d’Alembert operator

On a Lorentzian manifold $(M,g)$ of signature $(1,n-1)$ , the d’Alembert operator or wave operator is

\Box_g f=-\operatorname{tr}_g(\nabla df) =-\frac{1}{\sqrt{|\det g|}}\, \partial_\mu\!\left(\sqrt{|\det g|}\,g^{\mu\nu}\partial_\nu f\right).

It is the Lorentzian instance of the Laplace–Beltrami operator in the sign convention adopted here.

Minkowski form

On Minkowski spacetime with

g=\operatorname{diag}(-1,1,\ldots,1),

one has

\Box_g=\partial_t^2-\sum_{j=1}^{n-1}\partial_{x_j}^2.

Its principal symbol is

\sigma_2(\Box_g)(x,\xi)=-g_x^{-1}(\xi,\xi),

which vanishes exactly on null covectors. Thus $\Box_g$ is hyperbolic rather than elliptic.

Geometric role

The operator is normally hyperbolic. A hypersurface is characteristic for $\Box_g$ precisely where its conormal covector is null. This local symbol statement is distinct from the global Cauchy theorem and the existence of advanced and retarded Green operators, which require global hyperbolicity.

Sign convention

Many authors define $\Box_g=\operatorname{tr}_g\nabla d$ . With the same metric signature their operator is the negative of this one. The displayed Minkowski formula, not the symbol $\Box$ by itself, fixes the convention.

References

Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §§1.5 and 3.2.
Lars Hörmander, The Analysis of Linear Partial Differential Operators III, Springer, 1985. Publisher record. Relevant: Chapter XXIII.