  A timelike space is a Hausdorff topological space equipped with a partial
order relation $<$ and a distance function $\rho$ satisfying a collection of
axioms including a set of compatibility conditions between the partial order
relation and the distance function. The distance function is defined only on a
subset of the product of the space with itself that contains the diagonal,
namely, $\rho(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between
pairs of distinct points in a triple $x,y,z$, whenever these distances are
defined, satisfy the so-called \emph{time inequality}, which is a reverse
triangle inequality $\rho(x,y)+\rho(y,z)\leq \rho(z,y)$. In the 1960s, Herbert
Busemann developed an axiomatic theory of timelike spaces and of locally
timelike spaces. His motivation comes from the geometry underlying the theory
of relativity, and he tried to adapt to this setting his geometric theory of
metric spaces, namely, his theory of $G$-spaces (geodesic spaces). The
classical example he considers is the $n$-dimensional Lorentzian space. Two
other interesting classes of examples of timelike spaces he introduced are the
timelike analogues of the Funk and Hilbert geometries. In this paper, we
investigate these two geometries, and in doing this, we introduce variants of
them, in particular the timelike relative Funk and Hilbert geometries, in the
Euclidean and spherical settings. We describe the Finsler infinitesimal
structure of each of these geometries (with an appropriate notion of Finsler
structure) and we display the interactions among the Euclidean and timelike
spherical geometries. In particular, we characterize the de Sitter geometry as
a special case of a timelike spherical Hilbert geometry. The final version of
this paper will appear in Differential Geometry and its Applications.
