Landau The Classical Theory of Fields

4. Auflage 2013
ISBN: 978-1-4832-9328-8
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, Band Volume 2, 417 Seiten

Reihe: COURSE OF THEORETICAL PHYSICS

Translated from the 6th Russian edition, this latest edition contains seven new sections with chapters on General Relativity, Gravitational Waves and Relativistic Cosmology, where Professor Lifshitz's interests lay. The text of the 3rd English edition has been thoroughly revised and additional problems inserted

Autoren/Hrsg.

Landau, L D

Inhaltsverzeichnis

1;Front Cover;1
2;The Classical Theory of Fields;4
3;Copyright Page;5
4;Table of Contents;6
5;Excerpts from the Prefaces to the First and Second Editions;10
6;Preface to the Fourth English Edition;11
7;Editor'S Preface to the Seventh Russian Edition;12
8;Notation;14
9;CHAPTER 1. THE PRINCIPLE OF RELATIVITY;16
9.1;1 Velocity of propagation of interaction;16
9.2;2 Intervals;18
9.3;3 Proper time;22
9.4;4 The Lorentz transformation;24
9.5;5 Transformation of velocities;27
9.6;6 Four-vectors;29
9.7;7 Four-dimensional velocity;37
10;CHAPTER 2. RELATIVISTIC MECHANICS;39
10.1;8 The principle of least action;39
10.2;9 Energy and momentum;40
10.3;10 Transformation of distribution functions;44
10.4;11 Decay of particles;46
10.5;12 Invariant cross-section;49
10.6;13 Elastic collisions of particles;51
10.7;14 Angular momentum;55
11;CHAPTER 3. CHARGES IN ELECTROMAGNETIC FIELDS;58
11.1;15 Elementary particles in the theory of relativity;58
11.2;16 Four-potential of a field;59
11.3;17 Equations of motion of a charge in a field;61
11.4;18 Gauge invariance;64
11.5;19 Constant electromagnetic field;65
11.6;20 Motion in a constant uniform electric field;67
11.7;21 Motion in a constant uniform magnetic field;68
11.8;22 Motion of a charge in constant uniform electric and magnetic field;70
11.9;23 The electromagnetic field tensor;75
11.10;24 Lorentz transformation of the field;77
11.11;25 Invariants of the field;78
12;CHAPTER 4. THE ELECTROMAGNETIC FIELD EQUATIONS;81
12.1;26 The first pair of Maxwell's equations;81
12.2;27 The action function of the electromagnetic field;82
12.3;28 The four-dimensional current vector;84
12.4;29 The equation of continuity;86
12.5;30 The second pair of Maxwell equations;88
12.6;31 Energy density and energy flux;90
12.7;32 The energy-momentum tensor;92
12.8;33 Energy-momentum tensor of the electromagnetic field;95
12.9;34 The virial theorem;99
12.10;35 The energy-momentum tensor for macroscopic bodies;101
13;CHAPTER 5. CONSTANT ELECTROMAGNETIC FIELDS;104
13.1;36 Coulomb's law;104
13.2;37 Electrostatic energy of charges;105
13.3;38 The field of a uniformly moving charge;107
13.4;39 Motion in the Coulomb field;109
13.5;40 The dipole moment;112
13.6;41 Multipole moments;113
13.7;42 System of charges in an external field;116
13.8;43 Constant magnetic field;117
13.9;44 Magnetic moments;119
13.10;45 Larmor's theorem;121
14;CHAPTER 6. ELECTROMAGNETIC WAVES;124
14.1;46 The wave equation;124
14.2;47 Plane waves;126
14.3;48 Monochromatic plane waves;130
14.4;49 Spectral resolution;134
14.5;50 Partially polarized light;135
14.6;51 The Fourier resolution of the electrostatic field;140
14.7;52 Characteristic vibrations of the field;141
15;CHAPTER 7. THE PROPAGATION OF LIGHT;145
15.1;53 Geometrical optics;145
15.2;54 Intensity;148
15.3;55 The angular eikonal;150
15.4;56 Narrow bundles of rays;152
15.5;57 Image formation with broad bundles of rays;157
15.6;58 The limits of geometrical optics;159
15.7;59 Diffraction;161
15.8;60 Fresnel diffraction;166
15.9;61 Fraunhofer diffraction;169
16;CHAPTER 8. THE FIELD OF MOVING CHARGES;174
16.1;62 The retarded potentials;174
16.2;63 The Lienard-Wiechert potentials;176
16.3;64 Spectral resolution of the retarded potentials;179
16.4;65 The Lagrangian to terms of second order;181
17;CHAPTER 9. RADIATION OF ELECTROMAGNETIC WAVES;186
17.1;66 The field of a system of charges at large distances;186
17.2;67 Dipole radiation;189
17.3;68 Dipole radiation during collisions;193
17.4;69 Radiation of low frequency in collisions;195
17.5;70 Radiation in the case of Coulomb interaction;197
17.6;71 Quadrupole and magnetic dipole radiation;204
17.7;72 The field of the radiation at near distances;206
17.8;73 Radiation from a rapidly moving charge;210
17.9;74 Synchrotron radiation (magnetic bremsstrahlung);214
17.10;75 Radiation damping;220
17.11;76 Radiation damping in the relativistic case;225
17.12;77 Spectral resolution of the radiation in the ultrarelativistic case;228
17.13;78 Scattering by free charges;232
17.14;79 Scattering of low-frequency waves;236
17.15;80 Scattering of high-frequency waves;238
18;CHAPTER 10. PARTICLE IN A GRAVITATIONAL FIELD;241
18.1;81 Gravitational fields in nonrelativistic mechanics;241
18.2;82 The gravitational field in relativistic mechanics;242
18.3;83 Curvilinear coordinates;245
18.4;84 Distances and time intervals;249
18.5;85 Covariant differentiation;252
18.6;86 The relation of the Christofiel symbols to the metric tensor;257
18.7;87 Motion of a particle in a gravitational field;259
18.8;88 The constant gravitational field;263
18.9;89 Rotation;269
18.10;90 The equations of electrodynamics in the presence of a gravitational field;270
19;CHAPTER 11. THE GRAVITATIONAL FIELD EQUATIONS;274
19.1;91 The curvature tensor;274
19.2;92 Properties of the curvature tensor;277
19.3;93 The action function for the gravitational field;283
19.4;94 The energy-momentum tensor;285
19.5;95 The Einstein equations;289
19.6;96 The energy-momentum pseudotensor of the gravitational field;295
19.7;97 The synchronous reference system;301
19.8;98 The tetrad representation of the Einstein equations;306
20;CHAPTER 12. THE FIELD OF GRAVITATING BODIES;310
20.1;99 Newton's law;310
20.2;100 The centrally symmetric gravitational field;314
20.3;101 Motion in a centrally symmetric gravitational field;321
20.4;102 Gravitational collapse of a spherical body;324
20.5;103 Gravitational collapse of a dustlike sphere;331
20.6;104 Gravitational collapse of nonspherical and rotating bodies;336
20.7;105 Gravitational fields at large distances from bodies;345
20.8;106 The equations of motion of a system of bodies in the second approximation;352
21;CHAPTER 13. GRAVITATIONAL WAVES;360
21.1;107 Weak gravitational waves;360
21.2;108 Gravitational waves in curved space-time;362
21.3;109 Strong gravitational waves;365
21.4;110 Radiation of gravitational waves;368
22;CHAPTER 14. RELATIVISTIC COSMOLOGY;373
22.1;111 Isotropie space;373
22.2;112 The closed isotropic model;377
22.3;113 The open isotropic model;381
22.4;114 The red shift;384
22.5;115 Gravitational stability of an isotropic universe;391
22.6;116 Homogeneous spaces;396
22.7;117 The flat anisotropic model;402
22.8;118 Oscillating regime of approach to a singular point;405
22.9;119 The time singularity in the general cosmological solution of the Einstein equations;409
23;INDEX;414

Leseproben

CHAPTER 1

THE PRINCIPLE OF RELATIVITY

Publisher Summary

This chapter presents the principle of relativity. According to this principle, all the laws of nature are identical in all inertial systems of reference. In other words, the equations expressing the laws of nature are invariant with respect to transformations of coordinates and time from one inertial system to another. This means that the equation describing any law of nature, when written in terms of coordinates and time in different inertial reference systems, has one and the same form. The combination of the principle of relativity with the finiteness of the velocity of propagation of interactions is called the principle of relativity of Einstein, formulated by Einstein in 1905, in contrast to the principle of relativity of Galileo, which was based on an infinite velocity of propagation of interactions. The idea of an absolute time is in complete contradiction to the Einstein principle of relativity. For this, it is sufficient to recall that in classical mechanics, based on the concept of an absolute time, a general law of combination of velocities is valid, according to which the velocity of a composite motion is simply equal to the (vector) sum of the velocities which constitute this motion. This law, being universal, should also be applicable to the propagation of interactions. The principle of relativity leads to the result that time is not absolute. Time elapses differently in different systems of reference.

§ 1 Velocity of propagation of interaction

For the description of processes taking place in nature, one must have a By a system of reference we understand a system of coordinates serving to indicate the position of a particle in space, as well as clocks fixed in this system serving to indicate the time.

There exist systems of reference in which a freely moving body, i.e. a moving body which is not acted upon by external forces, proceeds with constant velocity. Such reference systems are said to be

If two reference systems move uniformly relative to each other, and if one of them is an inertial system, then clearly the other is also inertial (in this system too every free motion will be linear and uniform). In this way one can obtain arbitrarily many inertial systems of reference, moving uniformly relative to one another.

Experiment shows that the so-called is valid. According to this principle all the laws of nature are identical in all inertial systems of reference. In other words, the equations expressing the laws of nature are invariant with respect to transformations of coordinates and time from one inertial system to another. This means that the equation describing any law of nature, when written in terms of coordinates and time in different inertial reference systems, has one and the same form.

The interaction of material particles is described in ordinary mechanics by means of a potential energy of interaction, which appears as a function of the coordinates of the interacting particles. It is easy to see that this manner of describing interactions contains the assumption of instantaneous propagation of interactions. For the forces exerted on each of the particles by the other particles at a particular instant of time depend, according to this description, only on the positions of the particles at this one instant. A change in the position of any of the interacting particles influences the other particles immediately.

However, experiment shows that instantaneous interactions do not exist in nature. Thus a mechanics based on the assumption of instantaneous propagation of interactions contains within itself a certain inaccuracy. In actuality, if any change takes place in one of the interacting bodies, it will influence the other bodies only after the lapse of a certain interval of time. It is only after this time interval that processes caused by the initial change begin to take place in the second body. Dividing the distance between the two bodies by this time interval, we obtain the

We note that this velocity should, strictly speaking, be called the velocity of propagation of interaction. It determines only that interval of time after which a change occurring in one body to manifest itself in another. It is clear that the existence of a maximum velocity of propagation of interactions implies, at the same time, that motions of bodies with greater velocity than this are in general impossible in nature. For if such a motion could occur, then by means of it one could realize an interaction with a velocity exceeding the maximum possible velocity of propagation of interactions.

Interactions propagating from one particle to another are frequently called “signals”, sent out from the first particle and “informing” the second particle of changes which the first has experienced. The velocity of propagation of interaction is then referred to as the

From the principle of relativity it follows in particular that the velocity of propagation of interactions is the in inertial systems of reference. Thus the velocity of propagation of interactions is a universal constant. This constant velocity (as we shall show later) is also the velocity of light in empty space. The velocity of light is usually designated by the letter and its numerical value is

=2.998×1010cm/sec. (1.1)

(1.1)

The large value of this velocity explains the fact that in practice classical mechanics appears to be sufficiently accurate in most cases. The velocities with which we have occasion to deal are usually so small compared with the velocity of light that the assumption that the latter is infinite does not materially affect the accuracy of the results.

The combination of the principle of relativity with the finiteness of the velocity of propagation of interactions is called the (it was formulated by Einstein in 1905) in contrast to the principle of relativity of Galileo, which was based on an infinite velocity of propagation of interactions.

The-mechanics based on the Einsteinian principle of relativity (we shall usually refer to it simply as the principle of relativity) is called In the limiting case when the velocities of the moving bodies are small compared with the velocity of light we can neglect the effect on the motion of the finiteness of the velocity of propagation. Then relativistic mechanics goes over into the usual mechanics, based on the assumption of instantaneous propagation of interactions; this mechanics is called or The limiting transition from relativistic to classical mechanics can be produced formally by the transition to the limit 8 in the formulas of relativistic mechanics.

In classical mechanics distance is already relative, i.e. the spatial relations between different events depend on the system of reference in which they are described. The statement that two nonsimultaneous events occur at one and the same point in space or, in general, at a definite distance from each other, acquires a meaning only when we indicate the system of reference which is used.

On the other hand, time is absolute in classical mechanics; in other words, the properties of time are assumed to be independent of the system of reference; there is one time for all reference frames. This means that if any two phenomena occur simultaneously for any one observer, then they occur simultaneously also for all others. In general, the interval of time between two given events must be identical for all systems of reference.

It is easy to show, however, that the idea of an absolute time is in complete contradiction to the Einstein principle of relativity. For this it is sufficient to recall that in classical mechanics, based on the concept of an absolute time, a general law of combination of velocities is valid, according to which the velocity of a composite motion is simply equal to the (vector) sum of the velocities which constitute this motion. This law, being universal, should also be applicable to the propagation of interactions. From this it would follow that the velocity of propagation must be different in different inertial systems of reference, in contradiction to the principle of relativity. In this matter experiment completely confirms the principle of relativity. Measurements first performed by Michelson (1881) showed complete lack of dependence of the velocity of light on its direction of propagation; whereas according to classical mechanics the velocity of light should be smaller in the direction of the earth’s motion than in the opposite direction.

Thus the principle of relativity leads to the result that time is not absolute. Time elapses differently in different systems of reference. Consequently the statement that a definite time interval has elapsed between two given events acquires meaning only when the reference frame to which this statement applies is indicated. In particular, events which are simultaneous in one reference frame will not be simultaneous in other frames.

To clarify this, it is instructive to consider the following simple example.

Let us look at two inertial reference systems...