Schwarzschild-de Sitter spacetime in regular coordinates with cosmological time

TL;DR

This work derives two horizon-regular FRW-based coordinate systems for Schwarzschild-de Sitter spacetime in the presence of a positive cosmological constant $\Lambda>0$. By solving the Einstein equations under FRW-like conditions and enforcing $\dot a^2/a^2=\Lambda a^2/3$, the authors obtain the metric family $ds^2_{\pm}=a^2(\eta)[-A_{\pm}d\eta^2+A_{\pm}^{-1}dr^2+r^2d\Omega^2]$ with $\ell=ar$ and $A_{\pm}=\tfrac12\big(f(\ell)\pm\sqrt{f(\ell)^2+4\ell^2(\Lambda/3)}\big)$, where $f(\ell)=1-2m/\ell-\Lambda\ell^2/3$. They explicitly construct coordinate transformations to Kottler coordinates, $\pm d\tilde t = a d\eta + \sqrt{\Lambda/3}\;\ell/(fA_{\pm})d\ell$, proving these are valid SdS descriptions, and show a separate mapping to Lake-Israel coordinates, while clarifying that these are not maximal de Sitter extensions. The analysis reveals the two branches are equivalent descriptions related by a time-space swap and discusses their causal structure, horizon-crossing properties, and Newtonian/limiting behaviors (including $\Lambda=0$ and $m=0$ limits). The results extend cosmological and local coordinate frameworks for SdS, providing tools for studying the exact interplay between black holes and cosmological backgrounds and enabling explicit cross-checks with standard SdS charts.

Abstract

Starting from the Einstein equations in Schwarzschild-de Sitter (SdS) spacetime and imposing Friedmann-Robertson-Walker coordinates at large distances, we find two coordinate systems with time-dependent metrics that are smooth across both the black hole and cosmological horizons. These coordinates require a positive cosmological constant for regularity, and thus they are not de Sitter extensions of the Kruskal-Szekeres or Israel coordinates. One of the coordinate systems was only found in 1999 (Abbassi coordinates), and it has led to conflicting interpretations in the literature, while the other was briefly commented on and promptly dismissed as unphysical or incompatible with SdS. We derive that the second solution is equivalent to the first one, and that both are indeed equivalent descriptions of SdS spacetime. We also derive explicit coordinate transformations linking these coordinate systems to the Kottler coordinates and the maximally extended Lake-Israel coordinates. Among other applications, these results, which extend the largely used cosmological and local coordinates, should be useful for further developments in understanding the exact interplay between black holes and the cosmological background, which has been the focus of a number of recent works.

Figures (3)

• Figure 1: The plot compares the functions $A_+$ (solid blue), $A_-$ (solid green) \ref{['eq:ealphapm']} and $f$ (solid yellow) \ref{['eq:f']}. They are written as functions of the distance $\ell$, which is expressed in units of the BH mass $m$. Only $f$ has roots, $A_+$ is always positive and $A_-$ is always negative. The figure also shows the large $\ell$ asymptotics for $A_+$ (blue dashed line, which is constant and equal to 1), and for $A_-$ (the green dot-dashed line, which decays proportionally with $- \Lambda \ell^{2}$). This plot was done with $\Lambda = 10^{-2} m^{-2}$.
• Figure 2: Penrose diagram for SdS spacetime centered at the "standard universe" (region I). The horizons are depicted by solid gray lines (without arrows). The diagram can be continued to the left or to the right Gibbons:1977muBousso:2002fqLake:2005bf. Region II is a white hole (WH) interior, region III is a BH interior, region IV is a future region beyond the cosmological horizon and region V is a past region beyond the cosmological horizon. The blue and green solid lines with arrows show the radial propagation of light: either away (blue) or toward (green) the singularity at $\ell=0$.
• Figure 3: Null geodesics of SdS spacetime using compactified versions of the $r$ and $\eta$ coordinates from \ref{['eq:ds^2']}. The coordinates compactification is done with $\bar{x} = 2 \arctan(x)/\pi$. Ingoing (purple curves with arrows) and outgoing (blue curves with arrows) geodesics are defined from Eq. \ref{['eq:detadlambda']}, with different boundary conditions at $\bar{r} = 0.5$. Dashed gray lines represent curves of constant $\ell$. The physical singularity at $\ell = 0$ lies along the axes $\bar{r} =0$ and $\bar{\eta} = -1$. The two bluish regions correspond to values where $f(\ell) < 0$; their boundaries with the central white region mark the horizons, which are coordinate singularities in Kottler coordinates. The upper (bluish) boundary corresponds to the cosmological horizon, while the lower (bluish) boundary corresponds to the white-hole horizon. This plot was done using $m^2 \Lambda = 10^{-2}$ and $\dot a > 0$.