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The Initial Value Problem for the Generalised Einstein Equations

Oskar Schiller

TL;DR

This work establishes the well-posedness of the initial value problem for the Generalised Einstein Equations (GEE) in Hitchin's generalized geometry under closed divergence, proving existence of a maximal globally hyperbolic development (MGHD) and geometric uniqueness. By reducing the GEE to a hyperbolic system through a local Einstein frame and a generalized Lorenz gauge for the B-field (with DeTurck-type gauge for the metric), the authors leverage Ringström's Einstein–scalar field methodology to construct local developments and patch them into a global MGHD, while carefully handling initial data and gauge-invariance across Einstein frames. The analysis provides a frame-dependent yet compatible framework for the string-frame and Einstein-frame formulations, including precise initial data constraints, gauge conditions, and the coalescence of B-field and dilaton dynamics with the metric. The result extends the classical Choquet-Bruhat–Geroch MGHD theory to Generalised Einstein Equations, offering a rigorous mathematical foundation for the IVP in generalized geometry and the NS-NS sector of type II supergravity, with a formulation amenable to adaptation to other matter models. Overall, the paper delivers a robust, gauge-fixed, hyperbolic treatment of the GEE IVP and confirms the existence and uniqueness of the MGHD in this generalized geometric setting.

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 Ringström 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.

The Initial Value Problem for the Generalised Einstein Equations

TL;DR

This work establishes the well-posedness of the initial value problem for the Generalised Einstein Equations (GEE) in Hitchin's generalized geometry under closed divergence, proving existence of a maximal globally hyperbolic development (MGHD) and geometric uniqueness. By reducing the GEE to a hyperbolic system through a local Einstein frame and a generalized Lorenz gauge for the B-field (with DeTurck-type gauge for the metric), the authors leverage Ringström's Einstein–scalar field methodology to construct local developments and patch them into a global MGHD, while carefully handling initial data and gauge-invariance across Einstein frames. The analysis provides a frame-dependent yet compatible framework for the string-frame and Einstein-frame formulations, including precise initial data constraints, gauge conditions, and the coalescence of B-field and dilaton dynamics with the metric. The result extends the classical Choquet-Bruhat–Geroch MGHD theory to Generalised Einstein Equations, offering a rigorous mathematical foundation for the IVP in generalized geometry and the NS-NS sector of type II supergravity, with a formulation amenable to adaptation to other matter models. Overall, the paper delivers a robust, gauge-fixed, hyperbolic treatment of the GEE IVP and confirms the existence and uniqueness of the MGHD in this generalized geometric setting.

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 , are the space-time metric, a scalar field known as the dilaton function, and a two-form known as the -field. We develop a generalisation of the Lorenz gauge which, applied to the -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 Ringström 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.
Paper Structure (25 sections, 32 theorems, 116 equations, 1 figure, 1 table)

This paper contains 25 sections, 32 theorems, 116 equations, 1 figure, 1 table.

Table of Contents

  1. Abstract
  2. Introduction
  3. The Cauchy Problem for General Einstein-Matter Systems.
  4. Acknowledgements.
  5. Notation and Conventions
  6. Preliminaries: Techniques Applicable to IVPs for Einstein-Matter Systems
  7. Quasi-Linear Hyperbolic PDEs with Metric Principal Symbol
  8. Uniqueness of Local Solutions.
  9. DeTurck's Gauge Condition
  10. The DeTurck gauge propagates well.
  11. Implementing the DeTurck gauge.
  12. Unions and Intersections of Globally Hyperbolic Subsets
  13. The Generalised Einstein Equations
  14. The Initial Value Problem in for the Generalised Einstein Equations
  15. The Initial Value Formulation
  16. ...and 10 more sections

Key Result

Theorem 2.4

Let $(\Sigma, g_0, k, H_0, h_0, \xi_0, x_0)$ be initial data for the string frame GEE (combinedgeneinsteineqs). Then there exists a globally hyperbolic string frame development of the data.

Figures (1)

  • Figure 1: An illustration of the data associated to a point $p$ of the initial hypersurface $\Sigma$ in the proof of Theorem \ref{['genEinsteinlocaluniquenessthm']}. On the left hand side is the gauge-fixed development $D$ of the initial data constructed in Theorem \ref{['genEinsteinlocalexistencethm']}. On the right hand side is an arbitrary development $M'$. The two developments are compared (in a local Einstein frame) via a diffeomorphism $\tilde{f}_p\colon \widetilde{W}_p \to \widetilde{W}'_p$.

Theorems & Definitions (70)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 60 more