The Principles of Electromagnetic Theory and of Relativity

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.