The Principles of Electromagnetic Theory and of Relativity

The aim of this work is to study the principles upon which the classical and relativistic theories of the electromagnetic and gravitational fields are based. Thus, the primary object of the book is to present a simple exposition of Maxwell's theory, of General Relativity and of the link between those two concepts, namely, Special Relativity. In the nineteenth century the notion of a continuous field gradually replaced the idea of action at a distance. The electromagnetic theory that was elaborated at that time covers a very large area of Physics, since it makes possible the description of permanent phenomena, electrostatics and magnetostatics, as well as of variable phenomena. It anticipates the existence of waves, and thereby the theory of light is annexed to this vast domain. It was discovered that Maxwell's equations changed their form when they were related to reference systems associated with two observers in rectilinear uniform motion with respect to each other and each endowed with the absolute time required by classical mechanics. This was a most remarkable fact. Indeed, as soon as attempts were made to verify the results of classical kinematics by means of experiments with the propa­ gation of light, there arose a whole series of contradictions.

Preface.- One/Electromagnetic Theory.- I/Electrostatics.- 1. The experimental laws. Coulomb’s law.- 2. The general laws of electrostatics.- 3. The first law. Gauss’ theorem.- 4. Applications. The electric field on the surface of a conductor. Electrostatic pressure.- 5. The second law. Definition of the potential.- 6. The solutions of Laplace’s and Poisson’s equations.- 7. Poisson’s equation and boundary conditions.- 8. Applications.- 9. Dielectrics.- 10. Dielectrics and dipoles.- 11. Polarization and displacement.- II/Magnetostatics.- 1. Permanent states. The experimental law of Biot and Savart.- 2. The general laws of magnetism.- 3. Magnetic dipoles.- 4. Magnetic media.- 5. The magnetic moment of a layer. Magnetic permeability and susceptibility.- III/Electromagnetism.- A. Electromagnetic Induction. Displacement Current.- 1. Faraday’s experimental law.- 2. Conduction current. Displacement current.- B. Maxwell’s Equations.- 3. Systems of units.- 4. Basic relations.- 5. The potential.- 6. Equations of propagation. Retarded potentials.- C. Electromagnetic Energy. Energy Flux.- 7. Electric and magnetic energy densities.- 8. Poynting vector and Poynting’s theorem.- D. Electromagnetic Waves.- 9. Equations for the propagation of fields.- 10. Plane waves.- 11. Wave-trains.- 12. Spherical waves.- E. Electromagnetic Equations Valid for Non-Magnetic Bodies in Slow Motion.- 13. Application of Maxwell’s theory to moving media.- 14. Motion of a conductor or an insulator in an electric field.- 15. Displacement of a conductor or an insulator in a magnetic field.- 16. Hertz’ and Lorentz’ hypotheses.- IV/Sources of the Electromagnetic Field. Lorentz’ Theory.- 1. “Microscopic” fields and potentials connected with an electron.- 2. “Structure” of the Lorentz electron.- 3. Potentials and fields created by a distribution of electrons.- 4. Equations for the mean values and Maxwell’s macroscopic theory.- 5. Interpretation of the fields and the inductions of Maxwell’s theory. Electromagnetic equations for the case of matter at rest.- 6. Lorentz’ theory and electrodynamics of moving bodies.- Two/Special Relativity.- V/The Principle of Relativity.- A. The Principle of Relativity before Einstein.- 1. The principle of relativity in classical mechanics.- 2. The principle of relativity in electrodynamics.- 3. Experimental possibilities of detecting absolute motion by optical means.- 4. First-order effects. The hypothesis of a partial dragging of light by transparent bodies.- 5. Lorentz’s theory of electrons and first-order effects. The hypothesis of a motionless ether.- 6. Second-order effects.- 7. The Fitzgerald-Lorentz hypothesis.- B. The Principle of Special Relativity.- 8. Einstein’s basic postulate.- 9. Critique of the concept of simultaneity.- 10. The Lorentz transformation.- 11. Consequences of the transformation formulas.- 12. Proper time.- 13. Geometrical representation of the Lorentz formulas.- 14. Other expressions of the special Lorentz transformation.- 15. The general Lorentz transformation. C. Møller’s method.- 16. Change of inertial system for a moving object. The clock paradox.- VI/Four-Dimensional Formalism of Special Relativity.- 1. The pseudo-Euclidean universe of Special Relativity.- 2. Notational conventions.- 3. Reduced forms of the ds2 in Special Relativity.- 4. Space-like four-vectors. Time-like four-vectors. Isotropie four-vectors.- 5. The invariance of the ds2 under the displacement group in four-dimensional Euclidean space.- 6. The general Lorentz transformation and the special transformation.- 7. Expression of the coefficients in the general Lorentz transformation.- 8. Application to the special Lorentz transformation.- 9. Examples.- 10. The addition of velocities and the general Lorentz transformation.- 11. Application. Case where one of the systems is a proper system.- VII/Relativistic Kinematics.- A. Relativistic Law of Addition of Velocities.- 1. The velocity four-vector.- 2. The modification of velocities in a Lorentz transformation.- 3. The Lorentz transformation and the general formula for the addition of velocities.- 4. Length and direction of the velocity vector.- 5. The limiting velocity.- 6. Asymmetry of the parts played by the “relative” velocity and the “coordinate displacement velocity”.- 7. The special case of the addition of parallel velocities.- B. Wave Propagation and Relativistic Kinematics.- 8. Propagation of a plane wave in refractive media moving uniformly with respect to each other.- 9. Huygens’ principle and Special Relativity 198 10. Phase velocity and propagation velocity.- VIII/Relativistic Dynamics.- A. Relativistic Dynamics of a Point-Mass.- 1. Momentum, energy and proper mass of a particle.- 2. Minkowski force. The basic law of relativistic dynamics.- 3. Equivalence of mass and energy.- 4. Modification of velocities and basic quantities (momentum, energy, force) of dynamics in a Lorentz transformation.- 5. Systems of free particles.- 6. Systems of bound particles.- B. The Relativistic Dynamics of Continuous Media.- 7. The non-relativistic equations of a fluid in a system of orthogonal coordinates.- 8. The relativistic equations of a continuous medium.- 9. The material energy-momentum tensor.- 10. The case of a perfect fluid.- C. Use of Curvilinear Coordinates.- 11. Trajectory of a material point expressed in any arbitrarily chosen system of coordinates.- 12. The basic law of the dynamics of a point.- 13. Motion of a homogeneous fluid. The matter tensor.- 14. Equations of conservation and equations of motion.- 15. A special case: the equations of conservation and of motion for a perfect fluid.- IX/Relativistic Electromagnetism.- A. The Covariant Form of Maxwell’s Theory.- 1. The electromagnetic field, a tensor of second rank.- 2. The electromagnetic potential.- 3. Maxwell’s equations and the general Lorentz transformation.- 4. The Lorentz electron theory. The energy momentum tensor.- 5. Lorentz equations and Maxwell’s equations.- 6. The energy-momentum tensors.- 7. Use of arbitrary curvilinear coordinates.- B. Extensions of Maxwell’s Theory.- 8. The deduction of Maxwell’s equations from a variational principle.- 9. Mie’s Theory 267 10. The theory of M. Born and L. Infeld.- X/The Experimental Verifications of Special Relativity.- A. The Retardation of Moving Clocks.- 1. The theory of the Doppler effect and the slowing-down of clocks.- 2. Ives and Stillwell’s experiments (1941).- 3. The mean lifetime of mesons.- B. The Variation of Mass with Velocity.- 4. The motion of a charged particle in an electromagnetic field.- 5. The deviation of charged particles subjected to the action of parallel electric and magnetic fields perpendicular to the initial velocity of the particles.- 6. The elastic collision of two particles.- 7. The Compton effect.- C. The Equivalence of Mass and Energy.- 8. Mass defect and nuclear energy.- 9. The balance of energy and momentum in nuclear reactions.- Three/General Relativity.- XI/General Relativity.- A. The Newtonian Law of Gravitation.- 1. The Newtonian law of gravitation and observational data.- 2. The gravitational potential and its properties. The equivalence of gravitational mass and inertial mass.- 3. Poisson’s law.- 4. Newton’s law and the principle of Special Relativity.- B. The Principle of Equivalence and the Introduction of a Non-Euclidean Universe.- 5. Accelerated reference systems and “fictitious” inertial forces. The limits of the principle of Special Relativity.- 6. The local equivalence of gravitational and inertial forces.- 7. Introduction of a non-Euclidean universe.- 8. Study of a special case: the problem of the rotating disc.- C. Einstein’s Law of Gravitation.- 9. The law of gravitation outside matter.- 10. The law of gravitation inside matter or in the presence of an electromagnetic field.- 11. The trajectories of a particle subjected to a gravitational field are the geodesics of a Riemannian space.- XII/The Development of General Relativity and Some of Its Consequences.- A. The Equations in Various Approximations.- 1. The gravitational potential in the Newtonian approximation.- 2. The equations of the gravitational field in a system of De Donder and quasi-Galilean coordinates.- 3. Application to a continuous material medium treated as a perfect gas.- 4. Equations of the exterior case.- 5. Equations of the field and motion of the sources.- B. Study of a Rigorous but Special Solution of the Field Equations: Schwarzschild’s Solution.- 6. The gravitational field created in the neighbourhood of a static mass possesssing spherical symmetry.- 7. The field created in the neighbourhood of a spherically symmetric charged particle.- 8. The trajectory of a neutral particle in the neighbourhood of a static mass having spherical symmetry.- 9. The experimental verifications of Schwarzschild’s solution.- XIII/Unified Theories of Electromagnetism and Gravitation Characteristics of a Pure Field Theory Unified theories and Non-Dualistic Theories.- A. Unified Theories.- 1. Unified theories until the advent of General Relativity.- 2. General Relativity and the construction of unified theories.- 3. Interpretation of the electromagnetic and gravitational fields proposed by unified theories.- 4. Classical unified theories and the possibilities of further predictions.- 5. Unified theories and quantum theories.- B. Non-Dualistic Theories.- 6. The field and its sources.- 7. Non-linearity and the characteristics of a pure field theory.- C. Unified and Non-Dualistic Theories.- Four/Mathematical Supplement.- XIV/Tensor Calculus In An Euclidean Vector Space.- A. Rectilinear Axes.- 1. Covariance and contravariance.- 2. The norm of a vector. The scalar product of two vectors.- 3. Transformation of rectilinear axes.- 4. Invariants, four-vectors and tensors.- 5. Symmetry and antisymmetry.- 6. Transformation of the metric tensor. A special case : Utilization of orthogonal frames of reference.- 7. The rotations of axes in a four-dimensional Euclidean space.- B. Use of Arbitrary Curvilinear Coordinates.- 8. Passage from one system of curvilinear coordinates to another in an Euclidean vector space.- 9. Differential relations between the components of the metric tensor.- 10. Covariant differentiation.- 11. Tensor densities.- XV/Tensor Calculus in a Non-Euclidean Metric Manifold. Application to a Riemannian Space.- 1. Metric space and tangent Euclidean space.- 2. Affine connection.- 3. Representation of the first order.- 4. Representation of the second order.- 5. Vectors and tensors associated with a metric manifold.- 6. Covariant derivation.- 7. The parallel transport of a vector.- 8. The conditions of integrability and the structure of space.- 9. The curvature of Riemannian space. The Riemann-Christoffel tensor.- 10. Properties of the Riemann-Christoffel tensor.- 11. The geodesics of Riemannian space as analogues of the straight lines of Euclidean space.