  We study Lorentzian manifolds with a weight function such that the
$N$-Bakry-\'Emery tensor is bounded below. Such spacetimes arise in the physics
of scalar-tensor gravitation theories, including Brans-Dicke theory, theories
with Kaluza-Klein dimensional reduction, and low-energy approximations to
string theory. In the "pure Bakry-\'Emery" $N= \infty$ case with $f$ uniformly
bounded above and initial data suitably bounded, cosmological-type singularity
theorems are known, as are splitting theorems which determine the geometry of
timelike geodesically complete spacetimes for which the bound on the initial
data is borderline violated. We extend these results in a number of ways. We
are able to extend the singularity theorems to finite $N$-values $N\in
(n,\infty)$ and $N\in (-\infty,1]$. In the $N\in (n,\infty)$ case, no bound on
$f$ is required, while for $N\in (-\infty,1]$ and $N= \infty$, we are able to
replace the boundedness of $f$ by a weaker condition on the integral of $f$
along future-inextendible timelike geodesics. The splitting theorems extend
similarly, but when $N=1$ the splitting is only that of a warped product for
all cases considered. A similar limited loss of rigidity has been observed in
prior work on the $N$-Bakry-\'Emery curvature in Riemannian signature when
$N=1$, and appears to be a general feature.
