Analytical expression for the cosmological constant in the f(R) black hole setup

Derive a closed-form analytical expression for the cosmological constant Λ in the f(R) gravity model defined by fR(R) = R − 2Λ/3 for the charged black hole metric given in equation (34), by solving the implicit algebraic relation connecting the horizon radius r and Λ obtained in equations (48)–(50), so that Λ is expressed explicitly in terms of the constants M, Q, and R without dependence on r.

Background

Within the paper’s f(R) gravity analysis, the authors introduce a specific form fR(R) = R − 2Λ/3 and apply it to a charged black hole metric (equation (34)). Integrating the modified field equations yields an implicit algebraic relation (equations (48)–(50)) that ties the horizon radius r to the cosmological constant Λ, with M, Q, and R treated as constants.

Because the resulting expression involves r on both sides, the authors note that the functional form of Λ is highly complex and that they cannot obtain an explicit analytical expression for Λ. This leaves the task of isolating Λ in closed form—without implicit dependence on r—as an unresolved problem, motivating reliance on numerical or semi-analytical methods.