Numerical Study on the Causal Set Theory Path-Geodesic Corresondence hypothesis

Given a set of points $\mathcal{C}$ in a manifold $\mathcal{M}$ and a causal relation $\preceq$ one can describe a path between two points of $\mathcal{C}$ as a sequence of points $x_0\preceq x_1 \preceq\dots \preceq x_N$ for $x_i\in\mathcal{C}$. The Path-Geodesic correspondence hypothesis suggests that in the continuum limit (in which the number point density approaches infinity); the maximal length path between two points will approaxh the trajectory followed by a geodesic along the manifold $\mathcal{M}$. This project (alongside its accompanying research paper ) aims to test this hypothesis using a (1+1) FLRW background given by the differential line element $\text{d}s^2=\text{d}t^2-a(t)\frac{\text{d}r^2}{1-\kappa r^2}$.

To do this, the following (python) classes have been created, specifying the metric, the causet parameters and the geodesic trajectory

MetricTensor(kappa, a1)
CausetSimulation(Metric, TimeRange, SpaceRange, PointNumber, Divisions)
ContinuumSimulation(Metric, source, SpacialVelocity, tau_span, g_type)

Some example scripts are provided, which returned the following results.

Causet and corresponding Hasse Diagram on a Minkowskian background (a(t)=1 & κ=0).

Causet and corresponding Hasse Diagram on a Hyperbolic background (a(t)=1 & κ=-1).

Continuum (Blue) and Discrete (Red) geodesics in an Spherical background (a(t)=1 & κ=1).

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