Consider the wave equation initial value problem in
speed, denoted by
My question: Is there a way to explicitly write down a solution similar to Kirchhoff’s formula for
the case of constant speed
Intuitivly, i would expect the solution
values of spherical integrals of
dependent metric.
This is related to a similar question concerning a wave equation in 1 dimension: Wave propagationWav e propagation with v ariable wav e speed and Spherical means (Kirchoff’s formula) for v ariable speed wav e equation
There is an extensive literature on wave equations with variable coefficients, which are equivalent
to wave equations on curved spacetime - as in the monograph by Friendlander. Depending on what
you are willing to accept as ‘explicit’, the answer is that yes, there is an explicit generalisation of the
Kirchhoff integral. However, it is very difficult to calculate the coefficients of the data
integral, which are determined by the retarded Green function
of the spacetime with line element
action
The retarded Green’s function satisfies
us to explicitly write down the value of a solution
the future of an initial data hypersurface
where
These things (Green’s function, waves in curved spacetime) are of great interest in General
Relativity, and the GR literature is a good place to look for more details. In particular, I
recommend starting with Poisson’s Living Review article which covers the geometrical background
in a very readable way. You’ll find details here (and in the citations) on how to calculate
which is required for practical applications of the Kirchhoff formula.