Exotic spherically-symmetric Lambda-vacuum in the four-dimensional Starobinsky model

TL;DR

This paper presents an exact spherically symmetric vacuum solution in the Starobinsky $f(R)$ gravity model, revealing a pathological boundary with unphysical features and extreme fragility to perturbations.

Contribution

The authors derive a novel exact solution at a fine-tuned parameter boundary in the Starobinsky model, illustrating its pathological nature and physical instabilities.

Findings

Existence of a degenerate vacuum solution with a geometric Reissner-Nordstrom hair

Discontinuous collapse to Schwarzschild-de Sitter under perturbations

Identification of severe physical pathologies including ghost instabilities

Abstract

We introduce an exact, two-parameter family of static, spherically-symmetric, constant-curvature $\Lambda$-vacuum solutions within the four-dimensional Starobinsky $f(R)=R+\alpha R^2+2\Lambda$ model. When the bare cosmological constant is precisely fine-tuned to $\Lambda = 1/(8\alpha)$, the scalar curvature is rigidly fixed such that the derivative $f'(R)=1+2\alpha R$ identically vanishes. Because this derivative acts as the effective multiplier for the standard curvature terms in the modified field equations, its global vanishing mathematically erases the normal rules of gravitational dynamics, demonstrating that the family represents a pathological boundary to the space of viable physical geometries. This exact decoupling of the field equations permits the existence of a fundamentally unconstrained $1/r^2$ integration constant in the metric, which functions as a purely geometric Reissner-Nordstrom hair mimicker. However, any infinitesimal classical deviation from this exact boundary instantaneously destroys the degeneracy, rigorously forcing the geometric hair to vanish and discontinuously collapsing the spacetime back into the standard, dynamical Schwarzschild-de Sitter solution. We provide the exact derivation of this spacetime and methodically highlight its physical pathologies, including the identically vanishing Wald entropy of the associated black hole horizons, the strict divergence of the effective gravitational coupling, the complete breakdown of the test-particle approximation, and the onset of severe ghost instabilities. Ultimately, this exact solution functions as a ``do not enter'' sign within the Starobinsky model, pedagogically illustrating the extreme fragility and physical hostility of degenerate, purely mathematical solutions in highly non-linear $f(R)$ gravity theories.