  Timelike sectional curvature bounds play an important role in spacetime
geometry, both for the understanding of classical smooth spacetimes and for the
study of Lorentzian (pre-)length spaces introduced in
\cite{kunzinger2018lorentzian}. In the smooth setting, a bound on the sectional
curvature of timelike planes can be formulated via the Riemann curvature
tensor. In the synthetic setting, bounds are formulated by comparing various
geometric configurations to the corresponding ones in constant curvature
spaces. The first link between these notions in the Lorentzian context was
established in \cite{harris1982triangle}, which was instrumental in the proof
of powerful results in spacetime geometry \cite{beem1985toponogov,
beem1985decomposition, galloway2018existence}. For general semi-Riemannian
manifolds, the equivalence between sectional curvature bounds and synthetic
bounds was established in \cite{alexander2008triangle}, however in this
approach the sectional curvatures of both timelike and spacelike planes have to
be considered. In this article, we fill a gap in the literature by proving the
full equivalence between sectional curvature bounds on timelike planes and
synthetic timelike bounds on strongly causal spacetimes. As an essential tool,
we establish Hessian comparison for the time separation and signed distance
functions.
