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SPEAKER 1: Welcome, everybody, to this spring's Bethe lectures. The Bethe lectures are the physics department's premiere events every semester. They have started in 1977 to honor the retirement of Professor Hans Albrecht Bethe, who was a giant of 20th century physics and without doubt the most influential physics faculty member at Cornell.
Bethe was born in 1906 in Strasbourg-- that was in Germany at the time. He did his PhD in Munich with Arnold Sommerfeld. He went to Rome in 1931, worked with Fermi, where he developed the Bethe ansatz. He came to Cornell in 1935, where he stayed for the rest of his career, with an important gap during the war. His initial salary when he started at Cornell was $3,000 per year. OK?
About four years after he arrived, he explained how energy is produced in stars by burning hydrogen to helium, and he got an unshared Nobel Prize for that in 1967. During World War II, he was the head of the theory division at Los Alamos during the Manhattan Project, developing the first nuclear weapon. While he did help develop the hydrogen bomb after the war, he became a forceful advocate for the limited test ban treaty and later also argued against the Star Wars program in the Reagan era. He refused to testify against Oppenheimer in the McCarthy era in the infamous security clearance hearing.
In physics after the war, he was the first to calculate the Lamb-shift in 1947. He also brought along fame and [INAUDIBLE] Morrison to Cornell from Los Alamos to enhance the physics department here. In his later years, he focused on astrophysics and astronuclear physics. For example, at age 85, he wrote one of the important papers that helped solve the solar neutrino problem and also had important contributions to the field of supernova explosions.
At age 90, Kip Thorne approached Bethe and Jerry Brown, his collaborator on supernova, to discuss LIGO physics, and in particular, mergers of neutrons, stars, and black holes. And as a result, Bethe wrote an important paper on LIGO physics, which is titled "The Evolution of Binary Compact Objects which Merge," in 1998, and he was 92. So this gives us a nice segue into today's Bethe lecture with Professor Saul Teukolsky, who is a renowned expert in LIGO physics, numerical relativity, and general relativity.
Professor Teukolsky did his undergrad in South Africa, and he did his PhD in 1973. He got his PhD in 1973 from Caltech with Kip Thorne. Then he did a short post-doc at Caltech, and he joined the Cornell faculty in 1974. And very fittingly, he is the Hans A. Bethe professor here at Cornell since 1999.
He was elected to the American Academy of Arts and Sciences in '96, the National Academy of Sciences in 2003. He is a fellow of the APS, he was a Sloan fellow, a Guggenheim fellow. He got the forefront of large-scale computation award in 1990, and most importantly, he shared them at the 2016 special breakthrough prize in fundamental physics for the discovery of gravitational waves. He's a pioneer of numerical relativity, also famous for coauthoring the book, Numerical Recipes, and of course, many contributions for the physics and general relativity. So let's welcome Professor Teukolsky.
[APPLAUSE]
SAUL TEUKOLSKY: Thank you, [INAUDIBLE], for a very generous introduction. So it's really an overwhelming honor for me to be invited to give these lectures, not only because my whole career has been here at Cornell but also because these lectures are named for Hans. When I arrived as a beginning assistant professor, Hans was actually in the process of retiring. In those days, there was mandatory retirement by age, and perhaps that was a good thing. It sort of forced him-- he started a whole new career in astrophysics, which went on for over 30 years.
And one of my-- I mean, there are so many stories you can tell about Hans. One that affected me in particular was, in about 1985, I coauthored a textbook on black holes and white dwarfs and neutron stars with Stu Shapiro, who was a faculty member here, and the book had 20 chapters. And while we were writing it, we lined up 20 colleagues around the country, with the idea that we would send each one a chapter on a subject that they were expert in and get feedback.
And we found all the colleagues and they said, yes, yes, we'll do it for you. Of course. It's very important, I'm sure. You know? And you can imagine what happened. We sent the 20 chapters out. Some people sent us back maybe a paragraph-- you know, which didn't have much-- you know, "nice job" kind of thing. Not very useful. Kind of thing. Some said, oh, I'm sorry. You know, I really intended to do this, but I just don't have time.
Hans was different. Hans got a hold of us and said, you know, we must make an appointment, come to his office. So we show up at 2 o'clock on a Tuesday afternoon, and we sit down, and he says-- I'm not going to try and imitate the accent, but he says, "OK," he says, "On line three, you used 220 for the compressibility. Didn't you see the paper last year? It's now 180. Line six--"
You know, finally at about 5 o'clock, Stu and I were completely exhausted. You know, dragged ourselves out of the office. I think if we would have stayed there, he would have given us some more notes and so on. So that was Hans. I mean, any job that needed to be done, everything was done precisely, and he was completely encyclopedic in his knowledge of physics, especially in these areas that he was working on.
So in thinking about what I should talk about today, as you know, one of the biggest discoveries in physics in the last 50 years has been the detection-- the direct detection-- of gravitational waves from orbiting black hole systems that merge. So I thought I would take that as a starting point and look to the future. All right?
What are we going to be able to do as a result of this discovery that was not possible beforehand? And so that's the reason for this title, Testing General Relativity with LIGO. And the motivation is that, up until now, the tests that we've been able to do of general relativity-- for example, tests in the solar system. The famous perihelion precession of mercury. And even tests that have involved pulse stars, neutron stars.
So for example, the famous binary pulsar system that Hulse and Taylor won the Nobel Prize for is an example of a system where the pulsar, which is a very precise clock, is orbiting in the gravitational field of its companion neutron star, and so we can probe the effect of this curvature of space time on this very precise clock. And in particular, the observations have shown that the orbital period is decreasing as the system loses energy to gravitational waves. And the measurements are precise enough that the orbital decay agrees with the prediction of Einstein's famous quadrupole formula-- the antilog of the [INAUDIBLE] formula in electromagnetism.
But if you ask about strong fields-- for example, black holes, the ultimate strong field in general relativity-- there, we've had no tests. Here we have an example in physics of these bizarre predictions, right? This one-way membrane. This event of the horizon, where if you happen to cross in, you can never get out. Where space and time are completely distorted-- curved-- in the neighborhood of this object.
These kinds of things have been the subject of textbooks and graduate student homework problems, but really, quite divorced from reality. We've had no good probes of that. And so the importance of the LIGO detections, from my perspective, is that, number one, we've detected the gravitational waves themselves, but number two, they provide us a probe directly of the strong field region.
OK, so the questions are, are the observed black holes that LIGO sees, do they agree with what general relativity predicts? And is general relativity in general correct for ultra-strong gravitational fields? So let's think about alternatives, right? What if general relativity is wrong? What kind of things should we be looking for?
So for example, in the first column-- this one is probably better-- in the first column, here are some properties of general relativity. For example, gravity is described completely by just one field quantity. This field quantity is the metric tense. It describes the measurements that you can make of space and time. So an alternative would be to have more than this object-- to have other fields. For example, there are scalar tensor theories. Most low-energy limits of quantum gravity attempts involve these additional scalar fields. Could we detect their presence?
In general relativity, gravity travels at the speed of light, or in quantum language, the graviton is massless. There are alternative theories where the graviton would have a mass, presumably a small mass. The equations of Einstein that describe general relativity come from an action principle where the curvature tensor enters in the simplest possible way-- just linearly.
Well, you could imagine more complicated theories that involve, say, quadratic combinations of the scalar curvature or some other more complicated function. General relativity is set in a four-dimensional arena of space and time. Well, according to string theory or other alternative ideas about fundamental physics, we don't really live in a four-dimensional world, right? There's higher dimensions, maybe, that are extremely small.
In some of these theories, the small dimensions are not on the scale of a Planck length-- 10 to the minus 33 centimeters-- but are what are called large, right? Not necessarily cosmological scales, but big enough that they might show up in some kind of a non-quantum mechanical context.
So why should one consider alternatives if general relativity so far is good enough? And there are quite a few reasons, so first of all, this idea that, when we try to look at quantum theories that include general relativity as a low-energy limit, these theories have an annoying habit of not giving only general relativity but giving these other maybe scalar fields or things like that.
And then there are two observations that don't fit very neatly into our previous picture of just general relativity. One is that the universe today is not just expanding but that expansion is speeding up. Despite the fact that gravity is attractive, something is making the acceleration speed up. And we don't know what the cause of this is. Is it some kind of new energy, which we call dark energy because we don't know what it is, or is it some modification that's required to gravity to general relativity on very large scales?
Similarly, in the early universe, the simple big bang picture which gives us such remarkable agreement with cosmological observations is quite unsatisfying from a theoretical point of view. You have to put in a lot of initial conditions which seem rather specialized in order to get the observed universe. And the attractive thing about the inflationary model is, you replace all of these fine-tuning kinds of ideas with just one idea-- namely, there's this inflation. Some kind of field or energy density that produces a large inflation in the early universe. And then that naturally accounts for the big bang that we see.
Well, then you ask the question, you know, why inflation? Where does this come from? Could this be a prediction of some alternative theory of gravity?
So there are these kinds of ideas that have been floating around for a long time, and so theorists, being underemployed, tend to make long lists of alternative theories. And it's actually, I find it quite irritating to go to the archive, go to the general relativity section, and then look at the papers, and more than half of the papers are on alternatives and less than half are actually on the theory that we think is right-- or we think so far is right. But, luckily, we're going to try to get rid of some of these now.
All right, so let me emphasize how powerful the new probe is. So, in this plot, this is a plot of gravitational potential in dimensionless units, so high potential on this end, and this is a plot of curvature. So you can just think of it as strong curvature at the top, weak curvature at the bottom.
So here are the solar system tests of general relativity. The perihelion precession, lunar, laser rating, and so on. Here are the neutron star tests-- the binary pulsar that I described-- and here are the first two LIGO detections, way up in the corner, right? So strong gravitational potential and high curvature.
So this is the strong field aspect being probed for the first time, but also, dynamical. So here is a plot of-- this is the radiation reaction time scale. The time for gravitational radiation to change the properties of the system. And here again, the curvature, right? So curvature is strong at the top.
And so again, you see the solar system way down here, even the binary pulsar, right? This is 10 to the minus 8 and 10 to the 16 here. This is in seconds. And here are the LIGO events, right? 10 to the minus 4 seconds and high curvature. So it's a completely new experimental regime.
OK, so this is a famous figure from Kip Thorne, showing the three regimes that we use to describe what happens when two compact objects merge because of losing gravitational-- using energy to gravitational waves. So there's the Inspiral phase. So here, the gravitational wave signal-- this is the strain of the gravitational wave signal-- this is known in the sense of what used perturbation theory to predict this. This is weak field and relatively slow velocity, so you can make a post-Newtonian expansion to predict this waveform.
When the black holes have merged, you get a single black hole that sort of oscillates a little bit, emits some few dense harmonic perturbations that take away the last of the energy, and then you're left with a static or a stationary rotating single black hole. And this part can also be treated analytically by perturbation theory. In fact, that was my PhD thesis. Right?
And then in between, you have no small quantities, and you have to resort to numerical solution. So that's this mess showing there. So the Hulse-Taylor binary pulsar is in this stage, and if only we could wait about 100 million years, it would get to this stage. All right.
So here's the first LIGO detection, and the conclusion is, general relativity is actually pretty good, all right? So let me try to make that quantitative. So here is the signal. This is the Figure 1 on the detection paper. So here's the gravitational wave strain, now the measured gravitational wave strain at the two detectors, so red and then blue is at Livingston, and then they've taken the red one and flipped it on and superimposed it here. So even by eye, without any fancy Bayesian statistics or anything like that, you can tell this is the same event. OK?
Here, they've taken the signal, done some smoothing on it, and superposed in red a gravitational wave calculated by the full numerical solution of Einstein's equation. OK? So this comes from our collaboration and was used to show, even again by eye, you can just see the very good agreement between the signal and the theory. So if you subtract the signal from the theory, you get this residual, and this is consistent with noise at the level of about 4%. So if there's some deviation from general relativity present in this kind of an event, we already can state just from this very first event that it's got to be small at that limit.
You can also do other things-- consistency checks. For example, you can use the post-Newtonian model to fit the early inspiral. That gives you the masses and spins of the two black holes. You can then plug those into your numerical relativity code, predict the outcome-- the final mass and spin. You can then use black hole perturbation theory to say, if I have a black hole-- a single rotating black hole of this mass and spin-- and I give it a little perturbation, what frequency does it ring at?
So these are called quasi-normal modes. They have a frequency, but it's a damped sinusoid because the energy is lost very rapidly. So it's a low-cue system.
And you can again sort of see by eye a roughly exponential damping here after the merger, and so if you fit that to the predicted frequency and decay time, it's consistent with the first part of the wave. So you can already see, just from a simple thing like this, how, if you have some alternative theory of gravity, it's now much more difficult for you to get away with saying, oh, I predict the perihelion procession of mercury. You have to predict this waveform, and you have to predict at least sort of with some constraints these things like the rapid damping at the end and so on.
AUDIENCE: [INAUDIBLE]
SAUL TEUKOLSKY: It's much better than this. Some people have a philosophical prejudice against black holes. They would prefer anything else, even a traversable wormhole, it turns out, right? Right. So there have been proposals for alternative explanations, where astronomers have invoked black holes to explain some astronomical observation. People have proposed other things.
And these things are consistent with these things like x-ray binaries, where you have a sort of a stationary system where people have invoked black holes. The problem is, these things typically are not embedded in a real theory. They're just some proposal for an object. So can't really calculate a waveform like this because often there isn't really a theory. When you impose on stars, you can actually calculate a wave.
The problem with most of these proposals is it's very hard to have a physical system that's made of matter and have it damp away in a light crossing time. Typically, it's the sound speed that governs material properties like damping times and so on. And to get this to decay away on the light travel time rather than the sound speed of some material, you can basically cross out most of these things. So the way I would put it is, we've raised the bar. If you have some other proposal, first of all, you need a real theory so that we can calculate it, and then you have to explain how this energy disappears in a light crossing time.
OK, let's turn to some other ideas. So, as soon as you start putting other fields into the theory in addition to the metric tensor, you often end up with a theory that violates Lorentz invariance. Now you might say, whoa, whoa, you can't do that. Right? Well it's not a matter of whether you can or can't do it. It's a question of, to what degree are you allowed to get away with it?
So it turns out that there's a whole enterprise devoted to this. If you go to these data tables, which are an extension of the famous particle properties tables, you'll find a whole parameterized framework for describing violations of Lorentz invariance and then all the experimental evidence that constrains all the parameters in this thing.
So I'm not going to get into anything as complicated as that. Let's just take a very simple example. Suppose that the graviton has a mass. So, according to Einstein, gravitational waves travel at the speed of light. The graviton is mass. But let's just explore what happens if you have a small mass.
All right, so the group velocity, if you differentiate this, now has a small correction where 1 is the speed of light. And if I replace the graviton mass by its Compton wave length and-- sorry, its de Broglie wavelength, and replace the energy by the frequency, then we could think of two different ways of testing this. So first of all, if we have a gravitational wave detection, then-- you saw the waveform from the binary black holes-- the frequency of the waveform goes up as you approach merger, right? Because the orbital period is decreasing as they get closer and closer. As they spiral into each other.
So the travel time towards the end of the waveform would be from a higher frequency component than the travel time at the beginning. So let's just make life simple. Let's assume that at high frequencies, it approaches the speed of light, but that at earlier times, at lower frequencies, we have to worry about the fact that the group velocity is not quite 1. All right? So you can show that this difference is-- D here is the distance that has to be traveled.
And then from this vg is this. Right? We replace this by the frequency. So solve this for the wavelength of the graviton, and you get that in terms of the frequency, and then f delta t, this time difference, this is just the phase difference across the wave.
So this delta phi is measured by the experiment to about 1 over the signal to noise-- say about a 10. So if you put in the numbers here, you can see you get a constraint on the wavelength of the graviton-- some big number. Or, if you re-express it in terms of the mass, 10 to the minus 22 electron volts. OK? You might think we're done. That's so small. Who cares, right? But wait, there's a problem.
So this is better than has been done previously from other measurements. Some people have argued that, if you have a mass of graviton, then the potential should be like a Yukawa potential at large distances, and so you can use that to try to set limits. That's actually not guaranteed. There are theories where, at large distances, even though there's a mass, it's not a Yukawa. So we won't worry about that.
I'll just point out that LISA, which is Laser Interferometer Space Antenna, which is a proposed space mission which should fly in the early 2030s, will improve this by about four orders of magnitude just because of better signal to noise. It will see supermassive black holes spiraling together. And also just because of the distances that it will be able to see to.
All right. Now another way of testing for a mass of graviton is if you see an electromagnetic counterpart, right? So this is time going to the right. Here's the time of merger. Here's the time on the earth when we receive the gravitational wave signal. And then here, we receive some electromagnetic-- some photon. OK?
So the gravitational wave-- let's use the average group velocity-- takes the time d/v to travel to the earth, and then there's a time here which is the difference between the arrival of the gravitational wave, and let's say it's a gamma ray, so some observed time. Meanwhile, the electromagnetic signal, let's suppose there is some delay between when the merger occurs and when that signal is actually emitted, and then the gamma ray travels at the speed of light. So the time is just equal to the distance. All right?
So if you equate this sum to that sum, then the difference between tg and t gamma is just this t delay, which is some theoretical quantity we are going to have to put in, and then the observed delay between the two arrivals. All right? And you can work out two options-- either gravity travels faster than one, so gravity catches up, or gravity travels slower, so the photons catch up.
And in the case where gravity travels faster, if you assume that the delay time-- this delay-- is some positive thing-- that whatever is causing the explosion that produced the gamma rays didn't occur before the objects merged-- then this has to be bigger than some number, and I'm going to use two seconds. And if you don't know the punch line, you can wait and see where two seconds comes from, right? I'm sure everybody knows the punch line.
On the other hand, if the group velocity is less than one, then you have to make some assumption about what's the maximum delay here. I'm going to use 10 seconds. It doesn't matter. You could use 100, 1,000 seconds, whatever. You put these numbers up, and then 1 minus vg is bounded by basically 10 to the minus 16. OK? Again, an incredibly small number, right? And just changing this from 10 to 100 would make it 10 to the minus 15.
All right, and here's the punch line. Last August, LIGO detected gravitational waves from two neutron stars instead of two black holes-- two neutron stars that were merging. And now you have hot matter smashed together at almost the speed of light. So here is the trace from LIGO. This shows time versus frequency. You see how the frequency goes up, and this is the merger event.
And this is from three different gamma ray detectors on two satellites in space, and you see this gamma ray burst, which if it weren't for this gravitational wave that had been detected here, this would have been considered a routine rather miserable gamma ray burst. Miserable because there's so many hundreds of others that are more impressive. The delay between these is 1.7 seconds, and so using our previous thing, we can now say that 1 minus vg is of order 10 to the minus 16 at most, and if you turn that into the mass of the graviton, that's 10 to the minus 25 electron volts. So this is a much better limit than just using pure gravitational wave.
AUDIENCE: [INAUDIBLE]
SAUL TEUKOLSKY: Is what?
AUDIENCE: [INAUDIBLE]
SAUL TEUKOLSKY: Yeah. Yeah, these go inversely. Now this makes life tough for a lot of theorists who have proposed theories, but if you were one of the people who's trying to explain the acceleration of the universe, typically what you're trying to look at is some kind of a scale that's cosmological that is the cosmological scale today. And that's the Hubble constant, and if you express it in energy units, it's 10 to the minus 33 eV. So there's still plenty of wiggle room here to try to explain the current expansion of the universe without running into trouble with this particular thing.
Of course, we may get lucky. We may see events where this delay time is shorter, but you're not going to close-- I don't think you can close this gap completely, at least within a short period of time. Of course, as gravitational wave detectors improve, we might get better limits.
All right. Another prediction of Einstein's theory is that gravitational waves are polarized and there are two polarization states, just like electromagnetic waves have two polarization states. Now, in a metric theory, that means you assume that the basic idea that space and time measurements are governed by some metric-- all you're arguing about is what are the equations and other fields that may be determining that metric-- then the driving forces on a detector like LIGO come through this Riemann curvature tensor, and in the low velocity limit that's applicable on the earth, you can think of this as just a symmetric tensor-- the i and j range over x, y, and z-- so there's six degrees of freedom.
So in principle, a gravitational wave in an arbitrary metric theory has six possible polarization states. You can classify these possible states. The best ways to do it via the helicity. So it turns out that three of these six are pure helicity states and the other three are mixed. So already, if you're a theorist who believes that quantum field theory is the right way to do physics, and in quantum field theory, quantum states are described by irreducible representations of symmetry groups, then you would already be ruling this out, but we'll take the agnostic point of view and say, experimenters should look for all six of these polarization states.
So here's a picture to show you what they look like. So, in the top row, the circle thing means that the wave is coming out of the screen toward you, and you imagine a circular ring of test particles, and in this polarization, that circle gets deformed, strained into an ellipse, and half a cycle later, the ellipse is going this way. So it's a quadripolar distortion like this.
So this plus and cross, these are the predictions of general relativity. These are helicity two states, and according to general relativity, that's all you should see. If you have a scalar field as well as a metric tensor, then you have this breathing mode, which is also transverse to the wave, but the circle just gets bigger and smaller.
The other three states have a longitudinal components, and now we've rotated ourselves, so z is the propagation direction to the right. So there's a kind of a scalar deformation as well-- this longitudinal mode. And then these modes that are called vector modes, which look like these but they're sideways, right? They have a longitudinal component.
All right, so if you want to measure polarization, you have eight unknowns, the six polarization amplitudes, plus you need to know the direction on the sky. So that's two more components. If the direction is known-- for example, there's an electromagnetic counterpart-- then in principle, a single detector that you've instrumented correctly to measure all the different possible deformations could tell you the polarization. If the direction is not known, then what you have to do is you have to use a network of detectors distributed around the globe in order to get the direction by the light travel times, differences, and so on. And it turns out, at least six detectors are needed in principle to do that.
Laser interferometers, it turns out that these two modes are actually degenerate with each other. You can't distinguish them. So you can only measure five polarization states. And there are experiments under way that have only set upper limits, for example, to look at the stochastic background of gravitational waves and try to set limits on the energy density in these tensor and vector and scalar modes, but with detections like the neutron star detection where the direction is known, and with the start of a network now that the Virgo detector is online in Italy, this is a field which I predict over the next few years will soon be giving very strong results.
AUDIENCE: [INAUDIBLE]
SAUL TEUKOLSKY: No, you need six if you want the direction and the polarization. It turns out, with six detectors, even though there are eight degrees of freedom, you can get all six-- well, five polarization states plus two direction angles.
AUDIENCE: [INAUDIBLE]
SAUL TEUKOLSKY: Yes. This single detector is if you know the direction and you can instrument it. If you're not using the gravitational wave detector to get the direction. OK.
What about the wave form itself? So what you can do is you can take, for example, this is the prediction of post-Newtonian perturbation theory for the phase of the signal, and I've written it in the frequency domain, so this stands for stationary phase approximation. So this is an approximation to the phase of the wave as a function of frequency.
So you see just the usual ft that you would have, say, for a sine wave, then there's a constant, and then this f to the minus 5/3, you can show that's what you get from the quadrupole formula, so some coefficient times that, that's quadrupole. And then the other things are higher-order corrections in v/c, including the possibility of logarithmic terms that show up. So general relativity makes definite predictions for the numerical values of these coefficients.
And then as an experimenter, you can just say, well, suppose I wanted to know, are there deviations from what general relativity predicts? I could fit my data to try and bound these delta psi's and that would give me a parameterized way of saying that any theory that predicts a delta psi bigger than such and such violates what I've measured in my experiment. And then you can get fancier-- pp is parameterised post-Einsteinian. So here, you do this kind of thing not just to the phase but to the amplitude.
The problem is, there are so many parameters in these possibilities that right now, this is not a very restrictive test, but again, if you look at the predictions of how gravitational wave detectors will improve in sensitivity, within about five years, I think that this will become a very powerful test of alternative theories of gravity.
OK, I wanted to look at a particular case study of what's involved and what needs to be done if we're really going to get serious about testing general relativity. And so when you look at an alternative theory, it's called dynamical Chern-Simons. So this is motivated again by trying to make a quantum theory of gravity, so either in string theory or in the alternative formalism called loop quantum gravity, this comes out naturally in the low-energy limit.
So here's the action principle. So there's the usual general relativity, which is the scalar curvature of a simple term. Now there's a scalar field-- actually, it turns out it's a pseudoscalar field. And there's an interaction term between the general relativity term and the new scalar field.
And this interaction term involves the Riemann tensor contracted with its dual. So in electromagnetism, the analog would be f mu nu and the dual of f mu nu, which is e.b when you write it that way. So you can think of this just like an e.b but with gravitational variables. And then it's coupled to the scalar field, and then there's a coupling constant here, which has dimensions of square of a length.
Now, if you took this seriously as a fundamental theory of physics, so you vary the action and you get the equations of motion, then this term will give you general relativity, but then this interaction term here has these high powers of curvature and you start getting things like third derivatives of the metric. And this means that, even if I wanted to do a numerical simulation to take this theory and calculate what its waveform is, I probably can't do it because it doesn't have a well-posed initial value problem. Right?
In order to-- we're so used to taking something like the wave equation and saying, if you tell me the initial value of the field and the initial value of its time derivative, I can put them on a computer and then use the wave equation to propagate forward in time and say what happens. And that's all that well-posed means, right? The mathematicians have fancy ways of saying that but that's what we mean.
Well, once you start getting these higher time derivatives in here, it turns out that, almost always, these things are not well-posed, right? You have to do special tricks in order to find a theory where it's well-posed. But that's OK. If you take the point of view that this is not a fundamental theory but it's the low-energy theory of some fundamental theory, we call that an effective field theory.
And the reason that's a reasonable thing to do is, we know that this theory-- if, say, this Chern-Simons thing is really the correct low-energy limit. It's got to pass all these other general relativity tests, like in the solar system and so on. So we're going to be close to general relativity. So this effective field theory, there should be a way to write this as general relativity plus a perturbation-- plus small correction. So as long as we accept that an effective field theory does not describe the arbitrarily small distances, then we can use perturbation theory, and that's well-posed.
All right, now you've seen this idea in other places in physics. The simplest example is radiation reaction, right? So here I've written down f equals ma for a charged particle, but then we've added the radiation reaction force with the Abraham Lorenz prescription. And when you come across this in the textbooks and so on, they go on for pages and pages about how terrible this equation is. If you set f equal to 0-- so there's no force, you just have an electron sitting on the table-- then according to this equation, there's a solution where the acceleration is growing exponentially because these terms are positive, and that's called a runaway solution, so the electron just decides, oh, I think I'll accelerate exponentially fast.
And then, to cure that, they introduced pre-acceleration-- the electron knows that it's going to have a force at noon tomorrow, so it starts moving now so that it will be just right so it won't have an exponential solution tomorrow. You need three boundary conditions because you've got a third time derivative of x here. Right? All kinds of things. OK. That's all nonsense, right? So tear those pages out of your textbook and run.
The point is, this is an approximate equation. There were some assumptions made in deriving it. In fact, they're essentially the assumptions that go into an effective field theory about the scales-- the separation of the length scales and the time scales. And the runaway solution is just a solution to an approximate equation that violates the approximations that you made. So it's clearly a spurious solution.
From the effective field theory point of view, the way to treat this is, this is a perturbation, right? So you separate the acceleration into a zeroth-order term and a first-order term on both sides. At zeroth order, m times a is f, and then at first order, m times the perturbation will be just-- and in here, for a dot, you insert the term that you found in the previous order, right? So you take a0, differentiate that, and that's what goes in here. OK? And so there's a very simple hierarchy if you wanted to go to higher order. You're always solving a second order differential equation for the right-hand side, as known.
So you do the same thing for this dynamical Chern-Simon. At first order in those equations of motion, you get the wave equation on the scalar field, and then it's got a source term which is based on the unperturbed metric for the black holes that are inspiraling and colliding. The metric itself doesn't change to first order because the stress energy produced by this perturbation goes like the square of the amplitude of the field. So you won't have any change.
So already at first order, you can take our evolution-- this is our spectral Einstein code that does black hole evolutions. You can do a binary inspiral with Einstein's equations for the metric, and then you can evolve on that background-- you can evolve the wave equation for the scalar field and find the perturbation produced by that.
And here are some examples. So this is the waveform for the metric, and these are just three multiples of the scalar field, and the black lines, which look essentially flat, are the predictions of a post-Newtonian treatment. So earlier people had done an analytic treatment. But now, with the full numerical solution of this equation-- so this is now solved exactly on this black hole background-- you see, at the merger, there's a burst of scalar radiation that you wouldn't have known if you had done this.
So this is work by Maria Okounkova, who is a graduate student working with me at Caltech, and Leo Stein, who was a post-doc previously here at Cornell. And the idea is, then, now that you have this, you can go to second order that will produce a change in the waveform, and you can now start looking at that. But already, from the first order effects, you can estimate that in the LIGO detection so far, you can bound this coupling constant-- the strength with which the scalar field perturbs the geometry-- to be of order, say, 10 kilometers, whereas before this thing, the current bounds, which came from solar system experiments, were seven orders of magnitude bigger. OK? So just even with this very simple thing, you can see the power of this kind of approach.
Now I mentioned earlier quasinormal modes. These have been used to reinforce this idea that in general relativity, a black hole is a very simple object. That, when you have a static black hole, after all the waves and dynamics are done, that it's described by a Kerr metric. The Kerr metric is an analytic solution of Einstein's equation, and it only has the mass and the angular momentum-- the spin of the black hole. In principle, there could be a charge, but we don't worry about that in an astrophysical setting. OK?
And so John Wheeler came up with this famous aphorism, "A black hole has no hair." What he meant by that was that the black hole was smooth. Just had to know the mass and the angular momentum. All the information-- this is the diagram from the Physics Today article showing all kinds of objects-- I'm sorry-- all kinds of objects going into the black hole. And it doesn't matter what they were, they all have multiple moments and everything gets erased by the gravitational-- carried off by the gravitational waves, and you end up with the Kerr metric.
And this theorem is not necessarily true in alternative theory. So one test you could do-- all right, this is supposed to be a black hole realizes it has hair-- you could measure-- in that exponential decay of the waveform, you could try to measure not just the slowest damped quasinormal mode, but you could try to measure two of them because they have different frequencies and different damping time. And so you'd have four quantities that you've measured, and they should be related. If the black hole has no hair, they should depend only on a single mass and a single spin, and so you have a test.
Now unfortunately, this is a difficult test to do right now, because the signal to noise in that damped exponential right at the end is a very small fraction of the total signal to noise that you get. So this is a test where you could try to improve signal to noise by having a few dozen events and trying to stack them in a suitable way to try to pick out the damped modes at the end, but really, I think that it's the next generation of LIGO-- something maybe 8 to 10 years out, where you might get measurements strong enough to do this. This is definitely a test that LISA should be able to do. So I think again, this is something that we can look forward to seeing.
Now another line of alternative theories has several motivations. One of them is this so-called hierarchy problem, right? Why is gravity so weak compared with the other fundamental forces?
And one idea that has been proposed by Randall and Sundram and other people in different kinds of frameworks, is that not only are there other dimensions but that at least one of them might be big-- meaning not on the Planck scale-- and it's only gravity that can feel the presence of that other dimension, or other dimensions if there's more than one. So in this diagram, this brain is supposed to be where we are with all our three normal forces-- fundamental forces that we know about. Gravity is here with us as well but also extends out onto this other brain. And then here's another picture, showing just this idea of a four-dimensional brain, with gravity leaking in some-- maybe not exactly like this, but some other model.
And how could you test this? Well one way of testing it with gravitational waves is that the amplitude of the wave no longer falls like 1/r. So let's see if we can understand why that is. So the easiest way-- now remember, we're thinking about some-- I don't know, 10-dimensional theory that has gravity and blah, blah, blah and so on. How can I estimate how the wave amplitude falls with radius?
So the answer is geometric optics, OK? That basically, any time you have a wave and you're in the regime where the wave is propagating in a slowly varying background, then geometric optics is a good approximation. So just to remind you, you express the wave as an amplitude and a phase. You have a wave vector, which is the gradient of the phase, and then there's also a polarization tensor that I've left out. So between a and e to the i theta, this would be right for a scalar wave. For an electromagnetic wave, I'd have to put a vector-- the polarization vector-- in for it and so on. OK?
And then you get the eikonal equation from taking the geometric optics limit of the wave equation, and I've written it here in a very nice form, which I'll show you in a minute has an interpretation as conservation of quanta. So even though this is a completely classical idea, it has a very nice interpretation in terms of quanta.
By the way, if your geometric optics is rusty and you want to see where these things come from, there's a very nice new textbook by Thorne and Blanford called Applications of Classical Physics. It's only 1,500 pages. It's all of Landau and Lifshitz's classical physics but in a comprehensible and modern form. And you can look at-- they derive this for you, if you've forgotten where that comes from.
OK. So here we are in eight dimensions, seven dimensions, doesn't matter. So you write out the divergence using a little identity where you could replace this covariant derivative operator with a partial derivative. And we'll assume for simplicity that the metric in this space has some kind-- well, you know, we're far from the source where the wave was generated, so it's become close to flat space time. And so I've written it here as the Minkowski metric, and here, we have some spherical line element, except this can be not just a two sphere but a three sphere-- whatever your fancy is. OK?
So if you take the determinant of this metric, you'll see you get an r squared for each coordinate that's sitting in the spherical coordinates here. So in four-dimensional space, this is a two-dimensional sphere, so when d is 4, I get r squared twice-- I get r to the 4. And then when I take the square root, I get r squared. So that's where this comes from. OK?
So you write out this equation now-- so put in the determinant of the metric. And you can see this interpretation is conservation of quanta. If you look at this term-- so you know that, for example, for the harmonic oscillator, energy over frequency is an adiabatic invariant to conserve quantity, so here I've written energy over omega, except I put in a constant-- Planck's constant.
So the energy of the wave goes like the square of the amplitude. So it's an a squared, but then I have to take the derivative, right? It's like the e-field is a derivative of the potential, so I have to square that. So I get a squared omega squared and divide it by omega, and there you see a squared omega, right? So this is just saying that this quantity will be conserved, so I can cross that out, and then this quantity gives me that this thing in parentheses is a constant.
So if I solve this for the amplitude, I have a prediction that h scales like this. And we check-- if you put d equal to 4, this says h goes like 1/r. But now we have a prediction of, if you had a higher dimensional space through which the gravitational wave was propagating, the amplitude should fall in this way. All right?
So this is, again, from that neutron star detection. Assume general relativity is correct-- so we won't tamper with how the wave was generated but just that the propagation is going like 1 over the distance. I've used here the fancy astronomer's luminosity distance because it's measured from the flux of the waves, right? So we'll assume it's going like this power that we got in the previous slide, and we'll compare that with the distance measured from the gamma ray burst, and that you can do just using Hubble's law. All right?
So we know that according to Hubble's law, for objects that are close to us, like this particular event, velocity is proportional to distance, with the proportionality being the Hubble constant. So here's the plot. This is probability against number of dimensions-- this d.
And here is four dimension. And these are two different curves, depending on two different choices for the Hubble constant. But you can see, it doesn't matter. These are the error bars. You can see this is well away from, for example, d equal to 5. OK?
So again, already from this result, this wave propagated from the galaxy-- NGC 49, 93, whatever it was-- in four space time dimensions, not five. Pretty clear, right? Of course, theorists are inventive. This tells you that the distance of this object was about 40 megaparsecs. When you put in the error bars, you can maybe drive all these things down maybe to 20 megaparsecs.
But if you have a theory where the dimensionality only becomes apparent when you go to a very large cosmological distances-- so these theories have a screening length. If you make your screening length much bigger than 20 megaparsecs, then you can't rule out one of these brain world scenarios. And if you're using the brain world to try to explain some cosmological phenomenon, you can probably still get away with it. OK?
So what we need is a similar kind of event-- I mean, this was only the first event of this kind detected by LIGO. So hopefully in upcoming years, there'll be events seen at larger distances, and one can do an even better job. And LISA in particular can see to cosmological distances.
OK, so let me conclude with a little summary, that gravitational waves detectors now allow us to probe gravity where we couldn't before-- in strong fields, in very dynamical situations. Already, there are some interesting results, but for me, the most interesting thing is, if we look at how the detectors are going to improve in the next few years, I think that we will be able to set very strong limits on alternative theories of gravity. Thank you.
[APPLAUSE]
SPEAKER 1: Thank you very much. Time for questions. Yes?
AUDIENCE: So when Kip Thorne was here, you talked about what the upgrades [INAUDIBLE]. I'm just curious, what is the event or what type of event do you most look forward to seeing?
SAUL TEUKOLSKY: So the one event that we have not seen so far is a neutron star spiraling into a black hole, and that would be very interesting not simply for the gravitational wave side of it but for the nuclear physics side of it, because when two neutron stars collide, the physics of that is so complicated that the modeling that we can do right now is not really very good in terms of trying to infer physics from what we see. We're working on it. Give us a few years, we'll do better.
But with a neutron star-- with a single neutron star and a black hole, there are some very clear kinds of predictions that if we're lucky enough-- for example, depending on the mass ratio and the spin of the black hole, the neutron star, instead of just falling into the black hole as a single object, can be completely torn apart before the matter goes in. And that would show up in the waveform. We could see that gravitationally, because the waveform would suddenly-- from the inspiral-- would suddenly cut off.
And so from knowing the frequency that that occurs at, and again, if we're lucky, we should see some kind of electromagnetic display associated with this, we can learn a lot about nuclear physics-- about the equation of state of the matter and so on, without a lot of complicated modeling. So that would be my sort of number one thing that I think is quite possible in the next-- the next observing run for LIGO will be later this year, toward the end of the year, and maybe we'll-- we've been lucky twice so far. You know, maybe a third time.
SPEAKER 1: So for those of us who didn't fit into the room when Kit was here, can you remind us what the milestones will be? So how many updates there will be?
SAUL TEUKOLSKY: So we've had two observing runs on the current what's called advanced LIGO, so in the first one, the two black hole events were seen. Now, a total of about half a dozen black hole events have been seen all together. And then in the second observing run, which finished last August, just like two weeks before the end of the run, this neutron star event was seen. We've seen one of those.
The next observing run will start at the end of this year. Will last, again, about approximately a year. And the hope is that that will have-- you know, you can't pin these guys down about how much better it will be, but they just say it will be better in terms of-- especially the low frequency performance will be better. It will shut down again and then there will be another upgrade, and then hopefully LIGO will be at design sensitivity, which is a factor of three better than it is today. So just to remind you, 3 cubed is 27, so the event rate should be 27 times roughly what it is today.
SPEAKER 1: And how far off is that, roughly, in time?
SAUL TEUKOLSKY: Probably three years. And then there is a proposed upgrade after that, which is relatively cheap as these things go. I think it's got a good chance of going. That will give you another factor of, I don't know, 1.7 or something like that.
But then after that, there is a concept called Voyager, which is probably about at least eight years out, and that requires the NSF to really lay out-- you know, it takes the existing infrastructure-- so no new tunnels or anything like that-- but does sort of p best you could possibly do in that thing. But that's, I don't know, 100 million plus dollars, and so hopefully [INAUDIBLE].
SPEAKER 1: More questions? Yes?
AUDIENCE: So you had a slide about the time delay between the optical signal and the gravitational signal, in seconds. Is that delay just due to [INAUDIBLE] density of, like, [INAUDIBLE]?
SAUL TEUKOLSKY: Well, no, it's because it's defined as the time between the peak of the gravitational wave signal and the arrival of the gamma ray burst. So, presumably, that once the neutron stars actually are in context-- no longer moving like this-- presumably, the signal is already starting to decay. So there's some time between that moment of contact and this fireball being produced, and then, as you said, for the gamma rays to actually get out.
And there are lots of models of how that might happen, but we're not able at the moment to do a complete calculation. But we start with two neutron stars and produce a gamma ray burst. We don't know how to do that. So it's empirical. That's 1.7 seconds.
SPEAKER 1: Yes? Go ahead.
AUDIENCE: If you look at the gravitational wave signal, there's only about, I don't know, 10 oscillations or something, so you're very bandwidth-limited on one end. If you could go anywhere in frequency, where else is interesting?
SAUL TEUKOLSKY: Yeah. So I didn't talk at all about other ways of detecting gravitational waves. So I mentioned LISA a little bit, so that's a space-based experiment. So that would have an arm's length-- instead of four kilometers, would have an arm length of an order of millions of kilometers. And it would be sensitive, then, to much lower frequency gravitational waves. For example, very big black holes-- a million, even 100 million solar masses spiraling into each other. And should have signal to noise values of, like, 100 times stronger than even LIGO at design sensitivity.
Then if you go a little step lower in frequency, there are experiments in which astronomers here at Cornell are heavily involved in, and that's to try to detect gravitational signals by their effect on pulsar. So pulsars are very good clocks, and when we time the arrival of the pulses at the earth, if a gravitational wave goes by anywhere between us-- you can think of it as, when it passes the earth, it shakes the earth, and when it passes the pulsar, it shakes the pulsar. It shows up as the distortion in the arrival time of the pulses.
And so the idea is, you monitor a few dozen of the best pulsars as accurately as you can, distributed around the sky, and then you look for a correlated pattern as this wave would be traveling and you try to detect waves that way. And they're close. I mean, I would not be surprised if any time in the next three to five years, they may announce something.
And then the last way is to look for waves of cosmological scale. So these would be presumably waves that come from the early universe-- maybe the leftover signature of inflation-- the imprint of inflation on the big bang. And these would show up as affecting the polarization of the microwave background. So if you looked in different areas of the sky.
The problem that you probably are aware, there was an announcement of such a detection about a year or two ago and then that was later softened as being probably due to not subtracting out the foreground polarization effects adequately. But I think that that's obviously a very hot area and there are a lot of teams trying to work on that.
SPEAKER 1: Any last questions? Yes?
AUDIENCE: What have we learned just from the frequency of the events?
SAUL TEUKOLSKY: So the frequency ultimately is what sets that mass scale. That's how we know, for example--
AUDIENCE: [INAUDIBLE]
SAUL TEUKOLSKY: Oh, of that frequency. Yeah. I think we learned that we were lucky. Right? I mean, you know, six in a few months of observing added together is a very nice human scale number, right? And if you look at what the theoretical predictions were, going back in time, I mean, when LIGO was up for funding, there were congressional hearings of it because this was going to be a new line item in the NSF budget, and there were famous astronomers who testified to Congress that this should not be built. That there was no way that the event rate would be anything interesting and at the sensitivity that would be needed to see something. It was hopeless. Right? And it still got through Congress and it got funded.
SPEAKER 1: All right. On that optimistic note, let's thank Saul again.
[APPLAUSE]
General relativity is a very successful theory of gravity; so far no discrepancies with experiment are known. Yet at a fundamental level the theory is incomplete, since it does not include quantum effects. Modern attempts to make a quantum description of gravity predict small deviations from the predictions of general relativity. The detection of gravitational waves from colliding black holes has opened up new possibilities for finally testing general relativity in the strong-field regime.
In a Department of Physics Colloquium on March 27, 2018, part of the Bethe Lecture Series, Saul Teukolsky describes how LIGO is already able to perform interesting tests and discusses the future of such tests, both with ground-based and space-based detectors. Teukolsky is the Hans A. Bethe Professor of Physics and Astrophysics.
