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–(anti) de Sitter 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 nonuniversal, reducing to the familiar Kerr-Newman bound only in the limit of vanishing background curvature. We identify an ultraextremal 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.

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• Alternative gravity theories
• General relativity