Dominant energy condition and spinors on Lorentzian manifolds

TL;DR

This paper investigates the properties of the Dirac--Witten operator on Lorentzian spin manifolds under the dominant energy condition, using index theory to analyze the topology of initial data sets and the kernel of the operator.

Contribution

It introduces a Lorentzian analogue of Hitchin's alpha-invariant and extends index theoretical methods to weak energy condition cases, linking kernel properties to manifold topology.

Findings

Invertibility of Dirac--Witten operator under strict dominant energy condition

Extension of index methods to weak energy condition cases

Characterization of kernel elements as restrictions of parallel spinors on extended manifolds

Abstract

Let $(\overline M,\overline g)$ be a time- and space-oriented Lorentzian spin manifold, and let $M$ be a compact spacelike hypersurface of $\overline M$ with induced Riemannian metric $g$ and second fundamental form $K$. If $(\overline M,\overline g)$ satisfies the dominant energy condition in a strict sense, then the Dirac--Witten operator of $M\subseteq \overline M$ is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's $\alpha$-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac--Witten operator may be non-invertible, and we will study the kernel of this operator in this case. We will show that the kernel may only be non-trivial if $\pi_1(M)$ is virtually solvable of derived length at most $2$. This allows to extend the index theoretical methods to spaces of initial data, satisfying the dominant energy condition in the weak sense. We will show further that a spinor $\phi$ is in the kernel of the Dirac--Witten operator on $(M,g,K)$ if and only if $(M,g,K,\phi)$ admits an extension to a Lorentzian manifold $(\overline N,\overline h)$ with parallel spinor $\bar\phi$ such that $M$ is a Cauchy hypersurface of $(\overline N,\overline h)$, such that $g$ and $K$ are the induced metric and second fundamental form of $M$, respectively, and $\phi$ is the restriction of $\bar\phi$ to $M$.