  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 $M$.
