By A. Lichnerowicz
For seventy years, we've recognized that Einstein's thought is largely a idea of propagation of waves for the gravitational box. Confusion enters, notwithstanding, during the incontrovertible fact that the note wave, in physics, implies occasionally repetition and occasionally no longer. This confusion is usually elevated via he use of Fourier transforms, during which a disturbanse which seems to be with no repetition is resolved into periodic wave-trains with all frequencies. yet, in a basic curved space-time, now we have not anything similar to Fourier transforms. the following, we think about systematically waves resembling the propagation of discontinuities of actual amounts describing both fields (essentially electromagnetic fields and gravitational field), or the movement of a fluid, or jointly, in magnetohydrodynamics, the adjustments in time of a box and of a fluid. the most equations, for the various studied phenomena, represent a hyperbolic method and the examine of a proper Cauchy challenge is feasible. We name usual waves the case during which the by-product of more suitable order showing within the method are discontinuous on the traverse of a hypersurface, the wave entrance ; we name surprise waves the case the place the derivatives of an order inferior by means of one are discontinuous on the traverse of a wave entrance. XI xii PREFACE From 1950, many recognized scientits (Taub, Synge, Choquet-B ruhat, etc.) have studied the corresponding equations for various actual phenomena : platforms linked to the electromagnetic and gravitational fields, to hydrodynamics and to magnetohydrodynamics.
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For seventy years, we have now recognized that Einstein's thought is basically a idea of propagation of waves for the gravitational box. Confusion enters, despite the fact that, during the indisputable fact that the note wave, in physics, implies occasionally repetition and occasionally now not. This confusion is usually elevated through he use of Fourier transforms, through which a disturbanse which seems to be with out repetition is resolved into periodic wave-trains with all frequencies.
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Additional info for Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time
1 1) 0 0 . ~ (Cauchy data). Moreover, if an electromagnetic field system satisfies ( 113. 3. 11) outside r . 5). 3. 3. 10) gives the values over a0 and the three derivatives r of the three derivatives F0 i . It follows that, r, o0 Fjk for a solution of Maxwell's equations, the first derivatives of Foc~ may not be discontinuous at the traverse of r (si nee g00 ~ 0 ) . We see that the characteristic manifolds of the Maxwell system under consideration are the hypersurface tangent to the elementary cones.
1. 1), where (Ill. 4) where the coefficients of the pseudoriemannian connection are denoted by 5) satisfying over L: (y 0 = 0) the condition Lorentz (£F) 0 = J 0 is a solution of the problem and the potential vector for the Maxwell ex satisfies the equation Lorentz condition identically. 5. Ordinary electromagnetic waves. We have supposed that the electromagnetic field F is (C 0 , piecewise C2 ). We suppose here that the electric current J is continuous. 1) continuous) such that the derivatives of F are regu Jar Jy discontinuous at the traverse of a hypersurface L: which is called the wave front.
5) satisfying over L: (y 0 = 0) the condition Lorentz (£F) 0 = J 0 is a solution of the problem and the potential vector for the Maxwell ex satisfies the equation Lorentz condition identically. 5. Ordinary electromagnetic waves. We have supposed that the electromagnetic field F is (C 0 , piecewise C2 ). We suppose here that the electric current J is continuous. 1) continuous) such that the derivatives of F are regu Jar Jy discontinuous at the traverse of a hypersurface L: which is called the wave front.