The present book is devoted to certain mathematical aspects of wave propogation in curved space-times. In the
centre of the considerations are those phenomena, whose
physical laws can be described by or reduced to linear,
hyperbolic differential equations of second order P[u]
with metric principal part. The propagating quantity u
can be a scalar or tensor field, a density — generally
speaking, a section of a real or complex vector bundle E
over the underlying spacetime (M,g}. The source of the
propogation process is given by the inhomogeneous term f
and is, in general, a distribution. The hyperbolic character
of the differential equation leads to a finite propogation
velocity. Let us assume that the source acts only in a small
spatial domain and only during a short time interval
[to,to+At]. An observer situated at a space point x
receives then the arising wave at a later time t 1  t O and
during a time interval of length ~ At. The question is,
what happens after t 1 + At at x? There are certain
special operators P, for which always u{t,x} = O, if
t > t 1 + At, independent of the nature of the source and the
position x of the observer. For the other operators P
one has, in general, u{t,x} M O, if t > t 1 + At; i.e. the
Preface
observer at x receives a rest wave during a long period.
In the first case one says' P is a Huygens' operator, (or:
Huygens' principle is valid for P). Obviously, Huygens'
operators are suitable for a sharp relay of signals. In
mathematical language we can formulate' for a Huygens'
operator P the support of the forward solution of
P[u] = f is concentrated at those (t,x) which can be
joined with a {to,Xo} of supp f by a geodesic null line
{bicharacteristic} .