[Paper Review] Stability of black holes in f(R) gravity

Yun Soo Myung|arXiv (Cornell University)|Mar 12, 2015

Cosmology and Gravitation Theories18 references3 citations

TL;DR

This paper investigates the stability of Schwarzschild and Kerr black holes in the viable $f(R) = R + R^2/(6M^2)$ gravity model by mapping it to the Starobinsky scalar-tensor theory in the Einstein frame. Using this reformulation, the authors find that the Schwarzschild black hole is stable, while the Kerr black hole exhibits superradiant instability due to trapped massive scalar modes in a potential well, confirmed via quasibound state analysis and Hod’s instability criteria.

ABSTRACT

We investigate the stability of black holes in the viable model of $f(R)=R+R^2$ gravity which was known to be the best fit for inflation. These include Schwarzschild and Kerr black holes. Instead of studying the fourth-order linearized equation around the black hole background, we use the corresponding tensor-scalar theory of the Starobinsky model to perform their stability. The Schwarzschild black hole is stable, while the Kerr black hole is unstable because of superradiant instability.

Motivation & Objective

• To assess the stability of Schwarzschild and Kerr black holes in the $f(R) = R + R^2/(6M^2)$ gravity model, a viable inflationary theory.
• To overcome the technical difficulty of solving fourth-order field equations in $f(R)$ gravity by mapping the theory to a scalar-tensor formulation.
• To determine whether the Kerr black hole is unstable due to superradiance when coupled to massive scalar perturbations in the Starobinsky model.
• To apply Hod’s instability criterion—requiring both an ergoregion and a trapping potential well—for identifying superradiant instability in rotating black holes.

Proposed method

• Transform the $f(R) = R + R^2/(6M^2)$ gravity into a scalar-tensor theory via the auxiliary field $\psi$, leading to the Starobinsky model in the Jordan frame.
• Perform a conformal transformation to switch to the Einstein frame, where the action takes the form of the Starobinsky inflation model with a scalar field $\phi$ and a potential $V(\phi)$.
• Analyze linear perturbations of the Ricci scalar in the $f(R)$ theory by treating it as a massive spin-0 graviton in the Kerr background.
• Apply the superradiance condition $\omega < m\Omega$ and analyze quasibound states using boundary conditions: regularity at the horizon, decay at infinity, and $\omega^2 < M^2$.
• Use Hod’s argument to identify instability: existence of an ergoregion and a potential well between the ergoregion barrier and the mass barrier at infinity.
• Derive the instability condition $M < \sqrt{2}m\Omega$ by combining superradiance and quasibound state constraints.

Experimental results

Research questions

• RQ1 Is the Schwarzschild black hole stable in the $f(R) = R + R^2/(6M^2)$ gravity model?
• RQ2 Does the Kerr black hole exhibit superradiant instability when coupled to massive scalar perturbations in this $f(R)$ model?
• RQ3 Can the fourth-order nature of $f(R)$ gravity be circumvented by mapping to a scalar-tensor theory for stability analysis?
• RQ4 What are the necessary conditions for superradiant instability in the Kerr black hole within the Starobinsky model framework?
• RQ5 How does the potential well structure in the effective potential influence the formation of quasibound states responsible for instability?

Key findings

• The Schwarzschild black hole is stable under linear perturbations in the $f(R) = R + R^2/(6M^2)$ gravity model, confirmed via stability analysis in the Starobinsky scalar-tensor formulation.
• The Kerr black hole is unstable due to superradiant amplification of massive scalar modes trapped in a potential well between the ergoregion and the mass barrier.
• Superradiant instability occurs when the scalar mass satisfies $M < \sqrt{2}m\Omega$, where $\Omega$ is the angular velocity of the Kerr horizon.
• The instability is triggered by the existence of quasibound states with $\omega^2 < M^2$ and $\frac{M^2}{2} < \omega^2 < M^2$, ensuring a trapping well outside the ergoregion.
• The analysis confirms that the Kerr black hole satisfies both conditions of Hod’s instability criterion: an ergoregion for amplification and a potential well for mode trapping.
• The $p=2$ case of $f_p(R) = R + \lambda R^p$ leads to a regular Klein-Gordon-type equation, unlike $1<p<2$ or $p>2$, which produce non-integer power mass terms and complicate stability analysis.