# Geonic black holes and remnants in Eddington-inspired Born–Infeld gravity

## Abstract

We show that electrically charged solutions within the Eddington-inspired Born–Infeld theory of gravity replace the central singularity by a wormhole supported by the electric field. As a result, the total energy associated with the electric field is finite and similar to that found in the Born–Infeld electromagnetic theory. When a certain charge-to-mass ratio is satisfied, in the lowest part of the mass and charge spectrum the event horizon disappears, yielding stable remnants. We argue that quantum effects in the matter sector can lower the mass of these remnants from the Planck scale down to the TeV scale.

## Keywords

Event Horizon Curvature Divergence Matter Sector Determinantal Form Black Hole Remnant## 1 Introduction

Historically, the taming of singularities in classical field models has driven a great deal of research. A particularly elegant example is the nonlinear extension of Maxwell electrodynamics introduced by Born and Infeld [1] to remove the divergence of both the Coulomb field and the self-energy of point particles. In this *determinantal* form of the classical action, the modified field (the *BIon* [2]) is everywhere bounded but generated by a distributional source. This specific form of nonlinear electrodynamics arises in the low-energy limit of certain string theories [3, 4, 5].

Recovering the idea of the determinantal form of the gravitational action suggested by Eddington [6, 7], an Eddington-inspired Born–Infeld action (EiBI) for the gravitational field has been introduced recently [8, 9]. In order to avoid troubles with higher-order derivatives and ghosts, EiBI gravity is formulated in the Palatini approach, which means that the metric and connection are regarded as physically independent entities [10]. This implies that the connection is determined by the field equations, not constrained a priori to any particular form.

The EiBI theory is a modification of the Einstein–Hilbert action which might allow one to remove the appearance of singularities, thus avoiding an undesirable feature of Einstein’s theory of general relativity (GR). The EiBI theory is expected to be in agreement with GR at energies well below the Planck scale, which represents the regime where quantum gravitational effects are expected to begin to become important and modify the classical description. The naturalness of EiBI gravity has been argued on the basis of canonical procedures to construct Lagrangian densities with second-order field equations [11]. Moreover, this theory is able to avoid cosmological singularities [12], has been employed to study properties of dark matter and dark energy [13, 14, 15], in the coupling to several kinds of fields [16], as an alternative to inflation [17], and to explore the structure of compact stars [18, 19]. When coupled to a perfect fluid with a given equation of state, it has been found that the theory can be interpreted as GR coupled again to a perfect fluid, but with a modified equation of state [20, 21].

Though in its determinantal form the EiBI theory may appear as lacking an intuitive motivation, here we show that, when applied to elementary systems such as electric fields generated by point-like sources (or elementary particles), the theory boils down to a simple quadratic extension of GR. This simplification occurs when the stress-energy tensor of the matter possesses certain algebraic properties [22], namely, when it has two double eigenvalues. We take advantage of this property to explore in detail the internal structure of the electrovacuum solutions of the theory and find that the central singularity is generically replaced by a wormhole supported by the electric field. The wormhole structure turns out to be crucial to regularize the total energy stored in the electric field, which occurs in a way that resembles the original Born–Infeld electromagnetic theory. Among the solutions of the theory, there exist a family (characterized by a certain charge-to-mass ratio) for which curvature invariants are finite everywhere. These solutions, whose mass exactly coincides with the energy contained in the electric field, lose the event horizon when the number of charges drops below a critical value, \(N_q^c\sim 16\), yielding remnants which are not affected by Hawking’s quantum instability. The mass spectrum of these remnants can be lowered from the Planck scale (\(\sim 10^{19}\) GeV) down to the TeV scale if quantum matter corrections are considered. These results are derived in a four-dimensional scenario.

## 2 Theory and field equations

From (7) one clearly sees that the metric \(q_{\mu \nu }\) satisfies a system of second-order differential equations with the matter sources on the right-hand side (recall from (6) that \(\hat{\Sigma }=\hat{\Sigma }[\hat{T}]\)). Since \(q_{\mu \nu }\) is algebraically related to \(g_{\mu \nu }\) through (5), it follows that \(g_{\mu \nu }\) also satisfies second-order equations. On the other hand, it is easy to see that in vacuum, \({T_\mu }^\nu =0\) and \(|\hat{\Sigma }|=\lambda ^4\), \(g_{\mu \nu }\) and \(q_{\mu \nu }\) are identical up to a constant conformal factor and that \({R_\mu }^\nu (q)=\frac{(\lambda -1)}{\lambda \epsilon }{\delta _\mu }^\nu \), which is equivalent to \(R_{\mu \nu } (g)=\frac{(\lambda -1)}{\epsilon }g_{\mu \nu }\), thus confirming that \(\frac{(\lambda -1)}{\epsilon }\) plays the role of an effective cosmological constant in the full theory. Since the vacuum theory is equivalent to GR with a cosmological constant, no ghost-like instabilities are present in the theory, which is a rather general property of Palatini theories.

## 3 Electrovacuum solutions

## 4 EiBI as quadratic gravity

*exactly*coincide with those corresponding to the quadratic Palatini theory

^{1}. For \(z\rightarrow 1\), however, the geometry strongly departs from the low-energy limit represented by GR. To understand the relevance of this region, one should look at the behavior of the function \(z(x)=r(x)/r_c\):

## 5 Charge without charges and mass without masses

^{2}. As first shown by Misner and Wheeler [31], an electric flux through a wormhole can define by itself an electric charge

^{3}without the need for sources of the electric field. Therefore, the charge \(q\) appearing in our solutions is entirely given by the electric flux through any two-surface \(\mathcal {S}\) enclosing one of the sides of the wormhole, i.e.,

*geon*(self-consistent solutions of the sourceless gravito-electromagnetic field equations) originally introduced by Wheeler [33].

## 6 Horizons and remnants

*critical number of charges*and plays an important role in the existence or not of event horizons [see paragraph below], \(m_P\) is the Planck mass, and \(l_P\) is the Planck length. It is worth noting that defining the length scale \(l_\beta ^2\equiv (4\pi /\kappa ^2 c\beta ^2)\), we can write the expression for \(\mathcal {E}_{BI}\) given above as

A very important aspect of the \(\delta _1=\delta _1^*\) solutions is that, as can be graphically verified (see [28] for details), for \(N_q\le N_q^c\) there is no event horizon, which implies stability of that sector of the theory against Hawking radiation. For \(N_q>N_q^c\) the horizon exists and its location almost coincides with the prediction of GR. This means that black holes, understood as objects with an event horizon, can be continuously connected with horizonless configurations lying in the lowest part of the charge and mass spectrum. Such states can be naturally identified as black hole remnants and their existence could have deep theoretical implications for the information loss problem and the Hawking evaporation process [34]. Note, in addition, that for these remnants the surface \(z=1\) is timelike and \(S=2M_0c^2\int \mathrm{d}t\) coincides with the action of a point-like particle at rest, which suggests that they possess particle-like properties.

## 7 Coupling of BI matter

## 8 Conclusions and outlook

We have shown that for spherically symmetric charged systems EiBI theory recovers the GR predictions for \(r\gg r_c\). The theory, however, changes the microstructure of the space-time replacing the GR singularity by a wormhole. Though curvature divergences may exist at \(r=r_c\), their role is uncertain, since they affect neither the properties of the flux through the wormhole nor the finiteness of the total electric energy. The theory makes definite predictions as regards the existence of black hole remnants and their mass spectrum, with non-trivial implications for the Hawking evaporation process, the information loss problem, and potentially new dark matter candidates [35].

## Footnotes

- 1.
It has been reported recently that stellar models with certain polytropic equations of state may develop curvature divergences at their

*surface*[19], where the interior geometry is matched to an external Schwarzschild metric. Similar problems were already found in the context of Palatini \(f(R)\) theories (see [29] for a discussion). We support the view that when curvature divergences arise, a refined (microscopic) description of the troublesome region might help better understand their physical significance. In this sense, the absence of such pathologies in elementary charged systems, as found here, suggests that the results of [19] might be an artifact of the approximations employed in the continuum description of statistical/macroscopic systems. In fact, since a star is made out of elementary particles, our results indicate that nothing special should happen as the outermost regions are approached, where the effective separation between particles increases and the “isolated-particle description” of its constituents becomes more and more accurate. The average energy density and gradients in those regions cannot be larger than in the region close to an individual particle because the volumes involved differ by orders of magnitude. In our view, therefore, a microscopic description of a stellar surface, seen as a collection of elementary particles, seems to be free of the pathologies described in [19]. As another way out of this problem, it has been recently argued [30] that when the gravitational backreaction on the matter dynamics at the star surface is considered, the effective equation of state gets modified with the consequence that surfaces are no longer singular. - 2.
Note that in the context of GR it is sometimes stated that the charge and mass are concentrated at a point of zero volume at the center (the singularity). However, the fact is that there is no mathematically well-defined source able to generate the Reissner–Nordström solution [32].

- 3.
This charge is a very primitive concept that does not require for its existence neither the definition of metric nor affine structures on the manifold and, as such, is insensitive to the presence of curvature divergences.

## Notes

### Acknowledgments

Work supported by project FIS2011-29813-C02-02 (Spain), the Consolider Program CPANPHY-1205388, the JAE-doc program and i-LINK0780 project of CSIC, the CNPq projects 561069/2010-7 and 301137/2014-5 (Brazil), and the Erwin Schrödinger fellowship Nr. J3392-N20 of FWF (Austria).

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