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For a Lorentzian space measured by $\mathfrak{m}$ in the sense of Kunzinger,
S\”amann, Cavalletti, and Mondino, we introduce and study synthetic notions of
timelike lower Ricci curvature bounds by $K\in\boldsymbol{\mathrm{R}}$ and
upper dimension bounds by $N\in[1,\infty)$, namely the timelike
curvature-dimension conditions $\smash{\mathrm{TCD}_p(K,N)}$ and
$\smash{\mathrm{TCD}_p^*(K,N)}$ in weak and strong forms, and the timelike
measure-contraction properties $\smash{\mathrm{TMCP}_p(K,N)}$ and
$\smash{\mathrm{TMCP}_p^*(K,N)}$, $p\in (0,1)$. These are formulated by
convexity properties of the R\’enyi entropy with respect to $\mathfrak{m}$
along $\smash{\ell_p}$-geodesics of probability measures.

We show many features of these notions, including their compatibility with
the smooth setting, sharp geometric inequalities, stability, equivalence of the
named weak and strong versions, local-to-global properties, and uniqueness of
chronological $\smash{\ell_p}$-optimal couplings and chronological
$\smash{\ell_p}$-geodesics. We also prove the equivalence of
$\smash{\mathrm{TCD}_p^*(K,N)}$ and $\smash{\mathrm{TMCP}_p^*(K,N)}$ to their
respective entropic counterparts in the sense of Cavalletti and Mondino.

Some of these results are obtained under timelike $p$-essential nonbranching,
a concept which is a priori weaker than timelike nonbranching.

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