I was reading about the finite propagation speed of the wave equation on a Riemannian manifold. I was wondering if instead of the Laplace-Beltrami operator $\Delta$ we consider the equation $$\partial\_t^2 u = Lu$$ where $L$ is a negative-definite self-adjoint elliptic operator of order 2, would we still have finite speed of propagation property? Thanks a lot for your help.



Finite propagation speed of second order operators
Your example is a special case of an operator with a wave-like principal symbol on a Lorentzian manifold. Namely, if the coefficients of the second order derivatives in $L$ are $L = h^{ij} \partial\_i \partial\_j + \cdots$, then $h^{ij}$ is an inverse Riemannian metric and it is always possible to write $L = \Delta\_h + B$, with $\Delta\_h$ the $h$-Laplacian and $B$ a first order differential operator. Your equation then becomes $\Delta\_g \psi = B\psi$, where $\Delta\_g = \partial\_t^2 - \Delta\_h$ is the $g$-Laplacian and $g=\mathrm{diag}(1,-h)$ is a Lorentzian metric. In other words, $\Delta\_g$ is a wave operator. Thus, you get the usual finite speed of propagation results for your equation, because the lower order operator $B$ does not affect them.


An excellent treatment of equations with wave-like principal symbols can be found in [Wave Equations on Lorentzian Manifolds and Quantization](http://arxiv.org/abs/0806.1036) by
Christian Baer, Nicolas Ginoux and Frank Pfaeffle.


