Simulating, Visualizing and Playing with de Sitter and anti de Sitter spacetime

TL;DR

The paper addresses how to convey relativistic non-Euclidean spacetimes, specifically $de\ Sitter$ and $anti-de Sitter$ spaces, through playable computer simulations. It develops a relativistic engine based on the Minkowski hyperboloid model and the universal cover to simulate $\\widetilde{ad\\mathbb{S}^d}$ and $d\\mathbb{S}^d$, using shift-based isometries and tessellations to manage geodesic motion and numerical precision. The main contributions are two prototype games, visualizations, and implementation details that enable intuition about time dilation, length contraction, and spacetime symmetries, with extensibility to education and research. The work demonstrates playable, visually rich tools for exploring complex relativistic geometry and motivates further multi-dimensional and interactive extensions for broader impact.

Abstract

In this paper we discuss computer simulations of de Sitter and anti de Sitter spacetimes, which are maximally symmetric, relativistic analogs of non-Euclidean geometries. We present prototype games played in these spacetimes; such games and visualizations can help the players gain intuition about these spacetimes. We discuss the technical challenges in creating such simulations, and discuss the geometric and relativistic effects that can be witnessed by the players.

Figures (4)

• Figure 1: On the left, the {7,3} hyperbolic tessellation in Poincaré disk model. The Poincaré disk model is conformal: it does not distort small shapes, so all heptagons look close to regular; however, it distorts scale: all the heptagons shown are of the same size. On the right, the same scene in HyperRogue viewed from two points. In the Poincaré model, straight lines are projected as circular arcs orthogonal to the disk boundary; moving the center shows the player that the walls (orange) are indeed straight lines.
• Figure 2: Relative Hell. $\widetilde{ad\mathbb{S}^2}$ is displayed in the Poincaré disk model on the left, and the Beltrami-Klein disk model in the center. $d\mathbb{S}^2$ on the right, in stereographic projection.
• Figure 3: Anti-de Sitter spacetime: past light cone view (left), present (middle), and future light cone view (right). Note the stretched missile in the future light cone view. Taken from a replay; the red circle is the boundary of the light cone relative to a future position of the ship. Poincaré disk model.
• Figure 4: De Sitter spacetime: present in stereographic projection (left), spacetime view (center), and $\mathbb{H}^3$ view (right). Unfortunately, the spacetime and $\mathbb{H}^3$ views are difficult to capture in a still image (and also in video due to the compression).