1
00:00:12.140 --> 00:00:17.060
I am very glad to be here in Lindau for the third time
2
00:00:17.060 --> 00:00:25.060
and to have this opportunity of talking to you about problems that I have been working on recently.
3
00:00:25.340 --> 00:00:30.820
This time I would like to talk to you about the theory of gravitation.
4
00:00:30.980 --> 00:00:34.720
Rather different from my previous topics.
5
00:00:35.740 --> 00:00:43.740
You all know that a theory of gravitation was first put forward by Newton more than 200 years ago.
6
00:00:44.280 --> 00:00:47.320
Newton's theory was a very good theory.
7
00:00:47.820 --> 00:00:52.580
It survived unchallenged for more than 200 years.
8
00:00:53.120 --> 00:01:00.360
And then in the present century a new theory of gravitation was put forward by Einstein.
9
00:01:00.820 --> 00:01:06.200
Einstein's theory was connected with his principle of relativity
10
00:01:06.200 --> 00:01:15.460
and he showed how gravitation could be explained as an effect arising from the curvature of space and time.
11
00:01:16.520 --> 00:01:22.340
Einstein's theory was a very beautiful theory mathematically.
12
00:01:22.340 --> 00:01:32.760
And also it was found to be in good agreement with observation and so it became generally accepted in the world of science.
13
00:01:33.780 --> 00:01:38.340
Einstein's theory aroused an enormous scientific interest.
14
00:01:38.440 --> 00:01:45.420
Very many people worked very intensively on this theory for a good many years.
15
00:01:46.100 --> 00:01:52.420
And then the interest in the Einstein theory of gravitation rather died away.
16
00:01:52.420 --> 00:01:56.200
People found the equations difficult to work with.
17
00:01:56.200 --> 00:02:02.020
They found other subjects of interest, largely quantum theory.
18
00:02:02.260 --> 00:02:09.560
And for a while one did not hear so much about the Einstein theory in the world of science.
19
00:02:10.420 --> 00:02:19.400
But in recent times, since the war, there has been a revival of interest in the Einstein theory of gravitation
20
00:02:20.140 --> 00:02:24.540
and at present there are more and more people working on it.
21
00:02:25.080 --> 00:02:32.000
This revival of interest can be explained, I think, or accounted for by two reasons:
22
00:02:32.320 --> 00:02:38.700
Partly there have been new mathematical methods developed for dealing with it.
23
00:02:38.700 --> 00:02:48.800
And partly people have been continually getting new observations about the very distant parts of our universe.
24
00:02:49.220 --> 00:02:55.080
They have been getting these observations with the help of the very big telescopes, which are now available.
25
00:02:55.400 --> 00:02:59.940
And also with the help of the new technique of radio astronomy.
26
00:03:00.700 --> 00:03:05.840
So there is this revival of interest in Einstein's theory of gravitation.
27
00:03:05.840 --> 00:03:18.560
And as a consequence, there are now international conferences on gravitation, which are held regularly every two years.
28
00:03:19.340 --> 00:03:26.540
The third of these international conferences was held only last week near Paris.
29
00:03:27.560 --> 00:03:36.760
At these conferences people who are working on the theory of gravitation from all countries come and meet together
30
00:03:37.220 --> 00:03:45.660
and present reports on their resent researches and discuss the problems, which still remain to be solved.
31
00:03:46.960 --> 00:03:52.520
I think that the people who come to these conferences, the people who work on gravitation
32
00:03:52.520 --> 00:04:04.160
can very clearly be divided into three classes: There are the mathematicians, the physicists and the cosmologists.
33
00:04:04.680 --> 00:04:11.580
The mathematicians are concerned with getting exact solutions of Einstein's equations.
34
00:04:11.940 --> 00:04:16.020
They are interested in all kinds of exact solutions,
35
00:04:16.020 --> 00:04:22.460
independently of whether these solutions have anything to do with our actual world or not.
36
00:04:22.460 --> 00:04:24.660
(Laughter).
37
00:04:24.660 --> 00:04:29.300
The other two classes are concerned with the actual world.
38
00:04:29.920 --> 00:04:35.280
The physicists are concerned with studying the gravitational field
39
00:04:35.280 --> 00:04:41.700
as a physical field and finding out the physical effects of gravitational forces.
40
00:04:42.040 --> 00:04:48.160
And they hope to be able to detect these effects with their instruments.
41
00:04:49.120 --> 00:04:53.720
The cosmologists are concerned with the universe as a whole.
42
00:04:54.380 --> 00:05:00.040
They are dealing with what the universe is like at extremely large distances
43
00:05:00.040 --> 00:05:05.480
and their main problem is whether the universe is closed up or whether it is an open universe.
44
00:05:07.240 --> 00:05:13.180
I want to talk to you today only about the point of view of the physicist.
45
00:05:14.740 --> 00:05:18.480
For the physicist there is a fortunate circumstance
46
00:05:18.480 --> 00:05:24.880
in that one does not need to use the exact equations of the Einstein theory.
47
00:05:25.380 --> 00:05:27.860
One can work with a certain approximation.
48
00:05:28.420 --> 00:05:35.860
The effects of the gravitational field are attributed to a curvature in spacetime.
49
00:05:35.860 --> 00:05:42.000
And for the physicist one can count this curvature as extremely small.
50
00:05:42.000 --> 00:05:47.440
In the space of the physicist, this curvature is certainly extremely small
51
00:05:47.440 --> 00:05:53.100
and it is sufficient to work with the approximations of the Einstein theory
52
00:05:53.100 --> 00:05:57.860
applying to the case when the curvature of spacetime is extremely small.
53
00:05:57.860 --> 00:06:04.580
That would not do for the cosmologists, because this approximation would not be a good approximation,
54
00:06:04.580 --> 00:06:07.840
if one were concerned with extremely large distances.
55
00:06:07.840 --> 00:06:12.560
Distances comparable with the distances between disparent nebulae for instance.
56
00:06:12.560 --> 00:06:18.880
But for the distances, which interest the physicist this approximation is certainly a valid one.
57
00:06:20.480 --> 00:06:27.320
Now the exact equations of the Einstein theory is the equation (1) in these notes.
58
00:06:27.740 --> 00:06:32.400
I think a good many of you have the German translation of these notes with you,
59
00:06:32.400 --> 00:06:37.080
so that I need not write down all the equations but can just refer to them.
60
00:06:37.080 --> 00:06:41.400
And I will write down on the blackboard only the more important equations.
61
00:06:41.900 --> 00:06:48.780
If we take the approximate form of the Einstein theory, when it is applied to space,
62
00:06:48.780 --> 00:06:52.860
which is nearly flat, we have this as our basic equation.
63
00:06:55.460 --> 00:07:02.640
We have an equation, which involves a quantity h-mu-nu, which is introduced in this way:
64
00:07:03.720 --> 00:07:13.500
The exact theory of Einstein is based on a certain tensor, which is written like this, G with two suffices mu and nu.
65
00:07:14.160 --> 00:07:21.300
And mu and nu take on the four values 0, 1, 2, 3.
66
00:07:21.560 --> 00:07:27.520
This tensor describes the gravitational field, describes the curvature of spacetime.
67
00:07:27.520 --> 00:07:27.580
And it also fixes the system of coordinates.
This tensor describes the gravitational field, describes the curvature of spacetime.
68
00:07:27.580 --> 00:07:30.760
And it also fixes the system of coordinates.
69
00:07:31.680 --> 00:07:42.160
Now for space, which is nearly flat, this tensor differs only by a small quantity from its value for flat spacetime.
70
00:07:42.540 --> 00:07:49.640
For flat spacetime these different elements all have the values 1 or -1 or 0.
71
00:07:50.300 --> 00:08:00.600
And the differences from flat spacetime we denote by h-mu-nu and we count h-mu-nu as small and neglect quantities,
72
00:08:00.600 --> 00:08:03.080
which are of the second order of smallness.
73
00:08:04.420 --> 00:08:09.460
We then have this as our basic equation of the Einstein theory.
74
00:08:10.560 --> 00:08:18.340
The Laplacian operator, which is denoted by the symbol square applied to h-mu-nu.
75
00:08:19.660 --> 00:08:24.940
First I will write down all the equation and then explain it.
76
00:08:25.580 --> 00:08:35.160
dv-mu by dx-mu plus dv mu by dx mu equal 16 pi gamma rho-mu-nu.
77
00:08:35.160 --> 00:08:43.680
This rho-mu-nu is constructed from the tensor, which describes any matter which is present.
78
00:08:43.680 --> 00:08:49.800
And in particular, this rho vanishes when there is no matter.
79
00:08:51.220 --> 00:09:00.720
Gamma here is the gravitational constant and it counts as very small in the approximation with which we are working.
80
00:09:00.720 --> 00:09:10.220
This v-mu is a certain quantity, which is constructed from the first derivatives of h-mu-nu.
81
00:09:10.220 --> 00:09:15.640
And you will find the expression for v-mu written down in equation (3).
82
00:09:16.180 --> 00:09:19.160
And I need not say more about that.
83
00:09:19.160 --> 00:09:22.820
This is then our fundamental equation.
84
00:09:22.820 --> 00:09:33.220
Now for dealing with this equation people usually choose a system of coordinates, which makes this quantity v-mu vanish.
85
00:09:34.280 --> 00:09:37.660
It's quite a nice condition to impose on the coordinates.
86
00:09:37.660 --> 00:09:45.540
And when people are working with these coordinates, they say that they are working with harmonic coordinates.
87
00:09:46.020 --> 00:09:52.060
This harmonic condition on the coordinates, is one which is used very extensively
88
00:09:52.060 --> 00:10:01.840
and it results in a big simplification in the equation, because with these harmonic coordinates these two terms just vanish.
89
00:10:01.840 --> 00:10:06.160
And we are left with this equation with just those two terms.
90
00:10:07.100 --> 00:10:13.740
Now if we apply that equation to a region of space and time, where there is no matter present,
91
00:10:13.740 --> 00:10:24.600
we have this term also vanishing and we have just this equation left, square h-mu nu equals 0.
92
00:10:24.600 --> 00:10:30.160
And that is just the well-known equation for wave propagation,
93
00:10:30.160 --> 00:10:38.340
the equation, which we have for all kinds of fields when there are waves, which propagate with the speed of light.
94
00:10:39.880 --> 00:10:45.000
So that we can say that in this approximation of weak fields,
95
00:10:45.000 --> 00:10:52.260
the theory of Einstein leads to these waves in this quantity h-mu-nu.
96
00:10:53.040 --> 00:10:56.840
In those regions of space and time where there is no matter.
97
00:10:57.500 --> 00:11:05.840
Now an important feature of Einstein's theory is that it is valid for all systems of coordinates.
98
00:11:06.980 --> 00:11:11.500
We are working with a case when the field is weak.
99
00:11:12.000 --> 00:11:18.900
And the natural thing to do under those conditions is to work with a system of coordinates,
100
00:11:18.900 --> 00:11:22.760
which is approximately Cartesian.
101
00:11:22.760 --> 00:11:29.480
We cannot say that it is exactly Cartesian because there is still a little curvature in our space,
102
00:11:29.480 --> 00:11:34.380
which prevents one from giving a precise meaning to Cartesian coordinates.
103
00:11:34.380 --> 00:11:38.340
But still, we can take coordinates, which are approximately Cartesian
104
00:11:38.340 --> 00:11:44.220
and that is what we are doing when we introduce these quantities h-mu-nu.
105
00:11:44.220 --> 00:11:48.920
But even with these coordinates, which are approximately Cartesian
106
00:11:48.920 --> 00:11:57.260
and even with the harmonic condition there is still some arbitrariness left in our system of coordinates.
107
00:11:57.780 --> 00:12:03.920
And because of this arbitrariness which is still left in our system of coordinates,
108
00:12:04.400 --> 00:12:10.020
we cannot be very sure about the meaning of these waves.
109
00:12:10.020 --> 00:12:13.720
Whose existence is shown by this equation.
110
00:12:13.860 --> 00:12:19.920
We cannot be sure whether these waves are really something physical
111
00:12:19.920 --> 00:12:25.580
or whether they are just connected with our system of coordinates.
112
00:12:25.580 --> 00:12:32.760
Now that is really the main difficulty all the time when one is working with the Einstein theory.
113
00:12:32.760 --> 00:12:43.320
It is the difficulty of separating what is real and physical, from what depends simply on our system of coordinates.
114
00:12:43.880 --> 00:12:48.920
And that will be our main problem of discussion today.
115
00:12:49.760 --> 00:12:59.200
Now in order to fix our ideas rather more precisely, let us suppose that we have some actual physical problem.
116
00:12:59.880 --> 00:13:08.000
We have some masses coming together, perhaps even with high speeds, interacting with each other in some way.
117
00:13:08.000 --> 00:13:11.160
And we have gravitational forces between them.
118
00:13:11.160 --> 00:13:14.720
And we want to discuss exactly what happens.
119
00:13:14.720 --> 00:13:18.920
I shouldn't say exactly what happens, I should say we want to discuss what happens
120
00:13:18.920 --> 00:13:23.360
in this approximation of weak gravitational fields.
121
00:13:24.500 --> 00:13:29.380
We then have to look for a solution of this first equation here.
122
00:13:29.380 --> 00:13:34.740
Now solutions to that equation are quite familiar to physicists,
123
00:13:34.740 --> 00:13:41.440
because this equation itself is very similar to the equation, which we have in electrodynamics.
124
00:13:42.240 --> 00:13:49.600
We can look upon this right hand side as generating waves in this quantity h,
125
00:13:49.980 --> 00:13:56.640
in the same way as electric charges and currents generate electromagnetic waves.
126
00:13:57.280 --> 00:14:02.300
And from our familiarity with a solution of electromagnetic equations,
127
00:14:02.680 --> 00:14:08.360
we can immediately get information about the solution of this equation here.
128
00:14:08.900 --> 00:14:15.080
We know that there is a solution when we are given the value of rho.
129
00:14:15.080 --> 00:14:20.260
There is a solution in terms of retarded potentials.
130
00:14:20.260 --> 00:14:28.960
And this will be the solution, which is the important one - physically - if there are no incoming gravitational waves.
131
00:14:29.360 --> 00:14:37.820
The general solution one gets from this retarded solution by adding on to the retarded solution
132
00:14:37.820 --> 00:14:40.640
certain arbitrary incoming waves.
133
00:14:41.160 --> 00:14:46.240
But it will be mainly the retarded solution that I want to talk to you about.
134
00:14:46.800 --> 00:14:56.400
With this retarded solution we have the h-mu-nu at larger distances proportional to one over r.
135
00:14:56.400 --> 00:14:58.600
r being the distance.
136
00:14:59.340 --> 00:15:03.940
The h-mu-nu are rather like the electromagnetic potentials
137
00:15:03.940 --> 00:15:07.260
and to get something which corresponds to the electromagnetic field,
138
00:15:07.260 --> 00:15:14.560
something which we can count as the gravitational field, we must differentiate this h-mu-nu once.
139
00:15:15.180 --> 00:15:24.540
If we differentiate something, which is of the form of one over r for great distances, then we get two terms appearing:
140
00:15:24.540 --> 00:15:34.020
One term depending on one over r^2 and the other term depending on omega over r.
141
00:15:34.620 --> 00:15:43.300
I should say a term of the order of omega over r, where omega is the frequency of oscillations,
142
00:15:43.300 --> 00:15:47.060
which are occurring in this distribution of matter.
143
00:15:47.060 --> 00:15:55.120
For distances, which are not too large the one over r^2 term is the important term.
144
00:15:55.740 --> 00:16:01.920
That one over r^2 term gives you the Coulomb force in electrodynamics.
145
00:16:01.920 --> 00:16:06.640
And it gives you the Newtonian force in gravitation.
146
00:16:07.400 --> 00:16:16.160
But for much larger distances than that the omega over r term is the dominant term.
147
00:16:17.180 --> 00:16:21.100
And this term corresponds to waves.
148
00:16:21.100 --> 00:16:25.740
So that this will be the important term for our talk today.
149
00:16:25.740 --> 00:16:29.780
This term which dominates the solution at very large distances.
150
00:16:31.060 --> 00:16:38.500
Let us now fix our attention on the waves, which come out in one particular direction.
151
00:16:39.020 --> 00:16:45.740
Let us say the waves, which come out in the direction of the axis x3.
152
00:16:46.220 --> 00:16:48.460
Suppose this is the axis x3.
153
00:16:48.460 --> 00:16:54.320
And then we want to examine the solution of our field equation,
154
00:16:54.320 --> 00:17:00.860
four points out there where x3 has some large and positive value.
155
00:17:00.860 --> 00:17:04.100
And x1 and x2 are small.
156
00:17:04.760 --> 00:17:11.460
In this region out here, we shall have waves moving radially outward.
157
00:17:11.460 --> 00:17:16.160
And those will be the dominant part of our solution.
158
00:17:16.160 --> 00:17:28.220
To examine the solution in that region of space, we must put d by dx1 and d by dx2 equal to zero.
159
00:17:28.560 --> 00:17:36.000
But we must also put d by dx3 equal to minus d by dx0.
160
00:17:36.700 --> 00:17:44.700
There is an error in the paper, which has been distributed; this minus sign has been omitted so please insert it.
161
00:17:45.620 --> 00:17:54.940
We can no see what is the effect of putting in these conditions into the solution of our field equations.
162
00:17:57.120 --> 00:18:02.900
If we examine the harmonic conditions in that region of space and time,
163
00:18:02.900 --> 00:18:08.420
we get a set of equations, which is written down in the notes, equation (6).
164
00:18:09.320 --> 00:18:18.740
Now as I mentioned before, even with the harmonic conditions there is still some arbitrariness in our system of coordinates.
165
00:18:19.600 --> 00:18:25.920
So we can take this question, let us make a change in our system of coordinates,
166
00:18:25.920 --> 00:18:31.020
a change, which preserves the harmonic conditions.
167
00:18:31.420 --> 00:18:37.040
I don't want to make a general change in the coordinates, which is going to disturb the harmonic conditions.
168
00:18:37.040 --> 00:18:41.420
But I'm going to make a change, which preserves the harmonic conditions.
169
00:18:41.840 --> 00:18:46.940
And such a change is described by the equation (5) in the notes.
170
00:18:47.220 --> 00:18:52.620
Where a-mu is a field function which fixes the change.
171
00:18:53.020 --> 00:18:59.640
And a-mu must satisfy the wave equation, which is written immediately after equation (5).
172
00:19:00.680 --> 00:19:11.060
Well the effect of making this change is to bring in certain changes in all the ten quantities, h-mu-nu.
173
00:19:11.700 --> 00:19:14.100
And these changes are given in the notes.
174
00:19:14.540 --> 00:19:20.380
I don't need to describe them in detail, but I will just say what the important result is:
175
00:19:21.140 --> 00:19:34.880
We find that when we make this change in coordinates six of the h-mu-nu remain invariant
176
00:19:35.560 --> 00:19:43.740
and the other four of these ten quantities can get changed and they can be changed arbitrarily.
177
00:19:44.740 --> 00:19:49.580
Things, which can be changed arbitrarily when we make a change in our system of coordinates,
178
00:19:49.580 --> 00:19:52.420
cannot have any physical meaning.
179
00:19:52.420 --> 00:20:01.120
So that these four components of h-mu-nu which gets changed arbitrarily, will not have any physical significance.
180
00:20:01.980 --> 00:20:07.520
There are the six invariant ones and of these six invariant ones,
181
00:20:08.460 --> 00:20:13.780
four must be zero because of the harmonic conditions themselves.
182
00:20:14.700 --> 00:20:21.620
And that leaves only two, which are invariant and which are not restricted to be zero.
183
00:20:21.620 --> 00:20:33.580
Those two are the components h12 and h11 minus h22.
184
00:20:34.180 --> 00:20:40.680
These two components we may therefore expect to have a physical meaning.
185
00:20:40.980 --> 00:20:46.200
For waves which are moving in the direction of the axis x3.
186
00:20:47.060 --> 00:20:52.320
They are invariant under any transformations, which we can make,
187
00:20:52.320 --> 00:20:55.940
which preserve the harmonic conditions and they are not restricted to be zero.
188
00:20:56.500 --> 00:21:03.060
Well this means then that we should expect that we have these gravitational waves,
189
00:21:03.460 --> 00:21:09.800
which are physical and that we have these two kinds of polarisation
190
00:21:09.800 --> 00:21:14.060
for gravitational waves moving in the direction of the axis x3.
191
00:21:14.920 --> 00:21:20.940
The question remains: "Should these waves really be counted as something physical?"
192
00:21:20.940 --> 00:21:26.920
And that is rather to the question: "Do these waves carry energy?"
193
00:21:27.340 --> 00:21:34.160
So that brings us to the discussion of the question of the energy of the Einstein field of gravitation.
194
00:21:35.160 --> 00:21:45.140
For this discussion of energy one can set up a certain tensor or tensor density, T-mu-nu,
195
00:21:45.720 --> 00:21:53.300
which has the physical significance that its components are connected with stresses and energy density, momentum density.
196
00:21:54.060 --> 00:22:01.860
And one of its components, the T-0-0 component, can be interpreted as the energy density.
197
00:22:02.780 --> 00:22:10.400
One finds that this T-mu-nu added on to a suitable tensor describing the matter,
198
00:22:10.400 --> 00:22:15.900
satisfies the conservation law, which is written down in the notes there.
199
00:22:16.340 --> 00:22:25.960
So that if we define the energy density as T-0-0, we get exact conservation of energy.
200
00:22:26.700 --> 00:22:30.660
But there is some trouble with this little t.
201
00:22:31.340 --> 00:22:36.440
I talked about it as a tensor density, but it is not really a tensor density,
202
00:22:36.500 --> 00:22:40.780
it is something, which is called a pseudo tensor density.
203
00:22:40.780 --> 00:22:49.860
Because when we make a change in coordinates, it does not transform correctly to be a tensor density.
204
00:22:49.860 --> 00:23:01.440
And that means that if we use this T-0-0 as the energy density and we work out the energy in a certain region.
205
00:23:01.440 --> 00:23:07.480
Then if we make a change in our system of coordinates we shall get a different energy.
206
00:23:07.480 --> 00:23:13.180
Now energy ought to be something, which is physical, we want it to be a really physical thing.
207
00:23:13.180 --> 00:23:16.780
And it should be independent of our system of coordinates.
208
00:23:16.780 --> 00:23:23.000
This T-0-0 is really the best thing, which we can do for discussing energy density.
209
00:23:23.420 --> 00:23:27.020
And we have here a real difficulty.
210
00:23:27.580 --> 00:23:32.460
This difficulty has bothered people for very many years.
211
00:23:32.740 --> 00:23:42.440
And it has led to a procedure in practice, when people want to discuss energy in connection with the Einstein theory,
212
00:23:42.880 --> 00:23:46.560
they adopt some nice system of coordinates.
213
00:23:46.880 --> 00:23:53.000
And they assume that if the energy is calculated with this nice system of coordinates,
214
00:23:53.000 --> 00:23:55.900
the result will have some physical meaning.
215
00:23:56.560 --> 00:24:02.860
But that of course is not a very logical process, it's not logical at all and it is unreliable.
216
00:24:02.860 --> 00:24:09.320
And on account of that there has been much discussion for very many years
217
00:24:09.320 --> 00:24:14.760
as to whether these gravitational waves really do carry energy or not.
218
00:24:15.120 --> 00:24:18.480
Well with the development of the theory of gravitation,
219
00:24:18.480 --> 00:24:24.180
which has taken place in recent times, this question has been cleared up.
220
00:24:24.940 --> 00:24:28.720
One of the main lines of this recent development
221
00:24:28.720 --> 00:24:37.260
has been the expression of the equations of the Einstein theory in the Hamiltonian form.
222
00:24:38.020 --> 00:24:46.200
Now the Hamiltonian form of writing equations is a form, which has very great mathematical power.
223
00:24:46.720 --> 00:24:50.600
It was discovered more than 100 years ago by Hamilton,
224
00:24:50.600 --> 00:24:55.660
who worked it out simply because of the mathematical beauty connected with it.
225
00:24:55.660 --> 00:25:02.000
And Hamilton himself did not realise the great importance of his form of equations.
226
00:25:02.000 --> 00:25:12.460
But we see now that his form is really of fundamental importance in nature because his form of equations is the form,
227
00:25:12.460 --> 00:25:17.700
which lends itself naturally to a passage to the quantum theory.
228
00:25:18.100 --> 00:25:24.960
Just working from the Newtonian form of equations in motion, one has not got any good way of passing to the quantum theory.
229
00:25:24.960 --> 00:25:30.280
But working from the Hamiltonian form we have well defined rules,
230
00:25:30.280 --> 00:25:34.060
which have been applied successfully in very many cases
231
00:25:34.060 --> 00:25:41.720
for passing from any classical field theory or classical theory of particles to the corresponding quantum theory.
232
00:25:42.280 --> 00:25:47.020
A good deal of the recent interest in the theory of gravitation has been concerned with
233
00:25:47.020 --> 00:25:51.420
obtaining a quantum theory of gravitation.
234
00:25:51.820 --> 00:25:59.020
And for that purpose one must first put the classical theory into Hamiltonian form.
235
00:25:59.840 --> 00:26:07.640
Now, with the Hamiltonian form of the equations one deals with the state at a certain time.
236
00:26:07.640 --> 00:26:16.520
Now the state at a certain time means the state for all values of the coordinates x1, x2, x3,
237
00:26:16.900 --> 00:26:20.820
but for one particular value of the coordinate x0.
238
00:26:21.380 --> 00:26:31.000
Now you see when we discuss the state at a certain time, we are introducing a dissymmetry between the four coordinates.
239
00:26:31.000 --> 00:26:39.400
One of the great features of Einstein's theory, was the fact that we had asymmetry between the four coordinates,
240
00:26:39.400 --> 00:26:42.740
the three coordinates of space and the one time coordinate.
241
00:26:43.100 --> 00:26:52.540
And for a long time people were interested only in developing the Einstein theory in a form, which preserved this symmetry.
242
00:26:53.240 --> 00:26:57.880
It is just within the last few years that people have found,
243
00:26:57.880 --> 00:27:02.360
that they can get a lot of new results by departing from this symmetry
244
00:27:02.360 --> 00:27:09.460
and in particular by working with this concept of the state at a certain time.
245
00:27:09.460 --> 00:27:14.160
Where we go entirely away from this four-dimensional symmetry
246
00:27:14.160 --> 00:27:21.120
and we go back to the old idea of a three-dimensional world changing with a time coordinate.
247
00:27:22.060 --> 00:27:30.920
With the development of the Hamiltonian form, we get this work in which we destroy the four-dimensional symmetry.
248
00:27:30.920 --> 00:27:36.740
And of course in a way it's a pity to destroy the four-dimensional symmetry, everyone would agree with that.
249
00:27:36.740 --> 00:27:41.120
But there are these compensations that one has great mathematical power
250
00:27:41.120 --> 00:27:49.160
and one finds some new features of the equations, which are not so obvious when one keeps to the four-dimensional symmetry.
251
00:27:49.800 --> 00:28:00.060
With the Hamiltonian form one's dynamical variables are all paired off into dynamical coordinates and conjugate momenta.
252
00:28:01.060 --> 00:28:04.240
So far as concerns the gravitational field,
253
00:28:04.240 --> 00:28:12.920
we have the G-mu-nu for all values of x1, x2, x3 appearing as dynamical coordinates.
254
00:28:13.280 --> 00:28:21.780
And we have then momentum variables P-mu-nu appearing as the conjugates of these dynamical coordinates.
255
00:28:22.780 --> 00:28:33.260
Now one of the first things one found when one started to put the theory into Hamiltonian form one got a result,
256
00:28:33.260 --> 00:28:34.920
which was rather unexpected.
257
00:28:35.260 --> 00:28:46.060
Which was of the ten quantities G-mu-nu and their conjugate P-mu-nu,
258
00:28:46.840 --> 00:28:53.540
four of the P-mu-nu and their conjugates drop out from the Hamiltonian equations of motion.
259
00:28:53.540 --> 00:28:59.660
Namely these four: G-mu-0, P-mu-0.
260
00:29:00.080 --> 00:29:07.640
If one of these indices takes on the value 0, either one, either the first or the second because it's symmetrical,
261
00:29:07.640 --> 00:29:11.140
then we get these quantities here.
262
00:29:11.140 --> 00:29:14.900
And these quantities drop out from the Hamiltonian equations
263
00:29:14.900 --> 00:29:24.260
and we are left with Hamiltonian equations involving only the variables G-r-s, P-r-s.
264
00:29:24.820 --> 00:29:31.140
Now these roman letters r and s take on the values 1, 2 and 3
265
00:29:31.540 --> 00:29:38.180
and they are to be sharply distinguished from the Greek letters, which take on the values 0, 1, 2, 3.
266
00:29:38.680 --> 00:29:46.080
We have here just six G-r-s's and six P-r-s's instead of the ten G-mu-nu's and P-mu-nu's.
267
00:29:47.880 --> 00:29:51.500
And that means that with the Hamiltonian formulation,
268
00:29:51.500 --> 00:29:57.160
we start off expecting to have ten degrees of freedom for each point of space.
269
00:29:57.160 --> 00:30:05.600
But four of the degrees of freedom drop out and we're left with just six degrees of freedom for each point of space.
270
00:30:05.600 --> 00:30:16.140
And that is a big simplification and this simplification, which brings out the advantages of the Hamiltonian formalism.
271
00:30:17.220 --> 00:30:21.840
Now this simplification ought not to surprise one too much.
272
00:30:22.900 --> 00:30:32.500
One might have expected it if one just looked into what is really needed for describing the state at a certain time.
273
00:30:33.320 --> 00:30:41.460
The state at a certain time means the state for all regions of space for a certain value of x0.
274
00:30:42.300 --> 00:30:49.080
And that is to be pictured in spacetime as a three-dimensional hypersurface.
275
00:30:49.460 --> 00:30:52.220
The hypersurface x0 equals constant.
276
00:30:52.520 --> 00:30:56.660
Which is to be pictured as existing in four-dimensional spacetime.
277
00:30:57.820 --> 00:31:04.840
Now to describe such a hypersurface, we need only the six G-r-s's.
278
00:31:04.840 --> 00:31:11.940
They are sufficient to describe the geometry of the hypersurface and the coordinate system in the hypersurface.
279
00:31:11.940 --> 00:31:18.980
And these G-mu-0's are needed only to describe the relationship of this hypersurface to a neighbouring hypersurface.
280
00:31:19.520 --> 00:31:28.480
But if you're interested only in describing the state at one particular time, then we only need these six G-r-s's
281
00:31:28.480 --> 00:31:31.040
and we also need their dynamical conjugates.
282
00:31:32.500 --> 00:31:39.380
So that from rather general arguments, arguments of a geometrical and kinematical nature,
283
00:31:39.700 --> 00:31:46.400
one can see that these six degrees of freedom are all that is really necessary.
284
00:31:47.560 --> 00:31:56.300
If we ask this physical question, how should we set up the energy at a certain time,
285
00:31:56.900 --> 00:32:03.960
then it seems clear that this energy should not depend on any variables,
286
00:32:03.960 --> 00:32:08.300
which are not needed for describing the state at that time.
287
00:32:08.960 --> 00:32:14.340
So the energy at a certain time or the energy density in the region at a certain time
288
00:32:14.580 --> 00:32:19.220
should not depend on these variables G-mu-0, P-mu-0.
289
00:32:19.520 --> 00:32:28.360
Now if you look at the energy density given by the pseudo tensor, this T-0-0, that we had before,
290
00:32:28.360 --> 00:32:34.580
and you work it out, you see that T-0-0 does depend on G-mu-0.
291
00:32:35.600 --> 00:32:45.120
T-0-0 thus involves some quantities which are not really relevant for describing the state at a certain time.
292
00:32:45.560 --> 00:32:50.500
It involves certain things, which are concerned only with the coordinate system.
293
00:32:51.400 --> 00:32:57.360
Now that is quite a bad feature in this energy density, this pseudo energy density.
294
00:32:58.040 --> 00:33:05.120
And we can improve upon this feature by taking a modified expression for the energy density.
295
00:33:06.120 --> 00:33:11.660
We get a modified expression for the energy density by expressing this T-0-0
296
00:33:11.660 --> 00:33:21.260
in a suitable way and substituting for the G-mu-0's which occur in it, their values for flat spacetime.
297
00:33:21.260 --> 00:33:30.360
namely we substitute for G-0-0 the value -1 and for G-1-0, G-2-0, G-3-0, the values 0.
298
00:33:31.340 --> 00:33:36.480
By this procedure we can get an improved expression for the energy density.
299
00:33:37.140 --> 00:33:40.040
An improved expression, which I call W.
300
00:33:40.720 --> 00:33:45.100
Now W still depends on our system of coordinates.
301
00:33:45.100 --> 00:33:48.620
Although not so badly as T-0-0.
302
00:33:50.620 --> 00:33:54.320
It means that we have made some improvement,
303
00:33:54.800 --> 00:33:59.240
with regard to this difficulty of the dependence of the energy on the coordinate system.
304
00:33:59.980 --> 00:34:02.820
But there is still some trouble left.
305
00:34:03.280 --> 00:34:11.160
There is still a dependence of W on the three coordinates x1, x2, x3.
306
00:34:12.160 --> 00:34:17.060
And further, if we want to look at things from the physical point of view,
307
00:34:17.060 --> 00:34:23.240
we should consider W defined on a certain hypersurface in spacetime.
308
00:34:23.980 --> 00:34:31.720
And we may ask ourselves: "What happens if we make just a small deformation in this hypersurface?"
309
00:34:32.300 --> 00:34:36.040
A small deformation of the order of gamma or gravitational constant.
310
00:34:36.740 --> 00:34:39.520
Which is really an extremely small deformation.
311
00:34:40.160 --> 00:34:49.140
But we find that with this extremely small deformation W changes by a quantity of the same order of magnitude as itself.
312
00:34:53.020 --> 00:35:00.820
Well there is still this difficulty but there are some nice features about this expression for the energy density W.
313
00:35:02.080 --> 00:35:08.820
The gravitational part of this energy density can be divided into two terms.
314
00:35:08.820 --> 00:35:11.400
It falls very naturally into two terms.
315
00:35:11.400 --> 00:35:14.460
Which are given by equations (8) and (9).
316
00:35:15.800 --> 00:35:24.840
One of these terms, which I have written W suffice K, can be interpreted as kinetic energy
317
00:35:24.840 --> 00:35:28.300
because it is quadratic in the momentum variables p.
318
00:35:28.300 --> 00:35:32.160
It is quadratic and homogeneous in these momentum variables.
319
00:35:32.160 --> 00:35:35.820
And is just like any ordinary kinetic energy is in physics.
320
00:35:36.100 --> 00:35:39.800
So that can be very naturally interpreted as kinetic energy.
321
00:35:40.140 --> 00:35:44.260
The other term does not involve momentum variables at all.
322
00:35:45.080 --> 00:35:47.680
And we call that the potential energy.
323
00:35:48.320 --> 00:35:56.240
It is quadratic and homogenous in the field quantities, which we get by taking the first derivatives of the h's.
324
00:35:58.020 --> 00:36:02.660
So the gravitational part of the energy density divides into these two terms.
325
00:36:03.720 --> 00:36:12.740
The first of these terms is subject to an uncertainty, when we make a small deformation of the surface.
326
00:36:13.560 --> 00:36:18.480
But is not subject to any uncertainly, when we change the coordinates in the surface.
327
00:36:18.480 --> 00:36:22.600
It is of the correct tensor form with respect to the coordinates in the surface.
328
00:36:23.740 --> 00:36:28.460
The other part, the potential energy, is just the other way around.
329
00:36:28.880 --> 00:36:32.940
That behaves all right when we make a small deformation of the surface.
330
00:36:33.240 --> 00:36:37.940
But that gets disturbed when we change the coordinates in the surface.
331
00:36:39.860 --> 00:36:47.560
Well that is the situation with regard to this improved expression for the energy density.
332
00:36:47.840 --> 00:36:53.800
And that shows that there is still some uncertainly in the improved expression for the energy density.
333
00:36:53.800 --> 00:36:56.360
Depending on our system of coordinates.
334
00:36:56.700 --> 00:36:59.940
So that we are still in difficulties,
335
00:36:59.940 --> 00:37:05.260
with regard to the question of whether our gravitational waves really carry energy or not.
336
00:37:07.100 --> 00:37:12.620
However there is one example where these difficulties can be eliminated.
337
00:37:13.820 --> 00:37:22.240
And that is the example when we have waves moving only in one direction.
338
00:37:22.920 --> 00:37:31.780
If we apply these expressions for the energy density to the case when there are waves moving in only one direction.
339
00:37:32.320 --> 00:37:41.700
The direction of the axis x3, then we get the expressions written down by equations (12) and (13).
340
00:37:41.960 --> 00:37:44.640
That's what the potential and kinetic energy...(end).