The Wave Equation on a Curved Space-Time
This book was originally published in 1975. In Einstein's General Theory of Relativity the effects of gravitation are represented by the curvature of space-time. Physical processes occurring in the presence of gravitation must then be treated mathematically in terms of their behaviour in a curved space-time. One of the most basic of these processes is wave propagation, and this book gives a rigourous discussion of the local effects of curvature on the behaviour of waves. In the course of this discussion many techniques are developed which are also needed for a study of more general problems, in which the gravitational field itself plays a dynamical role. Although much of the book deals with four-dimensional space-time, the n-dimensional case is also treated, more briefly. The subject-matter is also of interest in other branches of mathematical physics and, as a fresh account of the classical work of Hadamard and M. Riesz, in the theory of partial differential equations.
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Characteristics and the propagation
Fundamental solutions page
Wave equations on ndimensional spacetimes
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adjoint analytic continuation bitensor C+(q C00 function called Cauchy data Cauchy problem causal domain caustic characteristic initial value compact set compact subset compact support components constant continuous linear map converges coordinate chart coordinate neighbourhood coordinate system coordinate transformation covector curves d'Alembertian defined denote derivatives differential equations differential operator distribution finite number follows fundamental solution future-compact future-directed G+(p geodesically convex domain Hence hypersurface identity implies initial value problem Lemma linear form locally integrable function manifold metric Minkowskian normal null cone null field null geodesies null semi-cone obtains obviously open set ordinary wave equation orientation parameter parametrix partition of unity proof of Theorem proved replaced satisfies second member semi-norm semi-norm estimate sequence space-like space-time supp support is contained Suppose tangent vector term test functions theory time-like time-orientation topology transport equations unique vanish vector field vector space wave equation zero