A note on the linear stability of black holes in quadratic gravity

Christian Dioguardi,Massimiliano Rinaldi

doi:10.1140/epjp/s13360-020-00935-2

Abstract

Black holes in f(R)-gravity are known to be unstable, especially the rotating ones. In particular, an instability develops that looks like the classical black hole bomb mechanism: the linearized modified Einstein equations are characterized by an effective mass that acts like a massive scalar perturbation on the Kerr solution in general relativity, which is known to yield instabilities. In this note, we consider a special class of f(R) gravity that has the property of being scale-invariant. As a prototype, we consider the simplest case f(R)=R^2 and show that, in opposition to the general case, static and stationary black holes are stable, at least at the linear level. Finally, the result is generalized to a wider class of f(R) theories.

Highlights

The challenging questions left unanswered by general relativity (GR), like the physical interpretation of singularities or the consistent quantization of the field equations call for exploring mRwihocerceriegstcehanelearErailRncs,toein.iens.t,rHuLcitl=iboenr√st .LgAafg(wrRaen)l.lg-TikahnniosLwwn=ideex√tceglnaRssisoisnofroetfphlGeaocRreideissbthyhaeas so-called f (R) gravity, generic function of the been explored in many contexts, from black hole physics to cosmology, especially inflation and dark energy.A very general feature of f (R) gravity is that it can be mapped to a standard scalar-tensor theory of gravity by means of a conformal transformation [3]
An instability develops that looks like the classical black hole bomb mechanism: the linearized modified Einstein equations are characterized by an effective mass that acts like a massive scalar perturbation on the Kerr solution in general relativity, which is known to yield instabilities
The extra degree of freedom is encoded in a dynamical scalar field with a potential that depends on the analytic form of f (R) and the equations of motion are again of second order at most

Summary

Introduction

The challenging questions left unanswered by general relativity (GR), like the physical interpretation of singularities or the consistent quantization of the field equations call for exploring mRwihocerceriegstcehanelearErailRncs,toein.iens.t–,rHuLcitl=iboenr√st .LgAafg(wrRaen)l.lg-TikahnniosLwwn=ideex√tceglnaRssisoisnofroetfphlGeaocRreideissbthyhaeas so-called f (R) gravity, generic function of the been explored in many contexts, from black hole physics to cosmology, especially inflation and dark energy (see [1,2] for general reviews). For the class of models of the form f (R) = Rn (n > 1) the effective mass vanishes if R = 0 for n = 2 and for any R for n = 2 Since this class contains the Kerr metric as exact solution, in this note we investigate whether the instability persists in this class of models. Such a class of modified gravity has attracted some attention, especially the case R2 since it is manifestly scale-invariant. If α1 = 0 [that is, there is no linear term in expansion (2)] and the solution has R = 0 the mapping is not possible In other words, these theories do not have a scalar-tensor interpretation.

Linearized equations in pure quadratic gravity

Stability of static spherically symmetric black holes

Stability of stationary black holes

Findings

Conclusions