  In this note, we consider perturbations of Minkowski space as well as more
general spacetimes on which the wave operator $\square_g$ is essentially
self-adjoint. We review a recent result which gives the meromorphic
continuation of the Lorentzian spectral zeta function density, i.e. of the
trace density of complex powers $\alpha \mapsto (\square_g-i
\varepsilon)^{-\alpha}$. In even dimension $n\geq 4$, the residue at
$\frac{n}{2}-1$ is shown to be a multiple of the scalar curvature in the limit
$\varepsilon\to 0^+$. This yields a spectral action for gravity in Lorentzian
signature.
