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| ISBN-13: | 9780486318400 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 03/25/2013 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 240 |
| File size: | 7 MB |
About the Author
It's All Relative
Around 1950, Hayward Cirker, Founder and President of Dover Publications, wrote to Einstein and asked his approval to proceed with a Dover paperback reprint of the 1923 collection of original papers on relativity by Einstein himself and others (H. A. Lorentz, H. Weyl, and H. Minkowski), which had originally been published in England. Einstein was reluctant, wondering how much interest there could possibly be in this relic of his work from 30 or more years earlier. Cirker persisted, and Einstein finally agreed — the Dover edition of The Theory of Relativity has been in print ever since and has been followed by many other Dover books on relativity.
The papers reprinted in this original collection will always be for the serious student the cornerstone of their Einstein library: Michelson's Interference Experiment (H. A. Lorentz); Electromagnetic Phenomena in a System Moving with any Velocity Less Than That of Light (H.A. Lorentz); On the Electrodynamics of Moving Bodies (A. Einstein); Does the Inertia of a Body Depend Upon its Energy Content? (A. Einstein); Space and Time (H. Minkowksi with notes by A. Sommerfeld); On the Influence of Gravitation on the Propagation of Light (A. Einstein); and The Foundation of the General Theory of Relativity (A. Einstein) found on pages 109–164 of this text; Hamilton's Principle and The General Theory of Relativity (A. Einstein); Cosmological Considerations on the General Theory of Relativity (A. Einstein); Do Gravitational Fields Play an Essential Part in the Structure of the Elementary Particles of Matter? (A. Einstein); and Gravitation and Electricity (H. Weyl).
In the Author's Own Words:
"How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?"
"What nature demands from us is not a quantum theory or a wave theory; rather, nature demands from us a synthesis of these two views which thus far has exceeded the mental powers of physicists."
"Do not be troubled by your difficulties with Mathematics, I can assure you mine are much greater." — Albert Einstein
Critical Acclaim for The Theory of Relativity:
"This book constitutes an indispensable part of a library on relativity." — Nature
Read an Excerpt
CHAPTER 1
MICHELSON'S INTERFERENCE EXPERIMENT
BY H. A. LORENTZ
1. As Maxwell first remarked and as follows from a very simple calculation, the time required by a ray of light to travel from a point A to a point B and back to A must vary when the two points together undergo a displacement without carrying the ether with them. The difference is, certainly, a magnitude of second order; but it is sufficiently great to be detected by a sensitive interference method.
The experiment was carried out by Michelson in 1881. His apparatus, a kind of interferometer, had two horizontal arms, P and Q, of equal length and at right angles one to the other. Of the two mutually interfering rays of light the one passed along the arm P and back, the other along the arm Q and back. The whole instrument, including the source of light and the arrangement for taking observations, could be revolved about a vertical axis; and those two positions come especially under consideration in which the arm P or the arm Q lay as nearly as possible in the direction of the Earth's motion. On the basis of Fresnel's theory it was anticipated that when the apparatus was revolved from one of these principal positions into the other there would be a displacement of the interference fringes.
But of such a displacement — for the sake of brevity we will call it the Maxwell displacement — conditioned by the change in the times of propagation, no trace was discovered, and accordingly Michelson thought himself justified in concluding that while the Earth is moving, the ether does not remain at rest. The correctness of this inference was soon brought into question, for by an oversight Michelson had taken the change in the phase difference, which was to be expected in accordance with the theory, at twice its proper value. It we make the necessary correction, we arrive at displacements no greater than might be masked by errors of observation.
Subsequently Michelson took up the investigation anew in collaboration with Morley, enhancing the delicacy of the experiment by causing each pencil to be reflected to and fro between a number of mirrors, thereby obtaining the same advantage as if the arms of the earlier apparatus had been considerably lengthened. The mirrors were mounted on a massive stone disc, floating on mercury, and therefore easily revolved. Each pencil now had to travel a total distance of 22 meters, and on Fresnel's theory the displacement to be expected in passing from the one principal position to the other would be 0-4 of the distance between the interference fringes. Nevertheless the rotation produced displacements not exceeding 0 02 of this distance, and these might well be ascribed to errors of observation.
Now, does this result entitle us to assume that the ether takes part in the motion of the Earth, and therefore that the theory of aberration given by Stokes is the correct one? The difficulties which this theory encounters in explaining aberration seem too great for me to share this opinion, and I would rather try to remove the contradiction between Fresnel's theory and Michelson's result. An hypothesis which I brought forward some time ago, and which, as I subsequently learned, has also occurred to Fitzgerald, enables us to do this. The next paragraph will set out this hypothesis.
2. To simplify matters we will assume that we are working with apparatus as employed in the first experiments, and that in the one principal position the arm P lies exactly in the direction of the motion of the Earth. Let v be the velocity of this motion, L the length of either arm, and hence 2L the path traversed by the rays of light. According to the theory, the turning of the apparatus through 90° causes the time in which the one pencil travels along P and back to be longer than the time which the other pencil takes to complete its journey by
Lv2/c3.
There would be this same difference if the translation had no influence and the arm P were longer than the arm Q by 1/2Lv2/c2. Similarly with the second principal position.
Thus we see that the phase differences expected by the theory might also arise if, when the apparatus is revolved, first the one arm and then the other arm were the longer. It follows that the phase differences can be compensated by contrary changes of the dimensions.
If we assume the arm which lies in the direction of the Earth's motion to be shorter than the other by 1/2Lv2/c2, and, at the same time, that the translation has the influence which Fresnel's theory allows it, then the result of the Michelson experiment is explained completely.
Thus one would have to imagine that the motion of a solid body (such as a brass rod or the stone disc employed in the later experiments) through the resting ether exerts upon the dimensions of that body an influence which varies according to the orientation of the body with respect to the direction of motion. If, for example, the dimensions parallel to this direction were changed in the proportion of 1 to 1 + δ, and those perpendicular in the proportion of 1 to 1 + ε, then we should have the equation
ε - δ = 1/2 v2/c2 (1)
in which the value of one of the quantities δ and ε would remain undetermined. It might be that ε = 0, δ = - 1/2v2/c2, but also [epsilob] = 1/2v2/c2, δ = 0, or ε = 1/4v2/c2, and δ = - 1/4v2/c2.
3. Surprising as this hypothesis may appear at first sight, yet we shall have to admit that it is by no means far-fetched, as soon as we assume that molecular forces are also transmitted through the ether, like the electric and magnetic forces of which we are able at the present time to make this assertion definitely. If they are so transmitted, the translation will very probably affect the action between two molecules or atoms in a manner resembling the attraction or repulsion between charged particles. Now, since the form and dimensions of a solid body are ultimately conditioned by the intensity of molecular actions, there cannot fail to be a change of dimensions as well.
From the theoretical side, therefore, there would be no objection to the hypothesis. As regards its experimental proof, we must first of all note that the lengthenings and shortenings in question are extraordinarily small. We have v2/c2 = 10-8, and thus, if ε = 0, the shortening of the one diameter of the Earth would amount to about 6.5 cm. The length of a meter rod would change, when moved from one principal position into the other, by about 1/200 micron. One could hardly hope for success in trying to perceive such small quantities except by means of an interference method. We should have to operate with two perpendicular rods, and with two mutually interfering pencils of light, allowing the one to travel to and fro along the first rod, and the other along the second rod. But in this way we should come back once more to the Michelson experiment, and revolving the apparatus we should perceive no displacement of the fringes. Reversing a previous remark, we might now say that the displacement produced by the alterations of length is compensated by the Maxwell displacement.
4. It is worth noticing that we are led to just the same changes of dimensions as have been presumed above if we, firstly, without taking molecular movement into consideration, assume that in a solid body left to itself the forces, attractions or repulsions, acting upon any molecule maintain one another in equilibrium, and, secondly — though to be sure, there is no reason for doing so — if we apply to these molecular forces the law which in another place we deduced for electrostatic actions. For if we now understand by S1 and S2 not, as formerly, two systems of charged particles, but two systems of molecules — the second at rest and the first moving with a velocity v in the direction of the axis of x — between the dimensions of which the relationship subsists as previously stated; and if we assume that in both systems the x components of the forces are the same, while the y and z components differ from one another by the factor [square root of (1 - v2/c2)] then it is clear that the forces in S1 will be in equilibrium whenever they are so in S2. If therefore S2 is the state of equilibrium of a solid body at rest, then the molecules in S1 have precisely those positions in which they can persist under the influence of translation. The displacement would naturally bring about this disposition of the molecules of its own accord, and thus effect a shortening in the direction of motion in the proportion of 1 to [square root of (1 - v2/c2)], in accordance with the formulae given in the above-mentioned paragraph. This leads to the values
δ = - 1/2 v2/c2, ε = 0
in agreement with (1).
In reality the molecules of a body are not at rest, but in every "state of equilibrium" there is a stationary movement. What influence this circumstance may have in the phenomenon which we have been considering is a question which we do not here touch upon; in any case the experiments of Michelson and Morley, in consequence of unavoidable errors of observation, afford considerable latitude for the values of δ and ε.
CHAPTER 2
ELECTROMAGNETIC PHENOMENA IN A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT
BY H. A. LORENTZ
§1. The problem of determining the influence exerted on electric and optical phenomena by a translation, such as all systems have in virtue of the Earth's annual motion, admits of a comparatively simple solution, so long as only those terms need be taken into account, which are proportional to the first power of the ratio between the velocity of translation v and the velocity of light c. Cases in which quantities of the second order, i.e. of the order v2/c2, may be perceptible, present more difficulties. The first example of this kind is Michelson's well-known interference-experiment, the negative result of which has led Fitzgerald and myself to the conclusion that the dimensions of solid bodies are slightly altered by their motion through the ether.
Some new experiments, in which a second order effect was sought for, have recently been published. Rayleigh and Brace have examined the question whether the Earth's motion may cause a body to become doubly refracting. At first sight this might be expected, if the just mentioned change of dimensions is admitted. Both physicists, however, have obtained a negative result.
In the second place Trouton and Noble have endeavoured to detect a turning couple acting on a charged condenser, the plates of which make a certain angle with the direction of translation. The theory of electrons, unless it be modified by some new hypothesis, would undoubtedly require the existence of such a couple. In order to see this, it will suffice to consider a condenser with ether as dielectric. It may be shown that in every electrostatic system, moving with a velocity v, there is a certain amount of "electromagnetic momentum." If we represent this, in direction and magnitude, by a vector G, the couple in question will be determined by the vector product
[G. v] (1)
Now, if the axis of z is chosen perpendicular to the condenser plates, the velocity v having any direction we like; and if U is the energy of the condenser, calculated in the ordinary way, the components of G are given by the following formulae, which are exact up to the first order,
[MATHEMATICAL EXPRESSION OMITTED]
Substituting these values in (1), we get for the components of the couple, up to terms of the second order,
[MATHEMATICAL EXPRESSION OMITTED]
These expressions show that the axis of the couple lies in the plane of the plates, perpendicular to the translation. If a is the angle between the velocity and the normal to the plates, the moment of the couple will be U(v/c)2 sin 2a; it tends to turn the condenser into such a position that the plates are parallel to the Earth's motion.
In the apparatus of Trouton and Noble the condenser was fixed to the beam of a torsion-balance, sufficiently delicate to be deflected by a couple of the above order of magnitude. No effect could however be observed.
§ 2. The experiments of which I have spoken are not the only reason for which a new examination of the problems connected with the motion of the Earth is desirable. Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelson's negative result, the introduction of a new hypothesis has been required, and that the same necessity may occur each time new facts will be brought to light. Surely this course of inventing -special hypotheses for each new experimental result is somewhat artificial. It would be more satisfactory if it were possible to show by means of certain fundamental assumptions and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system. Some years ago, I already sought to frame a theory of this kind. I believe it is now possible to treat the subject with a better result. The only restriction as regards the velocity will be that it be less than that of light.
§ 3. I shall start from the fundamental equations of the theory of electrons. Let D be the dielectric displacement in the ether, H the magnetic force, ρ the volume-density of the charge of an electron, v the velocity of a point of such a particle, and F the ponderomotive force, i.e. the force, reckoned per unit charge, which is exerted by the ether on a volume-element of an electron. Then, if we use a fixed system of co-ordinates,
[MATHEMATICAL EXPRESSION OMITTED] (2)
I shall now suppose that the system as a whole moves in the direction of x with a constant velocity v, and I shall denote by u any velocity which a point of an electron may have in addition to this, so that
vx = v + ux, vy = uy, vz = uz.
If the equations (2) are at the same time referred to axes moving with the system, they become
[MATHEMATICAL EXPRESSION OMITTED]
§ 4. We shall further transform these formulae by a change of variables. Putting
c2/c2 - v2 = β2, (3)
and understanding by l another numerical quantity, to be determined further on, I take as new independent variables
x' = βlx, y' = ly, z' = lz, (4)
t' = 1/β t - βl v/c2x (5)
and I define two new vectors D' and H' by the formulae
[MATHEMATICAL EXPRESSION OMITTED]
for which, on account of (3), we may also write
[MATHEMATICAL EXPRESSION OMITTED] (6)
As to the coefficient l, it is to be considered as a function of v, whose value is 1 for v = 0, and which, for small values of v, differs from unity no more than by a quantity of the second order.
The variable t' may be called the "local time"; indeed, for β = 1, l = 1 it becomes identical with what I formerly denoted by this name.
If, finally, we put
[MATHEMATICAL EXPRESSION OMITTED] (7)
[MATHEMATICAL EXPRESSION OMITTED] (8)
these latter quantities being considered as the components of a new vector u', the equations take the following form : —
[MATHEMATICAL EXPRESSION OMITTED] (9)
[MATHEMATICAL EXPRESSION OMITTED] (10)
The meaning of the symbols div' and curl' in (9) is similar to that of div and curl in (2); only, the differentiations with respect to x, y, z are to be replaced by the corresponding ones with respect to x', y', z'.
§ 5. The equations (9) lead to the conclusion that the vectors D' and H' may be represented by means of a scalar potential φ' and a vector potential A'. These potentials satisfy the equations
[MATHEMATICAL EXPRESSION OMITTED] (11)
[MATHEMATICAL EXPRESSION OMITTED] (12)
and in terms of them D' and H' are given by
[MATHEMATICAL EXPRESSION OMITTED] (13)
H' = curl' A' (14)
(Continues…)
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Table of Contents
I. Michelson's Interference Experiment. By H. A. Lorentz, 1-7,
II. Electromagnetic Phenomena in a System Moving with any Velocity less than that of Light. By H. A. Lorentz, 9-34,
III. On The Electrodynamics of Moving Bodies. By A. Einstein., 35-65,
IV. Does the Inertia of a body depend upon its energy-content? By A. Einstein, 67-71,
V. Space and Time. By H. Minkowski, 73-91,
VI. On the Influence of Gravitation on the Propagation of Light. By A. Einstein, 97-108,
VII. The Foundation of the General Theory of Relativity. By A. Einstein, 109-164,
VIII. Hamilton's Principle and the General Theory of Relativity. By A. Einstein, 165-173,
IX. Cosmological Considerations on the General Theory of Relativity. By A. Einstein, 175-188,
X. Do Gravitational Fields Play an Essential Part in the Structure of the Elementary Particles of Matter? By A. Einstein, 189-198,
XI. Gravitation and Electricity. By H. Weyl, 200-216,