读书笔记: 变系数波方程

The Motion of Point Particles in Curved Spacetime

Consider the wave equation initial value problem in R3\mathbb R^3 with spatially variable wave
speed, denoted by

∂2∂t2u(x,t)−c2(x)Δu(x,t)u(x,0)∂∂tu(x,0)=0in R3×(0,∞),=f(x),=g(x).

\begin{align*} \frac{\partial^2}{\partial t^2}u(x,t)-c^2(x)\Delta u(x,t)&=0\hspace{.2in}\text{in }\mathbb R^3\times(0,\infty),\\ u(x,0)&=f(x),\\\frac{\partial}{\partial t}u(x,0)&=g(x). \end{align*}
My question: Is there a way to explicitly write down a solution similar to Kirchhoff’s formula for
the case of constant speed c(x)=c0c(x)=c_0?
Intuitivly, i would expect the solution u(x0,t0)u(x_0,t_0) for fixed space and time to be dependent on
values of spherical integrals of ff and gg as in the constant speed case, but in a sound-speed
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 f,gf,g in this
integral, which are determined by the retarded Green function GRG_R of the wave operator:

□=∂2t−c(x)Δ.

\square = \partial_t^2 - c(x)\Delta. This Green function is intimately related to the geometry
of the spacetime with line element ds2=dt2−c(x)dx⃗ ⋅dx⃗ ds^2=dt^2-c(x)d\vec{x}\cdot d\vec{x}. In order to calculate
GRG_R, you essentially need to fully solve the geodesic equations obtained by extremising the
action ∫ds\int ds.
The retarded Green’s function satisfies

□GR(t,x;t′,x′)=−4πδ4(t,x;t′,x′),

\square G_R(t,x;t',x')=-4\pi\delta_4(t,x;t',x'), where
δ4\delta_4 is the 4-dimensional Dirac distribution. Then the generalized Kirchhoff formula allows
us to explicitly write down the value of a solution Ψ\Psi of the wave equation at a point (t,x)(t,x) to
the future of an initial data hypersurface Σ′={(t′,x′):x′∈R3}\Sigma^\prime=\{(t',x'):x'\in\mathbb{R}^3\}:

Ψ(t,x)=−14π∫Σ′(GR(t,x;t′,x′)g(x′)−f(x′)∂t′GR(t,x;t′,x′))dΣ′,

\Psi(t,x)=-\frac{1}{4\pi}\int_{\Sigma^\prime}(G_R(t,x;t',x')g(x')- f(x')\partial_{t'}G_R(t,x;t',x'))d\Sigma^\prime,
where dΣ′=c3/2(x′)d3x′d\Sigma^\prime=c^{3/2}(x')d_3x' is the volume element on Σ′\Sigma^\prime.
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 GRG_R,
which is required for practical applications of the Kirchhoff formula.

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