I feel very honored to be giving this lecture and connected with you, Yervant, because of my enormous enthusiasm and respect for the astronomy program here that you have shaped so beautifully over the years and the great times I have had here. I've been a visitor here many, many, many times over the decades since the early 1970s. And it's, in some sense, sort of a home away from home. And I'm grateful to Chuck for endowing this lectureship and making this possible. Thank you.
I'm going to talk today about probing what I like to call the warped side of the universe with gravitational waves and numeric relativity, neither of which tools we had as useful, workable tools a decade ago. And so this is a whole new subject, to a whole new area of science made possible by breakthroughs in computing technology-- well, numeric relativity technology. It's not just computing, but it's algorithms, it's understanding the Einstein equations much more deeply mathematically. And by the gravitational wave technology that this superb team that I'm associated with has developed.
By the warped side of the universe, I mean phenomena and objects that are made from warped space and time entirely or in considerable measure. So in reality, although I don't list it here on the next slides, I would include neutron stars there, for example, because the neutron star, their gravitational potential energy is a good 15% or so of the total energy. So gravity is a major component-- warped space time is a major component in a neutron star as well. But I will focus to a great extent on warped space time without matter. And later I'll discuss neutron stars and things that do have matter.
But the phenomena I mean are things like black holes. I'll show an icon can here that I'll explain later. The Big Bang singularity, in which the universe was born. Naked singularities, which might or might not exist-- probably not, but we will be searching for them with gravitational waves, as I'll explain. Our universe is our brain living in a higher-dimensional bulk as occurs in the movie Interstellar and as occurs according to string theory.
But also the idea that there may be macroscopic extra dimensions, as Lisa Randall and Raman Sundrum suggested some years ago. Cosmic strings, which, David Chernoff was just bending my ear about a few minutes ago, and which I'll talk about briefly later on. And gravitational waves. So here is a list of things that are basically made from warped space and time. And then there are also things like neutron stars where the warping is very important, but not necessarily dominant.
And let me begin with black holes. A spinning black hole shown in this little icon. This is meant to be the horizon of the black hole. As it spins, it drags space into a whirling motion around itself. And I'm going to focus a lot of attention on that whirling motion of space and what it does to phenomena related to black holes.
We visualize the shape of space around a black hole, the warping of space, by imagining taking an equatorial slice through the black hole that's a two-dimensional surface. It's warped because Einstein says that space time is warped by the matter and energy present in space time. And so if you take it out and embed it in a flat space, it has to bend down in the flat space in order to accommodate itself to the flat space. And so we get diagrams like this famous diagram that the warped space around a spinning black hole looks like a trumpet horn.
And so you have the warped space. You have the horizon down here, which if we were to store one more spatial dimension-- changed it from being a circle into being a spheroid-- we have the color coding showing the slowing of time, as I discussed in my public lecture yesterday. Yellow time flows at 1/10 the speed that it is far away, and black, it's time slows to a halt at the horizon. It's seen by anybody who hovers there refusing or managing not to fall through.
And we have the dragging of space into motion, a whirling motion with an angular velocity that is proportioned to the length of the white arrows. And in the language of general relativity, you have three metric that's encoded in this shape. You have the so-called lapse function, which is encoded in the color coding, the slowing of time. And you have the so-called shift function, which is the length of those arrows. And so there's some precise mathematics that underlies this.
Now, the question, of course, arises what warps the space and time? And the answer is, the energy that's contained in the warping, that the black hole is held together by non-linear self interaction. It's what we call a gravitational soliton. Objects that hold themselves together by nonlinear interactions are often called solitons.
And so the status of black hole research in 2006 when my story begins is that we knew about the theory of quiescent spinning black holes. We'd never seen compelling evidence at that time for spinning black holes. We had compelling evidence for black holes, but at that time, by contrast with today, we didn't have compelling evidence that black holes spin or spin by substantial speeds.
And wildly dynamical black holes, as when black holes collide, we understood nothing-- almost nothing about them, except order of magnitude estimates. And of course, they had never been observed. And as you are well aware, that has all changed now-- on the theoretical side, through computer simulations, on the observational side, through gravitational wave observations. And that is then the essence of my story, and where that is leading us, and what we have learned by these two new tools.
But it is it was as though in 2006 we had only seen the surface of the ocean on a calm day-- quiescent ocean. We'd never seen the ocean in a storm or with crashing ocean waves, and we wanted to know how it does, in this case now, not the ocean, but space time, which can be warped according to Einstein, how does it behave in a storm-- in a storm, in particular, for example, created by collisions of two black holes? This then it is the topic of the nonlinear dynamics of curved space time, and it does involve some of the deepest issues in the nature of space and time and some of the most interesting astrophysical phenomena.
It was actually John Wheeler, who was my PhD adviser, who, back in the era when I was a graduate student, gave us exhortations, almost religious exhortations, that you should go in, and you should figure out how curved space behaves when it's highly dynamical. He called this geometrodynamics. And this is Johnny lecturing a 60th birthday celebration for Willie Fowler, who's sitting in here-- those of you who knew Johnny or Willie-- and lecturing about geometrodynamics.
And so we tried, and we failed. Everybody fell flat on their face. Nobody could do it. You couldn't solve the equations analytically. You could make a order of magnitude estimates. You could try to figure out some things by perturbation theory. That taught us a little bit, but not very much. And so we had to wait until-- part of the community worked very hard to develop the tools-- but we had to wait for the tools and numerical simulations and gravitational wave observations that epitomized for the numerical simulations by the SXS, Simulating Extreme Space Times Project, which is a joint Cornell, Caltech, and now other universities project, and gravitational waves of observations by the LIGO project. So we now have those tools, and they've been teaching us a lot.
So I want to begin with numerical simulations. And this is a small piece of what's done in these numerical simulations, but it's a piece that has driven my interest the most. Of course, numerical simulations have been under development since the 1960s just like gravitational wave technologies. They've been the same period of time. The first attempts to do numerical numerical solutions of Einstein's equations that I know of that got anywhere were by Bryce DeWitt in the 1960s.
He was working at Livermore at the time. He was involved in some work on design of bombs. And as they needed to work in curvilinear coordinates, he carried curvilinear coordinates over into that particular field. And then he said to himself, OK, well, if I can use curvilinear coordinates for this, why don't I use it to solve Einstein's equations? And so he initiated that. And parallel with Ray Weiss, and and before him, Joel Weber initiating gravitational wave experiment.
But the big success in numerical simulations has only come in the last 10 years, and they are beginning to teach us about geometrodynamics. And so the context in which I'll talk about this mostly, though I'll discuss other contexts as well later, is black hole collisions. This is a diagram that was drawn by an artist following a sketch by me when we were just starting the LIGO project of what we wanted to see-- two black holes going around each other, colliding and merging.
And we knew before these simulations began that the power output in gravitational waves, in order of magnitude, was 100 times the luminosity of the universe, 100 times larger than the total power output of all the stars in the universe put together. And that that was true regardless of the masses of the black holes. Just if the masses were low, this would happen quick, and you would get out a few percent of the rest mass-- a few percent of the masses of the black holes would go into gravitational wave energy. If the black holes were a huge, then the collision would take longer, and you would still get out a few percent of the total mass of the black holes.
So we knew that. We knew there would be no electromagnetic waves emitted whatsoever unless there happened to be an accretion disk around the two black holes close enough to the black holes to be perturbed by the merger. And that's a long shot, and there has been no evidence of that with an electromagnetic counterpart in the first major gravitational waves detection that was announced in February.
So the energy comes off all in gravitational waves, but the key thing is the details of the collision, the geometrodynamics, are encoded in the gravitational wave's waveform. And this is my fanciful hope for what the waveforms might look like back in-- that was in the 1980s when we were embarking on LIGO as an organized project. It turned out to be less interesting than that so far, unfortunately. But nevertheless, encoded in the waves is an enormous amount of information that does tell us about geometric dynamics.
So an example of a numerical simulation-- identical black holes not spinning. This is the first simulation done by the SXS collaboration. Cornell and Caltech since expanded to CITA, CalState, Oberland, Washington State University. Do I have them all in there this time? And lead by Kidder, Pfeiffer, Scheel, and Teukolsky-- the key people, the primary authors of the spec code. Saul Teukolsky overseeing them and teaching everybody how do you do this. And then he's superb at turning the details over to his younger collaborators, and they do fabulous things.
And so what I'm showing you is not-- actually originally when I made that slide, I was going to show you the first simulation, but I slipped in there the simulation of the source of gravity that LIGO saw, and so it's a little different from the first simulation. But I showed this also in my popular talk, except I did not show the apparent horizons of the black holes, which are sitting up here going around and around. You see the warped space of the black holes. You see the shift function, or more precisely in the negative of the shift function that shows, in some sense, the dragging of space into motion, the lapse function that shows the slowing of time read down inside here. Time is enormously slow. The horizons are deep down inside there.
And we watch it go as we did last night. And it's just a pleasure for me to watch this. After all the decades of effort, to be able to simulate the generic situation, that these guys could do a simulation for what was observed and make these beautiful, beautiful then visualizations is really wonderful. So you'll see when the black holes emerge up at the top, you see the warped space time down in here. Now it's headed toward merger. It will be paused at merger.
These are apparent horizons. With apparent horizons, you see they two black holes original horizons sitting inside the common apparent horizon-- a little different than what's called an event horizon. But that's more technical than I want to get into. But it's really beautiful seeing this visualization.
But there are problems with this, in fact, that I did not discuss last night. This is an acknowledgement of the people and institutions involved in this. There's too little of this space geometry depicted in this way. Also-- I didn't write it down here-- but also, the actual details of the arrows and the shapes and the colors are rather gauge-dependent. They depend on how you slice up space time in the space plus time.
The computer does it in a manner that is guaranteed that the computer won't crash. And in some sense, that means it's doing it in a manner that will make things as smooth as possible and as calm as possible as unwild as possible. But nevertheless, the precise details depend on how you slice base time running up, space running horizontally-- just how do you slice space time to get this three dimensional surface?
And what I'm trying to tell you next, there's also some dependence on that, but not nearly to the same degree. The bigger issue is that too little of the information is depicted in this way. And so as these simulations started to work, as Saul and his colleagues started to have real success, I sat down with part of the team, some here at Cornell at the time, some at Caltech, Jeandrew Brink in South Africa, me-- this is taken from a video conference. We did all this collaboration by video link.
And we figured out new ways to visualize space time curvature, and I want to talk about that for a few minutes. So what I was showing you before was a visualization of the metric of space time. I'm going to show you now the visualization of what's called the Riemann curvature tensor, which is the local thing that produces forces, the tidal forces, for example, that raise the tides on the moon is the Riemann curvature tensor does that. I wanted to highlight that the key person who had the initial idea for this was actually Rob Owen, who was a post-doc here at Cornell at the time and in Saul's group, and then everything flowed from there.
So I'm beginning-- first I'm going to introduce the concept of Verizon horizon, or frame-dragging vorticity. This is the same word as used in fluid mechanics to describe something that's very similar. So if you have an observer above the black hole, she can be falling in or she can be hanging-- it doesn't matter-- or standing. As she looks at a gyroscope at her feet, her head will see the gyroscope going around and around in counter-clockwise direction. And if her feet look at a gyroscope at her head, her feet will see a gyroscope going around and around in what appears to be a counter-clockwise direction.
It's the same way as if you take a wet towel you wring the water out of it, your left hand will see your right hand going around counter-clockwise. Your right hand will see your left hand going around counterclockwise. So there's a twist to space in that sense. This is caused because the dragging of space in a motion is bigger near her feet than it is near her head, and so that's that differential dragging of space into motion. Or in technical language, there's a differential frame-dragging that causes this. But it is it actual twist, a physical twist of space, that shows up in gyroscopic motions.
And so I've painted red the regions that have counter-clockwise vorticity on the black hole's horizon, blue the regions that have clockwise vorticity. The vorticity is the angular velocity of the gyroscope here relative to that one there divided by this person's height. And so that's something that's independent of the person's height, and so it's something that mathematically embodies a piece of the Riemann curvature tensor. A horizon vortex then is a region with high vorticity, a high, large twist of space. So there are two horizon vortices, a red vortex, or counterclockwise, vortex here, a blue or counter-clockwise vortex down there.
Now, how about outside the black hole? Now I'm going to get a little bit technical. This is probably surely the most technical slide in the talk. But I want to explain just how this is. Done when we slice space time into space plus time-- so time's running up, we're in space running horizontally-- the electromagnetic field tensor, second rank anti-symmetric tensor, splits up into the electric field and the magnetic fields, which are much more familiar. Those are vector fields.
And we visualize those, but using field lines. So they field lines between the two ends of a bar magnet, for example, magnetic field lines, or between a positive charge and a negative charge-- again, that I call magnetic field lines-- or the magnetic field lines around the earth. And they're very familiar, and they're a powerful tool for visualizing electric and magnetic fields.
Similarly, this Riemann curvature tensor that embodies the local warping or curvature of space time, it splits up into two parts, and electric part and a magnetic part. But these are symmetric trace-free tensors mathematically. The electric part describes tidal gravity. It raises the tides on the earth when it is produced by the moon. The magnetic part describe this frame-dragging, this differential frame-dragging, the twist of space.
And we visualize then these and electric and magnetic part of the tidal field and frame-drag field by networks of field lines. And so, in particular, if I talk about vortex lines and their vorticities for the frame drag field, and integral curve of the eigenvector n of the frame drag field is called a vortex line, and it's eigenvalue is called its voticity. And so physically, the vorticity along in this vortex line is just what I said before, it's the angular velocity of space or of a gyroscope at her feet relative to one at her head divided by her height.
And so you have this physical vorticity, which is the dragging of space. And basically, space is twisting around the blue lines in a clockwise manner. It's twisting around the red lines in a counter-clockwise manner. And this depicts the frame-drag field, or the magnetic part of the Riemann tensor, in terms of field lines, and it's a pattern around a spinning black hole. We now understand that's what half of the Riemann curvature tensor looks like. It is this pattern of vortex lines around which space twists.
Regions of large twist, large vorticity, we call vortexes. And so there's a counter-clockwise vortex coming out of the north pole of this black hole, and a clockwise vortex coming out of the south pole of the black hole. So black holes have vortices, but vortices of twisting space, rather than vortices of whirling water like you're accustomed to in the fluid dynamics. But a rather similar kind of a concept.
Now, the interesting thing is, what happens to these vortices when two black holes collide? And so I'm going to show you-- actually the most interesting collision was the simplest one. It's a head-on collision between two black holes. So these two black holes, this one has its blue clockwise vortex sticking up, its red vortex sticking down. Those are the vortex lines. I'm going to remove the vortex lines and just watch what happens on the horizon. That's simpler. Watch the horizon vortices.
And these two black holes are going to collide and merge. And so let's watch that, and I'll pause the merging. So you now have one-- this is the absolute event horizon of the black hole. It's the point of no return. You fall through there, and you ever come out. And this black hole now has four vortices on it. It has a clockwise vortex here, a counter-clockwise one there. Just back behind here is a blue one, and back behind there is a red one. So there are four vortices on a black hole.
It turns out, black holes don't like to have four vortices. They only like to have two. And so this black hole is going to shed those extra two vortices in some manner, basically by them fighting with each other in what turns out to be a very fascinating manner that was first discovered in these numerical simulations. We now understand a little better from some analytic perturbation theory analyses. But a key point here is that once these vortices have been created, they each robustly retain their individuality.
So now we're going to watch these two black holes collide and see what happens to their vortices. And remember, the upper right is blue, the upper left is red to start out. So we'll just let this go, and watch the upper right, for example. So the vortices are, in some sense, fighting with each other and exchanging vorticity. And this oscillates and radiates away. The oscillation energy of the black hole is gravitational waves. And I will now explain how these gravitational waves are generated.
But it's this exchange of vorticity that's really quite interesting to see-- again, a discovery made through numerical relativity. We never understood this before. Yeah?
Did these start with people on opposite vertices?
Yes. Yes. So I've done the simplest case possible in order to have it be nice and pedagogical.
So a key thing that happens is that when this goes green, there are no no vortex lines sticking through the black hole. And so during these oscillations, what's happening is that the blue vortex lines and the red vortex lines are popping off of the black hole. They actually-- and so here you have the red ones. The red ones have reconnected when the black holes merge, and so the vortex lines go from here down to the red vortex on the backside. The blue vortex lines go from here around to the back side, and as a result of reconnection of vortex lines in the process of the collision, which is itself an interesting process.
But what happens is each time this goes green, the vortex lines pop off the black hole. they embrace each other, and they form a Taurus like a smoke ring. And you get a sequence of Tauri propagating outward at the speed of light. And these are gravitational waves. These are Tauri with the vortex lines winding around the Tauri. The color coding has been sort of lost on here. The vortex lines going in one direction should be blue, going the other direction should be red. Each of these actually contains both clockwise and counter-clockwise vortices.
But what happens is these vortices expand outward through an analog of Maxwell's equations, and the mathematics is very, very similar to Maxwell's equations. The moving vortices generate tendex lines, which stretch and squeeze. That's the other half of the Riemann curvature tensor, is tendex lines that stretch and squeeze, in the same way as a moving magnetic field and electromagnetism generates an electric field.
And so you wind up with these tendex lines, which are the inderal curves of the eigenvectors of the tidal field part of the Riemann tensor. They are wrapping around the vortices like the-- in this case, it's the stretching tendex lines, and the squeezing tendex lines are wrapped around the small part of the vortex.
You now have this structure intertwined vortex lights and tendex lines in the form of smoke rings traveling outward at the speed of light. And they are the gravitational waves that arrive at earth in this case and that LIGO detects. But LIGO fields only that tendex lines that stretch and squeeze. It does not feel the vortex lines. It's designed to feel the tendex lines. Not designed to feel the vortex lines.
And so the plane gravitational wave that heads toward earth, it has these stretching and squeezing tendex lines. It has the twisting vortex lines, and they;re superimposed on each other. And that's what the plane wave looks like when it's far from its source.
In an orbiting collation, which is a more generic situation than what we saw in LIGO, it's a little different. In that case, the vortex lines swing out around the black hole like water from a whirling sprinkler head. And so you have the clockwise vortex lines and the counter-clockwise vortex line swinging around as the black hole spins. And then they induce, through their motion, the tendex lines, which are intertwine with them and, again, produce gravitational waves. And this is the total pattern then at late times of the vortex lines themselves. Red ones coming off of this region-- part of the horizon of the black hole-- blue ones coming off of that part of the horizon. And it's an intertwined pattern of vortex lines, of vortices.
And near the black hole, these are basically attached to the black hole. And it's the near-zone vortices or vortexes, as we sometimes say, turning around and around to generate the gravitational waves. So it's rather beautiful to see this manner of production of gravitational waves. I'm now going to go on. I could say a lot more about that, but I want to discuss a number of other aspects of what we're learning through numerical simulations and gravitation wave observations.
And so I wanted to go on and just remark that when one does a simulation of a neutron star or black hole binary-- this is an early simulation by Francois Foucart, Matthew Duez, Larry Kidder, and Saul Teukolsky. The black hole is three times heavier than the neutron star. The neutron star is blue, the black hole is black. The black hole's spinning about that axis. And so the orbital plane is not the equatorial plane of the black hole.
It turns out that the effect of the vortices in this case is to drag the neutron star matter into the orbital plane, into the equatorial plane of the blackboard hole. And so you're going to watch what happens when the black hole's tendex lines, its tidal fields, reach out, tear the neutron star apart, and then the moving neutron star matter interacts with the vortex lines, which throw it up into this different plane out of the original orbital plane and up into the equatorial plane of the black hole.
And yes, you're dealing with 1 and 1/2 solar masses of nuclear matter being ripped apart by the tendex lines and being thrown up into a new plane by the vortex lines. Just seeing this, you get a sense that there's a lot of power in the Riemann tensor in both the tendex lines and vortex lines, so just let it go. It's already being thrown up out of-- this was the original orbital plane. It's now torn apart. A large fraction of the mass goes down the black hole, and some of the mass remains in an accretion disk around the black hole.
And of course, this is the foundation for what we call multimessenger astronomy. From this event, you've seen not only gravitational waves, but you see gamma rays and neutrinos produced by the hot nuclear matter that's been exposed when the neutron star is torn apart. Of course, we don't see it yet. But I think we're more or less guaranteed to be seeing it with advanced LIGO sometime in the next several years.
And so that leads me then to the second half of the talk. I want to talk for a little while about gravitational waves as probes of the warped side of the universe. I will return to simulations toward the end and remark about some other simulations that teach us in other contexts about the warped side of the universe. But I wanted to talk now for a little while about gravitational waves.
I wanted to make some general remarks that are familiar to most of you probably, but I think are well worth reminding ourselves at this era when we we're embarking on the gravitational wave astronomy as a whole new way to understand the universe. It's the contrast between electromagnetic and gravitational waves. Electromagnetic waves in the astrophysical context, well, they are oscillations of the electromagnetic field propagating through space time. Gravitational waves are oscillations of the fabric of space time itself. It could hardly be more different.
Electromagnetic waves astrophysically are almost always incoherent superpositions of emission from particles and atoms and molecules, whereas gravitational waves are emitted coherently by the boat motion. And I should have said of matter and energy in a black hole-- that's the energy that's tied up, or mass that's tied up in the warping of space time.
Electromagnetic waves are all too easily absorbed and scattered between their source and the astronomer. Gravitational waves are never significantly absorbed or scattered even near the Planck here near the beginning of the universe. And so they are a very powerful way of seeing things that cannot be seen electromagnetically.
Implications-- many gravitational wave sources won't be seen electromagnetically because of these enormous differences, and that's the case of the first gravitational wave discovery from the black hole binary. And big surprises are likely because of the radical difference between the two types of radiation. There are, as I remarked yesterday in the public lecture, there are four frequency bands that are likely to be open for gravitational waves in the next decade or two. The high frequency band, 10 to 10,000 Hertz oscillation or frequencies has already been opened by LIGO now. Virgo-- I'll discuss this a little bit-- there are other earth-based instruments that will come on board and participate in this soon.
LISA and, at some future time, what's called the Big Bang Observatory, are analogs of LIGO in space with spacecraft being tracked, tracking each other with light beams. And LISA and will operate in a frequency band between a tenth of a hertz and 10 to the minus 5 hertz. Again, here four orders of magnitude, frequency band, three orders of magnitude. So these are very, very broad bands. So I remind you the optical band is like a factor of 2 or so. It's a tiny band compared to what you're dealing with here.
Pulsar timing arrays, which are likely to succeed some time, I would say, in the next five years or so, operate in what I like to call the very low-frequency band, 10 to the minus 7 to 10 to the minus 9 hertz. The way this works is worth emphasizing. You look at a pulsar in the sky, it's a really good clock. It's a spinning neutron star. You have pulses that come in very regularly aside from some jitter that you can remove statistically by long-time observations.
And when the gravitational wave sweeps across the earth, it's not the stretching of space that does this. The gravitational wave also affects the rate of flow of time, and it affects all your clocks on the earth. And so you look at a pulsar in that direction, another one in this direction, another one in this direction. It's an array of pulsars, not an array of telescopes, and they all should speed up and slow down synchronously, but with relative amplitudes and phases that basically depend on the direction of the source. And so you can pull out the pattern, the gravitational wave pattern, from that as well. But the clocks on Earth are being affected for all of them. And this is a wonderful approach that Jim [INAUDIBLE] and his group were involved in.
And then there's the anisotropy of the polarization of the cosmic microwave background, which operates in the frequency band 10 to minus 18 to 10 to minus 16 hertz. And that also is very close to success. I mean, it's been seeing the particular anisotropy patterns, the so-called B-modes have been seen. But there is a background noise due to emission by intergalactic dust, which has to be pulled out and is not yet successfully pulled out of the data.
And just to give you some sense of this, here are the four frequency bands, and here are sources. The Big Bang singularity should be seeable in all of these frequency bands ultimately, but it will first be seen in the extremely low frequency band. Exotic physics in the very early universe, phase transitions associated with a change in physical laws, the transition from where the electromagnetic force is combined with the weak force-- what's called the electroweak force-- as the universe expands and cools, they come apart, the electromagnetic field is born, and the weak force is born from that coming apart.
And that's a phase transition that is-- it could well be first-order phase transition, produces gravitational waves that are concentrated in the LISA frequency band. There may be other phase transitions that produce gravitational waves and other frequency bands, but phase transitions in the early universe, cosmic strings that I'll talk about-- I won't talk about domain walls, but these are the walls of the bubbles of the new phase that around the old phase, what are called mesoscopic excitations, in some sense, an oscillation of the shape of the space of our universe-- in the fifth dimension to the movie Interstellar, in the higher dimensions, mesoscopic excitations.
So very speculative stuff, but very likely that some of this will occur, and we will see it with gravitational waves in due course. In the VLF band supermassive black holes bigger than a billion suns; In? The low frequency band, massive black holes, 300 to 30 million suns: a LIGOs band, 2 to 1,000 suns for black holes. Then you have binary stars, soliton stars, very speculative things; naked singularities, very speculative; neutron stars; compact binaries of various sorts; supernova explosions, and so forth.
So there's a great richness of things, some of which we are very confident we'll see, others that are quite speculative. There will be surprises. Some of the surprises will likely be, say, one or two of these speculative things. But there will be other things that we have missed completely.
So the gravitational wave interferometers that we work with on the ground in the high frequency band look like this. This is a standard theorist version. The gravitational wave comes along, and these are the tendex lines of the gravitational wave, so they're stretching along the red direction and squeezing along in the blue direction. So as this passes through the detector, these mirrors are pushed apart, those mirrors are pushed together. The fractional change in separation is the gravitational wave field h, which is a certain component of the second time integral of the Riemann curvature tensor it turns out that. But it's a dimensionless thing times L, so it's a piece of the Riemann curvature tensor integrated twice.
And I'll put in some numbers in a moment. But the key thing, and the whole purpose of creating the SXS project, this Cornell, Caltech, CITA, and so forth project, was to compute initially from black hole binaries, but now also from other sources, neutron star binary supernova explosions and so forth, compute the shapes of the waves for a given source, so those shapes or so-called waveforms can be used. This is stretching up, squeezing down. So it's the oscillating pattern, stretch and squeeze, so they can be used in the LIGO data analysis and later in the LISA data analysis and so forth.
And so that can be used both in searches and to extract information about the sources. So these are two wave forms-- there are two polarizations for gravitational waves as for electromagnetic waves, so there are two wave forms. And these are the particular wave forms for a black hole that is 60 solar masses, 10 solar mass black hole. This spends at 91% of the maximum speed, and this spins at 30% of the maximum speed. They're in circular orbits, and that's the direction to earth. And these are the waveforms.
A different direction, you'll have a slightly different wave forms. You change the spins, you change the masses, you'll have different waveforms. So there's a great richness of wave forms.
I'm just looking at the grid orientation of your tendexes there versus the orientation of the two arms of the interferometer. If those had came from a slightly different direction, when you say, 45-degree tilted, would that mean the interferometer would miss it?
Then the interferometer would miss this polarization. But as I mentioned, there's another polarization. And for other polarization, the tendex lines are at a 45-degree angle to these tendex lines. And so this is optimal for that polarization and gravitational wave that's rotated by 45 degrees, which is the other polarization.
That is the least optimal for this. And so you see one polarization or the other or are a mixture. So what you see is actually a mixture of these two waveforms that depends on the orientation of the detector, so there's additional richness there.
And so we set out to carry out-- we, I should say, Saul and his team. I'm the flaky theorist who doesn't do any of this science, but dreams about what you do with the output of the simulations-- set out to do 1,000 simulations to underpin the dictionary, and there are about 500 that have been done now. But as I mentioned yesterday, the SXS collaboration is now focusing in, once you see a source, focusing in on doing simulations in the vicinity of that source in order to optimally pin down the parameters of the source, the masses, the spins of the binary.
Numbers h was 10 to the minus 21 for the first signal that was seen. That was sort of the upper range of what we expected to see. Multiply it by 4 kilometer arm length, that's 4 times 19 to the minus 16 centimeters. Signal and noise ratio was 24. Divide by 24, and you basically have a noise level at 10 to the minus 17 centimeters within a factor of 2. And there's really a factor 2 missing here. Truly the noise level is at 10 to the minus 17 centimeters.
And so you ask, how is that possible? And this is just to remind, you even for a sophisticated physicist like some of us would believe we are, it's good to just go through this once in a while. There's a centimeter. Divide by 100, you get the thickness of a human hair. Divide by 100, you get the wavelength of the light that's used. Divide by 10,000, you get the diameter of an atom. Divide by 100,000, you get the diameter of the nucleus of an atom. Divided by 10,000, you get LIGO's current noise level.
When I first looked at numbers like this. I wrote in a famous line in a textbook called Misner, Thorne, and Wheeler. I wrote a sentence, "It's unlikely that this will ever succeed." And then I spent a big piece of my professional career helping make it succeed. I did not really appreciate what was possible, and it was Ray Weiss who convinced me, of course,
One key to this is you bounce the light back and forth in the arms for half a gravity wave period while the mirrors are separated, are stretched apart, say, along the horizontal direction, building up the signal by a factor of 1,000. So instead of 10 to the minus 17 centimeters for the noise level, you've built up to 10 to the minus 14 centimeters. I had the same noise ratio of 1, you would see a gravity wave that produces a 10 to the minus 17 centimeter a displacement. It's built up an equivalent phase shift. It's such with a displacement of 10 to the minus 14 centimeters. And so that's what you have to measure.
And with one photon, if you're really clever, you can go down to about the wavelength of light divided by 2 pi as a reduced wavelength. With end photons, because of photon statisticsm in principle, you can go down by the square of N. If you use 10 to the 18 photons, square n is 10 to the 9. There's 2 pi, so that gets you down another factor of 10 and this turns out to be 10 of minus 14.
The issue is, how do you ever do it in practice? And this says in principle, it's possible. The photon counting statistics don't have to get in the way. But how do you do it in practice is the lifetime work of a large number of superb experiments. I'm going to skip over other problems and talk quickly now about the network of ground-based gravitational wave interferometers that we work with.
There are LIGO in Hanford, Washington, LIGO in Livingston, Louisiana, basically separated across the North American continent. We currently have a LIGO detector called GEO600 in Hanover, Germany. It's a shark detector that was used for developing advanced technology for LIGO. Much lower sensitivity because of the shorter arms, but still in operation, and it will help out if we got a sufficiently strong source. And it is left running whenever these instruments are off, just so you have-- it's sitting there in case there's a supernova in our own galaxy. It would see that.
The virgo detector, a 3 kilometer arm is very similar technology in Pisa, Italy. It's a French, Italian, Dutch collaboration with now some participation from other European countries. One 3 kilometer interferometer will go into operation with the advanced technology around the end of this year.
Then under construction, this says LCGT-- the name change since I wrote this-- it's now called KAGRA, K-A-G-R-A, in Japan inside the Kamioka Mine where Kamiokande detectors, neutrino detectors are. It would go into operation around the end of this decade. It should be taking data in concert with LIGO, say, by 2019, I would hope. And then a LIGO interferometer sitting in a box, in boxes, ready to be installed in India when the facilities are ready in India, but that's not probably going to be up and running at comparible sensitivity to the rest of the network until 2022.
But a worldwide network can because we need multiple interferometers to be confident that what we see is real. Because there is noise in these instruments that nobody understands. They are extremely complex. They illustrate the complexity. There are something like 80,000 channels of data coming off the interferometer, all except one are housekeeping data telling you what's going wrong or what might be going wrong, monitoring every aspect of the interferometer and the environment. But it's unbelievable an number of data [INAUDIBLE].
So for a conference of detection to extract the wave forms, both wave forms, and determine the direction to the source by triangulation, the time delay, and the arrival time of the signals in or from different locations. So LIGO is a collaboration of 1,000 scientists at 75 institutions in 15 nations. David Reitze is the current director and Gabrielle Gonzalez, the spokesperson.
These are, as seen from the air, the two interferometers in Hanford, Washington, Livingston, Louisiana. In our 1989 proposal to build LIGO, we said that we would do this in a two-step-- that we would build initial interferometers as a sensitivity where we would have to be very lucky to see anything, but it would be possible, then the advanced interferometers at a sensitivity where we're likely to see a lot of things.
And so that's what we did. It was not easy to get into the buy-in when we were telling NSF that we would have to take two steps, and we had a lot of opposition from some powerful members of the astronomy community. But a lot of enthusiastic backing from some other powerful members of the astronomy community. This thing was controversial in that area.
But to its credit, NSF bought in, Congress bought it, and stuck with us all the way. From NSF, from 1990 onward, Congress, from 1992 onward. The initial interferometers were built 2001 to 2010. They could see 10 solar mass black hole binaries out to 300 million light years. Didn't see anything but interesting limits on. And then of course, the advanced interferometer installed between 2010 and 2015.
And they were turned on in late 2014-- one of them, the other in early 2015-- and brought down to within a factor of three of design sensitivity, and began their search a few days after the first gravity wave was detected.
And they discovered the gravitational wave during an engineering run when the team was making sure that everything was ready for the gravitational wave search. The configuration of the detector was frozen immediately when the signal came in, and that was what was the configuration used then for this first run.
First ran from September 2015 to January 2016 at 1/3 of design sensitivity, so it would see [INAUDIBLE] or mass black hole binaries out to a billion light years. But the 30 solar mass black hole binary that was seen would have been seeable out to closer to about eight billion light years. It was seen at a distance of 1.3 billion light years. So heavy ones can be seen farther. Design sensitivity will be reached by 2019 or sooner.
And if today, at the present time, as you could guess from the data in the first discovery paper, the event rate for black hole binaries reason might be roughly once a month just because there are two events in there, a really strong event and a rather weak event. So that's plausible, then. Seeing three times farther, it's three cubed, or approximately 30 times larger volume of the universe. The event rate goes up to something like one a day. So I think that's roughly what we are likely to be seeing in a few years.
From the initial interferometers to the advanced interferometers is an interesting set of steps. The initial interferometers have the mirror hanging from overhead support by a steel wire. It's an 11-kilogram mirror. To reduce the seismic noise so we can move from operating above 40 Hertz to operating all the way down to 10 Hertz, the mirrors are cascaded. It's a cascaded set of four pendulums,f and each pendulum gives you isolation form ground motion. And the fiber is changed from steel wire to fused silica quartz, which has much lower thermal noise in it, lower random forces in it.
And then we had 10 kilowatts of circulating light power between these mirrors in initial LIGO. In advanced LIGO, a design sensitivity about a megawatt, but it will probably be more like 200 watts together with squeezing. And I won't talk about squeezing, but this is equivalent to increasing the light power. That's a very impressive and frightening level of light power to deal with in practice.
You add onto the first interferometers a new mirror called a signal a recycling mirror, which I won't go into details, but it becomes a tool for being able to reshape the noise curve. Just by moving the location of the signal of a recycling mirror, you can change the noise curve from looking like that to looking like this. It's a function of frequency. This is the noise in the interferometer [INAUDIBLE].
And so you have a tool for tuning the interferometer to go after particular sources. You're basically taking photons that previously were giving you a lot of information down around a few Hertz and moving them all, in this case, up to give you information around 600 Hertz. So it's a versatile instrument in that sense.
And then some new vibration isolation, some active vibration isolation that helps with the low frequency noise. And then just to remark that in advanced LIGO, one of the experimental challenges is that you have to more modern motions of 40-kilogram mirrors to 10 to the minus 17 centimeters. And that is the size where the center of mass degree of freedom of the mirror starts to behave quantum mechanically.
And so the very act of making your measurements perturbs the interferometer by an amount dictated by the Heisenberg Uncertainty Principle. And you have to figure out how to deal with that and have that not interfere with the measurements.
So for first time, humans will see human-sized objects, in this case, 40-kilogram objects-- it's the weight of your children-- behave quantum mechanically. And what's called Quantum Nondemolition Technology is built into advanced LIGO to deal with this. Quantum Nondemolition Technology is technology and techniques designed to get a classical gravitational wave signal through a quantum mechanical particle or particles with mirrors without losing the information in the signal, without demolishing the signal. So a phrase invented by Vladamir Braginsky.
And this technology is a piece of modern quantum information science, Quantum Nondemolition Technology, built into advanced LIGO. And we began the work planning for this in 1980 spearheaded by Braginsky in Moscow, and then by my research group, particularly Carlton Caves, and then later, Alice [INAUDIBLE] and Yanbei Chen in my group.
The improving LIGO sensitivity, well, that was initial LIGO, was up here. This is the noise level in what is called strain per root Hertz. You multiply by the square root of the bandwidth that you're operating. If it's a broadband observation, that bandwidth would be the frequency itself. So multiply that to get the dimensionless noise.
So this is sitting, today, down at about two or three times 10 to the minus 23 multiplied by the square of 100. So it's a few times 10 to the minus 22. And getting down the design-- oh, I'm sorry. That was initial LIGO. Advanced LIGO is better by a factor of about three or four. And then today-- and that's the design noise curve for advanced LIGO.
It's remarkable that the team turned on, and within a few months after turning on, they were down to within a factor of three or four design sensitivity. That's a real tribute to the superb experimental work.
AUDIENCE: What is the spikiness [INAUDIBLE]?
KIP THORNE: So these are resonances, mechanical resonances in the system. So every wire by which a mirror hangs has a resonant frequency somewhere up in the hundreds of Hertz region. And you just remove them from the data with a notch filter. And so you've got a lot of things hanging and a lot of wire resonances. That's what most of these things are.
And so you should just ignore them in practice. You just notch them out in the data analysis. The gravity wave discovery is well-known now. The signal came in in Hanford, Washington as I remarked yesterday. This is the raw signal. We never expected to see a signal so strong that you would see it by eye. But all that's done here-- and it was just standard-- removed stuff below 35 Hertz because at present, the noise below 35 Hertz is quite bad, at that point will be 10 Hertz [INAUDIBLE] design sensitivity. Remove it above 350 Hertz because the noise is getting rather bad there. And so do a bandpass filter, and that's all that's been done. And this is the raw data bandpass filter coming in.
And what you see here, the thin line, is the best fit template from SXS numerical relativity simulations also bandpass filtered. So it's not quite the signal you expect to see. It's distorted by this bandpass filter. And so this is what came in in LIGO Hanford. Seven milliseconds later this came in in Livingston, superposed some beautiful agreement in there.
And then as I remarked yesterday, the SXS numerical relativity activity waveform in red. The gray is the measured waveform with noise removed and beautiful, beautiful, beautiful agreement.
AUDIENCE: Is that just one polarization?
KIP THORNE: It's the one polarization that measured by the detectors, which are parallel to each other to the extent that the curvature of the Earth allows, and so they're sensitive to one polarization. It's a particular combination of the incoming polarizations. But it's that combination that the detectors are sensitive to.
AUDIENCE: I'm curious because the question was asked before and I had the impression it was a black line and a red line, which was a numerical prediction of the two different polarizations.
KIP THORNE: But what you see with these detectors, because they have parallel arms, is you see a particular combination of those two curves.
AUDIENCE: OK, so they have to fit the combination.
KIP THORNE: Yeah, yeah. That's right. That's fit as well. That's fit as well. But this is what I want to focus on here. The green shows the relative velocity of the black holes. It's coming in at about 33% of the speed of light, relative velocity headed up toward about 60% of the speed of light relative velocity when the merger occurs. This is their separation, in units, of the sum of the radii of the two black holes.
But the thing I really care about is the post-Newtonian analytic waveforms models break down around 30% of the speed of light. This particular source, basically the entire signal has to depend on numerical relativity. And almost all, a very large fraction, of the signal power is in the highly non-linear regime of geometric dynamics.
And so this is really the beginning of observational geometric dynamics. And in an auxiliary paper, the team goes in and analyzes, searches for ways that general relativity might be failing and puts some very interesting limits on breakdowns of general relativity in this highly non-linear geometric dynamic region.
The beautiful agreement with the numerical relativity wave forms is a wonderful tool for the first time to test general relativity with good precision, in fact. You were never before able to do any testing, but now tested with good precision in the fully geometric dynamical regime. I remarked about the masses and so forth yesterday.
When LIGO-India comes online, we will improve the sizes of the error boxes with regard to location enormously. The error box on that first source was huge. It was sort of like this because we don't yet have Virgo. When Virgo comes online, a source at this location, you would localize it within this long, elongated error box. So that's LIGO plus Virgo a year or so from now. When LIGO-India comes online, these error boxes get much, much smaller.
So this is crucial for localizing sources in order to do multimedia in order to look for electromagnetic counterparts and neutrino counterparts.
And so I think I'm going to top there. I was going to talk also about other numerical relativity simulations that have taught us other things about nonlinear dynamics of curved space time. But I've used up my time. I think I've talked about probably the most important topic to this point. Let me stop there.
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Renowned astrophysicist Kip Thorne, co-founder of LIGO (Laser Interferometer Gravitational Wave Observatory) and executive producer / scientific advisor on the movie 'Interstellar,' spoke at Cornell April 7, 2016 as part of the Yervant Terzian Lectureship Series.