1
00:00:00,250 --> 00:00:19,229
Some the. Gravitational waves and gravitational radiation is a rather new subject.
2
00:00:19,230 --> 00:00:23,160
It was a subject that didn't even exist ten years ago.
3
00:00:23,370 --> 00:00:30,750
The first direct evidence after kind of a century after its prediction came in 2016.
4
00:00:31,230 --> 00:00:41,879
And since then it's become almost an indispensable tool in astrophysics to learn about populations
5
00:00:41,880 --> 00:00:51,690
of stars and the early universe in ways that we really don't have any other accessible pathway.
6
00:00:51,720 --> 00:00:58,620
So I'm going to give you in the lead talk kind of an overview, remind you of what gravitational radiation is,
7
00:01:00,210 --> 00:01:09,390
how we figure out how much energy is in a gravitational wave, and talk about the methods of detection.
8
00:01:10,720 --> 00:01:15,130
But to begin with, let's talk about the languages of gravity.
9
00:01:15,160 --> 00:01:22,989
Gravity really is spoken in three different languages and like different languages.
10
00:01:22,990 --> 00:01:27,130
It's almost mutually incomprehensible to go from one to the other.
11
00:01:27,880 --> 00:01:30,880
So don't worry about the equations directly.
12
00:01:32,120 --> 00:01:39,860
They're just meant for decoration. If they if they mean something to you, so much the better.
13
00:01:40,100 --> 00:01:44,610
So I'll talk you through this. For many hundreds of years.
14
00:01:44,610 --> 00:01:49,020
The theory of gravity was, of course, Isaac Newton's theory of gravity.
15
00:01:49,350 --> 00:01:52,410
F equals GMR over R squared.
16
00:01:53,040 --> 00:01:58,469
And what you see here is simply a more rigorous form To write down.
17
00:01:58,470 --> 00:02:02,459
The potential energy of an assemblage of matter.
18
00:02:02,460 --> 00:02:10,650
Rho is the mass density, and you're adding up all the little bits of GM over R to get to a total potential.
19
00:02:11,220 --> 00:02:19,590
And this language of gravity is sufficient to talk about just about everything in astrophysics.
20
00:02:19,590 --> 00:02:27,960
This is the language of gravity that 99.9% of astrophysicists use in their day to day work.
21
00:02:28,560 --> 00:02:40,139
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,
22
00:02:40,140 --> 00:02:49,650
it instantly turns into a change in the gravitational force all over the universe in principle by hand.
23
00:02:50,220 --> 00:02:55,550
If a point mass moves, then the potential changes instantaneously everywhere.
24
00:02:55,560 --> 00:03:01,800
In other words, it doesn't incorporate the notion of causality or special relativity.
25
00:03:02,790 --> 00:03:10,199
And to make a theory compatible with relativistic ideas is no easy task.
26
00:03:10,200 --> 00:03:15,600
So that was first done by Albert Einstein in his general theory of relativity.
27
00:03:15,900 --> 00:03:19,799
The general theory of relativity is sometimes thought of as well.
28
00:03:19,800 --> 00:03:28,680
Special relativity tells you how to go from one reference frame to another reference frame, moving at a constant velocity general relativity.
29
00:03:28,830 --> 00:03:35,010
You know, it's more general what general relativity really is.
30
00:03:35,040 --> 00:03:39,700
It's a theory of gravity. And I'll talk a little bit more about that.
31
00:03:39,710 --> 00:03:46,130
But anyway, this is just the field equation for Einstein's theory of gravity.
32
00:03:46,610 --> 00:03:55,280
And we'll come back to this. And then the last one, yet more incomprehensible is what I refer to as and identify with.
33
00:03:55,280 --> 00:03:56,810
The name is Richard Feynman,
34
00:03:57,230 --> 00:04:07,130
in the sense that this is the kind of final step where we try to make gravity compatible not only with relativity and causality,
35
00:04:07,520 --> 00:04:10,250
but with the notions of quantum mechanics.
36
00:04:10,850 --> 00:04:19,850
So there are things like propagate as we talk about individual spins of particles, the little bits of gravity have spin two,
37
00:04:20,420 --> 00:04:34,730
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.
38
00:04:35,120 --> 00:04:38,120
And we certainly have no quantum theory of gravity yet.
39
00:04:38,130 --> 00:04:42,440
We don't even have a consensus on the best way to proceed.
40
00:04:43,870 --> 00:04:54,640
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.
41
00:04:55,360 --> 00:05:01,870
Now, of course, electromagnetism. And certainly in the history of the theory of gravity,
42
00:05:01,870 --> 00:05:07,810
we would practitioners would go back to electromagnetism to try to learn what they should be doing.
43
00:05:08,320 --> 00:05:12,520
And we have a model in front of us that we can copy.
44
00:05:13,060 --> 00:05:16,330
Electromagnetism also comes in three different languages.
45
00:05:16,750 --> 00:05:21,400
There is it's not Colombian, but Colombian.
46
00:05:22,780 --> 00:05:30,010
That is to say, cool Ohm's Law, where we talk about electrostatic potentials associated with charges.
47
00:05:30,550 --> 00:05:34,090
That's kind of the analogue of Newtonian gravity for electricity.
48
00:05:34,690 --> 00:05:44,410
And then the fully developed classical field theory, Maxwell's equations, the first equations in physics to be fully relativistic,
49
00:05:45,100 --> 00:05:52,660
tells us how to go from static configurations to pretty much any configuration where the charges are moving around.
50
00:05:53,140 --> 00:06:01,240
And then finally, in the 1940s, the final step was taking the find the fine money in step where we developed
51
00:06:01,240 --> 00:06:08,980
a quantum theory of fully relativistic quantum theory of spin one photons.
52
00:06:09,370 --> 00:06:21,520
And pretty much now any process whatsoever at any level which simply involves ordinary particles and photons, can be described to arbitrary accuracy.
53
00:06:21,550 --> 00:06:26,800
So in this case, we understand how to combine one, two and three seamlessly.
54
00:06:26,830 --> 00:06:31,570
So if there were no other reason for studying quantum gravity.
55
00:06:32,820 --> 00:06:35,940
Or for studying gravitational radiation.
56
00:06:35,940 --> 00:06:46,290
I should say gravitational radiation is absolutely key to our theoretical understanding of a complete theory of gravity,
57
00:06:46,290 --> 00:06:49,950
and that's reason enough to study gravitational radiation.
58
00:06:50,340 --> 00:07:00,180
But there's yet more because gravitational radiation, like electromagnetic radiation, turns out to have practical applications.
59
00:07:00,690 --> 00:07:08,700
I don't think you'll be paying commercial licensing fees for gravitational wave broadcasts anytime soon,
60
00:07:09,060 --> 00:07:15,540
but it is nevertheless a very useful and very interesting astrophysical tool.
61
00:07:18,170 --> 00:07:23,110
So this is a summary of what I've said. We have three languages I should call this.
62
00:07:23,120 --> 00:07:29,990
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.
63
00:07:30,500 --> 00:07:34,640
And we're at the stage where we understand 1 to 2.
64
00:07:35,180 --> 00:07:37,970
And then there's a big question mark with number three.
65
00:07:40,890 --> 00:07:48,630
So let's look at things in a little bit more detail and we'll see how the notion of gravitational radiation arises.
66
00:07:49,380 --> 00:07:58,770
So this is the Poisson equation for static potential theory, and this tells me how to compute my gravitational potential.
67
00:07:59,010 --> 00:08:02,550
I have a bunch of little I can think of them as point masses,
68
00:08:03,030 --> 00:08:11,570
and I add up all the little tiny g m over hours from all my constituent masses in my body.
69
00:08:11,580 --> 00:08:20,300
So here's the observer out here. Here is my origin and the r prime is a vector within the body itself.
70
00:08:20,310 --> 00:08:26,880
Capital R is between a particular point mass within the body and where I'm locating.
71
00:08:27,090 --> 00:08:33,630
And this is a relatively simple formula to compute the gravitational potential and from the gravitational potential,
72
00:08:33,990 --> 00:08:37,890
the gravitational force from any kind of configuration.
73
00:08:39,000 --> 00:08:41,280
Now, what about a time dependent theory?
74
00:08:42,780 --> 00:08:53,230
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.
75
00:08:53,260 --> 00:08:56,940
And that would be nice. And in fact, that works incredibly well.
76
00:08:57,150 --> 00:09:07,720
That's basically the way. People do calculations of things moving through galaxies and through evolving systems.
77
00:09:08,020 --> 00:09:16,900
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.
78
00:09:17,140 --> 00:09:21,010
Then instantaneously a distance capital are away.
79
00:09:21,340 --> 00:09:26,530
The gravitational field changes. And that works.
80
00:09:27,970 --> 00:09:33,580
In practice. Very well. But it can't be exactly correct.
81
00:09:35,460 --> 00:09:42,600
Because gravitational field simply can't propagate instantaneously across the universe.
82
00:09:43,860 --> 00:09:52,080
So we can take a big clue from studying how electric and magnetic fields work.
83
00:09:52,620 --> 00:09:56,520
And we use Maxwell's equations.
84
00:09:57,390 --> 00:10:05,280
So if we go back to the fundamental Maxwell in for equations,
85
00:10:05,280 --> 00:10:13,290
then it turns out we can always write the equation for the electromagnetic potential in this form.
86
00:10:13,290 --> 00:10:16,500
And you may recognise this sort of group of terms over here.
87
00:10:16,860 --> 00:10:19,920
This is the standard, quote wave equation.
88
00:10:21,050 --> 00:10:24,050
You have a second derivative with respect to time.
89
00:10:24,050 --> 00:10:31,670
And then this Dell squared is the partial derivatives, the second quarter partial derivatives with respect to space.
90
00:10:32,180 --> 00:10:34,879
And it's a simple linear equation.
91
00:10:34,880 --> 00:10:44,930
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.
92
00:10:45,470 --> 00:10:50,030
In fact, it's identical, except there's a prime here.
93
00:10:51,190 --> 00:10:57,550
Otherwise it's exactly the same mathematical form, but that little prime.
94
00:10:57,580 --> 00:11:04,410
And what does that? We have to come down here. So T prime is actually T minus.
95
00:11:04,420 --> 00:11:12,190
And now we have this retarded time, Capital R, which depends on our prime itself, divided by the speed of light.
96
00:11:12,700 --> 00:11:22,120
So the potential of time t depends on the superposition of what the source was in all its little individual bits.
97
00:11:22,690 --> 00:11:27,140
A time T minus are over c ago.
98
00:11:27,430 --> 00:11:34,040
And it's of course not the same r. For each point because they're located at different locations.
99
00:11:34,040 --> 00:11:37,670
It depends upon our prime. So suddenly it gets a lot more complicated.
100
00:11:38,210 --> 00:11:44,750
It looks simple when you write it down this way. But there's actually a huge amount of information that is hidden there.
101
00:11:46,180 --> 00:11:55,810
And that's what really happens. So just to make it more explicit now.
102
00:11:57,160 --> 00:12:01,120
This little black dot represents some kind of a particle in my body.
103
00:12:01,360 --> 00:12:11,530
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.
104
00:12:11,530 --> 00:12:17,890
T But at this point, if I want to calculate what the potential is at this distance, ah,
105
00:12:18,190 --> 00:12:26,620
I calculate the charge density at our prime at a time capital R oversea before the current
106
00:12:26,620 --> 00:12:31,990
time T and that'll be a different number for everywhere in the body gets more complicated.
107
00:12:34,300 --> 00:12:37,510
Let's see what the consequences of this are.
108
00:12:38,840 --> 00:12:43,580
So here is. A point charge.
109
00:12:44,090 --> 00:12:51,530
And these are the lines of force. And you notice the lines of force are all pointing directly to where the point charge is.
110
00:12:51,980 --> 00:13:00,290
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.
111
00:13:00,350 --> 00:13:04,040
I mean, what's what's going on? You can't get you can't make better slides, Professor.
112
00:13:04,550 --> 00:13:09,140
Well, I'm shooting a little bit. This is the actual diagram.
113
00:13:11,220 --> 00:13:14,470
And what we have here is a more complicated situation.
114
00:13:14,490 --> 00:13:19,140
This is a charge which has been sitting at this location for a while,
115
00:13:19,530 --> 00:13:25,829
and then it's accelerated up to this other location with these little points coming off.
116
00:13:25,830 --> 00:13:35,030
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
117
00:13:35,030 --> 00:13:41,810
meaning of that movement has been encoded in the actual electric field lines.
118
00:13:43,010 --> 00:13:48,890
So you notice the distant field lines. If I all draw them, they're all they haven't gotten the news yet.
119
00:13:49,660 --> 00:13:55,820
We are all oriented to where the charge was before it started accelerating.
120
00:13:57,340 --> 00:14:05,380
And then there is this transition zone during the acceleration process itself.
121
00:14:09,160 --> 00:14:19,780
Which forms a kind of a coherent structure unto itself and is moving outward, that transition zone.
122
00:14:20,080 --> 00:14:23,140
And then, of course, within the transition zone,
123
00:14:23,410 --> 00:14:30,220
the news has arrived as to where the charges and those field lines are pointed in a different direction.
124
00:14:33,610 --> 00:14:44,019
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.
125
00:14:44,020 --> 00:14:50,710
It causes this kink to appear, and that kink propagates outward at the speed of light.
126
00:14:50,950 --> 00:14:55,430
In fact, that's not even a good way to say it. That kink is like.
127
00:14:57,470 --> 00:15:04,000
That transverse kink is what our retina records as light.
128
00:15:04,130 --> 00:15:08,890
That's what excites our cells. That is light itself.
129
00:15:08,900 --> 00:15:18,340
It really is the effect. Light is the effect of that retarded time in the solution to the mathematical equations.
130
00:15:18,350 --> 00:15:22,690
That's what radiation is. And gravity.
131
00:15:23,100 --> 00:15:29,700
Gravitational radiation is actually very similar in its underlying principles.
132
00:15:31,490 --> 00:15:41,450
Now it's a little bit more complicated. Gravity is a theory, a geometrical theory, and Einstein's theory of gravity.
133
00:15:41,470 --> 00:15:44,570
The idea is that we live in a space.
134
00:15:44,590 --> 00:15:49,120
We live in a Minkowski space. And it's not a space.
135
00:15:50,190 --> 00:15:55,800
That's particularly intuitive, although we've lived in it all our lives pretty much.
136
00:15:56,250 --> 00:16:02,550
It's not a three dimensional Euclidean space. Looks like three dimensional Euclidean space, but you're being deceived.
137
00:16:03,150 --> 00:16:07,080
We live in four dimensions and one of the dimensions.
138
00:16:08,120 --> 00:16:16,429
When we try and compute, you know, the Pythagorean theorem C squared equals H squared plus B squared and so on.
139
00:16:16,430 --> 00:16:21,890
Well, you have to bring in a minus sign. So mathematicians love that kind of stuff.
140
00:16:23,550 --> 00:16:28,290
The rest of us are kind of wondering what does that actually mean?
141
00:16:28,290 --> 00:16:40,020
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.
142
00:16:40,410 --> 00:16:44,220
You know, it's this odd dimension, this thing that comes in with the minus sign.
143
00:16:44,550 --> 00:16:47,970
That's what our consciousness experiences as time.
144
00:16:47,970 --> 00:16:50,970
But we should be thinking of it as just another part of the space.
145
00:16:52,050 --> 00:16:58,560
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.
146
00:16:59,520 --> 00:17:06,060
And the other oddity here. This w w de w is normally huge.
147
00:17:06,090 --> 00:17:09,510
We do vast leaps of.
148
00:17:10,750 --> 00:17:19,030
Intervals in the W direction. When we do teeny tiny the axis as we sort of go through our existence.
149
00:17:19,750 --> 00:17:20,200
So.
150
00:17:22,200 --> 00:17:33,330
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.
151
00:17:33,690 --> 00:17:41,730
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.
152
00:17:42,830 --> 00:17:46,400
But that's me. That's the world we really live in.
153
00:17:47,000 --> 00:17:55,100
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.
154
00:17:55,500 --> 00:17:59,270
Well, yes, you do. You've lived in it your entire life.
155
00:18:00,200 --> 00:18:03,750
This is hyperbolic space. Get over it.
156
00:18:10,980 --> 00:18:19,810
So. The interesting thing about this space is that in Einstein's theory of gravity, gravity itself is not thought of as a force.
157
00:18:19,840 --> 00:18:25,540
There are other forces that are present electricity and magnetism, but gravity is not a force.
158
00:18:26,960 --> 00:18:33,350
Gravity is actually a distortion of that Minkowski space or curvature.
159
00:18:33,590 --> 00:18:40,310
As the mathematicians, I don't like the word curvature because you can have something that's curved like a cylinder.
160
00:18:40,970 --> 00:18:47,420
And it turns out the properties of a curved cylinder are pretty much exactly the same as a flat piece of paper.
161
00:18:47,600 --> 00:18:52,920
That's why you can turn one into the other. But that's the kind of term that's often used.
162
00:18:52,940 --> 00:18:59,210
It's really a specific type of distortion. The surface of a sphere is truly mathematically curved.
163
00:18:59,540 --> 00:19:08,390
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.
164
00:19:08,990 --> 00:19:12,860
So that's what gravity does. Gravity creates that kind of curvature.
165
00:19:13,930 --> 00:19:21,620
And so what that does mathematically is it changes the form of this space time interval.
166
00:19:21,640 --> 00:19:26,140
So what I put up here is how things change when you have a black hole.
167
00:19:26,710 --> 00:19:33,520
So six square deep squared acquires a coefficient to GM over RC squared.
168
00:19:34,240 --> 00:19:38,950
And then I'm going to switch from Cartesian now to spherical coordinates.
169
00:19:39,610 --> 00:19:45,010
So here is d r squared and that has the same term now in the denominator.
170
00:19:45,340 --> 00:19:55,090
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.
171
00:19:56,490 --> 00:20:00,870
And that's unchanged. But that is what gravity does.
172
00:20:00,870 --> 00:20:05,580
It takes that Minkowski space and it distorts it.
173
00:20:06,210 --> 00:20:13,800
And the amazing thing is you can recover all of Newtonian gravity in the right limit from this approach.
174
00:20:14,490 --> 00:20:18,540
Newtonian gravity doesn't go away in general relativity.
175
00:20:18,780 --> 00:20:26,610
It simply becomes ensconced in a more general, more beautiful theory.
176
00:20:27,030 --> 00:20:33,570
As Einstein said, it's the ultimate fate for a theory that's not quite correct.
177
00:20:33,960 --> 00:20:38,970
It's to find itself a home in some limit in a more general theory.
178
00:20:39,000 --> 00:20:41,370
And that's what happens with Newtonian gravity.
179
00:20:42,510 --> 00:20:48,780
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.
180
00:20:49,290 --> 00:20:56,460
So we demand that the difference between those two pieces of my interval, those two pieces of the metric,
181
00:20:56,790 --> 00:21:06,690
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.
182
00:21:06,720 --> 00:21:18,180
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.
183
00:21:21,060 --> 00:21:26,880
So how is this formalised? Here's the world of special relativity.
184
00:21:27,540 --> 00:21:33,180
We give this its own name. CE square, Deke House squared is this whole combination.
185
00:21:33,840 --> 00:21:39,629
And you notice that in cases where of course, where the X, Y, and Z are zero,
186
00:21:39,630 --> 00:21:45,630
if I happen to be moving along in my with my coordinate system so that I have only the change in time,
187
00:21:45,810 --> 00:21:52,410
but not the change in space, then detail and the T are the same thing in that case.
188
00:21:52,680 --> 00:21:56,220
So the power people like to think of as co moving time.
189
00:21:57,580 --> 00:22:02,810
And we write it this way mathematically. So the trick here is alpha and beta.
190
00:22:02,830 --> 00:22:10,510
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.
191
00:22:10,690 --> 00:22:14,710
When you do it, you have to worry about it. But we don't have to worry about it this morning.
192
00:22:15,610 --> 00:22:19,989
And a zero means time. The zero is CDP.
193
00:22:19,990 --> 00:22:24,370
And then one, two, three, simply refer to X, Y, and Z.
194
00:22:24,670 --> 00:22:30,700
And the rule is, if an index is repeated, then you sum over all values.
195
00:22:31,660 --> 00:22:36,910
That's the Einstein summation convention. Einstein got to be a little bit of a mathematician.
196
00:22:36,910 --> 00:22:44,080
Mathematicians didn't do that before Einstein. So Einstein's contribution was said, I'm not going to write the summation sign.
197
00:22:44,590 --> 00:22:50,110
If the index is repeated, then you sum over it, unless I tell you not to.
198
00:22:51,940 --> 00:22:58,419
But without. So that's the idea. And so this is a very compact way of writing this expression.
199
00:22:58,420 --> 00:23:02,450
And you can think of alpha and beta as a nice little matrix that looks like this.
200
00:23:02,470 --> 00:23:06,610
It's mostly zeros, except along the diagonal as shown.
201
00:23:07,390 --> 00:23:14,440
So that's the world of special relativity. Now, more generally, when matter is present.
202
00:23:16,660 --> 00:23:21,190
My C squared. The Tao is written this way.
203
00:23:21,760 --> 00:23:28,540
The notation people like to use is G. Alpha beta that is called the metric tensor.
204
00:23:28,540 --> 00:23:33,520
And that unlike ETA alpha, beta, eta alpha beta is very simple.
205
00:23:33,730 --> 00:23:42,460
G alpha beta can be in principle anything. Pretty much anything for you know, subject to.
206
00:23:43,680 --> 00:23:52,379
Certain smoothness, requirements and so forth. But think of it as anything, and it can be anything depending upon what coordinates I use.
207
00:23:52,380 --> 00:23:59,730
I can take Minkowski space which looks very simple and turn it into something that looks really ugly.
208
00:24:00,000 --> 00:24:06,479
If I use properly Spheroidal coordinates to describe it, which I'm allowed to do.
209
00:24:06,480 --> 00:24:07,740
That would be part of this.
210
00:24:08,780 --> 00:24:19,700
But it also can get complicated because the space itself is more complicated as well as difficult or more abstract curvature properties.
211
00:24:20,270 --> 00:24:24,770
So this is now the realm of general relativity by G alpha.
212
00:24:24,770 --> 00:24:29,060
Beta is the same, is it alpha beta? That's the world of special relativity.
213
00:24:29,480 --> 00:24:35,200
And here this morning for this lecture, it doesn't get too much more complicated than that.
214
00:24:35,210 --> 00:24:42,290
I'm going to let G. Alpha beta, equal eight, alpha, beta, plus a little tiny bit left over h, alpha beta.
215
00:24:42,680 --> 00:24:48,790
And that little tiny bit is going to be what gravitational radiation is.
216
00:24:48,800 --> 00:24:52,100
So that's the how we think of gravitational radiation.
217
00:24:52,430 --> 00:24:57,110
Gravitational radiation is causing flutters in Minkowski space.
218
00:24:58,050 --> 00:25:00,030
That's really the way to think of it.
219
00:25:00,940 --> 00:25:12,370
And when I apply my rule to find the minimum time distance between two intervals, that will tell me exactly how the particles move.
220
00:25:13,270 --> 00:25:17,000
So that's the world of gravitational radiation that we study.
221
00:25:18,040 --> 00:25:23,030
Now, happily. It turns out that those little ages.
222
00:25:24,050 --> 00:25:27,500
Those satisfy? Exactly. You can always write them.
223
00:25:27,620 --> 00:25:36,770
Choose your coordinates. It's always possible to write them in such a way that they satisfy exactly the wave equation That.
224
00:25:38,150 --> 00:25:45,800
The potential. And although I didn't say it, the victor potential also does in max rally in theory.
225
00:25:46,130 --> 00:25:54,020
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.
226
00:25:55,850 --> 00:26:03,440
But this is very nice because we can take a lot of what we know about solutions to the, quote wave equation.
227
00:26:04,060 --> 00:26:06,530
To give you just call this the wave equation.
228
00:26:06,890 --> 00:26:14,990
It gives it a rise pretty much almost any time you have some kind of a simple form of wave propagation, the wave equation.
229
00:26:15,890 --> 00:26:24,020
And we know that it has solutions. Here's a wave propagating in the Z direction, cosine a Z minus omega t.
230
00:26:24,380 --> 00:26:34,430
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.
231
00:26:34,640 --> 00:26:43,670
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.
232
00:26:45,710 --> 00:26:54,650
Now, the interesting thing is that when you have a wave propagating in the Z direction, almost all the H's are zero.
233
00:26:55,100 --> 00:27:00,260
That makes life simple. And there's only two.
234
00:27:01,360 --> 00:27:05,920
Independent agents that you need to worry about. There's h.
235
00:27:06,010 --> 00:27:15,630
X. X. And one mode has h x x equals minus h, y y, and nothing else.
236
00:27:15,960 --> 00:27:22,710
Two modes of propagation and then another motor propagation has only an h x y present.
237
00:27:22,710 --> 00:27:29,370
And that's the same as in h y x h is going to be a symmetric tensor in its indices.
238
00:27:29,370 --> 00:27:32,490
It doesn't matter what order you write them in and that's it.
239
00:27:34,270 --> 00:27:40,660
That is good enough to describe any kind of superposition of gravitational waves.
240
00:27:41,050 --> 00:27:45,790
You just need two different modes of polarisation.
241
00:27:46,180 --> 00:27:51,280
So what do they look like? They look. They look like this. Let's see if I can make this move.
242
00:27:53,180 --> 00:27:56,420
On my screen. Yeah.
243
00:28:00,730 --> 00:28:06,610
So if I have a ring of particles and a gravitational wave is passing through the screen.
244
00:28:07,650 --> 00:28:16,700
This is the way the forces would operate. They would squeeze in either this kind of sense.
245
00:28:16,710 --> 00:28:26,220
We call that the plus polarisation. It looks like a plus sign or one rotated by 45 degrees, which was the polarisation.
246
00:28:27,860 --> 00:28:38,180
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.
247
00:28:38,990 --> 00:28:45,680
So if I is equal to one, this is simply the x in the x direction and a wave comes by.
248
00:28:46,310 --> 00:28:51,640
This is how it changes. It has this interesting looking formula.
249
00:28:51,660 --> 00:28:54,990
It depends upon the separations in other directions.
250
00:28:56,090 --> 00:29:06,440
When away passes by. When I calculate what my new X separation is, and that's what gives rise to these.
251
00:29:07,530 --> 00:29:11,359
Funny. Sort of ten surreal motions.
252
00:29:11,360 --> 00:29:17,120
You squeeze along one axis and you expand along the axis at 90 degrees to it.
253
00:29:17,510 --> 00:29:23,730
That's what the effect of gravitational radiation is. And gravity can come in plane waves.
254
00:29:24,730 --> 00:29:35,570
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.
255
00:29:35,860 --> 00:29:43,719
Otherwise, it's very similar. And then we identify the plus polarisation I'm using now spherical coordinates.
256
00:29:43,720 --> 00:29:49,660
So I don't want to talk about X and X, but I talk about theta and theta and Phi and Phi,
257
00:29:49,660 --> 00:29:57,130
the two angles on the surface of a sphere theta being the CO latitude and Phi being the azimuthal angle.
258
00:30:01,180 --> 00:30:07,930
What about. Here's an interesting question now. What if? What about the energy in a gravitational wave?
259
00:30:08,350 --> 00:30:12,670
How do I compute something like that? How hard is it to bend space?
260
00:30:16,960 --> 00:30:22,990
Can't. Can't. I can't get a handle on it. So there's an interesting way to do that.
261
00:30:23,020 --> 00:30:25,510
Let's start with the wave equation itself.
262
00:30:26,380 --> 00:30:35,050
This box is called the deadline version operator, and it simply combines this del squared with the DX by the P square.
263
00:30:36,010 --> 00:30:44,499
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,
264
00:30:44,500 --> 00:30:51,010
alpha beta, d, t, and because alpha and beta are repeated, I sum over them.
265
00:30:51,190 --> 00:30:55,960
Remember, they're all zero except when alpha and beta are X or Y.
266
00:30:57,750 --> 00:31:04,590
And if I do that and I kind of, you know, play with my calculus, I integrate by parts.
267
00:31:04,770 --> 00:31:10,349
You do the things that you did when you learned your calculus. So, for example, the first term here,
268
00:31:10,350 --> 00:31:21,030
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.
269
00:31:21,480 --> 00:31:27,950
I can write the equation in this form, in that. May trigger something from your days.
270
00:31:29,060 --> 00:31:36,410
When you played with these kinds of equations in fluids or in electricity and magnetism,
271
00:31:37,190 --> 00:31:44,720
because that form of a of the wave equation, when I multiply, I take its first moment.
272
00:31:44,780 --> 00:31:52,070
That's what you describe what I've just done. This has the form of an energy conservation equation.
273
00:31:53,340 --> 00:31:58,770
It's a time derivative of a D by d t of some kind of a density.
274
00:32:00,020 --> 00:32:06,140
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.
275
00:32:06,590 --> 00:32:09,890
So it's still a plus the divergence of some kind of a flux.
276
00:32:11,140 --> 00:32:17,260
And so this is what turns out to be an energy density in a wave and.
277
00:32:18,330 --> 00:32:21,510
And energy flux. Now, there's one important difference.
278
00:32:22,080 --> 00:32:26,910
This is the zero over here. I can multiply this by any constant I choose.
279
00:32:27,060 --> 00:32:32,190
So I have kind of the form, but I don't know the overall constant.
280
00:32:32,580 --> 00:32:35,730
How do I get the constant? Well, that takes a little more work.
281
00:32:35,850 --> 00:32:42,390
So I won't do it here this morning. But what you need to do is keep those source turns on the right side.
282
00:32:43,260 --> 00:32:46,800
And then after you do that multiplication by D by d t,
283
00:32:46,830 --> 00:32:58,410
you can write the right hand side as the rate at which the gravitational forces do work back on their sources.
284
00:32:59,340 --> 00:33:06,900
So it's kind of the power lost. And by doing that, you can determine what the overall constant is.
285
00:33:07,640 --> 00:33:11,270
But just to get the form of these is a very, very simple operation.
286
00:33:11,690 --> 00:33:15,230
To get the precise constant, you have to do a little, little bit more work.
287
00:33:16,160 --> 00:33:22,190
So that, I think, is the best way to understand how you calculate the energy in a gravitational wave.
288
00:33:22,430 --> 00:33:26,400
And this is what it looks like constant turns out to be.
289
00:33:26,430 --> 00:33:33,809
Turns out a C to the fourth over 64 PI out in front and varies the energy flux with
290
00:33:33,810 --> 00:33:39,140
the minus sign retain the energy flux in particular has a nice very simple form.
291
00:33:39,440 --> 00:33:49,280
And if you plug in your nice cosine function's cosine k z minus omega three, you can show the way fluxes behave.
292
00:33:49,640 --> 00:33:55,700
The flux is really this energy density times the speed of light for a plane wave.
293
00:33:56,480 --> 00:34:08,300
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.
294
00:34:09,140 --> 00:34:21,560
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.
295
00:34:22,540 --> 00:34:30,820
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.
296
00:34:32,440 --> 00:34:36,790
For a simple sum of cosine and sine wave.
297
00:34:37,120 --> 00:34:44,740
These two terms contribute equally. So in fact, there's another 32 here that comes in and I can take one or the other.
298
00:34:45,340 --> 00:34:56,550
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.
299
00:34:58,100 --> 00:35:02,780
Then what people like to do. This is all written as a function of time.
300
00:35:03,380 --> 00:35:08,960
But you can also write it as a function of frequency like Fourier transforming it.
301
00:35:09,850 --> 00:35:13,750
And so cosmologists prefer that they like to use the frequencies.
302
00:35:14,920 --> 00:35:20,410
So this average value of this sum here is written in this way.
303
00:35:20,440 --> 00:35:27,429
Each of the D by d ts when you differentiate a cosine, the DBP brings frequency.
304
00:35:27,430 --> 00:35:30,550
That's what f is. And.
305
00:35:31,470 --> 00:35:37,260
Then all the rest of this is a some an integral over some other quantity.
306
00:35:37,560 --> 00:35:44,760
The noted S-H, which is known as the power spectral density, it's a basically the same thing.
307
00:35:44,760 --> 00:35:54,090
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.
308
00:35:54,420 --> 00:35:58,440
You know, you get omega is two pi times F,
309
00:35:58,920 --> 00:36:05,940
so you get some of that coming in and I'm integrating from zero to infinity because I just want to have positive frequency.
310
00:36:06,210 --> 00:36:11,790
So there's another factor of two. It's that kind of thing. So don't worry about this eight pi squared.
311
00:36:12,060 --> 00:36:22,290
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.
312
00:36:22,500 --> 00:36:26,490
So you'll see that a bit later. I wanted to touch upon it in my own talk.
313
00:36:29,860 --> 00:36:35,190
So for a sum of playing waves, we write back in the form row.
314
00:36:35,290 --> 00:36:40,810
GW So row it's kind of like a the equivalent mass.
315
00:36:41,980 --> 00:36:53,080
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.
316
00:36:53,380 --> 00:36:57,250
But I'm always going to write it in the form of rho GW times C squared.
317
00:36:57,550 --> 00:37:01,960
That's the energy density in the waves and that that's what we've just done.
318
00:37:02,290 --> 00:37:11,920
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.
319
00:37:12,760 --> 00:37:17,500
And as I say, don't worry about the details now you'll see it a second time.
320
00:37:18,250 --> 00:37:21,760
But when Barry talks about it, it won't be the first time that you're saying it.
321
00:37:23,160 --> 00:37:31,410
And so what I do now is I form a ratio when I'm interested in when I do cosmology is how much energy
322
00:37:31,410 --> 00:37:39,420
is there in the gravitational waves compared to the critical energy density kind of in our universe,
323
00:37:39,840 --> 00:37:43,170
which is a universe which is exactly flat.
324
00:37:44,010 --> 00:37:49,260
It's exactly between being a closed universe and an open universe.
325
00:37:49,740 --> 00:37:52,500
What is the ratio of those two things?
326
00:37:52,500 --> 00:38:02,490
How much gravitational wave energy is there relative to the critical energy density in our universe, which is given by this expression?
327
00:38:03,090 --> 00:38:06,270
H0 Is the Hubble constant, the rate of expansion.
328
00:38:07,240 --> 00:38:19,380
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.
329
00:38:20,010 --> 00:38:25,380
So think of it now is an integral now from zero up to some particular frequency.
330
00:38:25,890 --> 00:38:31,890
And I write down this ratio omega g as a function of F.
331
00:38:32,910 --> 00:38:38,280
So rather than row GW, I form this kind of combination of derivatives.
332
00:38:39,380 --> 00:38:43,850
So f times a d by the f, we'll give you something which is of order.
333
00:38:45,160 --> 00:38:49,570
The GW to begin with, but it now depends upon frequency.
334
00:38:49,960 --> 00:38:54,430
So I can ask the question how much energy is in this frequency band?
335
00:38:54,700 --> 00:38:59,170
So this is what this is measuring and this is the expression that you'll see later.
336
00:38:59,260 --> 00:39:03,070
Okay, I'm going to leave it there and come back to some.
337
00:39:04,650 --> 00:39:09,150
Other interesting applications of the energy and gravitational waves.
338
00:39:10,150 --> 00:39:14,380
It's actually pretty simple to calculate in practice. Here's what you do.
339
00:39:15,580 --> 00:39:19,420
You need to evaluate these moments of inertia.
340
00:39:19,420 --> 00:39:25,590
Ten steps. So I j i j here can be one, two or three.
341
00:39:26,940 --> 00:39:35,310
Usually we'll talk about iron J being X and Y, and the gravitational wave will go off in the Z direction.
342
00:39:36,590 --> 00:39:40,180
So here is a moment of inertia tensor Roe.
343
00:39:40,220 --> 00:39:45,560
So this is the energy, the mass density. There's the volume x, i j.
344
00:39:46,280 --> 00:40:00,050
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.
345
00:40:00,740 --> 00:40:10,460
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.
346
00:40:10,610 --> 00:40:16,650
If there is an I that z. So that cake remember to some over that.
347
00:40:16,700 --> 00:40:20,330
So there is my trace Delta I j.
348
00:40:21,590 --> 00:40:26,960
This audience knows the LPGA. Surely that's your old friend Chronic or Delta.
349
00:40:27,620 --> 00:40:31,280
So it's zero everywhere. And Maci and Jay are the same.
350
00:40:32,800 --> 00:40:44,320
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.
351
00:40:44,710 --> 00:40:50,050
And then delta i j over three that when I take its trace, that goes to one.
352
00:40:50,530 --> 00:40:59,649
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.
353
00:40:59,650 --> 00:41:06,580
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.
354
00:41:06,760 --> 00:41:17,110
The double dot means I take the second derivative as usual, and this is a very simple looking relationship between the geometry.
355
00:41:18,330 --> 00:41:25,770
Of the gravitational wave and say the underlying moment of inertia tensor of a binary star.
356
00:41:26,640 --> 00:41:30,000
We're usually looking at merging black holes or something like that.
357
00:41:30,600 --> 00:41:33,600
And so that's how I go from one to the other.
358
00:41:34,590 --> 00:41:41,400
Here's a very famous formula that Einstein derived back in 1918, except he got it wrong.
359
00:41:43,050 --> 00:41:48,630
Einstein didn't write a paper on general I mean, on gravitational radiation without making a mistake.
360
00:41:48,720 --> 00:41:53,970
So if you find the subject a little confusing, you're in very good company.
361
00:41:55,230 --> 00:42:06,040
So actually, the bit of historical note, the person who got this right was Arthur Eddington, an astrophysicist, and it went for three years.
362
00:42:06,060 --> 00:42:09,210
I notice Einstein had a ten there instead of a five.
363
00:42:09,810 --> 00:42:13,170
No one checked the math. Until Addington.
364
00:42:13,500 --> 00:42:19,350
And if you know something about Addington, it won't surprise you to learn that he rolled up his sleeves and went through to
365
00:42:19,350 --> 00:42:23,940
make sure he understood every damn line in that paper before he made use of it.
366
00:42:24,540 --> 00:42:30,980
And he found the error. And much to his credit, he didn't say, Oh, Einstein got it completely wrong.
367
00:42:30,990 --> 00:42:34,350
You know, it's actually this. He just quoted the result.
368
00:42:34,380 --> 00:42:39,690
He said, Oh, you notice that actually should be a five instead of a ten, and that's it.
369
00:42:41,250 --> 00:42:46,709
So he was very generous. It's it's basically Einstein got all the hard stuff.
370
00:42:46,710 --> 00:42:49,920
Right. But you notice what's interesting here is there are three dots.
371
00:42:51,060 --> 00:42:55,889
So unlike in electromagnetism, it's the acceleration of the charges here.
372
00:42:55,890 --> 00:43:00,660
It's the what's the word? Is it jerk for three derivatives?
373
00:43:01,110 --> 00:43:04,860
It's the jerk of the moment of inertia tensor that comes in.
374
00:43:05,580 --> 00:43:10,020
And you can do the same thing with gravitational I'm sorry, with angular momentum,
375
00:43:10,590 --> 00:43:14,070
which you need to worry about when you want to worry about how orbits change.
376
00:43:14,250 --> 00:43:18,000
That's a more complicated looking formula. You have two dots here.
377
00:43:18,000 --> 00:43:23,840
You have three dots here. You're summing over and here, but you have an eye and an amp.
378
00:43:24,030 --> 00:43:30,570
And here. Oh, I mean, ask the audience what's the epsilon eye and what's the name of that symbol?
379
00:43:32,980 --> 00:43:36,690
Hmm. Totally symmetric goods. Well, that's right.
380
00:43:36,690 --> 00:43:40,290
The levitra beta. Right. Totally anti symmetric tensor. Very good form arms.
381
00:43:41,130 --> 00:43:51,060
So basically, if I am K is 1 to 3 or an even permutation of one, two, three, then it's plus one.
382
00:43:51,390 --> 00:43:58,800
If it's an odd permutation like three two then it's minus one if any two indices are the same and zero.
383
00:43:59,610 --> 00:44:03,770
So it's a way to do a cross product. So it's a kind of a cross product between.
384
00:44:03,840 --> 00:44:07,140
So this is now how you can calculate the angular momentum loss.
385
00:44:09,340 --> 00:44:14,840
And that's. All. That's kind of. You should take a picture of that, put it in your wallet,
386
00:44:15,290 --> 00:44:20,659
because that's kind of a nice pocket sized edition of pretty much everything
387
00:44:20,660 --> 00:44:24,710
you Need if you want to understand general relativity in a practical sense.
388
00:44:24,890 --> 00:44:32,450
It's the practical and the old. If this was like the 1930s, they would say The Practical Man's Guide to General Relativity.
389
00:44:32,840 --> 00:44:36,920
And it would be these equations. They're all you need to know.
390
00:44:38,190 --> 00:44:48,620
And they kind of burst upon the scene in 1974 when Hulse and Taylor found the binary pulsar.
391
00:44:48,630 --> 00:44:53,880
They found the system with two neutron stars, one of which was a pulsar.
392
00:44:54,300 --> 00:45:03,300
And you probably know pulsars send out very regular radio signals and they are fantastically accurate.
393
00:45:04,110 --> 00:45:10,200
So they are a gift from nature to astrophysics because they take the most accurate
394
00:45:10,200 --> 00:45:14,460
clocks in the universe and they put them in relativistic systems for us.
395
00:45:14,550 --> 00:45:22,110
It just couldn't be better. And so the first such binary pulsar was discovered in 1974.
396
00:45:22,410 --> 00:45:25,110
So here you see the two orbiting around one another.
397
00:45:25,680 --> 00:45:34,640
And by following the arrival time of those pulsars, you could learn how the period of the orbit changes.
398
00:45:34,650 --> 00:45:40,229
And if you remember your Kepler carrion mechanics, your laws of gravity,
399
00:45:40,230 --> 00:45:45,600
you know that the energy of the orbit can be written entirely in terms of the period.
400
00:45:49,100 --> 00:45:53,280
So here's actually what the orbit looks like. So my student has drawn this up.
401
00:45:53,690 --> 00:45:57,950
This is the current shape of the whole Stellar Pulsar.
402
00:45:58,220 --> 00:46:02,600
And we're going to go through 300 million years and 17 seconds.
403
00:46:04,610 --> 00:46:08,690
And here is the gravitational radiation carrying energy and angular momentum.
404
00:46:08,960 --> 00:46:12,680
Notice that Perry, astronaut of that is hardly changing at the beginning.
405
00:46:13,660 --> 00:46:21,100
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.
406
00:46:21,760 --> 00:46:27,460
So that's how the shape of the orbit. It gets smaller, of course, because it's losing energy,
407
00:46:28,240 --> 00:46:35,020
but it's e centricity also goes from point six to rather large centricity down to
408
00:46:35,020 --> 00:46:40,300
a perfect circle at the same time because of the loss of gravitational radiation.
409
00:46:42,030 --> 00:46:48,480
And here's kind of a colour coded picture in real time with equal time intervals between the two.
410
00:46:48,510 --> 00:46:52,200
You can see at the end it goes really very, very quickly.
411
00:46:53,220 --> 00:46:57,840
And more importantly, this is the shape.
412
00:46:58,200 --> 00:47:02,940
You can think of this pretty much. This is the cumulative shift of the power astron time.
413
00:47:03,570 --> 00:47:07,379
Think of it as the change in the orbital period. That's the best way.
414
00:47:07,380 --> 00:47:14,790
And then you see it decreasing with time. And this was the Discovery year 1974, and they followed it very, very closely.
415
00:47:15,120 --> 00:47:20,550
And then they got the Nobel Prize right here. It looks suspicious to you.
416
00:47:21,480 --> 00:47:26,760
And then round about 2000, they really should just go back and make sure.
417
00:47:27,330 --> 00:47:35,790
So it was a of course, an epical discovery because gravitational radiation was real.
418
00:47:36,840 --> 00:47:44,100
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.
419
00:47:44,100 --> 00:47:48,360
People just wondered whether it was some kind of a mathematical artefact.
420
00:47:48,660 --> 00:47:54,480
And there really was no such thing as an actual energy being carried off by gravitational waves anyway.
421
00:47:54,720 --> 00:48:02,940
All of that got laid to rest with this discovery. And then there's an even more amazing system that was discovered in 2004.
422
00:48:04,670 --> 00:48:11,840
Where we had not the binary but binary pulsars system with two pulsars in it,
423
00:48:12,110 --> 00:48:19,040
which is much closer, which is really nearly edge on and which had a very small eccentricity.
424
00:48:19,280 --> 00:48:26,570
And all of these combine the observers to be able to do fantastically accurate observations.
425
00:48:26,930 --> 00:48:30,620
So here there's no gap because there's no Nobel Prize to be won.
426
00:48:31,280 --> 00:48:37,430
And so the coverage is very, very thorough and this is not a fit to the data.
427
00:48:38,060 --> 00:48:42,310
This is a prediction of general relativity, and that is the data.
428
00:48:42,320 --> 00:48:48,140
I haven't seen data that good in astrophysics since, you know, the cosmic microwave background.
429
00:48:48,230 --> 00:48:54,500
It is .01 3% agreement with the gravitational radiation formula.
430
00:48:54,530 --> 00:48:59,110
So it's a beautiful, beautiful result. So this is all indirect.
431
00:48:59,120 --> 00:49:05,930
We're sort of looking at what happens to the orbits. What about the actual detection of gravitational radiation?
432
00:49:06,620 --> 00:49:10,550
What about these motions like that? Do we we ever see that?
433
00:49:11,450 --> 00:49:18,290
Yes, we do. They came in 2015 and it was quite a tour de force.
434
00:49:19,480 --> 00:49:24,670
So there are two interferometers. This is one logo.
435
00:49:25,420 --> 00:49:31,480
The is the acronym. It's in Hanford, Washington, four km long arms at 90 degrees.
436
00:49:31,810 --> 00:49:35,350
Think of the gravitational wave coming through gravitational waves, by the way.
437
00:49:35,800 --> 00:49:38,950
They don't care about anything. They don't care about Earth.
438
00:49:38,970 --> 00:49:43,310
They don't care about planets. Gravitational waves penetrate apps.
439
00:49:43,500 --> 00:49:48,270
If you could use them, if you could generate them, they'd be perfect for communicating with submarines.
440
00:49:49,440 --> 00:49:52,890
You'd need a big interferometer to detect them. The practical difficulties.
441
00:49:53,340 --> 00:50:01,399
But are essentially zero absorption. And the idea here is that this is an interferometer.
442
00:50:01,400 --> 00:50:03,800
So the length of these arms changes.
443
00:50:04,190 --> 00:50:15,140
And as I'll explain in a minute, the sort of precise cancellation of the optics gets changed when a gravitational wave passes by.
444
00:50:16,310 --> 00:50:20,270
So I think I'm going to well, yes, I'm going to pay attention to what's on the left here.
445
00:50:20,660 --> 00:50:24,110
So there's two of these in their original incarnation.
446
00:50:24,500 --> 00:50:27,740
There is Hanford, Washington, Livingston, Louisiana.
447
00:50:28,190 --> 00:50:35,209
There's there are two interferometers that are set up. And the idea is that if we have a real gravitational wave,
448
00:50:35,210 --> 00:50:43,220
we should see them at each of these facilities about 10 milliseconds apart, which is the light travel time between the two.
449
00:50:43,430 --> 00:50:48,500
That's how you know, it's a real signal. This is simply a measure of kind of the sense of sensitivity.
450
00:50:48,980 --> 00:50:52,460
But the idea is a little bit easier to grasp in this diagram.
451
00:50:52,940 --> 00:50:57,930
So here we have a wave coming down. And the idea is that I have a laser beam.
452
00:50:57,950 --> 00:50:59,570
It goes through a beam splitter.
453
00:50:59,930 --> 00:51:12,290
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.
454
00:51:12,650 --> 00:51:18,830
And then they recombine in the beam splitter and some of that is sent to the photodetector.
455
00:51:19,670 --> 00:51:26,840
If there's no gravitational wave, the experiment is set up so that there is precisely zero.
456
00:51:27,530 --> 00:51:30,530
There's destructive interference, is what I'm trying to say.
457
00:51:31,010 --> 00:51:38,360
In other words, the two arms of the interference interferometer are exactly 180 degrees out of phase.
458
00:51:39,970 --> 00:51:44,830
If there is the slightest gravitational wave, the slightest separation.
459
00:51:45,070 --> 00:51:51,880
When I say slight, I mean 1% of the mass of a proton excuse me, the diameter of a proton.
460
00:51:52,810 --> 00:51:56,290
If I'm off by that much, I get a big signal that's easily detectable.
461
00:51:56,950 --> 00:52:02,040
So that is the accuracy that they can deal with. And you can imagine this is for kilometres.
462
00:52:02,050 --> 00:52:09,220
They bounce them back and forth. They have an effective arm's length of like ten kilometres in every wiggle of.
463
00:52:10,280 --> 00:52:13,430
The laser light is precisely accounted for.
464
00:52:13,760 --> 00:52:18,800
They have that kind of phase coherence. That's the amazing thing about this experiment.
465
00:52:22,490 --> 00:52:27,230
And there's our old friend reminding you how it goes. So here's the idea.
466
00:52:28,460 --> 00:52:34,550
You remember how wave interference works. If I have to waves and praise, they add up to constructive interference.
467
00:52:35,000 --> 00:52:41,870
If they are exactly 180 degrees, then a peak is aligned with a trough and I get utter destructive interference.
468
00:52:42,290 --> 00:52:49,310
And I've done a little bit of mathematics at the bottom, which you can do yourself if you remember your trigonometry.
469
00:52:49,340 --> 00:52:55,910
If I have a cosine omega t and I add to it a cosine immediately plus five, some kind of a phase difference.
470
00:52:56,420 --> 00:53:01,180
It turns out you can write the result in a convenient formula.
471
00:53:01,190 --> 00:53:07,400
The face different comes in as to cosine of pi over two times another cosine function.
472
00:53:07,700 --> 00:53:16,340
And the way liger works. This phase difference will be here's the pi 180 degrees pi radians, 180 degrees out of phase.
473
00:53:16,940 --> 00:53:21,070
And so if x were zero, that would be complete cancellation.
474
00:53:21,170 --> 00:53:29,780
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.
475
00:53:30,080 --> 00:53:36,110
X doesn't have to be a constant here of course fact it won't be X itself can depend upon time.
476
00:53:36,590 --> 00:53:47,209
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.
477
00:53:47,210 --> 00:53:53,750
So in other words, in Legault, this omega is the laser frequency, very, very, very large number.
478
00:53:54,650 --> 00:54:02,480
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.
479
00:54:02,960 --> 00:54:10,910
That's the range of ligo's sensitivity. So what I will see is the envelope of the laser light.
480
00:54:11,630 --> 00:54:15,560
The envelope will be the gravitational wave. That's how it works.
481
00:54:16,100 --> 00:54:20,929
It's very simple idea. So here you see just a mathematical example.
482
00:54:20,930 --> 00:54:27,560
I worked out to show that in principle when X is equal to cosine t plus cosine of
483
00:54:27,590 --> 00:54:33,110
two t my laser frequency in this case I've written is $0.40 and you peel off.
484
00:54:34,400 --> 00:54:40,550
The actual gravitational wave from as the envelope of the carrier wave.
485
00:54:41,090 --> 00:54:44,900
And that's exactly what they did, and that's what was found.
486
00:54:45,800 --> 00:54:48,980
So here's the signal, pretty much almost raw.
487
00:54:49,130 --> 00:54:52,490
This was an incredibly clean first time experiment.
488
00:54:52,530 --> 00:54:55,910
I think that's what blew everybody away.
489
00:54:56,600 --> 00:55:01,640
It's not something where you had to rely on the statisticians to be able to draw this out.
490
00:55:02,000 --> 00:55:05,190
You could practically just take it out and look at it.
491
00:55:05,210 --> 00:55:09,470
So this was the signal in Livingston, The signal and Hanford.
492
00:55:09,480 --> 00:55:12,800
Here they are superimposed with a ten millisecond delay.
493
00:55:13,160 --> 00:55:15,830
So they are absolutely right on top of one another.
494
00:55:16,190 --> 00:55:24,950
This has exactly the gravitational wave form two merging black holes, which only ten years earlier we could not have calculated.
495
00:55:25,840 --> 00:55:32,950
Because it's too hard, even with a computer, to figure out how black holes actually merge into another black hole.
496
00:55:34,970 --> 00:55:41,570
That was done only relatively recently so we could get the wave form right through the entire pattern.
497
00:55:41,990 --> 00:55:48,570
It doesn't look very hard somehow, but I can't tell you how much work this this bit here was easy to do.
498
00:55:48,590 --> 00:55:54,110
This bit, it was easy somehow to do. The transition required a huge amount of work.
499
00:55:56,300 --> 00:56:04,720
And. Just to remind you, here is an actual calculation of merging black holes.
500
00:56:05,170 --> 00:56:10,420
And what you're looking at in the diagram here is the H,
501
00:56:10,420 --> 00:56:20,800
the colour coded hxx1 of the coefficients that appears in the metric tensor and the black dots are the black holes.
502
00:56:21,250 --> 00:56:31,240
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.
503
00:56:31,540 --> 00:56:35,320
And this isn't real, honest to goodness, general relativistic calculation.
504
00:56:35,640 --> 00:56:45,280
There you see it's actually going from the kind of Newtonian like orbit to emerge black hole and then the actual.
505
00:56:46,370 --> 00:56:49,489
A black hole can have any odd shape.
506
00:56:49,490 --> 00:56:51,770
A black hole, as they say, has no hair.
507
00:56:52,370 --> 00:57:00,920
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,
508
00:57:01,250 --> 00:57:04,430
they get radiated away as gravitational radiation,
509
00:57:04,790 --> 00:57:11,420
and if they get radiated away as gravitational radiation as part of the wave signal that's actually discovered.
510
00:57:13,800 --> 00:57:17,420
The latest. I'll conclude my talk here.
511
00:57:18,080 --> 00:57:22,730
There's another way to detect gravitational radiation. And this one is this one is pretty cool.
512
00:57:23,720 --> 00:57:29,780
So this is called this isn't pulsar timing array. So what we have here is the schematic.
513
00:57:29,810 --> 00:57:33,140
Here's the earth. And there you see space being rippled.
514
00:57:33,620 --> 00:57:36,860
And I don't have to worry about time when I do gravitational waves.
515
00:57:37,100 --> 00:57:45,739
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.
516
00:57:45,740 --> 00:57:48,500
So I can really think of this all occurring in space.
517
00:57:48,950 --> 00:57:57,620
And this these undulations are the wave passing through and these are pulsars and they're sending their radio signals to the earth.
518
00:57:58,370 --> 00:58:03,439
And the idea is that I measure the time the pulsars are so accurate we can measure their
519
00:58:03,440 --> 00:58:08,840
periods to 17 significant figures like knowing the age of the universe to one second.
520
00:58:10,460 --> 00:58:14,480
So we can measure the kind of tiny H's that we're talking about here.
521
00:58:15,260 --> 00:58:24,470
And then the idea is that you have a bunch of pulsars, very accurately known periods all over the galaxies.
522
00:58:25,100 --> 00:58:29,299
And here I have two pulsars and here is the yellow angle between them.
523
00:58:29,300 --> 00:58:41,600
I measure the change in the period due to the fact that the way that the pulsar signal has passed through a gravitational wave.
524
00:58:42,320 --> 00:58:45,870
And I have two pulsars and I correlate.
525
00:58:45,890 --> 00:58:49,850
So this introduces another idea, the idea of correlation.
526
00:58:50,240 --> 00:58:56,510
So if this one has a delay and this one has a delay, then that's positively correlated.
527
00:58:56,510 --> 00:59:01,280
If this one has an advance and this one has an advance, it's neg, it's positively correlated.
528
00:59:01,970 --> 00:59:05,780
If this is delay in advance, it's negatively correlated.
529
00:59:05,990 --> 00:59:14,990
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,
530
00:59:14,990 --> 00:59:19,890
the yellow with red, all of these or I'm sorry, I'm doing it the wrong way.
531
00:59:19,910 --> 00:59:23,229
I'm actually measuring the correlation as a function of angles.
532
00:59:23,230 --> 00:59:28,580
So this would be one pairwise correlation. But red would be another pairwise correlation.
533
00:59:28,970 --> 00:59:35,520
The Blue Star, it's the same pulsar, but two different. Two different other pulsars in two different angles.
534
00:59:36,530 --> 00:59:40,910
Amazingly enough, I can calculate how mathematically.
535
00:59:42,160 --> 00:59:49,180
What I just described in word. How good is the correlation as a function of the separation of my pulsar pairs?
536
00:59:49,840 --> 00:59:57,940
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.
537
00:59:58,540 --> 01:00:01,990
So this is called the Hollings and Downs Curve.
538
01:00:02,620 --> 01:00:07,660
This is the angle between the pulsars. It runs between zero and 180 degrees.
539
01:00:08,870 --> 01:00:13,459
And when it's zero and the oldest, the shape is what's important.
540
01:00:13,460 --> 01:00:18,620
The overall normalisation can change depending on exactly how you process the signal.
541
01:00:18,890 --> 01:00:23,150
But the shape of this is what's critical. It's a relatively simple function.
542
01:00:23,420 --> 01:00:28,430
And when the angle is zero, of course you get a maximum goes to negative and then it rises again.
543
01:00:29,930 --> 01:00:34,980
And in order to do this experiment, you need lots and lots and lots of pulsars.
544
01:00:35,000 --> 01:00:38,390
And it was not possible to do that until like yesterday.
545
01:00:38,900 --> 01:00:48,470
This is a very recent result. And these are the first initial data that have come in from so-called nano grab.
546
01:00:50,400 --> 01:00:58,200
Experiments. That's a funny name. Nano gravity comes in because the the frequencies are nano hertz.
547
01:00:58,230 --> 01:01:05,250
And that sounds fast at first, but nano hertz means ten to the minus nine hertz or like 1000000000 seconds.
548
01:01:05,910 --> 01:01:11,010
So these are periods of decades or years.
549
01:01:12,440 --> 01:01:19,970
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.
550
01:01:20,510 --> 01:01:25,650
These are wavelengths which are more like light years in size, distributed through the galaxy.
551
01:01:26,330 --> 01:01:30,710
And there's the data. Now, it's interesting because, of course,
552
01:01:31,280 --> 01:01:43,010
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.
553
01:01:43,940 --> 01:01:47,420
And in fact, everybody believes that the signal is real.
554
01:01:48,110 --> 01:01:57,820
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.
555
01:01:57,830 --> 01:02:03,220
So there's no question that they actually have something. And the question is what is this legal?
556
01:02:03,620 --> 01:02:09,620
It's like being in a restaurant and you hear individual conversations, individual sauces.
557
01:02:10,220 --> 01:02:20,480
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.
558
01:02:20,990 --> 01:02:30,139
It is probably it is probably black holes that are merging in galaxy sized collisions because they
559
01:02:30,140 --> 01:02:35,629
will be giving off for most of their lifetime gravitational waves at these kinds of periods,
560
01:02:35,630 --> 01:02:42,650
years to decades. But they could be more exotic things like gravitational waves from the Big bang itself.
561
01:02:43,220 --> 01:02:50,600
And so that's what has people excited. So that's a completely different way of detecting gravitational waves.
562
01:02:52,800 --> 01:02:58,670
So to conclude. This is an unresolved background.
563
01:02:58,880 --> 01:03:01,370
Legault is individual sources.
564
01:03:01,400 --> 01:03:09,740
The background probably consists of merging supermassive black holes in the centres of galaxies throughout the entire universe.
565
01:03:10,160 --> 01:03:13,610
We may we may see individual sources,
566
01:03:13,610 --> 01:03:24,170
perhaps from this pulsar timing array poke out as we get more and more sensitivity by getting more and more pulsars in the array.
567
01:03:24,830 --> 01:03:34,760
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.
568
01:03:35,420 --> 01:03:39,290
All right. So I think I'm going to stop there. Thank you very much for your attention.
569
01:03:39,380 --> 01:03:40,280
I think we have a great.