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Overview
This concise account of special relativity is geared toward nonspecialists and belongs in the library of anyone interested in the subject and its applications to both classical and modern physics. The treatment takes a historical point of view, without making heavy demands on readers' mathematical abilities; in fact, the theory is developed without the use of tensor calculus, requiring only a working knowledge of three-dimensional vector analysis. Topics include detailed coverage of the Lorentz transformation, including optical and dynamical applications, and applications to modern physics. An excellent bibliography completes this compact, accessible presentation.
Product Details
| ISBN-13: | 9780486844527 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 09/15/2019 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 128 |
| File size: | 15 MB |
| Note: | This product may take a few minutes to download. |
About the Author
C. W. Kilmister was Professor of Mathematics at the University of London.
Read an Excerpt
CHAPTER 1
THE LORENTZ TRANSFORMATION
1. Introduction
It is probably fair to say that at the turn of the nineteenth century an optimistic view was taken of the achievements of theoretical physics. Few problems, so it seemed, remained to be solved. There were, it is true, some details which marred this general picture. For example, the advance of the perihelion of Mercury exceeded the predicted amount by nearly 10%. Attempts to describe the interaction of radiation and matter led to a formula which disagreed with experiment. Certain problems in the optics of moving media were still unresolved. But few physicists would have thought at the time that these particular features would require for their explanation a complete revolution in physical ideas. The first two difficulties find their resolution in the general theory of relativity and in the quantum theory, respectively, and we shall not be concerned with them in the present book. The optical problems, however, are resolved by the special theory of relativity which is our immediate concern. Before describing these problems in detail (see § 4) we shall consider some of the additional difficulties of pre-relativity physics.
2. Newtonian Mechanics
In Newton' s hands mechanics was based on the concepts of absolute space and time. These arose in the following way:
The first law of motion states that a body under no forces continues in a state of rest or moves with uniform velocity in a straight line. In order that this law may be meaningful the coordinate-system with respect to which the velocity is measured must be specified. For example, consider two coordinate-systems S, S' related by the formula
r' = r + 1/2 gt2, t' = t,
where r, r' refer to the position-vectors of a point relative to S, S' and g is a constant vector. A straight line described with uniform velocity relative to S is defined by
r = a + ut,
where a, u are constant vectors. Hence we find
r' = a + ut + 1/2 gt2,
which shows the path relative to S' to be a parabola. Thus a particle under no forces and obeying the first law of motion relative to S will have to be described relative to S' either as under forces or as disobeying the first law. Newton made his law specific (Cajori, 1946) by supposing it to refer to an absolute coordinate-system with respect to which everything could in principle be measured. It was realised later that it was unnecessary to postulate this absolute coordinate-system. If we have any coordinate-system Sin which the first law is valid, then it is valid also in any coordinate-system S' whose origin is moving uniformly relative to S. For if
r' = r - vt, ' = t, (2.1)
where v is the constant relative velocity vector of the two coordinate systems, then when
r = a + ut, we also have r' = a + (u - v)t.
If we call a coordinate-system in which Newton's first law is valid an nertial frame (the first law is sometimes referred to as the law of inertia), we can say that the idea of an absolute coordinate-system can be replaced by that of the whole set of inertial frames, no one of which is pre-eminent. This replacement drew attention to the importance of considering coordinate-systems in relative motion. Neumann (1870) raised the problem of how to describe and specify this set of inertial frames, for which he introduced the name 'The Body Alpha'.
The situation in Newtonian dynamics at the end of the nineteenth century was therefore that the need for an absolute space had been replaced by that for a set of inertial frames. The problem of specifying the frames was still unsolved. Moreover, since in the above equations (e.g. in (2.1)) we always have t' = t, the concept of an absolute time (Cajori, 1946) has been left untouched.
3. The Wave Theory of Light
The undulatory theory of light acquired increasing certainty in the late nineteenth century from its connexion with Maxwell's electromagnetic equations. These equations have the form (in Gaussian units in which the field vectors have the same dimensions) :
[mathematical expression not reproducible] (3.1)
where c is the ratio of electromagnetic and electrostatic units of charge. If we consider only empty space, we have
D = E, B = H, (3.2)
and in the absence of charges and currents, therefore, we find
[mathematical expression not reproducible] (3.3)
Hence
[mathematical expression not reproducible]
so that H satisfies the wave equation
[mathematical expression not reproducible] (3.4)
with a similar result for E. It follows that disturbances in the fields will be propagated with velocity c. The constant c can, however, be determined experimentally. Some of the earliest experiments were performed by Weber and Kohlrausch (1856), who compared the capacity of a Leyden jar as measured by an electrostatic method, with that calculated from the effects of the current produced by discharging the jar. These experiments gave a value of c equal to 3.l x 1010 cm./sec., close to the speed of light in vacuo. More accurate experiments (Curtis, 1929) gave c = 2.99790 x 1010 cm./sec., which agrees very closely with a recent value of 2.99792 x 1010 cm./sec. obtained by Froome (1952). This suggests the identification of light and electromagnetic radiation, and such an identification gives a very satisfactory explanation of optical phenomena.
The wave equation (3.4), however, contains no reference to the velocity of the source of the light and this naturally suggests that the velocity of the light must be independent of the velocity of its source. This is in agreement with observation. For example, there exist certain ' double stars ' consisting of two stars moving in orbits about their common centre of gravity. At one point in the orbit one star will be travelling towards the earth, at another directly away. If the centre of gravity is at a distance h from the earth, the light will reach the earth at a time of order h/c after it has been emitted, where c is the velocity of light. For any small change Δc in c, we have a change Δt = - (h/c2)Δc in the time of arrival. This change would produce apparent irregularities in the motion of such stars. No such irregularities have been observed and we are forced to conclude that the velocity of light is independent of the velocity of the source. However, this independence of the velocity of the light and the velocity of the source poses th problem of the coordinate-system with respect to which c is to be measured. In the theory of sound a similar problem arises, but there it is easily resolved since the speed is to be measured relative to the still air. In the nineteenth century, it seemed reasonable to give a similar answer in the case of light. This required the postulation of an unobserved, all-pervasive medium — the aether — in which the wave motion took place. (The idea of such an aether was much older than the difficulties which led to its popularity. It seems first to have been introduced by Hooke (1667) though its properties were somewhat different.) After the postulation of the aether, it then became apparent that the aether might equally provide the standard required to specify the Body Alpha (§ 2). This would have the great advantage of linking the hitherto separated theories of mechanics and electromagnetism. A difficulty at once arises, however, since Maxwell 's equations should then hold in every one of the inertial frames. It is easy to see that this cannot be so, for if we write as a special case of the transformation (2.1), when v is directed along the common x-axis,
x' = x - vt, y' = y, z' = z, t' = t, (3.5)
the first component of the equation [mathematical expression not reproducible] (3.6)
However, there no other equation in the set linking [partial derivative]E1/[partial derivative]t' [partial derivative]E1/[partial derivative]x', so that it is impossible to transform the right-hand side, by transformations of the field vectors, in such a way that the transformed equation would read
[mathematical expression not reproducible]
This difficulty leads us to enquire whether (2.1) is the only possible transformation between inertial frames. We shall find later (§ 6) that it is possible to modify (2.1) in such a way that it remains a transformation between inertial frames, but leaves Maxwell' s equations unchanged in form. It is clear from (3.6) that this modified transformation must involve some change in the time coordinate, such that [partial derivative]t'/[partial derivative]x is non-zero. Accordingly, we shall find that the transformation also removes the concept of an absolute time, which, as we noted in § 2, had been left untouched by the introduction of the Body Alpha. Before this, however, we shall consider some further experimental evidence which played an important part in the development of the special theory of relativity.
4. Critical Experiments
The assumption of the existence of the aether as a medium of transmission of light raises a large number of further questions about the effects of motion relative to the aether of material through which the light passes. Michell (1784) had already posed the question of whether light rays from a star should be refracted differently from rays originating on the earth due to their different motions relative to the aether. It was later shown by Arago (1839, 1853) in a series of experiments that light refraction was apparently unchanged by this motion, although his results were only approximate. To account for this result, Fresnel produced a theory on the basis of certain assumptions about t he aether density and an 'aether drag' on moving bodies. This theory gave for the velocity of light V in a medium of refractive index n the expression
V = c/n + (1 - 1/n2)v, (4.1)
where v is the velocity of the medium. Fresnel's formula (4.1) was confirmed experimentally by Fizeau (1859), and more accurately by Zeeman (1914, 1915). It is clear in any case that if the motion of the medium is to affect the transmission of light at all, the velocity must be given by an expression of the type
V = c/n + f(n, v),
where f(l, v) = 0, since, when n = 1, we have a vacuum and the value of v is then irrelevant. This rules out the otherwise plausible possibility V = c/n + v, which would come from Newtonian mechanics using (3.5) to transform to a frame in which the medium is at rest. For this reason, the transformation (3.5) will be unsatisfactory when applied to problems related to the transmission of light in a moving medium.
The most serious question raised by the assumption of the aether is whether it is possible to detect the motion of the earth relative to it. Michelson (1881) carried out what is now known as the Michelson-Morley experiment (this was repeated later, with technical improvements, by Michelson and Morley (1887)). In this experiment (see Fig. 1), A, A are mirrors and B is a half-silvered mirror. Monochromatic light from a single source is split at the half-silvered mirror and travels over two paths at right angles, returning and interfering at the telescope. If at any time the apparatus happens to have a velocity v as shown, relative to the aether, we have for the times along the two paths
[mathematical expression not reproducible] (4.2)
The apparatus 1s now rotated through 90°, when the new times will be
[mathematical expression not reproducible] (4.3)
Hence
[mathematical expression not reproducible] (4.4)
This time difference produces a shift of m fringes in he interference pattern observed by the telescope, where mλ = Δ[bar.t] - Δt. It is possible, in any particular performance of the experiment, that the earth might be at rest relative to the aether at the time at which the experiment was carried out. To eliminate this possibility, the experiment was repeated after an interval of six months, when the velocity should be a maximum.
No shift was observed by Michelson, but the expected shift was near to the limits of experimental accuracy. The experiment was repeated on various occasions and no fringe shift was observed until Miller (1925) in a series of experiments claimed to find a shift corresponding to a velocity of the earth relative to the aether of about 10 km./sec. More recently, Synge (1952) has suggested a theory which would account for Miller's non-zero shift, and his theory suggested a certain experiment, similar to Michelson and Morley's, which would give a larger result. This experiment was carried out by Ditchburn and Heavens (l952), and gave a null result to a high order of experimental accuracy. We ari!forced to conclude that the residual shift in Miller's experiments is due to some unknown experimental error. Miller's results have been discussed recently by Shankland, McCuskey, Leone and Kuerti (1955) with the conclusion that no significant fringe shift exists.
Fitzgerald (Lodge, 1892) suggested, as a hypothesis to explain the Michelson-Morley experiment, that bodies moving relative to the aether are slightly contracted in their direction of motion. If the contraction ratio were [square root] (1 - v2/c2): 1, this would explain the null result (from (4.2) and (4.3)). This idea was taken up by Lorentz (1892-3). The Fitzgerald-Lorentz contraction would of course be unobservable, since it applies to all bodies including measuring rods.
Yet another experiment designed to test whether it is possible to measure the earth's motion through the aether is that of Trouton and Noble (1902, 1903). A charged condenser carried through the aether by the earth's relative motion should exhibit different behaviour depending on whether its plates are parallel to or perpendicular to its direction of motion. In any position, the charges on the plates constitute a current. The component of velocity parallel to the plates produces two equal and opposite currents, and a consequent magnetic field H in the dielectric parallel to the plates. The component of velocity perpendicular to the plates produces two currents almost neutralising each other. If we neglect the field produced by these currents, and remember that the electric field E is perpendicular to the plates, we find that we have an electromagnetic momentum, P = [E.sub.^] A H/4π, parallel to the plates. The transport of momentum in this way is equivalent to a change in angular momentum, since, if Ω is the angular momentum, we have
[mathematical expression not reproducible]
There is, therefore, a couple [v.sub.^] P acting on the condenser in a sense tending to rotate it into the longitudinal position. Since P is of order v/c2, the couple is of order v2/c2. No such couple was measured by Trouton and Noble. Later experiments (Tomaschek , 1926 and Chase, 1926) confirmed this result.
We shall consider later one other related experiment carried out by Wilson and Wilson (1913). It is convenient to defer this, however, until we have found the transformation, mentioned at the end of § 3, which leaves Maxwell's equations invariant.
5. Transformation of Maxwell's Equations — I
It is clear, firstly, that any transformation leaving Maxwell's equations unchanged will also leave unchanged the wave equations (3.4), namely
[mathematical expression not reproducible] (5.1)
For this reason, we begin by finding transformations under which (5.1) is invariant. We make the following assumptions:
(1) the transformations leave the law of inertia invariant, as with (2.1);
(2) if two events coincide in one coordinate-system, they will coincide in all;
(3) there is cylindrical symmetry about the line joining the origins;
(4) the transformations leave the wave equations (5.1) invariant.
(Continues…)
Excerpted from "Special Relativity for Physicists"
by .
Copyright © 1986 G. Stephenson and C. W. Kilmister.
Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Table of ContentsI. The Lorentz Transformation
II. Immediate Consequences of the Lorentz Transformation.
III. Optical and Dynamical Applications of the Lorentz Transformation.
IV. Applications to Modern Physics
Bibliography
Index.