Kerr-Newman-de Sitter black holes in $f(R)$ gravity with constant curvature: horizon structure and extremality

TL;DR

This work develops a unified analytic treatment of Kerr-Newman-de Sitter black holes in $f(R)$ gravity with constant curvature. By mapping constant-curvature $f(R)$ solutions to GR and solving the quartic horizon equation, the authors obtain closed-form horizon radii and explicit extremality relations, revealing how background curvature and charge modify the Kerr–Newman bound. They derive $a^2$ and $l^{-2}$ as functions of horizon location and charge, identify an ultra-extremal regime, and show the existence of a minimum rotation under significant curvature or for charged cases (e.g., $q=M/2$). Imposing a mass constraint further factorizes the horizon equation, yielding a chiral-like horizon structure where only outer–cosmological mergers occur, enriching the analytic understanding of extremality in de Sitter spacetimes with modified gravity.

Abstract

The theory of $f(R)$ gravity with constant curvature (i.e. constant scalar curvature) admits rotating and charged black hole solutions obtained from the Kerr-Newman-(A)dS metrics of general relativity through appropriate rescalings of the metric parameters. In this paper, we focus on the Kerr-Newman-de Sitter case and present a unified analytic treatment of the horizon structure and its physical properties, allowing for a transparent comparison between general relativity and $ f(R)$ gravity with constant curvature. We solve the quartic equation determining the horizon locations and derive closed analytic expressions for the horizon radii. Focusing on extremal configurations, we obtain analytic formulas for the squared rotation parameter $ a^2 $ and the inverse square of the curvature radius $ l^{-2} $ as functions of the horizon location and the electric charge. For generic values of these parameters, the extremality conditions are non-universal, reducing to the familiar Kerr-Newman bound only in the limit of vanishing background curvature. We identify an ultra-extremal configuration in which $ a^2 $ attains its maximal value at zero charge and decreases monotonically to zero as the charge approaches its limiting value, while $ l^{-2 }$ increases correspondingly. As an illustrative example, we show that black holes with charge $ q=M/2 $ necessarily possess a minimum rotation, which emerges naturally as an intersection point in our analytic description of $ a^2 $ and $ l^{-2 }$, when embedded in a universe characterized by a critical value of $ l^{-2} $ (equivalently, the scalar curvature or the cosmological constant). Finally, we demonstrate that when the mass satisfies $ M^2= (a^2+q^2)(1-a^2/l^2)$, the quartic horizon equation factorizes, leading in the extremal regime to a chiral-like horizon structure that allows only the outer-cosmological horizon merger.

Figures (3)

• Figure 1: Dependence of the squared rotation parameter $a^2$ and the inverse square of the curvature radius $l^{-2}$ on the electric charge $q$ of the black hole. As the electric charge increases toward its limiting value, the quantity $a^2$ decreases monotonically, while $l^{-2}$ increases monotonically.
• Figure 2: Behavior of the squared rotation parameter $a^2$ and the inverse square of the curvature radius $l^{-2}$ as functions of the horizon location $r$ for a charged black hole with $q=M/2$. The red points indicate the ultra-extremal values, while the orange point marks the intersection $a^2=l^{-2}$, corresponding to the minimal rotation of the black hole. The right panel shows $l^{-2}$ separately for improved visibility.
• Figure 3: Dependence of the squared rotation parameter $a^2$ and the inverse square of the curvature radius $l^{-2}$ on the horizon location $r$ of the black hole for vanishing electric charge. The left panel shows the simultaneous behavior of both quantities, while the right panel highlights the variation of $l^{-2}$ for clarity.