Discrete geometry of a small causal diamond

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance $⟨C_k⟩$ of $k$-element chains or total orders in the underlying causal set $\mathbf{C}$. We begin by obtaining the first-order curvature corrections to the flat spacetime expression for $⟨C_k⟩$ using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of $\mathbf{C}$ as well as the time-time component of its Ricci tensor in terms of the $⟨C_k⟩$. We also find a new dimension estimator for $\mathbf{C}$ which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

Figure 1

An Alexandroff interval $\mathbf{I}[p,q]$ in flat spacetime. $T_1$ is the proper time between the events $x_1$ and $q$ and $y_1$ is the midpoint of $\mathbf{I}[x_1,q]$.

Figure 2

The contributions to $⟨C_3^2⟩$ from no coincidences and one coincidence are shown.