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Overview
This is a systematic, covariant treatment of the classical theories of particle motion, fields, and the interaction of fields and particles. Particular attention is given to the interaction of charged particles with the electromagnetic field. The treatment throughout the book is relativistic; the author attempts, as much as possible, a coordinate free (or covariant) form of the equations both for particles and the fields. The book opens with an extensive discussion of space-time, Lorentz transformations, Lorentz-group and tensor and spinor fields. This material is essential to the understanding of many branches of theoretical physics, in particular relativistic quantum theory. Chapter II describes various relativistic forms of the fundamental problem of dynamics: describing the trajectories of particles for given external forces. The general dynamical principles to obtain the field equations and the important problems of the conservation laws are discussed in Chapter III. The second part of the book (Chapters IV-VI) is devoted to a lucid treatment of the interactions of fields and particles. Chapter IV deals with equations of motion and their solutions (the so-called Cauchy problem), focusing on the solution of field equations with Green's functions using Dirac formalism. The problem of feedback between particles and fields (radiation and radiation reaction) is taken up in Chapter V, as are questions concerning the limitations of classical field theories and classical dynamics. Dr. Barut concludes the book with an excellent exposition of the purely mechanical approach to the problem of the interactions of charged particles — the so-called action-at-a-distance formulation of electrodynamics. Problems, alternate proofs, and additional topics are included at the end of each chapter, where a useful bibliography is also provided. A general bibliography appears at the end of the book.
Product Details
| ISBN-13: | 9780486158716 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 04/02/2012 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 256 |
| File size: | 15 MB |
| Note: | This product may take a few minutes to download. |
Read an Excerpt
Electrodynamics and Classical Theory of Fields and Particles
By Asim O. Barut
Dover Publications, Inc.
Copyright © 1980 Asim O. BarutAll rights reserved.
ISBN: 978-0-486-15871-6
CHAPTER 1
LORENTZ TRANSFORMATIONS
1. The Physical Basis of the Lorentz Transformations
The physical laws governing the behavior of fields and particles are expressed in terms of space-time coordinates x, t and the functions of the coordinates. On the other hand, the physical laws describe permanencies in nature that are independent of any coordinate frame: a physical process will take place irrespective of what coordinate frame the observer may choose from which to observe it. The arbitrariness resulting from the choice of the coordinate frames must be eliminated from the formulation of the laws. Consequently, the transformations between different possible frames of reference and the quantities which are invariant under these transformations assume a fundamental importance. The physical laws will be written in such a way that their content and form is the same for a class of observers; for an objectively identical situation only the numerical values of observed quantities may change from observer to observer.
Frames of reference in which free particles move uniformly are called inertial frames of reference. This concept, formally, is the same both in classical mechanics and in the special theory of relativity.
In classical mechanics the correspondence between different inertial frames is expressed by the Galilean transformation law: two coordinate frames moving with respect to each other with the constant relative velocity w are related by
x' = Ox + wt, t' = t, (1.1)
where O is an arbitrary orthogonal transformation and expresses the relative position in space of the two coordinate frames at t = 0. The transformations, Eq. 1.1, which go hand in hand with Newton's laws, define the class of classical inertial frames. These frames are characterized therefore by the totality of the set (O, w) with w any vector, O any orthogonal transformation. Newton's laws are the same for all such inertial frames although the numerical values, say, of the coordinates of the particles are different:
Other or more general physical laws (or theories) may not be valid for the class of coordinate frames defined by Eq. 1.1. A difficulty was created for physics when an attempt was made to test the validity of light phenomena for all Galilean inertial frames defined by Eq. 1.1. In the case of the propagation of the electromagnetic waves it has been established experimentally that the velocity c of the propagation is constant and the same for all inertial frames, contrary to the Galilean rule of the addition of velocities as derived from Eq. 1.1. On the theoretical side, the phenomena of light propagation are correctly described by Maxwell's equations which are not valid for all Galilean inertial frames, or mathematically speaking, which are not invariant under the transformations Eq. 1.1.
This difficulty was solved by the special theory of relativity. The Lorentz transformations developed below will define a new class of inertial frames for electromagnetic phenomena and, because it is believed that the relations of physical phenomena to observers is the same for all physics, for all other phenomena until we are met with a new difficulty.
The study of the transformations, of the type of Eq. 1.1 adds a. new dimension to physical theories. On the one hand, one can study the allowed transformations for a given theory; on the other hand, one can try to construct, new theories which will allow a given set of transformations. All theories are associated with a set of transformation laws that define the class of observers for which the theory is valid. Relativistic theories of mechanics, hydrodynamics, thermodynamics, statistical mechanics, and so on, are all generalizations of the corresponding nonrelativistic theories with Eq. 1.1 replaced by the Lorentz transformations.
After this background we go on to the development of Lorentz transformations. Because the transformation laws, Eq. 1.1, are empirically valid in the low-velocity domain, the Lorentz transformations we are seeking must reduce to the form Eq. 1.1 in the limit of small velocities. To find the proper generalization, let us consider the example of light propagation in detail.
For a light signal propagating as a spherical wave the equation of the wave front is
c2t2 - x21 - x22 - x23 = 0.
The constancy of the velocity of light c implies now that in any other inertial frame to be defined, which at t = 0 coincided with the first, the equation of the wave front is
c2t'2 - x'21 - x'22 - x'23 = 0,
i.e. again a spherical wave with the same velocity c. This is only possible if t' ≠ t, because x'i ≠ xi. Thus, the transformation law (x', t') <-> (x, t) must be such that
x20 - r2 = 0 implies x'20 - r'2 = 0, and vice versa, (1.2)
where we have put x0 = ct. Furthermore, the transformations must have an inverse because we can interchange the roles of the coordinates (x, t) and (x', t') and map the origin into the origin, as we have assumed the two frames to be coinciding at t = 0. Consider the values of x0 and r such that x20 - r2 = a ≠ 0 (i.e., points not on the light front). The quantity a has the dimension of (length). For dimensional reasons x'20 - r'2 must be, therefore, a linear function of a, and by previous requirements actually equal to a. The transformations which are uniquely determined by the single relation
x20 - r2 = x'20 - r'2 = a, a real, (1.3)
are then the Lorentz transformations. We shall see that they will go over to Eq. 1.1 in the limit of small velocities. Eq. 1.3 also defines the set of inertial frames for electrodynamics and all special relativistic theories.
From an experimental point of view the constancy of the velocity of light is not enough to obtain Eq. 1.3. Two other experiments must be invoked to assert this, even for linear transformations. These are the Kennedy-Thorndike experiment showing the independence of the transit time of light on any closed path on the velocity of the earth, and the Ives-Stilwell experiment on the second-order Doppler shift, to be treated later.
One can also start, as did Lorentz and Poincaré,. from Maxwell's electrodynamics and see under what transformations these equations are invariant. One again arrives at Eq. 1.3. In any case, whether derived inductively or deductively, we take Eq. 1.3 as the defining equation of inertial frames. Correspondingly we define all transformations that leave the expression
x2 [equivalent to] x20 - x21 - x22 - x23
invariant as homogeneous Lorentz transformations. More generally theinhomogeneous Lorentz transformations are defined to be those which leave the expression
(x - y)2 [equivalent to] (x0 - y0)2 - (x1 - y1)2 - (x2 - y2)2 - (x3 - y3)2
invariant. These differ from the homogeneous transformations by a translation of space and time coordinates. The special theory of relativity has been introduced here in coordinate space, as was the case historically. One can equally well introduce it in momentum space starting from the frequency and propagation vector of light waves (see Sec. 4.C of this chapter).
A thorough knowledge of the properties of Lorentz transformations and their representations is necessary for the following chapters. The next sections are devoted mainly to these mathematical questions.
2. Mathematical Properties of the Lorentz Space
Notations
We introduce a 4-dimensional vector space of four-vectors x with components x0, x1, x2, x3, and define a real norm (or metric),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
and a scalar product of two four-vectors x and y,
xy [equivalent to] x0y0 - x1y1 - x2y2 - x3y3(1.5)
The vectors transformed under a Lorentz transformation L will be denoted by x' = Lx, y' = Ly, ··· From the postulated invariance of the norm; i.e., (Lx)2 = x2, we can deduce the invariance of the scalar product (1.5) under Lorentz transformations. For this purpose consider the vector x + y. Because
[L(x + y)]2 = (x + y)2,
we obtain, by squaring both sides,
x'y' = LxLy = xy.
There is a one-to-one correspondence between the vectors of the four-dimensional Euclidian space R4 and this Lorentz space L4.
Every four-vector x will have a positive definite Euclidian norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in R4 and an indefinite norm, Eq. 1.4, in L4. Eq. 1.4 can also be written within the framework of a positive definite metric as
x2 = (x, Gx); xy = (x, Gy), (1.4')
where G is the 4 x 4 matrix
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
with the matrix elements g00 = -gkk = 1, gμv = 0 if μ ≠ v. Thus we shall use, equivalently,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5')
where the so-called covariant components xμ of the coordinates are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The contravariant components of x are xμ.
These various notations for the norm and the scalar product are convenient and common. We must observe throughout the distinction between the covariant and the contravariant indices.
Vectors and Scalar Products
We divide the vectors x according to their norms into three groups:
x2 > 0, time-like vectors,
x2 = 0, light-like or null vectors,
x2< 0, space-like vectors.
All light-like points form the surface of a three-dimensional cone in the four-dimensional space, called the light cone. The light cone can be visualized as follows: a spherical light wave diminishing down to the origin (backward cone) and then expanding to infinity (forward cone), all, of course, with the velocity c. Vectors with x2 = a = constant lie on three-dimensional hyperboloids. The two-dimensional picture in Fig. 1 schematically illustrates these groups of vectors.
All time-like vectors are inside the light cone and the space-like ones are outside. The slope of the line connecting any two points indicates the velocity of the signals between the points. Thus space-like points cannot be reached from each other by signals with velocities equal or less than c. This is the so-called causality requirement which plays an important role in relativistic theories.
The invariant scalar product xy =(x, Gy) = xμyμ may be completely defined by the following two properties:
xy = yx, (αx + βy)z = αxz + βyz; α, β real numbers. (1.7)
We then obtain by simple algebra the following identities:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
Two four-vectors x and y are called orthogonal if xy = 0. Thus every light-like vector is orthogonal to itself. It follows from the above identities that the sum and the difference of any two light-like vectors are orthogonal and have equal and opposite norms. Two orthogonal light-like vectors are at most multiples of each other, for from xy = 0 and x2 = y2 = 0 it follows that x·y = (x2)1/2(y2)1/2; hence the spatial parts, and therefore the vectors, are proportional to each other. A simple way of constructing arbitrary orthogonal vectors is to take any two vectors of equal norm and form the sum and the difference of them.
We shall now prove some more useful relationships between the vectors.
(1) All vectors x orthogonal to a time-like vector z are space-like and form a three-dimensional space-like subspace, for from x0z0 = x·z, we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
because z is time-like, or, squaring both sides,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
hence x is space-like. By the same argument, every vector orthogonal to a light-like vector is space-like, except the previously mentioned multiple of the light-like vector itself. There are only two orthogonal space-like vectors, both orthogonal to the light-like vector. Obviously there are three mutually orthogonal space-like vectors. This completes the orthogonality properties of the four-vectors.
(2) Analogous to the decomposition of ordinary vectors into orthogonal components, we can represent every vector x in the form
x = y + τz, (1.9)
where z is a fixed time-like vector and y a unique space-like vector orthogonal to z determined by x, and τ is a real parameter. τ and y are then the components of x with respect to z. If z:(1, z) and y:(0, y), then τ is the time component of x. τ can be considered as an invariant time coordinate because the decomposition, Eq. 1.9, is invariant under Lorentz transformations. Similarly y will be the spatial, three-dimensional part of the four-vector z in invariant form.
(Continues...)
Excerpted from Electrodynamics and Classical Theory of Fields and Particles by Asim O. Barut. Copyright © 1980 Asim O. Barut. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
DOVER BOOKS ON PHYSICS,Title Page,
Copyright Page,
Dedication,
PREFACE TO THE DOVER EDITION,
PREFACE TO THE FIRST EDITION,
INTRODUCTION,
PART I - Relativistic Description of Fields and Particles,
I. - LORENTZ TRANSFORMATIONS,
II. - RELATIVISTIC DYNAMICS,
III. - RELATIVISTIC FIELD THEORY,
PART II - Interactions of Fields and Particles,
IV. - EQUATIONS OF MOTION AND THEIR SOLUTIONS,
V. - RADIATION AND RADIATION REACTION,
VI. - ACTION AT A DISTANCE ELECTRODYNAMICS,
RELATED MATHEMATICAL BOOKS,
AUTHOR INDEX,
SUBJECT INDEX,