# Formation and evaporation of an electrically charged black hole in conformal gravity

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## Abstract

Extending previous work on the formation and the evaporation of black holes in conformal gravity, in the present paper we study the gravitational collapse of a spherically symmetric and electrically charged thin shell of radiation. The process creates a singularity-free black hole. Assuming that in the evaporation process the charge *Q* is constant, the final product of the evaporation is an extremal remnant with \(M=Q\), which is reached in an infinite amount of time. We also discuss the issue of singularity and thermodynamics of black holes in Weyl’s conformal gravity.

## 1 Introduction

^{1}In the present paper, we will focus on Einstein’s conformal gravity, but we will also show how some of our results can be extended to the Weyl gravitational theory. In particular, Sects. 2 and 3 are devoted to Einstein’s conformal gravity, while in Sect. 4 we show that our results can be easily extended to Weyl’s conformal gravity.

Einstein’s gravity is invariant under general coordinate transformations. If some quantity is singular in a certain coordinate system and not in another one, then the singularity is not physical, but only an artifact of the choice of the reference frame. In this case, we speak about “coordinate singularity”, and it is not a true singularity of the spacetime. For example, in the Schwarzschild spacetime in Schwarzschild coordinates, the metric is singular at the spherical surface \(r = 2M\) describing the event horizon. However, the spacetime is perfectly regular there, and it is indeed possible to remove the singularity with a suitable change of coordinates.

Conformal gravity is invariant under both general coordinate transformations and conformal transformations. In conformal gravity, it is possible to remove the spacetime singularities present in Einstein’s gravity by finding a suitable conformal transformation \(\Omega =\Omega (x)\) [13, 14]. In this context, we should speak about “conformal singularities”, because, in analogy to the coordinate singularities in Einstein’s gravity, they are not true singularities of the spacetime, but only artifacts of the choice of the conformal gauge.

Note that we cannot use the same mathematical tools to study the spacetime singularities in Einstein’s gravity and in conformal gravity. For example, in Einstein’s gravity it is common to check the regularity of a spacetime by studying invariants like the scalar curvature and the Kretschmann scalar. However, in conformal gravity the scalar curvature and the Kretschmann scalar are not invariant any longer: they are invariant under general coordinate transformations, but not under conformal transformations, and therefore they are not associated to any physical quantity. Another example is represented by the study of time-like geodesics. In Einstein’s gravity, the spacetime is geodesically incomplete if the trajectory of a massive particle cannot be extended beyond a certain point. However, in conformal gravity there are no massive particles (at least in the usual way), because they would violate the conformal symmetry of the theory.

If conformal invariance is a symmetry in Nature, it must be broken, because the Universe around us is not conformally invariant. For example, in a conformally invariant theory we cannot measure lengths and time intervals, which is definitively not our case. Conformal invariance may be spontaneously broken, and in such a case Nature has selected one of the possible vacua. Conformal gravity is an appealing theoretical framework to solve the problem of spacetime singularities when we assume that Nature can only select a physical vacuum in the class of singularity-free metrics [1, 2, 15].

It is well known that for the theory in (2) conformal invariance is broken at the quantum level [16]. However, in any finite (i.e. without quantum divergences) quantum field theory of gravity, conformal invariance is preserved [13]. This is realized in a class of weakly nonlocal gravitational theories that turn out to be finite at the quantum level and perturbatively unitary [17, 18, 19].

*L*is a parameter with the dimension of length. It is natural to assume that

*L*is either of order the Planck length \(L_\mathrm{Pl} \sim 10^{-33}\) cm or of order the gravitational radius of the system. In the former case, observational tests may be out of reach. In the latter case, we have the constraint \(L/M < 1.2\) from the X-ray spectrum of astrophysical black holes [20, 21], and independent constraints may be obtained from future observations of gravitational waves [22].

Moreover, the Schwarzschild metric rescaled by (5) is certanly a regular black hole’s solution of the Weyl conformal gravity too. Indeed, any Ricci-flat spacetime in whatever Weyl invariant gravity theory turns out to be singularity-free for a suitable rescaling.

In the present paper, we continue our study on the formation and evaporation of black holes in conformal gravity extending our previous work [23, 24, 25, 26]. In particular, we study the gravitational collapse of a spherically symmetric and electrically charged thin shell of radiation. The outcome of the collapse is a spherically symmetric and electrically charged singularity-free black hole, which is the counterpart of the singular Reissner–Nordström solution in Einstein’s gravity. We study the evaporation process and discuss our finding considering also the results of our previous work [26].

## 2 Gravitational collapse

Massive particles are not conformally invariant. The simplest model of gravitational collapse in conformal gravity in the symmetric phase is represented by the collapse of a thin shell of radiation. In the present paper, we want to consider the possibility of a non-vanishing electric charge.

*M*(

*v*) and

*Q*(

*v*) are given, respectively, by

*v*is the ingoing null coordinate, \(M_{0}\) is the ADM mass, and \(\Theta \) is the unit step function (Heaviside function). The energy–momentum tensor of the thin shell of radiation is

*r*and all values of the null ingoing coordinate

*v*.

## 3 Black hole evaporation

The thermodynamical properties of a black hole depend on the metric on and outside its event horizon. We can thus study the evaporation process of the newly born black hole with the metric in (13) instead of the metric in (11), which significantly simplifies our calculations.

*t*, \(\theta \), and \(\phi \). With such a choice of the metric tensor, the surface gravity is [28]

*L*, and thus identical to the Hawking temperature of the standard Reissner–Nordström black hole. Such a result had to be expected because the surface gravity at the event horizon is conformally invariant. The surface area at the event horizon, \(A_\mathrm{H}\), is instead different if \(L \ne 0\)

*L*we change the value of \(A_\mathrm{H}\) too.

### 3.1 Canonical ensemble

*Q*is constant during the evaporation process. For example, this is the case if the black hole can only emit uncharged particles. In such a case, the final product of the evaporation process is an extremal remnant with \(M=Q\).

*L*is proportional to the black hole mass, say \(L=\alpha M\) where \(\alpha \) is some constant, we have

*L*should depend on the characteristic length scales present in the original (non-rescaled) solution of the theory. Here we choose it to be directly proportional to the mass of the black hole

*M*. Another possible choice would be to assume that it is proportional to the charge

*Q*or to some combination of the two.

### 3.2 Micro-canonical ensemble

*T*is fixed [31, 32, 33]. The evaporation rate in micro-canonical ensemble is [33]

*L*. The mass-loss rate is vanishing exactly at the extremal point in both canonical and micro-canonical ensembles. The area surface of the horizon is regular in this limit, while it is important for the canonical ensemble that the Hawking temperature scales like

More generally, if we have (or not) a remnant as the final state of the evaporation process and the expression for the mass-loss rate is an analytic function of the difference between the actual mass of the black hole and the mass of the remnant (or zero if there is no remnant to be left), then the evaporation time will always be infinite. If *dM* / *dt* was a function whose behaviour is with the power exponent on the difference smaller than 1 (so non-analytic), then the evaporation time would be finite.

We notice that in the conformal gravity considered in this section as well as in Einstein’s gravity the evaporation times of charged black holes not possessing naked singularity (\(M>Q\)) diverge in all ensembles here studied. This is an evidence for the analogue of the third law of thermodynamics for black holes since the extremal black holes (\(M=Q\)) have vanishing Hawking temperature \(T_\mathrm{H}=0\). To reach them by any thermodynamical process (like Hawking evaporation) requires infinite amount of time so they are unattainable thermodynamical states. This situation is changed in Weyl’s conformal gravity as we discuss below.

## 4 Black holes in Weyl’s conformal gravity

### 4.1 Singularity-free black hole

*B*(

*r*) is:

*k*to vanish. This is an exact solution for a static black hole with the charge

*Q*. The only drawback of this solution for the metric, understood in standard (non-conformal) general relativity, is that it possesses a singularity at \(r=0\), similarly to the case of the standard Reissner–Nordström spacetime. This is both curvature and geodesic singularity. However, in a theory explicitly enjoying conformal symmetry we can easily remove this singularity by exploiting the freedom given by the overall conformal factor in the metric. For example with the conformal factor

In conformal gravity there is no singularity in \(r=0\) because one can check that standard GR-invariants (like the Kretschmann scalar) are regular. Moreover, for any probe (both massless and massive: conformally or minimally coupled to gravity) the amount of an affine parameter on their geodesics needed to reach the point \(r=0\) is infinite. Hence, the point \(r=0\) is unreachable. This resolution of the singularities proceeds exactly the same way in any theory of conformal gravity [14].

### 4.2 Black hole evaporation

Since the conformal factor (45), which makes the solution completely regular, is a bit complicated, we would like to present here thermodynamical analysis for the simplified solution (43) when \(\tilde{v}=0\). (This case is the most similar, but still different from RN solution.) In this particular case to determine the position of the horizon we need to solve a condition \(B(r)=0\) for values of the radial coordinate *r*. With \(\tilde{v}=0\) we solve easily the linear equation and we find \(r_\mathrm{H}=-\nicefrac {u}{w}\). We assume that the radius \(r_\mathrm{H}\) takes a positive value. Here we note that the analysis for the case with non-vanishing *v* and *k* is hampered by the fact that we would have to solve quadratic or cubic equation respectively to determine the position of the horizon. Another important remark is that to understand the causal structure of the spacetime we need only to study the trajectories of light rays so only the conformal structure of spacetime matters.

*u*is an arbitrary negative parameter. The precise location of the spherical surface of the horizon is at the radius

*M*and the parameter

*u*, namely \(u=-2M\) (valid for \(v=0\)). The luminosity in the canonical statistical ensemble is computed as

*Q*and the parameter of the theory \(\alpha \).

## 5 Concluding remarks

In this paper, we analyzed the formation and evaporation process of a spherically symmetric and electrically charged black hole in conformal gravity. The black hole is created by the collapse of a spherically symmetric and electrically charged thin shell of radiation. We worked with a singularity-free spacetime in the spontaneously broken phase of conformal invariance. Curvature invariants like the scalar curvature, the square of the Ricci tensor, and the Kretschmann scalar remain finite over the whole spacetime. The center of the black hole at \(r=0\) can be reached neither by massive particles in a finite proper time, nor by massless particles with a finite value of their affine parameter.

We studied the evaporation process in the canonical and micro-canonical ensembles in details in Einstein’s conformal gravity, assuming the possibility of emission of only uncharged particles. In both ensembles, the evaporation process always produces an extremal remnant with \(M=Q\), which is approached in an infinite amount of time. In Weyl’s conformal gravity we studied thermodynamics of Mannheim’s exact black hole solutions. We found that the evaporation time is there finite in the canonical and infinite in the micro-canonical ensemble, qualitatively very similarly to the case of Schwarzschild black hole in Einstein’s gravity.

It is worth noting that conformal invariance can fix the singularity issue even in the presence of the Maxwell field in its own way. While the electric potential remains singular at the center \(r=0\), this is not a problem because no massive (massless) particle can reach the center in a finite proper time (finite value of their affine parameter). This point is thus unattainable and the singularity of the electric potential has no physical implications.

## Footnotes

- 1.
In this paper we employ natural units: \(c = G_N = \hbar =k_B= 1\) and a metric with signature \(( - + + \, + )\).

## Notes

### Acknowledgements

C.B. acknowledges support from the National Natural Science Foundation of China (Grant no. U1531117), Fudan University (Grant no. IDH1512060), and the Alexander von Humboldt Foundation. S.P. thanks the Department of Physics at Fudan University for hospitality during his visit.

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