Euclidean (A)dS spaces over $p$-adic numbers

Abstract

With the help of Wick rotation over $p$-adic numbers ${\mathbb{Q}}_p$, the $p$-adic version of Euclidean ${\text{dS}}_2$ space(noted as $p{\text{dS}}_2$) is obtained based on $p{\text{AdS}}_2$($p$-adic version of Euclidean ${\text{AdS}}_2$ space), the latter of which is already known. The corresponding embedding equations are also found. The distances $D(X,Y)$'s on $p{\text{(A)dS}}_1$ and $p{\text{AdS}}_2$ have intuitive explanations. On the graph representations of ${\mathbb{Q}}_p$ and ${\mathbb{Q}}_{p^2}$, namely Bruhat-Tits trees ${\text{T}}_p$ and ${\text{T}}_{p^2}$, $D(X,Y)$ is found to be the inverse of distance between a particular subgraph and the line connecting $X$ and $Y$.

10 pages, 3 figures

References in corpus (2)

• Yet another p-adic hyperbolic disc
• Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs