  Let $\mathcal{A}_0$ and $\mathcal{A}_1$ be two self-adjoint Fredholm
Dirac-type operators defined on two non-compact manifolds. If they coincide at
infinity so that the relative heat operator is trace-class, one can define
their relative eta function as in the compact case. The regular value of this
function at the zero point, which we call the relative eta invariant of
$\mathcal{A}_0$ and $\mathcal{A}_1$, is a generalization of the eta invariant
to non-compact situation. We study its variation formula and gluing law. In
particular, under certain conditions, we show that this relative eta invariant
coincides with the relative eta invariant that we previously defined using APS
index of strongly Callias-type operators.
