Some the. Gravitational waves and gravitational radiation is a rather new subject. It was a subject that didn't even exist ten years ago. The first direct evidence after kind of a century after its prediction came in 2016. And since then it's become almost an indispensable tool in astrophysics to learn about populations of stars and the early universe in ways that we really don't have any other accessible pathway.
So I'm going to give you in the lead talk kind of an overview, remind you of what gravitational radiation is, how we figure out how much energy is in a gravitational wave, and talk about the methods of detection. But to begin with, let's talk about the languages of gravity. Gravity really is spoken in three different languages and like different languages. It's almost mutually incomprehensible to go from one to the other. So don't worry about the equations directly.
They're just meant for decoration. If they if they mean something to you, so much the better. So I'll talk you through this. For many hundreds of years. The theory of gravity was, of course, Isaac Newton's theory of gravity. F equals GMR over R squared. And what you see here is simply a more rigorous form To write down. The potential energy of an assemblage of matter. Rho is the mass density, and you're adding up all the little bits of GM over R to get to a total potential.
And this language of gravity is sufficient to talk about just about everything in astrophysics. This is the language of gravity that 99.9% of astrophysicists use in their day to day work. But the problem with this theory of gravity from a theoretical physics point of view is that it assumes that if there's a little change in density, it instantly turns into a change in the gravitational force all over the universe in principle by hand.
If a point mass moves, then the potential changes instantaneously everywhere. In other words, it doesn't incorporate the notion of causality or special relativity. And to make a theory compatible with relativistic ideas is no easy task. So that was first done by Albert Einstein in his general theory of relativity. The general theory of relativity is sometimes thought of as well.
Special relativity tells you how to go from one reference frame to another reference frame, moving at a constant velocity general relativity. You know, it's more general what general relativity really is. It's a theory of gravity. And I'll talk a little bit more about that. But anyway, this is just the field equation for Einstein's theory of gravity. And we'll come back to this. And then the last one, yet more incomprehensible is what I refer to as and identify with.
The name is Richard Feynman, in the sense that this is the kind of final step where we try to make gravity compatible not only with relativity and causality, but with the notions of quantum mechanics. So there are things like propagate as we talk about individual spins of particles, the little bits of gravity have spin two, photons have spin one, and the there's a huge amount of activity going on, trying to understand how a quantum theory of gravity would work.
And we certainly have no quantum theory of gravity yet. We don't even have a consensus on the best way to proceed. So step one is to incorporate causality and then step to the ultimate step would be to incorporate quantum mechanics into a law of gravity. Now, of course, electromagnetism. And certainly in the history of the theory of gravity, we would practitioners would go back to electromagnetism to try to learn what they should be doing. And we have a model in front of us that we can copy.
Electromagnetism also comes in three different languages. There is it's not Colombian, but Colombian. That is to say, cool Ohm's Law, where we talk about electrostatic potentials associated with charges. That's kind of the analogue of Newtonian gravity for electricity.
And then the fully developed classical field theory, Maxwell's equations, the first equations in physics to be fully relativistic, tells us how to go from static configurations to pretty much any configuration where the charges are moving around. And then finally, in the 1940s, the final step was taking the find the fine money in step where we developed a quantum theory of fully relativistic quantum theory of spin one photons.
And pretty much now any process whatsoever at any level which simply involves ordinary particles and photons, can be described to arbitrary accuracy. So in this case, we understand how to combine one, two and three seamlessly. So if there were no other reason for studying quantum gravity. Or for studying gravitational radiation.
I should say gravitational radiation is absolutely key to our theoretical understanding of a complete theory of gravity, and that's reason enough to study gravitational radiation. But there's yet more because gravitational radiation, like electromagnetic radiation, turns out to have practical applications. I don't think you'll be paying commercial licensing fees for gravitational wave broadcasts anytime soon, but it is nevertheless a very useful and very interesting astrophysical tool.
So this is a summary of what I've said. We have three languages I should call this. I could call it Newtonian, the policy, and it's probably a better word since Laplace was the person who introduced the idea of the potential. And we're at the stage where we understand 1 to 2. And then there's a big question mark with number three. So let's look at things in a little bit more detail and we'll see how the notion of gravitational radiation arises.
So this is the Poisson equation for static potential theory, and this tells me how to compute my gravitational potential. I have a bunch of little I can think of them as point masses, and I add up all the little tiny g m over hours from all my constituent masses in my body. So here's the observer out here. Here is my origin and the r prime is a vector within the body itself. Capital R is between a particular point mass within the body and where I'm locating.
And this is a relatively simple formula to compute the gravitational potential and from the gravitational potential, the gravitational force from any kind of configuration. Now, what about a time dependent theory? So what people do in practice to turn a static theory into a time dependent theory is just put it here and then you're done for that. And that would be nice. And in fact, that works incredibly well.
That's basically the way. People do calculations of things moving through galaxies and through evolving systems. Whenever we have a potential. And we want to make it time dependent to by the source of my gravitational field is moving with time. Then instantaneously a distance capital are away. The gravitational field changes. And that works. In practice. Very well. But it can't be exactly correct. Because gravitational field simply can't propagate instantaneously across the universe.
So we can take a big clue from studying how electric and magnetic fields work. And we use Maxwell's equations. So if we go back to the fundamental Maxwell in for equations, then it turns out we can always write the equation for the electromagnetic potential in this form. And you may recognise this sort of group of terms over here. This is the standard, quote wave equation. You have a second derivative with respect to time.
And then this Dell squared is the partial derivatives, the second quarter partial derivatives with respect to space. And it's a simple linear equation. And then we have the source term on the right, and it's solution looks very much like the one that I just put up in my earlier slide. In fact, it's identical, except there's a prime here. Otherwise it's exactly the same mathematical form, but that little prime. And what does that? We have to come down here. So T prime is actually T minus.
And now we have this retarded time, Capital R, which depends on our prime itself, divided by the speed of light. So the potential of time t depends on the superposition of what the source was in all its little individual bits. A time T minus are over c ago. And it's of course not the same r. For each point because they're located at different locations. It depends upon our prime. So suddenly it gets a lot more complicated.
It looks simple when you write it down this way. But there's actually a huge amount of information that is hidden there. And that's what really happens. So just to make it more explicit now. This little black dot represents some kind of a particle in my body. And I'm interested now when I compute my electrostatic potential, not just to add up all the effects of all the different charges at some time.
T But at this point, if I want to calculate what the potential is at this distance, ah, I calculate the charge density at our prime at a time capital R oversea before the current time T and that'll be a different number for everywhere in the body gets more complicated. Let's see what the consequences of this are. So here is. A point charge. And these are the lines of force. And you notice the lines of force are all pointing directly to where the point charge is.
Now, you say, well, that's a pretty crummy slide that you made when you cropped off this part and then you craft it off here. I mean, what's what's going on? You can't get you can't make better slides, Professor. Well, I'm shooting a little bit. This is the actual diagram. And what we have here is a more complicated situation. This is a charge which has been sitting at this location for a while, and then it's accelerated up to this other location with these little points coming off.
And then it just sort of coasted after that. And what you see because of this effect of the retarded time is the history of the meaning of that movement has been encoded in the actual electric field lines. So you notice the distant field lines. If I all draw them, they're all they haven't gotten the news yet. We are all oriented to where the charge was before it started accelerating. And then there is this transition zone during the acceleration process itself.
Which forms a kind of a coherent structure unto itself and is moving outward, that transition zone. And then, of course, within the transition zone, the news has arrived as to where the charges and those field lines are pointed in a different direction. So if I isolate an individual field line and explore it a little bit more carefully, I see what the effect of including that retarded time does. It causes this kink to appear, and that kink propagates outward at the speed of light.
In fact, that's not even a good way to say it. That kink is like. That transverse kink is what our retina records as light. That's what excites our cells. That is light itself. It really is the effect. Light is the effect of that retarded time in the solution to the mathematical equations. That's what radiation is. And gravity. Gravitational radiation is actually very similar in its underlying principles.
Now it's a little bit more complicated. Gravity is a theory, a geometrical theory, and Einstein's theory of gravity. The idea is that we live in a space. We live in a Minkowski space. And it's not a space. That's particularly intuitive, although we've lived in it all our lives pretty much. It's not a three dimensional Euclidean space. Looks like three dimensional Euclidean space, but you're being deceived. We live in four dimensions and one of the dimensions.
When we try and compute, you know, the Pythagorean theorem C squared equals H squared plus B squared and so on. Well, you have to bring in a minus sign. So mathematicians love that kind of stuff. The rest of us are kind of wondering what does that actually mean? But that is the world that we lived in. We live in a four dimensional world, and the fact that we have a time is sort of an accident of that, really. You know, it's this odd dimension, this thing that comes in with the minus sign.
That's what our consciousness experiences as time. But we should be thinking of it as just another part of the space. That's the way to think of it. If you really want to do the calculations, it's just another dimension of space that comes in with the minus sign. And the other oddity here. This w w de w is normally huge. We do vast leaps of. Intervals in the W direction. When we do teeny tiny the axis as we sort of go through our existence.
So. It turns out that the best way to do that is to isolate the bigness of VW by a big number, a big constancy, which turns out to be the speed of light. And then what's left over the d. T is something that we can measure in units that we're happier with seconds, minutes and hours. But that's me. That's the world we really live in. If you want to say, Well, I don't have an intuitive feel of what this what this so-called hyperbolic space is really like.
Well, yes, you do. You've lived in it your entire life. This is hyperbolic space. Get over it. So. The interesting thing about this space is that in Einstein's theory of gravity, gravity itself is not thought of as a force. There are other forces that are present electricity and magnetism, but gravity is not a force. Gravity is actually a distortion of that Minkowski space or curvature.
As the mathematicians, I don't like the word curvature because you can have something that's curved like a cylinder. And it turns out the properties of a curved cylinder are pretty much exactly the same as a flat piece of paper. That's why you can turn one into the other. But that's the kind of term that's often used. It's really a specific type of distortion. The surface of a sphere is truly mathematically curved.
You can't wrap a piece of paper. You can't wrap the plane in a very easy way around the surface of a ball without distorting it. So that's what gravity does. Gravity creates that kind of curvature. And so what that does mathematically is it changes the form of this space time interval. So what I put up here is how things change when you have a black hole. So six square deep squared acquires a coefficient to GM over RC squared.
And then I'm going to switch from Cartesian now to spherical coordinates. So here is d r squared and that has the same term now in the denominator. And then there is a solid I should have put an R squared there, but that is the solid angle part of the what's called the metric. And that's unchanged. But that is what gravity does. It takes that Minkowski space and it distorts it. And the amazing thing is you can recover all of Newtonian gravity in the right limit from this approach.
Newtonian gravity doesn't go away in general relativity. It simply becomes ensconced in a more general, more beautiful theory. As Einstein said, it's the ultimate fate for a theory that's not quite correct. It's to find itself a home in some limit in a more general theory. And that's what happens with Newtonian gravity. Now, to actually calculate what happens, the way that you do that is that you have a minus sign here and a plus sign here.
So we demand that the difference between those two pieces of my interval, those two pieces of the metric, I want them to be a minimum of all the possible orbits to get from A to B, The one that minimises the difference is the one that nature chooses. So it's a beautiful way to derive the equations of motions and that gets you back to Newtonian gravity and of course, beyond Newtonian gravity. So how is this formalised? Here's the world of special relativity.
We give this its own name. CE square, Deke House squared is this whole combination. And you notice that in cases where of course, where the X, Y, and Z are zero, if I happen to be moving along in my with my coordinate system so that I have only the change in time, but not the change in space, then detail and the T are the same thing in that case. So the power people like to think of as co moving time. And we write it this way mathematically. So the trick here is alpha and beta.
Those are super scripts and subscripts. Don't. For this lecture, we don't have to worry about whether I write them on the bottom or on the top. When you do it, you have to worry about it. But we don't have to worry about it this morning. And a zero means time. The zero is CDP. And then one, two, three, simply refer to X, Y, and Z. And the rule is, if an index is repeated, then you sum over all values. That's the Einstein summation convention. Einstein got to be a little bit of a mathematician.
Mathematicians didn't do that before Einstein. So Einstein's contribution was said, I'm not going to write the summation sign. If the index is repeated, then you sum over it, unless I tell you not to. But without. So that's the idea. And so this is a very compact way of writing this expression. And you can think of alpha and beta as a nice little matrix that looks like this. It's mostly zeros, except along the diagonal as shown.
So that's the world of special relativity. Now, more generally, when matter is present. My C squared. The Tao is written this way. The notation people like to use is G. Alpha beta that is called the metric tensor. And that unlike ETA alpha, beta, eta alpha beta is very simple. G alpha beta can be in principle anything. Pretty much anything for you know, subject to.
Certain smoothness, requirements and so forth. But think of it as anything, and it can be anything depending upon what coordinates I use. I can take Minkowski space which looks very simple and turn it into something that looks really ugly. If I use properly Spheroidal coordinates to describe it, which I'm allowed to do. That would be part of this. But it also can get complicated because the space itself is more complicated as well as difficult or more abstract curvature properties.
So this is now the realm of general relativity by G alpha. Beta is the same, is it alpha beta? That's the world of special relativity. And here this morning for this lecture, it doesn't get too much more complicated than that. I'm going to let G. Alpha beta, equal eight, alpha, beta, plus a little tiny bit left over h, alpha beta. And that little tiny bit is going to be what gravitational radiation is. So that's the how we think of gravitational radiation.
Gravitational radiation is causing flutters in Minkowski space. That's really the way to think of it. And when I apply my rule to find the minimum time distance between two intervals, that will tell me exactly how the particles move. So that's the world of gravitational radiation that we study. Now, happily. It turns out that those little ages. Those satisfy? Exactly. You can always write them.
Choose your coordinates. It's always possible to write them in such a way that they satisfy exactly the wave equation That. The potential. And although I didn't say it, the victor potential also does in max rally in theory. So here I've written it in the simplest way. There's no sources now. So I'm looking at the propagation of these waves away from the sources. But this is very nice because we can take a lot of what we know about solutions to the, quote wave equation.
To give you just call this the wave equation. It gives it a rise pretty much almost any time you have some kind of a simple form of wave propagation, the wave equation. And we know that it has solutions. Here's a wave propagating in the Z direction, cosine a Z minus omega t. So we write H, alpha beta as some kind of an amplitude, and we need to put subscripts on alphabet on on the A in order to do that.
And then it's either a cosine or a sign. So it's very it's all the stuff you learned about waves that goes right over. Now, the interesting thing is that when you have a wave propagating in the Z direction, almost all the H's are zero. That makes life simple. And there's only two. Independent agents that you need to worry about. There's h. X. X. And one mode has h x x equals minus h, y y, and nothing else. Two modes of propagation and then another motor propagation has only an h x y present.
And that's the same as in h y x h is going to be a symmetric tensor in its indices. It doesn't matter what order you write them in and that's it. That is good enough to describe any kind of superposition of gravitational waves. You just need two different modes of polarisation. So what do they look like? They look. They look like this. Let's see if I can make this move. On my screen. Yeah. So if I have a ring of particles and a gravitational wave is passing through the screen.
This is the way the forces would operate. They would squeeze in either this kind of sense. We call that the plus polarisation. It looks like a plus sign or one rotated by 45 degrees, which was the polarisation. And in general, from a mathematical point, if I have two particles that are separated by x I so I will always be one, two or three. So if I is equal to one, this is simply the x in the x direction and a wave comes by. This is how it changes. It has this interesting looking formula.
It depends upon the separations in other directions. When away passes by. When I calculate what my new X separation is, and that's what gives rise to these. Funny. Sort of ten surreal motions. You squeeze along one axis and you expand along the axis at 90 degrees to it. That's what the effect of gravitational radiation is. And gravity can come in plane waves.
We're also interested in radiation, gravitational radiation, when it has this radio form and there's a one over our dependence in the amplitude. Otherwise, it's very similar. And then we identify the plus polarisation I'm using now spherical coordinates. So I don't want to talk about X and X, but I talk about theta and theta and Phi and Phi, the two angles on the surface of a sphere theta being the CO latitude and Phi being the azimuthal angle.
What about. Here's an interesting question now. What if? What about the energy in a gravitational wave? How do I compute something like that? How hard is it to bend space? Can't. Can't. I can't get a handle on it. So there's an interesting way to do that. Let's start with the wave equation itself. This box is called the deadline version operator, and it simply combines this del squared with the DX by the P square.
And I'm just going to do one thing. I'm going to start with the wave equation, and I'm going to multiply by minus the h, alpha beta, d, t, and because alpha and beta are repeated, I sum over them. Remember, they're all zero except when alpha and beta are X or Y. And if I do that and I kind of, you know, play with my calculus, I integrate by parts. You do the things that you did when you learned your calculus. So, for example, the first term here,
it turns out I can take that the HBP and stick it inside another D by D.T. And then with the pieces from the second term, I can write them this way. I can write the equation in this form, in that. May trigger something from your days. When you played with these kinds of equations in fluids or in electricity and magnetism, because that form of a of the wave equation, when I multiply, I take its first moment.
That's what you describe what I've just done. This has the form of an energy conservation equation. It's a time derivative of a D by d t of some kind of a density. Plus the divergence of a flux. Now this is a minus sign, so that doesn't mean I just stick the minus sign in here. So it's still a plus the divergence of some kind of a flux. And so this is what turns out to be an energy density in a wave and. And energy flux. Now, there's one important difference.
This is the zero over here. I can multiply this by any constant I choose. So I have kind of the form, but I don't know the overall constant. How do I get the constant? Well, that takes a little more work. So I won't do it here this morning. But what you need to do is keep those source turns on the right side. And then after you do that multiplication by D by d t, you can write the right hand side as the rate at which the gravitational forces do work back on their sources.
So it's kind of the power lost. And by doing that, you can determine what the overall constant is. But just to get the form of these is a very, very simple operation. To get the precise constant, you have to do a little, little bit more work. So that, I think, is the best way to understand how you calculate the energy in a gravitational wave. And this is what it looks like constant turns out to be.
Turns out a C to the fourth over 64 PI out in front and varies the energy flux with the minus sign retain the energy flux in particular has a nice very simple form. And if you plug in your nice cosine function's cosine k z minus omega three, you can show the way fluxes behave. The flux is really this energy density times the speed of light for a plane wave. So it's all relatively simple. Now, a little later in the day, we're going to hear about gravitational waves in a cosmological setting.
And so I'm going to sort of set up some of Barry's talk now, just to introduce some notation to save him a little bit of trouble later on. So the energy density, if I go back to my earlier slide, those two terms actually combine let me do that just to remind you. For a simple sum of cosine and sine wave. These two terms contribute equally. So in fact, there's another 32 here that comes in and I can take one or the other.
So that's what that is. And if I look at a bunch of superpositions of these cosine waves and evaluate this in some average sense. Then what people like to do. This is all written as a function of time. But you can also write it as a function of frequency like Fourier transforming it. And so cosmologists prefer that they like to use the frequencies. So this average value of this sum here is written in this way. Each of the D by d ts when you differentiate a cosine, the DBP brings frequency.
That's what f is. And. Then all the rest of this is a some an integral over some other quantity. The noted S-H, which is known as the power spectral density, it's a basically the same thing. It's the energy written as a function of frequency. The eight pi squared here is just a convenient normalisation factor, so don't worry about that.
You know, you get omega is two pi times F, so you get some of that coming in and I'm integrating from zero to infinity because I just want to have positive frequency. So there's another factor of two. It's that kind of thing. So don't worry about this eight pi squared. But the main thing is just this form of thinking of the energy as so much energy in this frequency and so much energy in the other frequencies. So you'll see that a bit later. I wanted to touch upon it in my own talk.
So for a sum of playing waves, we write back in the form row. GW So row it's kind of like a the equivalent mass. If I took the energy and divided by C square in the gravitational wave and used equals EMC squared rho GW would be the inertial mass of those waves. But I'm always going to write it in the form of rho GW times C squared. That's the energy density in the waves and that that's what we've just done.
And this is if I now take Ro GW and I put it in the C squared to use this formula, then this is the result that follows. And as I say, don't worry about the details now you'll see it a second time. But when Barry talks about it, it won't be the first time that you're saying it.
And so what I do now is I form a ratio when I'm interested in when I do cosmology is how much energy is there in the gravitational waves compared to the critical energy density kind of in our universe, which is a universe which is exactly flat. It's exactly between being a closed universe and an open universe. What is the ratio of those two things? How much gravitational wave energy is there relative to the critical energy density in our universe, which is given by this expression?
H0 Is the Hubble constant, the rate of expansion. And. The last thing about this is that we like the integral that I gave of the frequency was over, was a definite integral from zero to infinity. So think of it now is an integral now from zero up to some particular frequency. And I write down this ratio omega g as a function of F. So rather than row GW, I form this kind of combination of derivatives. So f times a d by the f, we'll give you something which is of order.
The GW to begin with, but it now depends upon frequency. So I can ask the question how much energy is in this frequency band? So this is what this is measuring and this is the expression that you'll see later. Okay, I'm going to leave it there and come back to some. Other interesting applications of the energy and gravitational waves. It's actually pretty simple to calculate in practice. Here's what you do. You need to evaluate these moments of inertia.
Ten steps. So I j i j here can be one, two or three. Usually we'll talk about iron J being X and Y, and the gravitational wave will go off in the Z direction. So here is a moment of inertia tensor Roe. So this is the energy, the mass density. There's the volume x, i j. Now this is the same moment of inertia tensor j i j except I have subtracted off so that it is what mathematicians call it is trace lists.
The trace of this is I set I equal to j and add them up i x x plus i y y plus i z z. If there is an I that z. So that cake remember to some over that. So there is my trace Delta I j. This audience knows the LPGA. Surely that's your old friend Chronic or Delta. So it's zero everywhere. And Maci and Jay are the same. So if I set eye equal to J and I sum over it, you'll notice here I get I k k or I, you know, i j j and I take care of the same thing.
And then delta i j over three that when I take its trace, that goes to one. So j i j is trace list and that turns out to be what we want when we calculate the H is in general relativity. We want the trace list form. Here's a pretty much an exact equation which tells me how to go from this moment of inertia. The double dot means I take the second derivative as usual, and this is a very simple looking relationship between the geometry.
Of the gravitational wave and say the underlying moment of inertia tensor of a binary star. We're usually looking at merging black holes or something like that. And so that's how I go from one to the other. Here's a very famous formula that Einstein derived back in 1918, except he got it wrong. Einstein didn't write a paper on general I mean, on gravitational radiation without making a mistake. So if you find the subject a little confusing, you're in very good company.
So actually, the bit of historical note, the person who got this right was Arthur Eddington, an astrophysicist, and it went for three years. I notice Einstein had a ten there instead of a five. No one checked the math. Until Addington. And if you know something about Addington, it won't surprise you to learn that he rolled up his sleeves and went through to make sure he understood every damn line in that paper before he made use of it.
And he found the error. And much to his credit, he didn't say, Oh, Einstein got it completely wrong. You know, it's actually this. He just quoted the result. He said, Oh, you notice that actually should be a five instead of a ten, and that's it. So he was very generous. It's it's basically Einstein got all the hard stuff. Right. But you notice what's interesting here is there are three dots. So unlike in electromagnetism, it's the acceleration of the charges here.
It's the what's the word? Is it jerk for three derivatives? It's the jerk of the moment of inertia tensor that comes in. And you can do the same thing with gravitational I'm sorry, with angular momentum, which you need to worry about when you want to worry about how orbits change. That's a more complicated looking formula. You have two dots here. You have three dots here. You're summing over and here, but you have an eye and an amp.
And here. Oh, I mean, ask the audience what's the epsilon eye and what's the name of that symbol? Hmm. Totally symmetric goods. Well, that's right. The levitra beta. Right. Totally anti symmetric tensor. Very good form arms. So basically, if I am K is 1 to 3 or an even permutation of one, two, three, then it's plus one. If it's an odd permutation like three two then it's minus one if any two indices are the same and zero.
So it's a way to do a cross product. So it's a kind of a cross product between. So this is now how you can calculate the angular momentum loss. And that's. All. That's kind of. You should take a picture of that, put it in your wallet, because that's kind of a nice pocket sized edition of pretty much everything you Need if you want to understand general relativity in a practical sense.
It's the practical and the old. If this was like the 1930s, they would say The Practical Man's Guide to General Relativity. And it would be these equations. They're all you need to know. And they kind of burst upon the scene in 1974 when Hulse and Taylor found the binary pulsar. They found the system with two neutron stars, one of which was a pulsar. And you probably know pulsars send out very regular radio signals and they are fantastically accurate.
So they are a gift from nature to astrophysics because they take the most accurate clocks in the universe and they put them in relativistic systems for us. It just couldn't be better. And so the first such binary pulsar was discovered in 1974. So here you see the two orbiting around one another. And by following the arrival time of those pulsars, you could learn how the period of the orbit changes.
And if you remember your Kepler carrion mechanics, your laws of gravity, you know that the energy of the orbit can be written entirely in terms of the period. So here's actually what the orbit looks like. So my student has drawn this up. This is the current shape of the whole Stellar Pulsar. And we're going to go through 300 million years and 17 seconds. And here is the gravitational radiation carrying energy and angular momentum.
Notice that Perry, astronaut of that is hardly changing at the beginning. And then it starts to move. Then then faster and faster. And then right at the very end it goes very quickly and boom, there's coalescence. So that's how the shape of the orbit. It gets smaller, of course, because it's losing energy, but it's e centricity also goes from point six to rather large centricity down to a perfect circle at the same time because of the loss of gravitational radiation.
And here's kind of a colour coded picture in real time with equal time intervals between the two. You can see at the end it goes really very, very quickly. And more importantly, this is the shape. You can think of this pretty much. This is the cumulative shift of the power astron time. Think of it as the change in the orbital period. That's the best way. And then you see it decreasing with time. And this was the Discovery year 1974, and they followed it very, very closely.
And then they got the Nobel Prize right here. It looks suspicious to you. And then round about 2000, they really should just go back and make sure. So it was a of course, an epical discovery because gravitational radiation was real. Having covered that part of the talk is don't really have the time. But gravitational radiation was very controversial for most of its existence. People just wondered whether it was some kind of a mathematical artefact.
And there really was no such thing as an actual energy being carried off by gravitational waves anyway. All of that got laid to rest with this discovery. And then there's an even more amazing system that was discovered in 2004. Where we had not the binary but binary pulsars system with two pulsars in it, which is much closer, which is really nearly edge on and which had a very small eccentricity. And all of these combine the observers to be able to do fantastically accurate observations.
So here there's no gap because there's no Nobel Prize to be won. And so the coverage is very, very thorough and this is not a fit to the data. This is a prediction of general relativity, and that is the data. I haven't seen data that good in astrophysics since, you know, the cosmic microwave background. It is .01 3% agreement with the gravitational radiation formula. So it's a beautiful, beautiful result. So this is all indirect.
We're sort of looking at what happens to the orbits. What about the actual detection of gravitational radiation? What about these motions like that? Do we we ever see that? Yes, we do. They came in 2015 and it was quite a tour de force. So there are two interferometers. This is one logo. The is the acronym. It's in Hanford, Washington, four km long arms at 90 degrees. Think of the gravitational wave coming through gravitational waves, by the way.
They don't care about anything. They don't care about Earth. They don't care about planets. Gravitational waves penetrate apps. If you could use them, if you could generate them, they'd be perfect for communicating with submarines. You'd need a big interferometer to detect them. The practical difficulties. But are essentially zero absorption. And the idea here is that this is an interferometer. So the length of these arms changes.
And as I'll explain in a minute, the sort of precise cancellation of the optics gets changed when a gravitational wave passes by. So I think I'm going to well, yes, I'm going to pay attention to what's on the left here. So there's two of these in their original incarnation. There is Hanford, Washington, Livingston, Louisiana.
There's there are two interferometers that are set up. And the idea is that if we have a real gravitational wave, we should see them at each of these facilities about 10 milliseconds apart, which is the light travel time between the two. That's how you know, it's a real signal. This is simply a measure of kind of the sense of sensitivity. But the idea is a little bit easier to grasp in this diagram. So here we have a wave coming down. And the idea is that I have a laser beam.
It goes through a beam splitter. Part of it bounces back and forth between these very carefully designed mirrors, the test masses, and then the same thing on the right side. And then they recombine in the beam splitter and some of that is sent to the photodetector. If there's no gravitational wave, the experiment is set up so that there is precisely zero. There's destructive interference, is what I'm trying to say.
In other words, the two arms of the interference interferometer are exactly 180 degrees out of phase. If there is the slightest gravitational wave, the slightest separation. When I say slight, I mean 1% of the mass of a proton excuse me, the diameter of a proton. If I'm off by that much, I get a big signal that's easily detectable. So that is the accuracy that they can deal with. And you can imagine this is for kilometres.
They bounce them back and forth. They have an effective arm's length of like ten kilometres in every wiggle of. The laser light is precisely accounted for. They have that kind of phase coherence. That's the amazing thing about this experiment. And there's our old friend reminding you how it goes. So here's the idea. You remember how wave interference works. If I have to waves and praise, they add up to constructive interference.
If they are exactly 180 degrees, then a peak is aligned with a trough and I get utter destructive interference. And I've done a little bit of mathematics at the bottom, which you can do yourself if you remember your trigonometry. If I have a cosine omega t and I add to it a cosine immediately plus five, some kind of a phase difference. It turns out you can write the result in a convenient formula. The face different comes in as to cosine of pi over two times another cosine function.
And the way liger works. This phase difference will be here's the pi 180 degrees pi radians, 180 degrees out of phase. And so if x were zero, that would be complete cancellation. I put phi equals pi in this formula. Cosine pi over two is zero and then I have a tiny bit left over that is the gravitational wave. X doesn't have to be a constant here of course fact it won't be X itself can depend upon time.
No reason why we can't include that in the formula. And if you do the small x expansion, what comes out of there is x times sine omega t. So in other words, in Legault, this omega is the laser frequency, very, very, very large number. This x is also time dependent and its frequency will be measured in something like 2/10 of a second or one hundredths of a second. That's the range of ligo's sensitivity. So what I will see is the envelope of the laser light.
The envelope will be the gravitational wave. That's how it works. It's very simple idea. So here you see just a mathematical example. I worked out to show that in principle when X is equal to cosine t plus cosine of two t my laser frequency in this case I've written is $0.40 and you peel off. The actual gravitational wave from as the envelope of the carrier wave. And that's exactly what they did, and that's what was found. So here's the signal, pretty much almost raw.
This was an incredibly clean first time experiment. I think that's what blew everybody away. It's not something where you had to rely on the statisticians to be able to draw this out. You could practically just take it out and look at it. So this was the signal in Livingston, The signal and Hanford. Here they are superimposed with a ten millisecond delay. So they are absolutely right on top of one another.
This has exactly the gravitational wave form two merging black holes, which only ten years earlier we could not have calculated. Because it's too hard, even with a computer, to figure out how black holes actually merge into another black hole. That was done only relatively recently so we could get the wave form right through the entire pattern. It doesn't look very hard somehow, but I can't tell you how much work this this bit here was easy to do.
This bit, it was easy somehow to do. The transition required a huge amount of work. And. Just to remind you, here is an actual calculation of merging black holes. And what you're looking at in the diagram here is the H, the colour coded hxx1 of the coefficients that appears in the metric tensor and the black dots are the black holes. And they are I think they're set up on kind of a circular orbit in this particular problem and they are losing energy and angular momentum.
And this isn't real, honest to goodness, general relativistic calculation. There you see it's actually going from the kind of Newtonian like orbit to emerge black hole and then the actual. A black hole can have any odd shape. A black hole, as they say, has no hair.
And so a black hole will settle down to a static configuration and all the irregularities in the shape, which is what you get initially, they get radiated away as gravitational radiation, and if they get radiated away as gravitational radiation as part of the wave signal that's actually discovered. The latest. I'll conclude my talk here. There's another way to detect gravitational radiation. And this one is this one is pretty cool.
So this is called this isn't pulsar timing array. So what we have here is the schematic. Here's the earth. And there you see space being rippled. And I don't have to worry about time when I do gravitational waves. The h, x, x and h, y, y are present, but there's no h0x, which is what I would have if there was a time. So I can really think of this all occurring in space.
And this these undulations are the wave passing through and these are pulsars and they're sending their radio signals to the earth. And the idea is that I measure the time the pulsars are so accurate we can measure their periods to 17 significant figures like knowing the age of the universe to one second. So we can measure the kind of tiny H's that we're talking about here. And then the idea is that you have a bunch of pulsars, very accurately known periods all over the galaxies.
And here I have two pulsars and here is the yellow angle between them. I measure the change in the period due to the fact that the way that the pulsar signal has passed through a gravitational wave. And I have two pulsars and I correlate. So this introduces another idea, the idea of correlation. So if this one has a delay and this one has a delay, then that's positively correlated. If this one has an advance and this one has an advance, it's neg, it's positively correlated.
If this is delay in advance, it's negatively correlated. And you can imagine over time there might be no correlation. So I have all these pairwise correlations that I measure the yellow with blue, the yellow with red, all of these or I'm sorry, I'm doing it the wrong way. I'm actually measuring the correlation as a function of angles. So this would be one pairwise correlation. But red would be another pairwise correlation.
The Blue Star, it's the same pulsar, but two different. Two different other pulsars in two different angles. Amazingly enough, I can calculate how mathematically. What I just described in word. How good is the correlation as a function of the separation of my pulsar pairs? That's the trick there. I can't get into the details of exactly how you do that in this talk, but it's a beautiful result. So this is called the Hollings and Downs Curve.
This is the angle between the pulsars. It runs between zero and 180 degrees. And when it's zero and the oldest, the shape is what's important. The overall normalisation can change depending on exactly how you process the signal. But the shape of this is what's critical. It's a relatively simple function. And when the angle is zero, of course you get a maximum goes to negative and then it rises again. And in order to do this experiment, you need lots and lots and lots of pulsars.
And it was not possible to do that until like yesterday. This is a very recent result. And these are the first initial data that have come in from so-called nano grab. Experiments. That's a funny name. Nano gravity comes in because the the frequencies are nano hertz. And that sounds fast at first, but nano hertz means ten to the minus nine hertz or like 1000000000 seconds. So these are periods of decades or years.
So this takes a long time to go. These are very the legal results are one hundredths of a second, 2/10 of a second. These are wavelengths which are more like light years in size, distributed through the galaxy. And there's the data. Now, it's interesting because, of course, you are helped by a HELLING down curve which has been drawn through the data are not long ago data, but that's pretty good. And in fact, everybody believes that the signal is real.
It is a bit messy, but already I have learnt that this is now like a month or so old and the data are now getting better with time. So there's no question that they actually have something. And the question is what is this legal? It's like being in a restaurant and you hear individual conversations, individual sauces. This is like being in the restaurant and you hear the background. And so this is the harm and we'd like to know what is causing the background.
It is probably it is probably black holes that are merging in galaxy sized collisions because they will be giving off for most of their lifetime gravitational waves at these kinds of periods, years to decades. But they could be more exotic things like gravitational waves from the Big bang itself. And so that's what has people excited. So that's a completely different way of detecting gravitational waves. So to conclude. This is an unresolved background. Legault is individual sources.
The background probably consists of merging supermassive black holes in the centres of galaxies throughout the entire universe. We may we may see individual sources, perhaps from this pulsar timing array poke out as we get more and more sensitivity by getting more and more pulsars in the array. But we may and this is kind of the hope. Also be taken by surprise and learn about sources that we in fact hadn't anticipated at all.
All right. So I think I'm going to stop there. Thank you very much for your attention. I think we have a great.