Linear hyperbolic equations in a double null foliation

TL;DR

This paper establishes existence and uniqueness results for linear hyperbolic systems in a double null foliation of spacetime, and explores their connection to the Bianchi equations in vacuum Einstein solutions.

Contribution

It proves a global existence and uniqueness theorem for linear hyperbolic systems in a double null foliation and introduces a new algebraic constraint relating these systems to the Bianchi equations.

Findings

Proved global existence and uniqueness for linear hyperbolic systems.

Derived a novel algebraic constraint for tensorfields satisfying linearized Bianchi equations.

Clarified the relationship between hyperbolic systems and Bianchi equations in vacuum spacetimes.

Abstract

The Bianchi identities for the Weyl curvature tensor of a spacetime $(M, g)$ solving the vacuum Einstein equations in a double null foliation exhibit a hyperbolic structure, which can be used to obtain detailed nonlinear estimates on the null Weyl tensor components. The aim of this paper is twofold. First we discuss existence and uniqueness for solutions of first-order linear hyperbolic systems of equations in a double null foliation on an arbitrary spacetime, with initial data posed on a past null hypersurface $\underline C_0 \cup C_0$. We prove a global existence and uniqueness theorem for these systems. Then we discuss the relationship between these systems, the Bianchi equations, and the linearized Bianchi equations (the linearized Bianchi equations are obtained from the usual Bianchi equations by replacing the null Weyl tensor components with unknown tensorfields). We derive a novel algebraic constraint which must be satisfied, at every point in the spacetime, by tensorfields satisfying the linearized Bianchi equations.