Retarded Green Function in Curved Spacetime

That's an odd, convoluted way of putting it.

There is nothing too unusual about the Green function being nonzero inside the lightcone. This even happens in flat space in 2+1, 4+1, 6+1, etc. dimensions (cf. ripples on a 2-d surface, like a pond).

I disagree with the paragraph's assertion that spacetime curvature alters the speed of a wave. The speed of propagation is still c, but the shape of the wave may change, including new wavefronts developing inside the lightcone.

 

Yes, but please don't make it sound like ripples on a pond have anything to do with the wave equation! Water waves derive from a solution of an elliptic equation, in fact the three-dimensional Laplace's equation. And they have no "light cone" to be inside of - the maximum velocity of propagation of water waves is infinite.

 

Could be wrong, but it looks to me from Eq (1.13) that the effect on the electromagnetic Green's function is due solely to Ricci curvature. That is, it doesn't happen in empty space.

On the other hand, the corresponding equation (16.4) for the gravitational Green's function contains the Riemann tensor. In both cases, the equation is no longer just the wave equation, it's more like the Klein-Gordon equation, i.e. dispersive. Discontinuities will still propagate with velocity c, but low frequency waves will not, so that wave packets will tend to develop a tail.

 

Do you have a reference for that? I don't see how Laplace's equation can allow traveling wave solutions.

 

The definitive book on the subject of water waves is by J J Stoker, and is available online here.

In a nutshell, the motion of the fluid is irrotational, which implies that v is derivable from a velocity potential, v = ∇φ. Also incompressible,∇·v = 0, which implies that ∇²φ = 0.

For small amplitude waves in water of infinite depth, the boundary condition at the surface z = 0 is φ_tt + gφ_z = 0, leading to wave solutions of the form φ = exp(ikx + kz - iωt) with ω² = gk.

 

The fact that ikx and iωt have opposite signs inside the exponential would indicate that the full equations, including time, are hyperbolic. The spacelike part of the wave equation is of course Laplace's equation (or Helmholtz's equation).

The Boussinesq equation for water waves is hyperbolic, for example:

http://en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves)

 

Actually, doesn't it just indicate whether the wave is traveling to the left or right? In fact I originally had written +iωt and went back and changed it.

Laplace's equation is the full equation, not an approximation to something hyperbolic. The velocity potential φ(x, t) is a solution of ∇²φ = 0, with arbitrary time dependence! Only the boundary condition constrains the time dependence.

Regarding the boundary condition, there are approximation schemes for the two opposite cases: deep water and shallow water. ("Deep" and "shallow" means compared to the wavelength.) For deep water, the boundary condition is linear, and the dispersion formula I gave, ω² = gk, applies to the simplest case of infinite depth. For shallow water (equivalently, "long waves"), the boundary condition is nonlinear. Boussinesq is an example of this. It's more complex, but also more interesting because the solutions include solitons.