# Hawking emission of charged particles from an electrically charged spherical black hole with scalar hair

## Abstract

A static spherically symmetric black hole usually turns out to be either a Schwarzschild black hole or a Reissner-Nordström black hole. This result was summarised by Ruffini and Wheeler as the so-called no hair conjecture which states that for a spherically symmetric black hole only the information about mass (*M*) and electric charge (*e*) of the black hole is available for an external observer. In this work, we calculate the emission rate of charged particles from an asymptotically flat charged spherically symmetric black hole endowed with a scalar hair using a semi-classical tunneling formalism. We observe that the total entropy of the black hole contains an energy-dependent part due to the scalar charge. The upper bound on the charge-mass ratio of the emitted particles is also observed to decrease with the scalar charge as well.

## 1 Introduction

Black holes are fascinating offshoots of general relativity, unmatched in their elegance and complexity. The recent detections of gravitational waves by LIGO and Virgo [1, 2, 3, 4, 5, 6, 7] and the recently obtained “image” of M87* black hole at the centre of the M87 galaxy by the Event Horizon Telescope [8] have quite conclusively established that black holes are not merely mathematical constructs but actual physical entities. In 1974, Hawking [9] showed that “black holes” are not “entirely black”; they spontaneously emit particles at a temperature proportional to their surface gravity. Since then various derivations of Hawking radiation have been proposed. In particular, Parikh and Wilczek [10] used the semi-classical tunneling formalism to study Hawking emission of massless uncharged particles from the Schwarzschild and Reissner-Nordström (RN) black holes. According to this formalism, the black hole loses energy due to radiation and thereby decreases its horizon radius. Thus, the outgoing particle creates a potential barrier by itself (see Refs. [11, 12, 13]). Using WKB approximation one can evaluate the emission rate from the imaginary part of the action of the outgoing particles. The tunneling method has since been used to study Hawking radiation from a wide variety of black holes. Zhang and Zhao [14] extended the tunneling method to study the emission of massive charged particles from Reissner-Nordström black holes. Later, Jiang and Wu [15] used tunneling formalism to study Hawking radiation of charged particles from Reissner-Nordström-de Sitter black hole. Tunneling of charged particles in the modified RN black hole was studied by Liu [16]. Jiang,Wu and Cai [17] studied the tunneling of particles from Kerr and Kerr-Newman black holes. Jiang, Yang and Wu [18] also studied the tunneling of charged particles from Reissner-Nordström black holes of arbitrary dimensions. Jiang and Wu [19] studied the tunneling of charged particles from Reissner-Nordström-de Sitter black holes with a global monopole. Sarkar and Kothawala [20] gave a generalised treatment of Hawking radiation as tunneling for asymptotically flat, spherically symmetric black holes. Tunneling mechanism has also been extensively used to study Hawking radiation from numerous other black holes (for example, see Refs. [21, 22, 23, 24]). Tunneling of Dirac particles from black rings was studied in Ref. [25]. Jiang, Chen and Wu [26] also studied the tunneling of massive particles from the cosmological horizon of a Schwarzschild-de Sitter black hole.The general outcome of all these investigations is the non-thermality of the Hawking emission spectrum. For a detailed review of the tunneling mechanism, we refer the reader to Ref. [27].

The question of information loss has also been addressed using the tunneling formalism. Using logarithmic correction to the Bekenstein-Hawking entropy, Chen and Shao [28] showed that in a black hole evaporation process unitarity is preserved and the information loss paradox can be successfully resolved. Singleton et al. [29] showed that if the back reaction and the quantum corrections are taken into account, the information can be carried away by the correlations of the outgoing radiation during complete evaporation of the black hole and the information loss paradox can be solved. Sakalli et al. [30] studied hawking radiation in linear dilaton black holes and showed that no information is lost during complete evaporation of the black hole.

In the present work, we employ the tunneling method to study Hawking radiation of charged particles from an asymptotically flat static spherically symmetric charged black hole, endowed with a scalar hair, dubbed as the scalar-hairy RN black hole [31]. The scalar charge comes as an additive correction to the square of electric charge and can survive even in the absence of the electric field. For negative values of the scalar charge in the absence of the electric charge, or even if the negative scalar charge exceeds in magnitude the square of the electric charge, the metric reduces to that of a mutated Reissner-Nordström spacetime, leading to an Einstein–Rosen bridge [32]. The quasinormal modes of scalar and Dirac fields for this spacetime have been studied by Chowdhury and Banerjee [33] for both positive and negative values of the scalar charge. The superradiant stability of this spacetime has been studied in Ref. [34]. The primary objective of the present work is to study the effect of the scalar hair on the transmission rate of charged Hawking quanta. Based on the non-negativity of mutual information between consecutive Hawking emissions (see Refs. [35, 36]), we also find the dependence of the charge to mass ratio of the emitted particles on the scalar charge *s* in each step.

We observe that the change in entropy of the scalar-hairy RN black hole due to the emission of charged particles contains an additional energy-dependent contribution which vanishes during emission of uncharged particles and also in the absence of the scalar hair. The tunneling rate for the emission of charged particles from the scalar-hairy RN black hole matches smoothly with that of the standard RN black hole, in the absence of the scalar hair.

The paper is organised as follows. We start with a brief description of the background spacetime in Sect. 2. Section 3 deals with the derivation of the emission rate of charged scalar particles from the scalar-hairy RN black hole. In Sect. 4, we show the dependence of the upper bound of the charge-mass ratio of the particles, emitted in each step, on the scalar charge. Finally, in Sect. 5 we discuss the results that we arrive at.

## 2 The scalar-hairy-Reissner-Nordström black hole

*s*[31],

*M*and

*e*are respectively the mass parameter and electric charge of the black hole. The total energy-momentum tensor of the electromagnetic field and the scalar field is given by

The energy-momentum tensor of the scalar-hairy RN spacetime satisfies both the dominant and strong energy conditions for \(s>-e^2.\) For this work, we will assume \(s>0\), so this condition is satisfied. Henceforth, we will also assume \(G=c=k_B=\hbar =1.\)

## 3 Tunneling rate of charged particles

*L*is the total Lagrangian of the matter-gravity system. When a particle of charge

*q*and mass \(\omega \) tunnels out of the event horizon the electric charge of the black hole changes from \(e\rightarrow e-q\) and the ADM mass changes from \(M_{ADM}\rightarrow M_{ADM}-\omega \). The imaginary part of the corresponding action can thus be written as

*M*as

*M*. However, in the limit of \(e\rightarrow 0\), \(M_{ADM}\) vanishes.

*r*- integral has a pole at \(r'_+\). Deforming the contour around this pole we obtain,

*A*is the area of the event horizon, \(\kappa \) is the surface gravity and \(\varPhi \) is the electric potential at the horizon,

*q*,

Figure 1 shows the plot of the black hole temperature with respect to the scalar and electric charges.

*s*but reaches a plateau quickly. For non-zero scalar charge, the black hole temperature rises with the electric charge, reaches a plateau and sharply falls to zero as the extremal value of the electric charge, \(e_{ext}={\sqrt{\sqrt{M_{ADM}^2 \left( M_{ADM}^2+4 s\right) }+M_{ADM}^2}}/{\sqrt{2}}\), is approached. At sufficiently small values of

*e*, \(T_{BH}\) becomes vanishingly small.

## 4 Charge-mass ratio bound from mutual information of successively emitted charged Hawking quanta

Figure 2 shows the variation of the upper bound on the charge-mass ratio with the scalar and electric charges of the black hole. We observe that at a fixed electric charge of the black hole, for smaller values of the scalar charge, the upper bound on the charge-mass ratio decreases with *s*; however, the rate of fall decreases with the increase in the scalar charge. On the other hand, for non-zero scalar charge, the maximum possible charge-mass ratio increases with the electric charge of the black hole and approaches that for the standard RN black hole.

## 5 Summary and discussion

In this paper, we studied the problem of Hawking emission of massive charged particles from a spherically symmetric charged black hole endowed with a scalar hair using the semi-classical tunneling formalism. The black hole is very similar to the standard Reissner-Nordström black hole but with an additional scalar hair (See Refs. [31, 33, 34]). The presence of the scalar hair gives rise to an effective Newtonian constant, \(\tilde{G}=G\left( 1+s/e^2 \right) \) and an ADM mass, different from the mass parameter *M* (see Eq. (20)). It is interesting to note that in the limit of vanishing electric charge, the ADM mass of the black hole goes to zero which is due to the fact that the effective Newtonian constant \(\tilde{G}\) blows up as \(e\rightarrow 0\). This suggests that the scalar-hairy RN black hole cannot radiate away its electric charge and settle down to a scalar-hairy (electrically) uncharged distribution. This is consistent with the following thermodynamic consideration. We note that for any positive value of the scalar charge, the black hole temperature becomes vanishingly small at sufficiently small values of the black hole electric charge (see Fig. 1) and thus according to the Third Law of black hole thermodynamics [44], it is impossible for the scalar-hairy RN black hole to radiate away its electric charge in any finite number of steps.

The total change in entropy of the scalar-hairy RN black hole due to the emission of the massive charged particle contains a \(\omega \)-dependent contribution due to the scalar charge (see Eqs. (30), (33) and (35)). The emission rate, (see Eq. (34)) matches smoothly with that for the standard Reissner-Nordström black hole (see Ref. [14]) in the limit of vanishing scalar charge *s*. The emission rate for uncharged massless particles obtained by putting \(q=0\) in Eq. (34) (see Eq. (36)) is also of the same functional form as that derived using the geodesic equation. We also note that the Hawking emission spectrum deviates from pure thermality. This is consistent with the generic results found in the investigations [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39] and the recent findings that the quantum process involved in Hawking radiation is unitary, meaning a pure state to pure state transition [28, 29, 30, 45, 46].

It is important to note that using the modified geodesic equation instead of the Eq. (10) as suggested by Pu and Han [47] also yields the same expression of the tunneling rate.

We also studied the mutual information stored in consecutive Hawking emission. Demanding non-negativity of the mutual information, we observed that the maximum allowed charge-mass ratio of the emitted particles decreases with the scalar charge of the black hole.

It will be interesting to study the effect of Lorentz symmetry breaking on quantum tunneling in other black hole solutions such as those in bumblebee gravity. A few investigations have already been done in this direction (see Refs. [48, 49, 50]).

## Notes

### Acknowledgements

The author is indebted to Professor Narayan Banerjee for valuable discussions and comments.

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