Dominant energy condition and spinors on Lorentzian manifolds

Let $(\bar{M},\bar{g})$ be a time- and space-oriented Lorentzian spin manifold, and let $M$ be a compact spacelike hypersurface of $\bar{M}$ 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 $M\subseteq \bar{M}$ 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 $\varphi$ is in the kernel of the Dirac–Witten operator on $(M,g,K)$ if and only if $(M,g,K,\varphi )$ admits an extension to a Lorentzian manifold $(\overline N,\overline h)$withparallelspinor$\bar\phi$suchthat$M$ is a Cauchy hypersurface of $(\bar{N},\bar{h})$, such that $g$ and $K$ are the induced metric and second fundamental form of $M$, respectively, and $\varphi$ is the restriction of $\bar{\varphi }$ to $M$.