Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time

For seventy years, we have known that Einstein's theory is essentially a theory of propagation of waves for the gravitational field. Confusion enters, however, through the fact that the word wave, in physics, implies sometimes repetition and sometimes not. This confusion is often increased by he use of Fourier transforms, by which a disturbanse which appears to be without repetition is resolved into periodic wave-trains with all frequencies. But, in a general curved space-time, we have nothing corresponding to Fourier transforms. Here, we consider systematically waves corresponding to the propagation of discontinuities of physical quantities describing either fields (essentially electromagnetic fields and gravitational field), or the motion of a fluid, or together, in magnetohydrodynamics, the changes in time of a field and of a fluid. The main equations, for the different studied phenomena, constitute a hyperbolic system and the study of a formal Cauchy problem is possible. We call ordinary waves the case in which the derivative of superior order appearing in the system are discontinuous at the traverse of a hypersurface, the wave front ; we call shock waves the case where the derivatives of an order inferior by one are discontinuous at the traverse of a wave front. XI xii PREFACE From 1950, many well-known scientits (Taub, Synge, Choquet-B ruhat, etc.) have studied the corresponding equations for different physical phenomena : systems associated to the electromagnetic and gravitational fields, to hydrodynamics and to magnetohydrodynamics.

Klappentext
For seventy years, we have known that Einstein's theory is essentially a theory of propagation of waves for the gravitational field. Confusion enters, however, through the fact that the word wave, in physics, implies sometimes repetition and sometimes not. This confusion is often increased by he use of Fourier transforms, by which a disturbanse which appears to be without repetition is resolved into periodic wave-trains with all frequencies. But, in a general curved space-time, we have nothing corresponding to Fourier transforms. Here, we consider systematically waves corresponding to the propagation of discontinuities of physical quantities describing either fields (essentially electromagnetic fields and gravitational field), or the motion of a fluid, or together, in magnetohydrodynamics, the changes in time of a field and of a fluid. The main equations, for the different studied phenomena, constitute a hyperbolic system and the study of a formal Cauchy problem is possible. We call ordinary waves the case in which the derivative of superior order appearing in the system are discontinuous at the traverse of a hypersurface, the wave front ; we call shock waves the case where the derivatives of an order inferior by one are discontinuous at the traverse of a wave front. XI xii PREFACE From 1950, many well-known scientits (Taub, Synge, Choquet-B ruhat, etc.) have studied the corresponding equations for different physical phenomena : systems associated to the electromagnetic and gravitational fields, to hydrodynamics and to magnetohydrodynamics.

Zusammenfassung
Here, we consider systematically waves corresponding to the propagation of discontinuities of physical quantities describing either fields (essentially electromagnetic fields and gravitational field), or the motion of a fluid, or together, in magnetohydrodynamics, the changes in time of a field and of a fluid.

Inhalt
I Tensor Distributions.- II Maxwell's equations and electromagnetic waves over a curved space-time.- III Einstein's equations and the Leray theorem.- IV Gravitational and electromagnetic shock waves.- V Relativistic hydrodynamics.- VI The equations of magnetohydrodynamics.- VII Magnetohydrodynamic shock waves.- VIII Hugoniot's function and applications.- Annex I Shock waves and alfven waves.- AI.1. Singular shocks.- AI.2. Compatibility between shock waves and Alfven waves.- Annex II Magnetosonic rays.- AII.1. Directions of the rays.- AII.2. Action of 6 on the direction of the ray.- Annex III Classical approximations of the relativistic shock equations.- AIII.1. The frame connected with the shock.- AIII.2. Classical approximation.- Note Approach to a quantum theory of fields for a curved space-time.- I Tensor propagators.- NI. 1. Orientations over a space-time.- NI.2. Global hyperbolicity.- NI.3. Bitensors and Dirac bitensors.- NI.4. Linear differential-tensor operators associated with g...- NI.5. Elementary Kernels and propagators.- NI.6. Tensor propagators relative to the space-time of.- Minkowski.- NI.7. Propagators relative to the operator (? + ?).- II Applications to quantization problems over a curved space-time.- NII.1. Commutator for vector Meson.- NII.2. Commutator for a free electromagnetic field.- NII.3. Commutator for a varying gravitational field with mass.- NII.4. Commutator for a varying gravitational field without mass term.- NII.5. Creation. Annihilation operators.- NII.6. Dirac field.- References for the Note.

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