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

TL;DR

This paper constructs $p$-adic analogs of Euclidean de Sitter and Anti-de Sitter spaces using Wick rotation over $ ext{Q}_p$, explores their embedding equations, and interprets distances via Bruhat-Tits trees.

Contribution

It introduces the $p$-adic Euclidean (A)dS spaces based on known $p$-adic AdS, derives their embedding equations, and provides geometric interpretations of distances on Bruhat-Tits trees.

Findings

Defined $p$-adic Euclidean $ ext{dS}_2$ space using Wick rotation.

Derived embedding equations for $p$-adic (A)dS spaces.

Connected distances on Bruhat-Tits trees to subgraph and line distances.

Abstract

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