Second Note Electrodynamics in the General Relativity Theory

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MP352: Special Relativity

The first and second exams cannot be retaken individually. Exercises: The homework problems are out on Mondays on this page. The solutions are returned into a locked metal box marked with the text "General Relativity" in the A corridor on the 2nd floor for grading by the following Monday lecture. Textbook: The closest textbook for the course is S. Carroll's Spacetime and Geometry Addison Wesley This book is in the reference library.

It is not necessary to buy the book; the lecture notes given below cover the content of this course and you could also read Carroll's lecture notes on which his book is based; the lecture notes are good too, but they are shorter and less polished than the book which are freely available on the net. Hobson, G. Efstathiou and A. Lasenby Cambdridge Other literature Three classic texts: S. Weinberg: Gravitation and Cosmology Wiley C. Misner K. Thorne, J. Wheeler: Gravitation Freeman R. Wald: General Relativity, The University of Chicago Press Two good short textbooks that do not cover the whole course, but which are easy to read: B.

Foster and J. More recent books, with a different approach: J. Schutz: Gravity from the Ground Up Cambridge The course begins with a review of special relativity to introduce some of the concepts and notation needed in general relativity. Knowledge of special relativity is assumed. It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get. It follows from this equation that from a composition of two velocities which are less than c , there always results a velocity less than c.

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For if we set , and being positive and less than c , then. It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain. We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics. Let the Maxwell-Hertz equations for empty space hold good for the stationary system K, so that we have.

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Now the principle of relativity requires that if the Maxwell-Hertz equations for empty space hold good in system K, they also hold good in system k ; that is to say that the vectors of the electric and the magnetic force— , , and , , —of the moving system k , which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations:—.

Evidently the two systems of equations found for system k must express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for system K.

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Since, further, the equations of the two systems agree, with the exception of the symbols for the vectors, it follows that the functions occurring in the systems of equations at corresponding places must agree, with the exception of a factor , which is common for all functions of the one system of equations, and is independent of and but depends upon v.

Thus we have the relations. If we now form the reciprocal of this system of equations, firstly by solving the equations just obtained, and secondly by applying the equations to the inverse transformation from k to K , which is characterized by the velocity - v , it follows, when we consider that the two systems of equations thus obtained must be identical, that.

Further, from reasons of symmetry 8 and therefore. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector X, Y, Z is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving system at least at the relevant instant , then the force acting upon it, measured in the moving system, is equal to the vector , ,.

Consequently the first three equations above allow themselves to be clothed in words in the two following ways:—. Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. In the system K, very far from the origin of co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations.

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Here , , and , , are the vectors defining the amplitude of the wave-train, and l , m , n the direction-cosines of the wave-normals. We wish to know the constitution of these waves, when they are examined by an observer at rest in the moving system k.

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This is Doppler's principle for any velocities whatever. When the equation assumes the perspicuous form. We see that, in contrast with the customary view, when. This equation expresses the law of aberration in its most general form. If , the equation becomes simply. We still have to find the amplitude of the waves, as it appears in the moving system.

If we call the amplitude of the electric or magnetic force A or respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain. It follows from these results that to an observer approaching a source of light with the velocity c , this source of light must appear of infinite intensity. Since equals the energy of light per unit of volume, we have to regard , by the principle of relativity, as the energy of light in the moving system.

But this is not the case. If l , m , n are the direction-cosines of the wave-normals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:—. We may therefore say that this surface permanently encloses the same light complex.

We inquire as to the quantity of energy enclosed by this surface, viewed in system k , that is, as to the energy of the light complex relatively to the system k. The spherical surface—viewed in the moving system—is an ellipsoidal surface, the equation for which, at the time , is.

If S is the volume of the sphere, and that of this ellipsoid, then by a simple calculation. Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and when measured in the moving system, we obtain. It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law. We seek for the pressure of light exerted on the reflecting surface, and for the direction, frequency, and intensity of the light after reflexion.

Let the incidental light be defined by the quantities A, , referred to system K. Viewed from k the corresponding quantities are. Finally, by transforming back to the stationary system K, we obtain for the reflected light. The energy measured in the stationary system which is incident upon unit area of the mirror in unit time is evidently. The energy leaving the unit of surface of the mirror in the unit of time is.

The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the product P v , where P is the pressure of light, we obtain. In agreement with experiment and with other theories, we obtain to a first approximation. All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body.

By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies. If we imagine the electric charges to be invariably coupled to small rigid bodies ions, electrons , these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies. We then obtain the equations. If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations.

Now, secondly, let the velocity of the electron at a given epoch be v. We seek the law of motion of the electron in the immediately ensuing instants of time.


Without affecting the general character of our considerations, we may and will assume that the electron, at the moment when we give it our attention, is at the origin of the co-ordinates, and moves with the velocity v along the axis of X of the system K. From the above assumption, in combination with the principle of relativity, it is clear that in the immediately ensuing time for small values of t the electron, viewed from the system k , moves in accordance with the equations.

With the help of these equations we transform the above equations of motion from system k to system K, and obtain. We write the equations A in the form. This force might be measured, for example, by a spring balance at rest in the last-mentioned system. With a different definition of force and acceleration we should naturally obtain other values for the masses.

This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously. We remark that these results as to the mass are also valid for ponderable material points, because a ponderable material point can be made into an electron in our sense of the word by the addition of an electric charge, no matter how small.

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We will now determine the kinetic energy of the electron. If an electron moves from rest at the origin of co-ordinates of the system K along the axis of X under the action of an electrostatic force X, it is clear that the energy withdrawn from the electrostatic field has the value. As the electron is to be slowly accelerated, and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion W of the electron.

Bearing in mind that during the whole process of motion which we are considering, the first of the equations A applies, we therefore obtain. Velocities greater than that of light have—as in our previous results—no possibility of existence. This expression for the kinetic energy must also, by virtue of the argument stated above, apply to ponderable masses as well.

We will now enumerate the properties of the motion of the electron which result from the system of equations A , and are accessible to experiment. This relationship may be tested experimentally, since the velocity of the electron can be directly measured, e. These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must move.

In conclusion I wish to say that in working at the problem here dealt with I have had the loyal assistance of my friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions.


Special relativity

Most of the papers in that collection are English translations by W. Perrett and G. Jeffery from the German Das Relativatsprinzip , 4th ed.

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All of these sources are now in the public domain; this document, derived from them, remains in the public domain and may be reproduced in any manner or medium without permission, restriction, attribution, or compensation. The English translation modified the notation used in Einstein's paper to conform to that in use by the s; for example, c denotes the speed of light, as opposed the V used by Einstein in This electronic edition was prepared by John Walker in November You can download a ready-to-print PostScript file of this document or the LaTeX source code used to create it from this site; both are supplied as Zip ped archives.

In addition, a PDF document is available which can be read on-line or printed.