Normally hyperbolic operator

Definition

Let $(M,g)$ be a Lorentzian manifold and $E\to M$ a smooth vector bundle. A second-order differential operator

P:\Gamma^\infty(E)\longrightarrow\Gamma^\infty(E)

is normally hyperbolic in the convention used here if its principal symbol is

\sigma_2(P)(x,\xi)=-g_x^{-1}(\xi,\xi)\operatorname{id}_{E_x}.

Equivalently, in local coordinates its second-order part is $-g^{\mu\nu}\partial_\mu\partial_\nu$ times the identity.

The principal-symbol condition is local. It implies the unique connection form of $P$ , but global existence and causal support require additional hypotheses. On a globally hyperbolic spacetime they are supplied by the Cauchy theorem and the Green-operator existence theorem.

Examples

The scalar d’Alembert operator is normally hyperbolic. Adding a mass term or any smooth endomorphism preserves normal hyperbolicity, so the Klein–Gordon operator is another example. The square of a Lorentzian Dirac-type operator is normally hyperbolic up to the convention-dependent overall sign.

Distinction from ellipticity

The symbol fails to be invertible on every nonzero null covector. A normally hyperbolic operator is therefore not elliptic on a Lorentzian manifold of dimension at least two. Elliptic boundary-value theory and hyperbolic initial-value theory are different analytic settings.

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 3.2–3.4.
Nicolas Ginoux, The Dirac Spectrum, Springer, 2009. Publisher record. Relevant: Appendix A.