Non-singular black holes and mass inflation in modified gravity

Abstract

We analytically derive a class of non-singular, static and spherically symmetric topological black hole metrics in $F\left(R\right)$-gravity. These have not a de Sitter core at their centre, as most model in standard General Relativity. We study the geometric properties and the motion of test particles around these objects. Since they have two horizons, the inner being of Cauchy type, we focus on the problem of mass inflation and show that it occurs except when some extremal conditions are met.

Introduction

The recent detection of gravitational waves from binary systems of black holes (BHs) and the “multimessenger” signals from the first observation of the collision of two relativistic neutron stars largely confirmed some predictions of the strong-field regime of General Relativity (GR) [1], [2], [3], [4]. Moreover, two years ago, the Event Horizon Telescope collaboration [5] published the first imagine of a supermassive black hole at the centre of M87. It is then fair to say that the existence of black holes is almost certain, which makes longstanding conceptual problems, such as the central singularity and the information paradox, even more pressing.

Usually, astrophysical black holes are described by the Kerr metric, which is a stationary, vacuum solution of the field equations of GR, with a ring-shaped singularity at the centre. Since space–time singularities are problematic, a lot of investigation has been dedicated to viable alternatives to the Kerr model.

Another well-known problem related to the Kerr model is the instability that occurs near the inner Cauchy horizon, where an infinite amount of energy might accumulate, forming in fact a new space–time singularity, although the tidal forces can be finite, as opposed to the case of the central singularity [6], [7], [8]. This problem is known under the name of “mass inflation”.

The experimental data may help, in the near future, to better understand the nature of the sources of gravitational waves that we are able to detect. The ringdown waveform of a black hole is completely determined by Quasinormal modes (QNMs), which depend only on the mass and angular momentum of the BH. Thus, every deviation from the standard result of GR may be associated to alternative theories of gravity or a different nature of the source with respect to the case of Kerr BH. In this respect, the possible presence of additional “echoes” in the ringdown waveform has been largely debated in the last years (see the exhaustive review in Ref. [9]) and alternatives to the BHs as gravastars [10], [11], bosonstars [12], or other exotic compact objects [13], [14], [15], [16], [17], [18]) have been investigated.

In this paper, we mainly study a class of non-singular topological BHs in $F\left(R\right)$-gravity, where the usual Einstein–Hilbert term in the gravitational Lagrangian is replaced by a smooth function of the Ricci scalar $R$. We are able to find the metric in analytic form and we can show that these black holes usually have two horizons, the inner one being of Cauchy type. However, the inner part of the black hole is a singularity-free region.

As mentioned above, a inner Cauchy horizon can trigger the mass inflation problem. It is not clear, a priori, whether this occurs also in regular black holes though. This issue has been recently investigated in regular black holes with a de Sitter core [19]. However, our solutions do not have this internal structure so we need to analyse the mass inflation problem again.

The structure of the paper is the following: in Section 2 we derive the analytic form of the non-singular black holes in $F\left(R\right)$-gravity and we study some of their properties. In Section 3 we investigate the problem of mass inflation for these solutions and we draw some conclusion in Section 4.