Relativistic Theories of Materials

1. Introduction.- § 1. On the Beginning of Relativity.- § 2. The Space-Time Structure of Special Relativity and First Basic Consequences.- § 3. On the Operational Aspect of Physical Concepts.- § 4. New Ideas on Mass and Energy, in Contrast with Classical Physics, Accepted on the Basis of Special Relativity Kinematics.- § 5. On Forces, Cauchy Equations of Continuous Media, and the First Principle of Thermodynamics in Special Relativity.- § 6. On Electromagnetism, Heat Conduction, and Constitutive Equations in Special Relativity.- § 7. Gravitation and Relativity.- § 8. On the Local Equivalence Principle and the Basic Local Laws of the Electromagnetic Field and Continuous Media, Other than the Poisson Equation, in General Relativity. A Criterion Connecting those Laws with Their Analogues in Classical Physics or Special Relativity.- § 9. On the Invariance of Physical Equations and on the Possible Physical Equivalence of the Frames in which these Equations have the Same Form. On a Privileged Absolute Concept of Event Point.- § 10. On Harmonic Coordinates and the Existence of General Frames not Physically Equivalent in General Relativity.- § 11. Some Distinctive Properties of General Relativity. On the Equivalence of General Frames in General Relativity.- § 12. What We Mean by General Theory of Relativity.- § 13. On the Development of General Relativity. Inclusion of Elasticity, Electromagnitostriction, Couple Stresses, and Hereditary Phenomena.- § 14. Scope and Plan of the Present Tract.- Footnotes to Chapter 1.- I. Basic Equations of Gravitation, Thermodynamics and Electromagnetism, and Constitutive Equations from the Eulerian Point of View.- 2. Space-Time Kinematics Including Masses.- § 15. On the Riemannian Relativistic Space-Time Metric Introduced as a Chronometry. Admissible Frames. Some Possible Axioms for Non-Cosmological Relativity.- § 16. On Tensors in Relativistic Space-Time.- § 17. On Tensors in S4 in Connection with a Moving Continuous Body C or an Ideal Fluid F Spatial Projections and Natural Decompositions of Tensors, Spatial Derivatives and Spatial Divergences.- § 18. The Spatial Metric $$ d{\frac{1}{s}^2} $$. Another Physical Meaning of the Chronometry ds2. Ordinary Units.- § 19. On Some Classical Analogues of Locally Natural Frames from a Physical Point of View.- § 20. Material Derivatives, the Spatial Ricci Tensor, and the Relative Rate of Change of Proper Volume Dealt with from the Eulerian Point of View.- § 21. On Gravitational Mass and Reference or Conventional Mass. The Continuity Equation.- § 22. Angular and Deformation Velocities. Convected and Co-Rotational Fluxes. On T?/?? for T[??]??? = 0.- Footnotes to Chapter 2.- 3. Gravitation and Conservation Equations. Fluids and Elastic Waves.- § 23. The Einstein Gravitation Equations and Basic Consequences.- § 24. The Case of Interacting Matter Capable of Heat Conduction.- § 25. On the Second Principle of Thermodynamics, the Clausius-Duhem Inequality and Fourier’s Law. A Relativistic Proof of the Symmetry of the Heat Conduction Coefficient.- § 26. On the Paradox of an Infinite Velocity of Heat Propagation from the Classical Point of View and the Relativistic One.- § 27. On the Local Spatial Physical Isotropy of S4.- § 28. Free Energy and Relativistic Thermodynamics for Possibly Viscous Fluids.- § 29. On Non-Viscous Fluids in the Presence of Heat Conduction and on Perfect Gases.- § 30. Acceleration Waves in Non-Viscous Fluids in the Absence of Electromagnetic Phenomena.- § 31. On Elastic Waves in Perfect Gases.- § 32. On the Importance of the Thermodynamic Tensor.- § 33. Some Historical Remarks on Relativistic Theories of Fluids and Hints at Some Further Results.- Footnotes to Chapter 3.- 4. Electromagnetism from the Eulerian Point of View. Polarizable Fluids.- § 34. Introductory Considerations. The Ohm Law and the Relations Between the Electric and Magnetic Fields and the Respective Inductions.- § 35. On the Maxwell Equations in Space Time.- § 36. On the Electromagnetic Energy Tensor E??. Some Requirements for it in the Absence of Polarization. Its Indeterminancy in the Presence of Polarization.- § 37. On Some Widely Used Instances of the Electromagnetic Energy Tensor E?? and Some Instances of Ex?/?.- § 38. On Isotropic Functions and Tensors.- § 39. Some Uniqueness Properties of the Electromagnetic Energy Tensor E?? On Its Arbitrariness in Connection with Heat Conduction.- § 40. Some Historical Hints. Basic General Energetic Properties of Minkowski’s Tensor and the Instances 5E?? to 7E?? of E??.- § 41. Some Versions of Poynting’s Theorem for Moving Media.- § 42. W as the Proper Density of Non-Material Electromagnetic Energy.- § 43. On the Equations of Gravitation and Energy Balance in the Presence of Electromagnetic Phenomena.- Footnotes to Chapter 4.- 5. On Media Capable of Electromagnetic Phenomena from the Eulerian Point of View. Magneto-Elastic Waves in Ideal Conductors.- § 44. Introduction.- § 45. Black Body and Absolute Temperature in Thermodynamic Equilibrium.- § 46. Polarizable Non-Viscous Fluids.- § 47. Polarizable Viscous Fluid.- § 48. The Cauchy Equations in the Presence of Heat Conduction and an Electromagnetic Field; Preliminaries for Ideal Conductors.- § 49. Dynamic Discontinuity Equations for Magneto-Elastic Acceleration Waves in Magnetizable Fluids.- § 50. Magneto-Elastic Acceleration Waves in Magnetizable Non-Viscous Fluids.- Footnotes to Chapter 5.- II. Materials from the Lagrangian Point of View.- 6. Kinematics and Stresses from the Lagrangian Point of View.- § 51. Historical Hints at Relativistic Theories of Elastic and More General Materials.- § 52. On the Representation of the Motion M of C.- § 53. Lagrangian Spatial Derivative and Absolute Derivative of a Double Tensor Field with Respect to the Motion M of C.- § 54. Polar Decomposition of the Position Gradient ?L? and Principal Axes of Strain.- § 55. Fermi Transport.- § 56. On the Dilation Coefficients for Line, Volume, and Surface Elements, and the Ratio dC/dC*.- § 57. The Vectors VL* and V*L for V? Spatial. Expressions of àL? and $$ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} _{LM}} $$ in Terms of u?/?.- § 58. New Determination of the General Solution for the Continuity Equation. Connection of DcV? and DcV? with DVL* and DVL* for V? Spatial, and Lagrangian Expression for the Electromagnetic Work d3?.- § 59. The First and Second Piola-Kirchhoff Stress Tensors K?M and YLM and Lagrangian Expressions for dl(i).- § 60. Connection Between $$ {X^{\rho \sigma }}{/_{\frac{1}{\sigma }}} $$ and K?M|M.- § 61. On ?LM? and the Lagrangian Expression of $$ \frac{1}{g}_\lambda ^\rho {X^{\lambda \sigma }}_{/\frac{1}{\sigma }} $$.- § 62. Explicit Form in Co-Moving Co-Ordinates for Some of the Preceding Lagrangian Formulas.- Footnotes to Chapter 6.- 7. Elasticity, Acceleration Waves, and Variational Principles for Simple Materials.- § 63. Foundations of Elasticity.- § 64. Some Theorems on Elastic Materials.- § 65. On Discontinuity Surfaces in Space-Time.- § 66. Dynamic Equations of Elastic Acceleration Waves.- § 67. Polarization and Inertial-Mass Quadrics. Acoustic Axes.- § 68. Pure Pressure States. Isotropic Elastic Materials. Comparison with the Classical Theory.- § 69. A Principle Concerning the Variation of the Metric Tensor of Riemannian Space-Time in the Adiabatic Elastic Case.- § 70. Variation of World Lines in the Adiabatic Elastic Case.- Footnotes to Chapter 7.- 8. Piezo-Elasticity and Magnetoelastic Waves from the Lagrangian Point of View.- § 71. Introduction.- § 72. Foundations of Piezo-Elasticity.- § 73. Extension of the Operations T...? T *..., T...?T*..., Dc and Dc to Tensors of Arbitrary Order.- § 74. On Rigid Motions in the Born Sense.- § 75. Born Rigidity and Stationary Tensors.- § 76. Some Invariance Properties of Ideal Conductors.- § 77. Dynamic Equations for Piezo-Elastic Ideal Conductors.- § 78. Magneto-Elastic Acceleration Waves in Piezo-Elastic Ideal Conductors 200 Footnotes to Chapter 8.- 9. Materials with Memory and Axiomatic Foundations.- § 79. Introduction to a Relativistic Theory of Materials with Memory.- § 80. Intrinsic Kinematic Histories. Total Geodesic Derivatives.- § 81. A Relativistic Version of the Principle of Material (Frame) Indifference.- § 82. Some Consequences of the Principle of Material Indifference.- § 83. On the Axiomatic Foundations of the Preceding Theory. Primitive Notions and First Axioms.- § 84. On Kinematic Axioms and the Notion of Physical Possibility.- § 85. Conservation Equations and Maxwell Equations in Our Axiomatic Theory.- Footnotes to Chapter 9.- 10. Couple Stresses and More General Stresses.- § 86. Introduction.- § 87. Contributions of Couple Stresses to the Expression of U?? and to the Equation of Energy Balance.- § 88. The Relativistic Cauchy Equations of Continuous Media in the Case of Couple Stresses.- § 89. The Non-Working Part of m???.- § 90. Some Commutation Formulas for Lagrangian Spatial Derivatives..- § 91. A Useful Expression for $$ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} _{LAB}} $$.- § 92. A Lagrangian Expression for the Work of Stress and Couple Stress in Special or General Relativity.- § 93. Elasticity with Couple Stress.- § 94. Hints at Non-Viscous Fluids Capable of Couple Stress and at Electromagnetoelasticity with Couple Stress.- § 95. Some Preliminary Variational Formulas Related to Second Order Lagrangian Kinematics and the Variation of Space-Time Metric.- § 96. A Variational Principle Involving Couple Stress and the Variation of Space-Time Metric.- § 97. A Variational Principle Involving the Variation of World Lines in the Presence of Couple Stresses. On Constitutive Equations.- § 98. On General Materials of Order n=2 in the Adiabatic Case. Variations of g?? and World Lines.- § 99. Variational Principles for Elastic Materials of any Order n ? 1, not Capable of Heat Conduction.- Footnotes to Chapter 10.- Appendix A. Double Tensors.- §A1. Definition of Double Tensors Related to Two Topological Spaces.- §A2. Partial Covariant Derivative and Total Covariant Derivative Based on a Mapping.- §A3. On Differentiation of Double Tensors, Functions of Double Tensors.- Case of Arguments Fulfilling Typical Regular Conditions.- Appendix C. On the Divergence of Spatial Vectors in Space-Time.- References.