes when the two laser beams interact. When a gravitational wave passes through them, the arm lengths change, and the patterns made by the interacting laser light also change.

  The LIGO detector is exquisitely accurate, and has a sensitivity that is about a million times greater than a Weber bar. The technical challenges required to reach this accuracy are phenomenal. The experimentalists have had to overcome all kinds of rogue signals that pollute the data. These range from seismic noise, in which the tiny internal motions of the Earth cause the re』ecting objects to vibrate, to the vibrations caused by a strong wind outside. In fact, the detectors are now becoming so precise that one of the limiting factors is currently the fact that laser light is made up from photons, so that a continuous signal from the reflecting objects is not possible.

  Despite these great problems, and despite the enormous financial, political, and engineering diculties it has taken to overcome them, the LIGO detector has ultimately been a great success. On 14 September 2015 it detected, for the first time in human history, direct evidence for the existence of gravitational waves from colliding black holes. It’s almost impossible to overstate the significance of this event, which will probably go down in history as one the greatest scientific achievements of the modern age. So let’s consider it in more detail.

  The observation of gravitational waves by LIGO

  At precisely 45 seconds past 9:50 a.m. (Greenwich Mean Time), on 14 September 2015, the LIGO gravitational wave detector at Livingston, Louisiana, detected a fluctuation in their interferometer. This signal lasted for only 0.2 seconds, and appeared to make the 4km-long arms of their interferometer change length by about 1/1,000 the width of proton. About 0.007 seconds later, a similar signal was detected by the detector in Hanford, Washington. This set alarm bells ringing, and the scientists who ran the experiment were soon in little doubt: they had detected a gravitational wave passing through the Earth.

  Before we think about the event that caused the wave, let’s consider the signal itself. If you were to look at the data from either LIGO site, it would appear as little more than a blip on the back of the constant noise that exists within the detectors. The signal, at its peak, was only ever about twice the amplitude of the random noise that pollutes the detectors』 output. This makes it dicult to see, and dicult to confirm it as a true detection. The reasons why the scientists were so convinced it really was a gravitational wave are twofold. First, the wobble in the two detectors, at the two diferent locations, were very similar. It might be possible to have a random tremor at one location on the Earth that caused something that looked like a gravitational wave, but to have exactly the same tremor at almost the same time at two diferent geographic locations is highly unlikely. Second, and just as importantly, the LIGO scientists knew what shape the wave should take. This allowed them to scan the data, in a process known as match filtering, to look for their signal. Taking these two facts into account means that the LIGO scientists are more than 99.999 per cent confident that the wobbles they observed really were due to a passing gravitational wave and not to any spurious source of noise.

  This result is clearly an enormous achievement for the experimental team at LIGO, who produced and operated two of the most sensitive pieces of scientific equipment that have ever existed. It is, however, also a huge achievement for a very large number of theoretical physicists. This is because the match filtering process I mentioned before, and which is crucial for the detection, requires the detailed modelling of some of the most extreme gravitational fields in nature: merging black holes. It is only by understanding the details of what happens when black holes collide that scientists were able to produce filters for the expected gravitational wave signals that LIGO might detect, and this is a lot more dicult than it sounds. There are a number of diferent ways that two black holes can merge, and an awful lot of mathematical and computational work has been needed to understand exactly what Einstein’s theory predicts for the gravitational waves that should be emitted from such systems.

  I won’t go into the lengthy details of any of the mathematics, but we should consider the actual astrophysical event that led to the emission of the waves. From the shape of the signal in the detectors, and the detailed modelling of merging black holes described earlier, it is thought that the waves were created by two black holes that spiralled towards each other before eventually merging, and settling down to form one large black hole. The two black holes that came together in this scenario were around twenty-nine and thirty-six times the mass of the Sun, and they settled down to a final end state in which the resulting black hole was about sixty-two times the mass of the Sun. This might sound rather peaceful, but it’s actually one of the most violent processes that can occur in nature.

  The astute reader will notice that twenty-nine plus thirty-six does not equal sixty-two. This is because the gravitational waves emitted during the merger carried away 3 solar masses of energy (remember that E = mc2, in Einstein’s theory). This is an extraordinary amount of energy for a system to lose, and goes some way to illustrating why it should be thought of as a violent event. To give some context to this number, consider the following: while these two black holes were emitting gravitational waves, they were giving out more energy than all the stars in the observable universe put together. They were also observable from 1.3 billion light years away, which is a sizable fraction of the entire observable universe. This, then, was a very extreme event, and a highly exciting one, from the point of view of gravitational physicists.

  Future prospects

  Although gravitational waves have now been detected, there’s no reason for us to rest on our laurels. The detection by LIGO, as well as being the end of a long quest, is also the start of a new type of astronomy. One part of the reason for this hopeful statement is that LIGO will, with any luck, continue to detect more and more of these events. Another, however, is due to the new generation of gravitational wave detectors that is being planned and built. This includes the possibility of building a LIGO site outside of the USA, with India being the current favoured location. An extra site of this type would increase the ability of LIGO to determine the direction on the sky from which a gravitational wave originated, and hence improve the prospect of using gravitational waves as a new tool for doing astronomy without light.

  Beyond LIGO, another possibility for the future detection of gravitational waves is a project called eLISA (the Evolved Laser Interferometer Space Antenna). The eLISA mission is a European Space Agency proposal to create a detector in space. This brings certain benefits over Earth-based gravitational wave detectors, the foremost of which is that it would be immune to seismic noise. This means it would be sensitive to a range of frequencies that is extremely hard to detect from the ground. It could also be much larger than Earth-based detectors, as the lasers can simply be fired between satellites, without any protective housing. The proposal for eLISA is to create a triangle of laser beams between three diferent satellites, each of which is separated from the others by a few million kilometres. When it comes to gravitational wave detectors, bigger is better’so eLISA is a great prospect for future detections.

  But while the arm lengths of eLISA can be huge, and the seismic noise is completely absent, it does face its own challenges. The environment in space is close to vacuum, but isn’t entirely empty.

  The charged material that the Sun throws of should be expected to interfere with any space-based detector of this kind, as should the cosmic rays that continuously bombard the Earth. On the ground we are shielded from this interference by the Earth’s atmosphere and magnetic field, but a gravitational wave detector in space would not be. And, of course, it is much more diicult to arrange and maintain an experiment when it’s in space. Nevertheless, there are high hopes that eLISA will be built, and that it will detect gravitational waves in space.

  One further way in which gravitational waves might be detected in the future is using cosmological observables. Cosmology is the study of the state and evolution of the Universe as a whole, and there are considerable prospects for finding the fingerprints of gravitational waves in a number of upcoming cosmology missions. We will discuss these in more detail in Chapter 5.