The Initial Value Problem for the Generalised Einstein Equations

TL;DR

This paper proves the existence and uniqueness of maximal globally hyperbolic developments for the Einstein equations in Hitchin's generalized geometry, extending classical results to a broader geometric framework involving scalar and B-fields.

Contribution

It establishes the well-posedness of the initial value problem for generalized Einstein equations, including gauge conditions and geometric uniqueness, in a novel geometric setting.

Findings

Existence of a maximal globally hyperbolic development (MGHD).

Development of a generalized Lorenz gauge for the B-field.

Geometric uniqueness of solutions up to diffeomorphism.

Abstract

We discuss the initial value problem for the Einstein equations in Hitchin's generalised geometry for the case of closed divergence (which correspond to the equations of motion in the bosonic part of the NS-NS sector in type II ten-dimensional supergravity) and establish the existence of a maximal globally hyperbolic development (MGHD). The dynamical fields, defined on a manifold of dimension $n+1$, are the space-time metric, a scalar field known as the dilaton function, and a two-form known as the $B$-field. We develop a generalisation of the Lorenz gauge which, applied to the $B$-field (and combined with a suitable gauge condition breaking diffeomorphism invariance), renders the system a wave equation with principal symbol given by the (dynamical) metric. Given initial data, we construct a development satisfying the gauge conditions. We show that all other developments are (in the appropriate sense) related to this development by a diffeomorphism, establishing geometric uniqueness. The existence of the MGHD follows then by a famous result by Choquet-Bruhat and Geroch. In showing existence and geometric uniqueness of developments, we follow an approach developed in detail by Ringstr\"om for the Einstein equations coupled to a scalar field. In a preliminary section, we present a formulation which is disentangled from the specific assumptions made on the matter, so that adaptation to other systems is straightforward.