Reissner-Nordström geometry counterpart in semiclassical gravity
Julio Arrechea, Carlos Barceló, Raúl Carballo-Rubio, Luis J. Garay
RReissner-Nordstr¨om geometry counterpart insemiclassical gravity
Julio Arrechea, Carlos Barcel´o, Ra´ul Carballo-Rubio and Luis J. Garay , Instituto de Astrof´ısica de Andaluc´ıa (IAA-CSIC), Glorieta de la Astronom´ıa, 18008 Granada,Spain Florida Space Institute, University of Central Florida, 12354 Research Parkway, Partnership 1,Orlando, FL, USA Departamento de F´ısica Te´orica and IPARCOS, Universidad Complutense de Madrid, 28040Madrid, Spain Instituto de Estructura de la Materia (IEM-CSIC), Serrano 121, 28006 Madrid, Spain
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We compute the Renormalized Stress-Energy Tensor (RSET) of a masslessminimally coupled scalar field in the Regularized Polyakov approximation, as well as itsbackreaction, on the classical Reissner-Nordstr¨om spacetime. The complete set of solutionsof the semiclassical self-consistent equations is obtained and compared with the classicalcounterparts. The semiclassical Reissner-Nordstr¨om family involves three kinds of geome-tries that depend on the charge-to-mass ratio of the spacetime. In the under-chargedregime, the geometry has its external horizon replaced by a wormhole neck that leads toa singular asymptotic region at finite proper distance. The over-charged regime reveals anaked singularity coated by a cloud of (infinite) mass coming from the quantized field. Inbetween both behaviours there is a separatrix solution reminiscent of the extremal blackhole classical geometry. As the RSET over an extremal horizon is finite, the semiclassi-cal backreaction does not get rid of the horizon. Nonetheless, we show that the resultinghorizon is singular. a r X i v : . [ g r- q c ] F e b ontents r = 0 23 In the search for the first quantum gravity effects accessible to our current understanding,the study of black-hole spacetimes subject to semiclassical effects have become a successfultest bench. Quantum field theory in curved spacetimes has transformed our understandingof the nature of black holes. Landmark predictions of quantum field theory in curvedspacetimes include black hole evaporation via the emission of Hawking quanta [1], theUnruh effect [2], and cosmological particle creation [3]. Furthermore, the interest raisedby the black hole evaporation paradigm [4] has been one of the main fuel in developingframeworks beyond general relativity. It has also been one of the main incentives for theanalysis of different deviations from general relativistic black holes (see e.g. [5–7]), togetherwith the possibility of detecting these deviations observationally [8, 9].Semiclassical gravity emerges as an implementation of quantum field theory in curvedspacetimes that treats spacetime classically while the material content giving rise to suchspacetime is of mixed, quantum and classical, nature. Thus, finding self-consistent semi-classical solutions amounts to solving the Einstein equations using the Renormalized StressEnergy Tensor (RSET) of quantum fields as an additional source of gravity. Before finding– 1 –olutions sourced by specific classical matter, it is convenient to understand the structuralcharacteristics of vacuum semiclassical solutions.Previous analyses [10–13] have tackled the problem of finding the semiclassical counter-part to the Schwarzschild spacetime, that is, the vacuum solutions of semiclassical generalrelativity. To that end, the RSET of a massless scalar field (the simplest model to analyze)was taken in the Boulware vacuum, the one vacuum state compatible with both staticand asymptotically flat geometries. The Boulware vacuum is singular at the Schwarzschildhorizon, indicating the presence of strong backreaction effects there. Indeed, the resultingself-consistent spacetimes—incorporating backreaction—have no horizon and are, insteadof black holes, asymmetric wormholes with one singular end. The resulting wormholes haveone asymptotically flat end, whereas the other end develops a null singularity located ata finite proper distance from the neck of the wormhole. As opposed to the Schwarwschildgeometry, its semiclassical counterpart exhibits a singularity which is not covered by anyevent horizon (for a discussion of potential lessons to be learned from this fact see [14]).In this work, we extend the analysis in [13] considering here the electrovacuum solutionsof semiclassical general relativity. Again, for the quantum sector, we compute the vacuumpolarization of a single massless scalar field minimally coupled to curvature. For the RSETwe use the so called Regularized Polyakov (RP-RSET) approximation described in detailin [13]. At the beginning of the next section we will discuss the reasons behind the use ofthis approximation and the associated benefits and limitations. Armed with this RSET,in this work we characterize the complete set of semiclassical electrovacuum solutions,which can be separated into three subclasses attending to whether the geometry is under-charged, over-charged or balanced. We numerically compute solutions belonging to eachof the subclasses, and we also find analytical approximations to the solutions in certainspacetime regions.The Reissner-Nordstr¨om geometry has been used in several analyses of quantum-induced phenomena. One notorious characteristic is that the classical sub-extremal Reisner-Nordstr¨om black hole shows a double horizon structure, with a future outer horizon and afuture inner horizon (we are using the notation first introduced in [15]). This structure issimilar to that exhibited by Kerr black holes, while preserving the benevolence of sphericalsymmetry. That is why the analysis of Reissner-Nordstr¨om geometries is also used as aproxy to understanding the more complicated Kerr case. The presence of a second internalhorizon adds some special features to these geometries. For example, different analysesshow that, as opposed to the outer horizon, the inner horizon is highly unstable underboth classical and quantum perturbations [16, 17]. Our analysis here offers an additionalperspective into these stability-related aspects: As we will see, the semiclassical correctionseliminate the outer horizon in such a way that the system never enters (at least in vacuum)a regime which explores the physics of an inner horizon. This could be taken as a sugges-tion that horizons are only present during transient regimes but are absent in genuinelystatic configurations.Another intereresting aspect of the Reissner-Norsdtr¨om family is the existence of anextremal solution, in which inner and outer horizons are spatially coincident. In the Boul-ware vacuum, the value at the RSET at the outer horizon diverges, but not if the horizon– 2 –s extremal [18–20]. This can be taken as an indication that extremal black holes are stableunder semiclassical effects. Perhaps surprisingly, this is not what we find. Semiclassicalcorrections displace the extremal horizon from its original position and transforms it into acurvature singularity, unveiling a narrow region beyond this singular horizon where semi-classical corrections become non-perturbative. This result exemplifies a situation wherecalculating the RSET over a fixed background geometry suggest that semiclassical correc-tions act perturbatively around the horizon, but the geometry incorporating back-reactionself-consistently develops singularities. In view of this, we point out the existence of regimesin which predictions based only on fixed-background computations should be taken withdue care.The previous results seem to indicate a strong tension between quantized fields livingin the Boulware state and event horizons of any kind. It has been argued that horizonswith a higher multiplicity may be compatible with a finite RSET [21]. As these geometriescannot be attained without the introduction of additional matter fields, we are leaving themoutside our discussion. On the other hand, by considering the fields to be in the Hartle-Hawking vacuum state, the local structure of the outer horizon of the Reissner-Nordstr¨omgeometry can be maintained. This is accomplished by the introduction of fluxes of energythat diverge at the horizon, thus cancelling the Boulware divergence. The inner portion ofthe geometry becomes unveiled and allows to explore how the Polyakov RSET backreactsnear the inner horizon. A detailed analysis on this topic will be presented elsewhere.Finally, let us mention that in order to analyze the structure of semiclassical chargedcompact objects, these must be matched at their surface with the semiclassical Reissner-Nordstr¨om geometry here depicted. The repulsion exerted by the electromagnetic fieldallows charged spheres of fluid to reach arbitrarily high compactness [22]. Hence, theseconfigurations could serve as models in which the effects of quantized fields in highlycompact scenarios could be analyzed.
In the same spirit as in [13], here we find the semiclassical Reissner-Nordstr¨om counterparts.The presence of the electromagnetic field gives rise to a broader spectrum of geometries.In our analysis we find three families of solutions. The first one, corresponding tospacetimes where the electromagnetic charge is below the mass, has a wormhole structuresimilar to that of the semiclassical Schwarzschild counterpart, and connects to that solutionin the vanishing charge limit. As charge is increased, electromagnetic repulsion makesthe wormhole neck shrink (as the classical horizon would do), but the structure of theasymptotic singularity at the other side of the neck remains unmodified, as the effects ofcharge decay with distance.The second family where charge surpasses mass resembles naked singularities, but thistime located at finite distance in the r coordinate. Contrary to the previous case wherethe RP-RSET backreacts to make the geometry devoid of horizons, the super-charged casecorresponds to a naked singularity in the classical regime. From this situation we learn howvacuum polarization backreacts on an already singular geometry, which turns out to resultin an increased strength of the singularity. The dominant contribution at short distances– 3 –omes from a runaway of vacuum polarization, stimulated, in part, by the blowing up of thePolyakov RSET as we approach the center of the geometry. The electromagnetic charge,however, retains its “undressed” value and does not contribute to increasing the strength ofthe singularity. In situations where the RP-RSET is sufficiently suppressed, the dominantdivergence contribution to the mass comes from terms proportional to the charge ratherthan those coming from vacuum polarization.In between both cases there is a separatrix solution, or “quasi-extremal” geometry,with a degenerate horizon where vacuum polarization is finite. The existence of the semi-classical extremal black hole has been subject of debate in the community [23–26]. Ithas been shown, through different approximations to the RSET, that its components arewell-behaved at the extremal horizon [18–20]. This suggests that backreaction will mod-ify mildly the horizon structure of the extremal black hole, but without destroying italtogether. With these considerations in mind we present a self-consistent semiclassical“quasi-extremal” black hole geometry in 3 + 1 dimensions and analyze some of its proper-ties. However, we observe that this “quasi-extremal” geometry develops a non-analyticityat the horizon, in the form of a cusp in the metric functions, and that this behaviour gen-erates a curvature singularity. This curvature singularity is not visible in curvature scalarinvariants. Instead, it appears when the Riemann curvature tensor is contracted with atetrad field parallel transported along an infalling geodesic trajectory. Backreaction on theextremal black hole is, in this sense, more benign than for spacetimes whose horizons havethe same local form as the Schwarzschild horizon. The horizon characteristic is preserved,though the effect of vacuum polarization makes this horizon singular. This result stronglysuggests the incompatibility between the Boulware vacuum state and regular horizons ofany sort.The paper is organised as follows. Section 2 reviews the classical Reissner-Nordstr¨omfamily and some of its properties, alongside a discussion of the Regularized Polyakov RSET.Section 3 sets up the analysis for the self-consistent electro-vacuum semiclassical equations.Sections 4, 5 and 6 analyze sub-charged, super-charged and quasi-extremal regimes, respec-tively. Finally, section 7 is reserved for discussion and conclusions. We consider the most general static and spherically symmetric line element ds = − e φ ( r ) dt + 11 − C ( r ) dr + r d Ω , (2.1)where e φ is the redshift function encoding the redshift suffered by escaping lightrays,and C is the compactness function, which can be written as C ( r ) = 2 m ( r ) /r where m ( r )is the Misner-Sharp mass contained inside spheres of radius r [27–29]. In vacuum, thetime component of the geometry equals the inverse of the radial one, but this relation nolonger holds in the presence of matter (note that the electromagnetic stress-energy tensor(SET) constitutes an exception). The Reissner-Nordstr¨om spacetime is the geometry that– 4 –esults from solving the Einstein-Maxwell equations under the assumptions of staticity andspherical symmetry. From the Maxwell equations we obtain the following form for theelectromagnetic SET, T νµ = diag( − , − , , Q πr . (2.2)Here, Q = Q + Q , where Q e and Q m denote the electric and magnetic charges, respec-tively. Greek letters denote spacetime indexes. The tt and rr components of the classicalfield equations are C + rC (cid:48) = Q r , − rψ + C (1 + 2 rψ ) = Q r , (2.3)where the (cid:48) denotes derivatives with respect to the r coordinate and ψ ≡ φ (cid:48) . Solvingequations (2.3) yields the Reissner-Nordstr¨om geometry ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r d Ω . (2.4)This solution shows several unique features due to the presence of charge. The zeroes ofthe redshift function determine the location of its two horizons r ± = M ± (cid:112) M − Q . (2.5)Therefore, depending on the charge-to-mass ratio of the geometry, it can exhibit two, one, orno horizons whatsoever. This splits the Reissner-Nordstr¨om family into three cathegories: • Sub-extremal (
Q < M ), which shows a timelike singularity covered by outer and innerhorizons at r − and r + respectively. The Schwarzschild black hole is the particularcase where Q = 0. • Super-extremal (
Q > M ), where the value of the charge surpassing that of the mass isunderstood as if some charged negative mass had entered the geometry. Ultimately,this gives rise to a naked singularity. • Extremal black holes ( Q = M ), for which outer and inner horizons become coincident,forming an extremal horizon at the end of an infinite neck. Whereas in flat spacetime infinite vacuum energies can be subtracted, this procedure be-comes intricate in curved spacetimes where, in order to account for the genuine contributionof vacuum energy to the spacetime curvature, the field stress-energy tensor undergoes arenormalization procedure [30]. Ideally, one would hope to have a single exact RSET, butin practice the situation is much more complicated. On the one hand, we have ambiguitiesin the renormalization procedure [31]. On the other hand, we have different approximationschemes which can be well-suited to different aspects of the problem.– 5 –oncerning the various approximation schemes, there is a range from basic to moresophisticated: stress-energy tensors that only inform about the s -mode of the fields, withor without backscattering [32, 33]; stress-energy tensors [34, 35] accounting for local contri-butions to curvature; and more elaborate expressions adapted to scalar fields of arbitrarymass and coupling in static spacetimes, where arbitrarily high multipoles are considered[36, 37]. Quantum corrections for the Reissner-Nordstr¨om spacetime have been calculatedin some of these approximations [36, 38, 39] and also making use of the celebrated traceanomaly [40, 41].It is worth mentioning the pursue for a complete calculation of the RSET by Andersonet al. [36]. Their RSET can be split into two independently conserved numerical andanalytical parts. The analytical portion includes both high-frequency, local contributionsand low-frequency, state-dependent terms [37]. Ideally, one would rather solve the fullbackreaction equations equipped with this analytical approximation to the RSET. Thecomplexity of the system of equations involved makes this task difficult, although solutionsdescribing symmetric wormhole spacetimes [42] have been found. Moreover, this analyticalapproximation to the RSET has been computed over the Reissner-Nordstr¨om spacetime[36], showing a pathological behaviour at the horizon that disappears after the addition ofthe numerical part.It is also important to stress that most of the aforementioned renormalized stress-energy tensor (RSET) approximations share the presence of higher-derivative terms thathinder a full self-consistent treatment of the semiclassical equations, where spacetime ge-ometry and material sources are computed simultaneously. In consequence, vacuum polar-ization is either computed on top of a fixed background, or its backreaction is consideredbut not in a complete self-consistent way [43]. Given the complexity of these approachesto implement and interpret self-consistent solutions, here we follow a different strategy. In1 + 1 dimensions, the renormalized stress-energy tensor of a massless scalar field acquiresa simpler form, obtainable via the point-splitting regularization method [32]. After renor-malization, the RSET is then transformed into a (3 + 1)-dimensional quantity by meansof the Polyakov approximation [44]. In this process we lose information about quantumfluctuations not living in the s-wave sector of the 4-dimensional spacetime. Conservationdemands adding a 1 / πr multiplicative factor (with r the coordinate denoting the arealradius) to the RSET components. The resulting tensor has vanishing angular componentsand diverges in the r → s -wave field modes in the RSET [33]. Unfortunately, we lack an approximation that in-corporates backscattering and that is regular at r = 0. Motivated by the search of RSETapproximations that are both regular at the center of spherically symmetric spacetimesand contains up to second-order derivatives of the metric functions, we follow [13, 48, 49]by adopting the Regularized Polyakov approximation, where the Polyakov multiplicative– 6 –actor is modified by the introduction of a regulator that acts as a cutoff to the magnitudeof the Regularized Polyakov RSET (RP-RSET) components. The RP-RSET is conserved,finite at the radial origin and has tangential pressures, features shared by the stress-energytensor found by Anderson et al. [36]. Note that these convenient properties are accom-plished simultaneously due to staticity. In dynamical scenarios [48, 49] the introductionof the regulator breaks conservation in such a way that cannot be compensated solely byadding angular components. Thus, in such cases, the regularized tensor no longes sat-isfies the covariant conservation condition. However, here we will only deal with staticconfigurations for which the Regularized Polyakov RSET is appropriate. Having reviewed the classical Reissner-Nordstr¨om family and some of its features, we nowturn to the realm of semiclassical gravity. The semiclassical Einstein equations G µν = 8 π (cid:16) T µν + (cid:126) (cid:104) ˆ T µν (cid:105) (cid:17) (2.6)can, in some particularly simple scenarios, be solved at the self-consistent level, wherespacetime geometry and matter contributions are determined simultaneously. For thispurpose, the quantum matter contribution (cid:104) ˆ T µν (cid:105) is constructed by means of the Polyakovapproximation. The Polyakov approximation involves a dimensional reduction to a (1 + 1)dimensional manifold described by the non-angular sector of the metric (2.1). Owing tothe fact that the wave equation for a scalar field propagating on top of a (1 + 1) spacetimeis conformally invariant, an exact expression for the RSET is obtained after point-splittingrenormalization [32], its components being (cid:104) ˆ T rr (cid:105) P2 = − l ψ (cid:104) SDT (cid:105) , (cid:104) ˆ T tr (cid:105) P2 = (cid:104) ˆ T rt (cid:105) P2 = 0 , (cid:104) ˆ T tt (cid:105) P2 = l e φ (cid:2) ψ (cid:48) (1 − C ) + ψ (1 − C ) − ψC (cid:48) (cid:3) + (cid:104) SDT (cid:105) . (2.7)Here, l P = (cid:126) / √ π and (cid:104) SDT (cid:105) denotes the state-dependent part of the Polyakov RSET,vanishing for the Boulware vacuum. The components (2.7) are then used in order toconstruct the temporal and radial sector of the (3 + 1) Polyakov RSET in the followingway: (cid:104) ˆ T µν (cid:105) P = 14 πr δ aµ δ bν (cid:104) ˆ T ab (cid:105) P2 , (2.8)with latin indexes taking the t, r values. The (1 + 1) components must be divided by thesurface area of the sphere to ensure (3 + 1)-dimensional conservation of the RSET. Thismultiplicative factor introduces a generic divergence in the components of the PolyakovRSET as these approach r →
0. Indeed, the Polyakov RSET has singular componentswhen computed over geometries with finite curvature invariants at r = 0. This comesin conflict with the idea that, in regular matter distributions of small compactness, thePolyakov RSET should amount to a tiny contribution throughout the whole spacetime.More explicitly, demanding that the Kretschmann scalar K = 4 C r + 2 C (cid:48) r + 8 ψ (1 − C ) r + (cid:2) ψC (cid:48) − ψ + ψ (cid:48) )(1 − C ) (cid:3) (2.9)– 7 –emains regular imposes the following behaviour for the metric functions φ ( r ) = φ + φ r + O ( r ) , C ( r ) = C r + O ( r ) , (2.10)where φ , φ , C are arbitrary, non-zero constants. Now, it can be easily checked that, forthe profiles (2.10), the semiclassical energy density ρ s = e − φ (cid:104) ˆ T tt (cid:105) P ∝ φ r + O ( r ) (2.11)diverges quadratically in r . This issue is aggravated by the fact that the semiclassicalequations, due to their nonlinear nature, have this divergence displaced to r = l P . Asa consequence, the Polyakov approximation breaks down at distances where we were notexpecting singularities to arise. Strictly speaking, the Polyakov approximation cannot betrusted at small enough scales [50]. In this work, our strategy is to regularize the Polyakovapproximation to avoid these types of divergences, and to analyze the resulting set ofsolutions. Along the way, we will comment on which characteristics of the solutions foundshould be independent of our final approximation scheme and which ones might not be.Inspired by the approach followed in [48], we provided a regularization scheme for thePolyakov RSET [13]. This procedure is carried out in two steps. First, we introduce acutoff in the non-angular sector of the Polyakov RSET by transforming the multiplicativefactor (cid:104) ˆ T ab (cid:105) DP → πr π (cid:0) r + αl (cid:1) (cid:104) ˆ T ab (cid:105) P , (2.12)where the super-index DP stands for Distorted Polyakov. Taking α > (cid:104) ˆ T ab (cid:105) DP at the radial origin. The second step consists in adding a Compensatory pieceto the Distorted Polyakov RSET so the sum of both terms gives a covariantly conservedtensor. The RP-RSET is then defined as T RP µν ≡ T DP µν + T C µν , (2.13)where the components of the compensatory tensor are assumed to be angular only forsimplicity, and come from algebraically solving ∇ µ T RP µr = 0 (2.14)Neither the choice of regulating factor (2.12), nor the assumption of (cid:104) ˆ T µν (cid:105) C having onlyangular components, are unique. Our choice is based on an attempt to modify the PolyakovRSET in the mildest way, while achieving the desired regularity properties. The angularcomponents of the RP-RSET have the form (cid:104) ˆ T θθ (cid:105) RP = (cid:104) ˆ T ϕϕ (cid:105) RP sin θ = − αr π (cid:0) α + r /l (cid:1) ψ (1 − C ) . (2.15)Self-consistent solutions of the semiclassical field equations are sensitive to the regularityof the RSET acting as a source, even in situations where the classical spacetime prior tothe backreaction has a physical singularity at r = 0. In [13] we obtained the semiclassical– 8 –ounterpart of the Schwarzschild geometry making use of the RP-RSET. We found that thecounterpart to the Schwarzschild black hole has its horizon replaced by a wormhole neck.This neck is always placed above the Schwarzschild radius of the geometry. The RP-RSETallowed to extend the space of solutions to those whose neck lies below the Planck length,something forbidden for the Polyakov RSET due to its inherently singular form.Before turning to the full semiclassical analysis, let us comment briefly on the regularityof the RP-RSET at horizons. To do so, let us calculate the semiclassical energy densitygenerated by a geometry with a Schwarzschild-like horizon. By Schwarzschild-like, we meana local behaviour of the form e φ ∝ r − r H r H + O (cid:18) r − r H r H (cid:19) , − C ∝ r − r H r H + O (cid:18) r − r H r H (cid:19) , (2.16)for the metric functions. Such profiles have a divergent RP-RSET with the energy densityand pressure behaving as ρ s = − p s ∝ − l r ( r − r H ) + O (cid:18) r − r H r H (cid:19) . (2.17)It is expected that, once (2.6) are solved at the self-consistent level, the backreaction of vac-uum polarization on Schwarzschild-like horizons will be non-perturbative, thus modifyingthe metric so as these horizons disappear. This is a defining characteristic of the Boulwarevacuum state, which is ill-defined at horizons (of the Schwarzschild kind). Had we con-sidered the Hartle-Hawking vacuum instead, then the local structure around the horizonwould have remained unspoiled, but at the cost of modifying the asymptotic regions.Now consider a geometry that has an extremal horizon, characterized by the metricfunctions having a double zero at r H . An extremal horizon has zero surface gravity and, inconsequence, the state-dependent terms of the Polyakov RSET (2.7) vanish for all vacuumstates, which become degenerate at the extremal horizon. We can easily check that thefollowing profiles e φ ∝ (cid:18) r − r H r H (cid:19) + O (cid:18) r − r H r H (cid:19) , − C ∝ (cid:18) r − r H r H (cid:19) + O (cid:18) r − r H r H (cid:19) , (2.18)provide finite semiclassical density and pressures ρ s = p s ∝ l r + O (cid:18) r − r H r H (cid:19) , (2.19)indicating that backreaction around an extremal configuration would, presumably, preservethe horizon. Indeed, later we will see that the horizon is preserved, although it receivesnon-perturbative quantum corrections that make it singular. The following sections will be devoted to writing down the semiclassical Einstein equations(2.6) having as sources the RP-RSET and the SET of the electromagnetic field, as well as– 9 –iscussing some of their most salient properties. By analyzing these expressions we are ableto determine the existence of three types of semiclassical solutions depending on the charge-to-mass ratio of the geometry (as in the Reissner-Nordstr¨om family), and reconstruct theshape of the solutions living in each of these three regimes. The tt and rr components ofthe semiclassical field equations are, respectively, C + rC (cid:48) = Q r + l r r + αl (cid:8)(cid:2) ψ (cid:48) + ψ (cid:3) (1 − C ) − ψC (cid:48) (cid:9) , (3.1) − rψ + C (1 + 2 rψ ) = Q r + l r r + αl ψ (1 − C ) . (3.2)Instead of working with these two equations simultaneously, we can solve algebraically for C in the second equation and plug the obtained expression into the first equation, whichresults into a first-order differential equation for the variable ψ : ψ (cid:48) = A + A ψ + A ψ + A ψ , with A ( r ) = Q D ( r ) ,A ( r ) = 2 r (cid:34) r − Q (cid:32) l (cid:2) r + αl (cid:3) (cid:33)(cid:35) D ( r ) ,A ( r ) = r (cid:34) (cid:0) r − Q (cid:1) (cid:32) l r (cid:2) r + αl (cid:3) (cid:33) + l (2 r − Q ) r + αl (cid:35) D ( r ) ,A ( r ) = r l r + αl (cid:34) r (cid:32) αl (cid:2) r + αl (cid:3) (cid:33) − Q (cid:32) l [ r + 2 αl ] (cid:2) r + αl (cid:3) (cid:33)(cid:35) D ( r ) , D ( r ) = − r + αl r ( r − Q ) (cid:2) r + l ( α − (cid:3) . (3.3)Once the solutions of this differential equation are analyzed, we can come back to thesecond equation in (3.1) in order to directly obtain the behavior of C .In view of the above expression, it becomes clear that taking α > / D vanishes at r = l P √ − α . By taking α >
1, we move this singularityoutside of the domain of the radial coordinate, as it picks up an imaginary component. Itis also interesting to note that (3.3) is, in principle, ill-defined at r = Q as well, as 1 / D vanishes for this radius too.The right-hand side of Eq. (3.3) is a cubic polynomial in ψ and can be factorized inroots that are functions of r . These roots are defined piecewise and can be matched alongdifferent intervals of the radial coordinate so that, when plotted, they appear as continuouscurves. We have adopted the following definition R = S r ≤ Q S Q < r ≤ r − i S r − i < r ≤ r div S r > r div , R = (cid:40) S r ≤ r − i S r > r − i , R = S r ≤ Q S Q < r ≤ r div S r > r div , – 10 –here the expressions for S i are, in terms of the coefficients in Eq. (3.3), S = − A A (cid:34) / (cid:0) A A − A (cid:1) A H − H / A (cid:35) , S , = − ± i √ (cid:18) A A (cid:19) R , (3.4)with H = (cid:20) − A + 9 A A A − A A + (cid:113)(cid:0) A − A A A + 27 A A (cid:1) − (cid:0) A − A A (cid:1) (cid:21) / . (3.5)The symbols r ± i in (3.4) mark the lower and upper limits of a region where the roots acquirea non-zero complex part. Particularly, R and R become complex conjugate roots withinthis interval. The occurrence of complex roots has no impact on solutions, since, as we willprove in the following sections, solutions of (3.3) with support in the interval r ∈ ( Q, r div )have to intersect certain fixed points.The roots {R i } i =1 will be of utility in constraining the shape of solutions, since theyindicate the turning points of the solution ψ . It is illustrative to compare with the analysispresented in [13] of the semiclassical Schwarzschild counterpart, which is obtained taking Q = 0 in the expressions above. For the Schwarzschild counterpart, one of the three rootsis trivial ( R = 0), while the others always take negative values. The presence of chargeintroduces a non-zero zeroth-order term term in (3.3), A (cid:54) = 0, modifying the shape of theroots (in particular, these now take positive values) and, in consequence, the domain ofthe solutions.In Fig. 1 we show a plot of the roots and two exact solutions (the meaning of whichis discussed right below) for a particular value of the charge parameter. The two specialexact solutions of equation (3.3) are ψ ± = − r + αl l r (cid:32) ± (cid:115) r + ( α − l r + αl (cid:33) . (3.6)The simplest way to find these exact solutions is analyzing the C → −∞ limit of thesystem of semiclassical equations; direct substitution shows that these are indeed actualexact solutions of (3.3). Remarkably, these are also exact solutions in the Q = 0 situationthat remain unchanged for nonzero values of the charge. In fact, we can rewrite (3.3) as ψ (cid:48) = F Sch ( r, ψ ) + G ( r, Q, ψ )( ψ − ψ − )( ψ − ψ + ) , (3.7)where F Sch corresponds to the right hand side of (3.3) evaluated at Q = 0, and G = l Q (cid:18) l rr + αl ψ (cid:19) ( r − Q ) (cid:2) r + ( α − l (cid:3) . (3.8)– 11 – .1 0.2 0.3 0.4 - - - - - R ψ R R ψ + ψ − r d i v Q ψ r Figure 1 . Plot of the roots R , , (continuous curves) and the exact solutions ψ ± (dashed lines).The roots are defined piecewise, and take negative values except for the positively diverging portionof R . Its asymptote at r = r div (vertical, dashed line) marks the separatrix solution between thesub-extremal and super-extremal regimes. The dotted vertical line is r = Q . In this figure, wehave taken α = 1 .
01 and Q = 0 . ψ describing awormhole of asymptotic mass M = 0 .
285 has been drawn in black. The procedure giving rise tosuch solution is detailed in later sections.
This shows explicitly why ψ ± can be exact solutions of (3.3) for arbitrary values of thecharge Q even if they do not depend on Q , given that the Q -dependent terms in (3.3)vanish identically for ψ = ψ ± .The presence of a double ± sign in (3.6) comes from the existence of two branches ofsolutions of the semiclassical equations. These arise as a consequence of the semiclassicalcorrections introducing terms quadratic in ψ in (3.2). The branch associated with the − sign (which we shall call “unconcealed”) returns the correct classical field equations in the l P → ψ jumps from + ∞ to −∞ , or the quantity G (3.8) vanishes at some finite radius. These conditions follow from the requirement thatthe discriminant of Eq. (3.2) (seen as a second-order polynomial for ψ ) vanishes.We wish to end this section by stressing that, since (3.3) is a first order differentialequation for ψ , we can apply the uniqueness and existence theorem wherever r (cid:54) = Q ;note that for r → Q the right-hand side of (3.7) diverges as ( r − Q ) − , which makes thedifferential equation singular there. This implies that solutions of this equation cannotintersect at finite r but at the surface r = Q . In turn, we will show that the exact solutions– 12 –3.6) act as boundaries for the remaining solutions. In addition, the roots (3.4) denotethe turning points of ψ . These features will allow us to characterize the solutions of (3.3)thoroughly. Let us start analyzing the behavior of (3.3) at large radial distances and imposing asymp-totic boundary conditions in order to select the solutions describing asymptotically flatspacetimes. The expectation value that leads to the RP-RSET is taken in the Boulwarevacuum state, which has no particle content for asymptotic stationary observers. Thisensures the existence of solutions in which semiclassical corrections decay sufficiently fastwith distance as to preserve asymptotic flatness.If we assume that the metric in (2.1) is asymptotically flat, ψ being the derivative oflog( g tt ) / ψ ∝ r − η , η ≥ . (3.9)Inserting this ansatz into (3.3) and expanding in the large r limit we obtain the followingleading-order expansion ψ (cid:48) ∝ − r − η − , (3.10)which implies that the semiclassical equations are compatible with solutions with a leadingbehavior η = 2, which are in fact associated with the leading terms in the asymptoticexpansion of the Schwarzschild metric, the mass M being the corresponding proportionalityconstant (up to numerical factors).Now, we allow the next subleading terms to enter expression (3.10), informing us aboutthe first Q -dependent corrections, ψ (cid:48) (cid:39) − ψr (1 + rψ ) + Q r . (3.11)Integrating this expression twice yields φ (cid:39) ψ + 12 ln (cid:34) cosh (cid:32) √ Qr + ψ (cid:33)(cid:35) , (3.12)which now displays the leading asymptotic behavior characteristic of the Reissner-Nordstr¨ommetric, in which e φ (cid:39) − Mr + Q r + O (1 /r ) . (3.13)In fact, in order to reproduce the above expression one just needs to choose the integrationconstants ψ and ψ as ψ = 12 ln [sech ( ψ )] , ψ = − arctanh (cid:32) √ MQ (cid:33) , (3.14)Note that the expression of the leading-order contribution (3.9) serves to determine whichterms in Eq. (3.3) enter the expansion (3.11) at the same order as the first Q -dependent– 13 –erm. Assuming that ψ obeys an expansion with additional terms decaying with r , as in Eq.(3.9), is not sufficient to find the correct asymptotic limit. Such an expansion for ψ fails toproperly approximate the asymptotic solution since, for the Reissner-Nordst¨om geometry, ψ equals a quotient between polynomials. At best, expanding ψ in powers of r succeeds inreturning the Schwarzschild geometry, but fails in determining the Reissner-Nordstr¨om ge-ometry, which involves subleading contributions. As a consequence, the correct asymptoticmetric comes from solving the approximate expression (3.11) in a self-consistent manner.In addition, the behaviour for C is found by replacing (3.12) inside (3.2) and takingthe leading order contribution for large r . We obtain (1 − C ) (cid:39) e − φ in the asymptoticlimit, so the Reissner-Nordstr¨om geometry is fully recovered.After fixing the asymptotic behavior of the different solutions, we proceed by integrat-ing the semiclassical equations inwards, towards smaller values of r . In doing so, severalscenarios can arise depending on the balance between the charge Q and the asymptoticmass M . We start our inwards integration from the asymptotic region with a positive ψ (this condition is equivalent to assuming M > ψ ensures positivity of C ) that situates the solution above all roots and analytical exactsolutions depicted in Fig. 1. Self-consistency of (3.3) ensures that ψ grows monotonicallyinwards unless it intersects one of the roots. In view of Fig. 1, there exist three posibilities:the solution ψ either diverges at some radius r > r div ; it grows sufficiently slow as to cross r div , encountering a maximum and extending to r (cid:39)
0; or it stays in between both regimes,diverging at r = r div at the same rate as the root R does. The solution will follow either ofthese paths depending on the relative values of Q, M and α . In turn, we can assure that, aslong as M > α >
1, there exists a critical value of the charge Q crit that correspondsto the separatrix solution. In the following we analyze these three cases individually. The first solutions we analyze are deformed continuously to the Schwarzschild case in thelimit Q →
0, and are valid up to a critical value of the charge Q crit that is determined bythe self-consistency of these solutions. As the first step in our analysis, we assume a positive value of ψ at a large initial radius.This starting assumption will be present in all the following sections, whereas the particu-larities of the ψ < ψ growsmonotonically inwards until it either reaches the root R , diverges exactly at r = r div , orit does so at some radius r > r div > Q . This last possibility is the one we explore in thepresent section. Assuming that the function ψ diverges at some finite radius r B > r div (the value r B stands for bouncing surface of the radial function, as we will see below), thedifferential equation (3.3) can be approximated, at leading order in ψ , by ψ (cid:48) (cid:39) A ψ . (4.1)– 14 –he term A is just the coefficient A in (3.3) evaluated in the r → r B limit, where ittakes a constant value A = − l r B (cid:34) r (cid:32) αl (cid:2) r + αl (cid:3) (cid:33) − Q (cid:32) l (cid:2) r + 2 αl (cid:3)(cid:2) r + αl (cid:3) (cid:33)(cid:35)(cid:0) r − Q (cid:1) (cid:2) r + l ( α − (cid:3) . (4.2)The sign of this constant depends on the value of the charge Q . Integrating the differentialequation (4.1) returns the following pair of solutions ψ (cid:39) ± (cid:115) k r − r B ) + O ( r − r B ) / , (4.3)where, for consistency with the notation in [13], we have introduced the redefinition k = − A . (4.4)Here, the integration constant has been fixed to preserve the divergence in ψ as r → r B .The ± signs indicate that there are two solutions, one per branch. For the moment, we takethe + sign since, in the inwards integration, ψ diverges towards positive infinity. Later, wewill give meaning to the − sign solution as well.For each of these two solutions to be real, the constant k must be positive, which inturn implies that | Q | < r B (cid:118)(cid:117)(cid:117)(cid:116) αl + (cid:0) r + αl (cid:1) l (cid:0) r + 2 αl (cid:1) + (cid:0) r + αl (cid:1) . (4.5)This is the semiclassical equivalent of the condition that guarantees the non-extremal natureof the classical Reissner-Nordstr¨om black hole. We will analyze in more detail the contentof this constraint on the charge below; for the moment let us keep describing the elementsof the metric.The redshift function follows from integrating Eq. (4.3) φ (cid:39) (cid:112) k ( r − r B ) + φ B + O ( r − r B ) / , (4.6)where the integration constant φ B denotes the value of the redshift function at r = r B .This parameter must be found by numerically integrating the solution from the asymptoticregion until the surface r B is encountered.As for the compactness function, replacing Eq. (4.3) inside (3.2) returns the compact-ness in the r → r B limit, which goes as C (cid:39) − k ( r − r B ) + O ( r − r B ) / , (4.7)with k = 4( r − Q )( r + αl ) l r k > . (4.8)– 15 –bserving the metric functions (4.6) and (4.7) it is noticeable that, at r = r B , the redshiftfunction is finite and nonzero, while compactness equals 1. As a consequence, the radialcomponent of the metric diverges at the spherical surface r B , while the time componentremains finite, indicating the pressence of a bouncing surface for the radial coordinate.Schwarzschild coordinates do not provide an adequate description of such surfaces, so weperform a coordinate transformation to the proper coordinate ldldr = ± (cid:112) k ( r − r B ) , (4.9)where the ± sign informs us about the branch of the radial coordinate in which observersare located. Integrating (4.9) yields l − l B = ± (cid:115) r − r B ) k . (4.10)Therefore, the radial coordinate has a minimum at the surface r B , while the l coordinateis continuous at that minimal surface. The + sign in (4.9) denotes the side of this minimalsurface where r grows with l , that is, the side that contains the asymptotically flat region.The − sign denotes the other side of r B , where r decreases as l grows. A jump betweenthe two branches of solutions in (4.3) takes place at l = l B . In terms of the l coordinate,the line element around r B is given by ds (cid:39) − e √ k k ( l − l B )+2 φ B dt + dl + (cid:20) k l − l B ) + r (cid:21) d Ω . (4.11)In the above expression, we have integrated (4.3) with the ± sign and performed thecoordinate change (4.10), matching the signs so that the line element is continuous anddifferentiable through l = l B . The line element (4.11) represents the metric in the near-neck region of an asymmetric wormhole. The l > l B region connects smoothly with theReissner-Nordstr¨om geometry sufficiently far from the neck, whereas the l < l B domaincorresponds to a new asymptotic region with a radically different asymptotic structure.As we already showed in [13], the semiclassical equations in the Boulware vacuum do notpermit the existence of non-extremal horizons, meaning horizons with a non-zero surfacegravity. Here we show that the charged case exhibits the same type of elimination ofthe horizon: substitution by a wormhole neck. The cubic dependence in ψ introducedby the RP-RSET in (3.3) is the ingredient that ultimately causes the occurrence of thesehorizonless solutions. In physical terms, this is a consequence of the huge backreaction thattakes place when non-extremal horizons and quantum fields living in the Boulware vacuumstate are forced to coexist. Understood as an integration from the asymptotic region,the mass contribution provided by the vacuum polarization of the scalar field results inthe compactness reaching 1 at a faster rate than vanishing of the redshift function. Thiscontribution amounts to a cloud of negative mass that extends to infinity. The role of thecharge here is, similarly to the classical situation, to produce a repulsive contribution thatslows down the inwards growth rate of the Misner-Sharp mass. Thus, the neck radius isdragged towards smaller values of r as Q increases.– 16 – .0 0.2 0.4 0.6 0.8 1.00.0000.0050.0100.0150.0200.0250.0300.035 Q crit − M M
Figure 2 . Plot of the quantity Q crit − M in terms of M . The parameter Q crit denotes the value ofthe charge that separates under-charged from over-charged spacetimes. In the Q → r B can only be found numerically, we are unable to provide an exact analytical boundfor Q in terms of the asymptotic mass M without resorting to numerical calculations. InFigure 2 we have plotted the first value of Q that takes the solution out of the wormholeregime for several (although quite small in Planck units) asymptotic masses. We find thatthe separatrix charge Q crit needs to surpass the asymptotic mass M . This is so becausethe vacuum polarization of the scalar field, by backreacting on the geometry, modifies themass of the spacetime, while charge stays constant throughout the geometry. In order tocompensate for this increase in mass as smaller radii are approached, Q crit lies above itsclassical value Q crit = M , as the expansion of (4.5) around its classical value shows | Q | (cid:46) r B (cid:34) − l r + O (cid:18) l P r B (cid:19) (cid:35) , (4.12)where r B > r + always. Since for classical sub-extremal black holes r + > M , we concludethat the smallest value of Q that does not obey (4.5) must be greater than M .We want to end this section with some remarks on the validity of our approximatesolutions. Two situations can be distinguished: for big wormhole throats ( r B (cid:29) √ αl P ),the metric (4.11) remains valid as long as r − r B (cid:28) l r B , (4.13)– 17 –hereas for small wormholes ( Q (cid:28) r B (cid:28) √ αl P ), r − r B (cid:28) r B (4.14)must hold. At distances where the above expressions fail to be good approximations, themetric should be matched with the Reissner-Nordstr¨om line element or with the asymptoticmetric at the inner side of the wormhole. The metric on the other side of the neck can be uniquely determined solely by the rootsand exact solutions ψ ± depicted in Figure 1. Relying in the fact that just beyond the neckthe solution is well-described by the − sign in (4.3), and that, necessarily, r B > r div , itfollows that ψ takes values below all the roots and exact solutions. By self-consistency of(3.3), ψ grows with r until it encounters the root R , which decreases almost linearly with r (check the bottom right portion of Figure 1). As the root is crossed, ψ begins decreasingmonotonically confined between R , which cannot be encountered for a second time, andthe exact solution ψ + , which cannot be crossed either from r B to ∞ , by virtue of thePicard-Lindel¨of theorem.The particular form of ψ deep inside the neck can be determined analytically assumingthat, asymptotically, the ψ function goes as ψ = ψ + + β ( r ) , (4.15)where β ( r ) is a function measuring the deviation from the exact solution. Replacing inside(3.3), keeping terms up to linear order in β , and taking the limit r → ∞ results in β (cid:48) (cid:39) − (cid:8) − r + 8 l (cid:2) Q + r (1 − α ) (cid:3) + l (5 − α ) (cid:9) l r β, (4.16)which integrated (4.16) gives β = β (cid:18) rl P (cid:19) − α e − r l (cid:20) − Q r − l (5 − α )8 r + O ( r − ) (cid:21) , (4.17)where β is an integration constant with dimensions of inverse of length. Inserting thisinside (4.15) and integrating to obtain the redshift function and compactness yields theasymptotic metric ds (cid:39) (cid:18) rl P (cid:19) − α e − r /l (cid:26) − a (cid:18) − l r (cid:19) dt + 2 β r l P (cid:20) − (9 − α ) l r (cid:21) dr (cid:27) + r d Ω . (4.18) Q -dependent terms enter this expression as subdominant contributions, so they do notalter the asymptotic form of the geometry, which is the same as in the Schwarzschild case.This becomes evident already in Eq. (3.3), where terms proportional to Q are subdominantwith respect to the “vacuum”, r -dependent contributions from the Schwarzschild sector.– 18 –oncerning the structure of the singularity, its main property is that it is a consequenceof the runaway of vacuum polarization, which makes the Misner-Sharp mass diverge to-wards negative infinity. By inspection it is possible to see that the r → ∞ limit is a nullsingularity. Moreover, this singularity is located at finite affine distance from the neckfor all geodesic observers. Figure 3 contains a numerical plot of the metric functions interms of the proper coordinate l for an internal region around the neck (the numericalintegration goes beyond the analytical approximation). Then, the geometry connects anasymptotically flat region on the right-hand side with an asymptotic null singularity onthe other end. From this perspective, there is no qualitative difference with respect to theSchwarzschild case ( Q = 0) discussed in [13]. To wrap up this discussion, let us stressthat whenever M > Q crit (cid:29) l P the results found would have been qualitatively the sameif we would have used instead just the Polyakov approximation. In fact, because of theappearance of a bounce at r B (cid:29) l P , the region close to r = 0 of the approximation is notexplored by most of the solutions found. As the Polyakov RSET is a good approximationto the RSET around horizons, we can expect that the presence of asymmetric wormholesolutions is a robust characteristic of the semiclassical electrovacuum equations. When r B ∼ l P we are entering into a much more slippery terrain. We cannot be sure whetherthe wormhole shape would be maintained in more refined approximations to the RSET.Notice, however, that this regime corresponds also to M ∼ l P , and that it is reasonable toexpect that the structure of black holes with Planckian size would surely lie outside theregime of validity semiclassical gravity.In the following sections, we will turn to other non-wormhole geometries with valuesof the charge greater than or equal to the separatrix value Q crit . We now turn to the case where solutions have no wormhole neck and thus extend all theway down to r = 0. There are two ways of accomplishing this behaviour: either consideringa negative mass ( M < Q beyond its critical value Q crit while M > Q increases. Similarly, the repulsion exerted by the electromagnetic field slows downthe growth rate of ψ , displacing the surface r B towards smaller radii. As Q increases, theroot R , which diverges (while remaining positive) at r div , is eventually intersected by thesolution. Indeed, the explicit expression for r div can be derived from solving the polynomialexpression A = 0 for r . From all the roots that factorize this expression, the one whichcorresponds to the surface of infinite R turns out to be r div = (cid:26) / Q + 2 Q (cid:104) Q V + l (cid:16) / U + α V (cid:17)(cid:105) / + (cid:104) Q V + 2 l (cid:16) / U + α V (cid:17)(cid:105) / +2 l (cid:20) / Q (3 + 2 α ) − α (cid:16) Q V + l (cid:104) / U + α V (cid:105)(cid:17) / (cid:21) + 2 / l α ( α − (cid:27) × (cid:104) Q V + 6 l (cid:16) / U + α V (cid:17)(cid:105) / , (5.1)– 19 – - - - - - r B r e φ C l
Figure 3 . Numerical plot of the wormhole geometry around the neck region. The yellow curvesrepresent the radial function r ( l ) (in units of r B ). The red curve is the redshift function e φ and theblue, dashed line represents the compactness function C . The right hand portion of the diagramconnects with the asymptotically flat region, where e φ = 1 − C = 1. The other side of thewormhole ends in a null singularity at r → ∞ . The proper distance between this null singularityand the neck is finite. The parameters from this simulation have been assigned small values equalto M = 0 . , Q = 0 .
05 and α = 1 .
01 for visualization purposes, but the overall features of thesolution remain unchanged for higher masses and charges. with U = (cid:8) Q (4 α −
1) + 2 l Q [ α (9 + 8 α ) −
2] + αl Q [12 + α (47 + 24 α )]+4 α l Q [ α (9 + 8 α ) −
3] + 4 α l (1 + α ) (cid:9) / , V = 2 Q + l Q (9 + 4 α ) + 2 αl (9 + α ) . (5.2)In the l P → r div (cid:39) Q + l Q + O (cid:18) l Q (cid:19) , (5.3)which imples that, in the classical limit, this surface lies at r = Q . Similar argumentsas those presented here are valid in the analysis of the classical equations and, in thatcase, the surface r = Q serves as a separatrix between sub-extremal Reissner-Nordstr¨omgeometries, in which ψ diverges at r = r + > Q , and super-extremal Reissner-Nordstr¨omgeometries, where ψ is bounded from above. The solution where ψ diverges exactly as( r − Q ) − corresponds to the extremal black hole. The introduction of the length scale l P related to quantum corrections displaces the separatrix surface outwards. Figure 4 showsa plot of the quantity r div − Q in terms of the charge Q .The crossing with R at r = r div marks a maximum in ψ that prevents the geometryfrom acquiring a wormhole shape. Instead, the coordinate r now extends to r = 0. In– 20 – r div − Q Q
Figure 4 . Plot of the quantity r div − Q for various values of the charge Q . Whereas for largecharges this quantity tends to zero, in the regime of small charges comparable, in magnitude, to l P ,this difference increases appreciably, going to 0 again in the Q → doing so, ψ has to cross r = Q , the surface where the differential equation (3.3) is singular(check Fig. 5). We can prove that the solution is regular there taking into account Eq.(3.7). Both ψ ± (3.6) are compatible with a regular behaviour of the differential equation at r = Q , as for the ψ ± solutions, the Q -dependent terms in (3.3) (which contain the possiblesingularities) vanish identically. Any other solution must intersect ψ + or ψ − at r = Q to avoid a singularity. Since the right-hand side of (3.7) is divergent, the Picard-Lindel¨oftheorem does not hold at r = Q , allowing solutions to intersect at that precise surface.Due to continuity, the solution we are describing in this section will intersect first ψ − . Nowlet us derive the form of the solution ψ around r = Q by assuming ψ = ψ − + ξ ( r ) , (5.4)where ξ is the function that measures the deviation between solutions. To guarantee thefiniteness of (3.8), ξ must vanish at least linearly as r approaches Q . Replacing (5.4) andits derivative in (3.7) and dropping terms beyond linear order in ξ , we obtain in the r → Q limit ξ (cid:48) ( r ) (cid:39) ξ ( r ) r − Q , (5.5)which upon integration returns ξ (cid:39) ( r − Q ) ξ , (5.6)where γ is an integration constant with dimensions of inverse of length squared. Thus, as ψ crosses r = Q , it does so in a manner that ensures that the right-hand side of (3.7) doesnot diverge. – 21 – - - - - - - - - - R R ψ ψ − r d i v ψ r Q r d i v Figure 5 . Numerical plot of an over-charged solution (black curve) alongside the roots R , andthe exact solution ψ − . Once it intersects R , the solution must cross ψ − at r = Q and becomestrapped between R and ψ − by self-consistency of the differential equation. If the maximum for ψ is reached closer to r div , then the solution decreases more abruptly and can intersect R zero,one or two times, but the behaviour at r = 0 remains unchanged. For this plot we have chosen M = 4 , Q = 1 . M and α = 1 .
01. The region around the intersection point r = Q has beendepicted in detail. After the solution crosses r = Q towards smaller radii, it can intersect R zero times,once or twice. In the first case, ψ stays confined between the exact solution ψ − and the root R , its value decreasing until it reaches −∞ , as depicted in Fig. 5. The situation in whichthere are two intersections with R occurs whenever ψ decreases sufficiently fast (aftercrossing r = Q ) so that it intersects the rightmost part of R . After the first intersection,it grows until R is encountered for a second time. In between these two situations thereis one in which R is touched once tangentially at the same point in which R reaches itsmaximum value, which becomes a saddle point of ψ . This qualitative description of thebehavior of the solutions is therefore exhaustive.The other posibility that we mentioned at the beginning of this section requires anegative asymptotic mass, M <
0. In this case, ψ takes negative values above ψ − atinfinity. By self-consistency of (3.3), the solution decreases monotonically inwards, goesacross r div , and diverges towards −∞ in the r → R and ψ − .Thus, the r (cid:39) r = 0, as it is analyzed in the next section.– 22 – .1 Singularity at r = 0The form of the solution close to the origin in the over-charged regime comes from a combi-nation between the (singular) electromagnetic field and the effects of vacuum polarization.The magnitude of the RP-RSET describing quantum fluctuations near r = 0 is stronglyaffected by the value of the regulator. Indeed, if the RP-RSET is sufficiently suppressedby considering a large value of α , the dominant source of compactness in the r → Q . In order to obtain the behaviour ofthe compactness function close to the radial origin we perform an asymptotic analysis of(3.1) in the r → ψ (cid:48) are theninserted in (3.1). By assuming ψ takes the form ψ = ar , (5.7)which is the only profile both divergent at r → r → a : a = − α α , a ± = − (cid:16) α ± (cid:112) α ( α − (cid:17) . (5.8)The first of these values is a solution that appears as a consequence of the introduction ofthe electric charge. We have not been able to relate it with any of the situations describedin this work, so chances are that this solution does not connect with an asymptoticallyflat region. The last two values correspond to evaluating the exact solutions ψ ± in the r → a − in (5.8), which has a well-defined classical limit. Terms subdominant with respect tothe leading order (5.7) (which would be linear in r ) can be obtained, but the leading-orderform of ψ is sufficient to illustrate our point in this section.After substitution of (5.7) and its derivative in (3.1), we obtain the following approx-imate linear differential equation for the compactness C (cid:48) = [(2 − a − ) a − − α ] C ( α + a − ) r [1 + O ( r )] + αQ ( α + a − ) r (cid:2) O ( r − ) (cid:3) . (5.9)In this equation, subleading terms divergent in r that accompany the compactness andthe charge can be neglected, since they contribute to the solution at subdominant order.Integrating Eq. (5.9) is straightforward, yielding C = c (cid:18) rl P (cid:19) b (cid:2) O (cid:0) r (cid:1)(cid:3) + c (cid:18) Qr (cid:19) (cid:2) O (cid:0) r (cid:1)(cid:3) , (5.10)where c is an arbitrary integration constant and b and c are known parameters whosevalues are shown in what follows. The value of c must be estimated by numerical integra-tion and will have some unknown dependence on M , Q , α and l P . Since, in view of Fig. 5,– 23 – approaches ψ − in the r → c <
0. The exponent of the first term is found to be b = − α ) (cid:104) − α + (cid:112) α ( α − (cid:105) α − , (5.11)whereas the second coefficient is c = − α (cid:110) (cid:112) α ( α − − α (cid:104) α + 1 − (cid:112) α ( α − (cid:105)(cid:111) − . (5.12)Therefore, in the r → α we essentially recover thebehaviour from the classical Reissner-Nordst¨om geometry, as expected for a fully sup-pressed RSET, whereas in the α → b diverges. Although for α > r = 0. An already singular geometry is being coated by a cloud of negative mass comingfrom vacuum polarization which nourishes from this singularity. Hence, it is expected thatthe leading divergence in C becomes more than polynomially strong in the α → α < (cid:0) √ (cid:1) , the first term in (5.10) dominates overthe second one. For large α , however, the electromagnetic charge carries the leading-orderdivergence.There exists an intermediate situation where both terms in (5.10) diverge at the samerate as r →
0. This occurs for the exact value α = (cid:0) √ (cid:1) , for which b = − α in (5.9) and integrating yields C = (cid:0) √ (cid:1) Q r log r + O (cid:0) r − (cid:1) . (5.13)This solution acts as the separatrix between two distinct behaviours of the compactnessfunction at the singularity. It describes a situation where the divergent accumulation ofvacuum polarization becomes comparable to that of the mass contribution coming fromthe (singular) electromagnetic source. Then, both contributions intertwine and give riseto a dominant divergence in C that depends on Q times a logarithmic contribution. Weconsider somewhat remarkable that a particular regulation of the strength of semiclassicalcorrections within the vicinity of the radial origin is capable to enhance divergent contribu-tions coming from classical sources, whose sole interaction with quantum fields is throughspacetime geometry.In summary, this section has illustrated that the over-charged regime, which guaranteesthe presence of a naked singularity, shows a hierarchy of divergent behaviours depending onhow we adjust the regulator. For small α , the dominant contribution to the compactnesscomes from the backreaction of vacuum polarization on the vicinity of the singularity.However, if the RP-RSET is sufficiently dampened, the contribution coming from thesingular charge-source giving rise to the electromagnetic field is uncovered. In betweenboth regimes there exists a separatrix solution where the dominant divergence becomes amix of quantum and classical contributions.– 24 – Quasi-extremal regime
Always starting from an asymtoticaly flat region, we have explored solutions where ψ diverges at some radius r > r div , and solutions where ψ encounters the root R , thus havinga maximum and avoiding the formation of a minimal surface, extending the solution to r = 0. In between both regimes there exists a separatrix solution, where ψ diverges atprecisely r div , reminiscent of the classical extremal black-hole solution. In the presentsection we will characterize this separatrix solution.The separatrix solution corresponds to the case in which the divergence in ψ takes placeat the same radius where R diverges, which is at r = r div . Furthermore, this divergencein ψ occurs at the same pace as R blows up, that is, as ( r − r div ) − . This can be verifiedby assuming the following behaviour for the ψ function ψ = 1 λ ( r − r div ) γ , with γ > . (6.1)The present separatrix behaviour has the particularity that taking the limit r → r div in(3.3) reveals that the coefficient A vanishes as A ∝ ( r − r div ). This can be checked byfactorizing the numerator of A (3.3), which is a polynomial of sixth degree for the radialcoordinate, and realizing that one of its roots equals r div (5.1). In virtue of (6.1) and itsderivative, the following approximate relation is derived from (3.3): − γλ ( r − r div ) − γ − = Bλ ( r − r div ) − γ +1 + A ( r div ) λ ( r − r div ) − γ + O [( r − r div )] − γ , (6.2)where B = A / ( r − r div ) | r = r div has a finite value at r = r div . The above expression enforces γ = 1 for consistency. Therefore, both the quadratic and cubic terms in ψ from Eq. (3.3)enter the differential equation for ψ at leading order in the r → r div limit. Simplifyingterms, the expression reduces to a quadratic equation for λλ + λA ( r div ) + B = 0 , (6.3)Solutions of Eq. (6.3) return two values of λ , which we denote as λ ± (the ± label heredenotes the sign of λ ). We have found the explicit analytical expressions of λ ± by usingMathematica and they correspond to very lengthy expressions which we avoid showing here.The coefficient λ − is negative and hence comes in conflict with our integration conditionfrom the asymptotic region. The presence of an outer horizon at r = r div , for which theredshift function vanishes at r = r div with a positive slope implies ψ must be positive anddivergent there (when approaching the horizon from r > r div ). Hence, only the constant λ + reproduces the required behavior of an outer horizon. Therefore, in the following wechoose λ = λ + in order to derive the approximate form of the geometry.Given that ψ = φ (cid:48) by definition, Eq. (6.1) with γ = 1 and λ = λ + leads to the redshiftfunction e φ (cid:39) f (cid:12)(cid:12)(cid:12)(cid:12) r − r div r div (cid:12)(cid:12)(cid:12)(cid:12) /λ + , (6.4)with f a positive integration constant. The exponent at which the redshift functionvanishes is modulated by λ + , which has an involved dependency on α, l P , and Q . For– 25 – /λ + Q Figure 6 . Plot of the exponent of the redshift function in terms of the charge Q for various valuesof α (from bottom to top, α takes the values 1 . , , Q → α while for large Q it approaches the extremal black holesolution from below. The extremal black hole is only recovered for | Q | → ∞ . For small Q theredshift function can have a Schwarzschild-like horizon. example, figure 6 shows a plot of the exponent 2 /λ + in terms of the charge Q for variousvalues of the regulator. We observe that, for Q (cid:29) l P , it goes as2 λ + = 2 − l Q + O (cid:18) l Q (cid:19) , (6.5)and the extremal Reissner-Nordstr¨om solution (for the r > M = Q geometric patch) isrecovered only in the limit of l P = 0. On the other hand, the compactness function can bederived from (3.2) C = 1 − κ (cid:18) r − r div r div (cid:19) + O (cid:2) ( r − r div ) (cid:3) , (6.6)with κ = λ ( r − Q )( r + αl ) l r > . (6.7)Given the classical limits of the quantities r div in Eq. (5.3) and λ + in Eq. (6.5), weobtain κ = 1 for Q (cid:29) l P , thus reproducing the extremal Reissner-Nordstr¨om compactnessfunction.The local form of the metric for r > r div in this solution has the form ds (cid:39) − f (cid:12)(cid:12)(cid:12)(cid:12) r − r div r div (cid:12)(cid:12)(cid:12)(cid:12) /λ + dt + (cid:20) √ κ (cid:18) r − r div r div (cid:19)(cid:21) − dr + r d Ω . (6.8)This metric has several interesting features, which we turn to describe. One characteristicthat may grab the attention is that the redshift function of the geometry goes to zero with a– 26 –maller exponent than in the classical extremal solution. The radial part of the geometry,however, retains the quadratic dependence on r − r div ; the affine distance between thehorizon r div and any point of the spacetime is infinite, as in ordinary extremal spacetimes.The RP-RSET has once more caused a clear distinction between both components of themetric. This is why we refer to these solution as “quasi-extremal”.Now we shall continue the solutions beyond r div . We could do that in a completelysymmetric fashion. However, this would imply, on the one hand, that the radius woulddiminish inwards, but moreover, that the solution is no longer a proper vacuum solutionas it would correspond to having a null shell localized at the horizon. On the contrary, wecan select a solution with a negatively diverging behaviour at r div , ψ = − /λ ( r div − r ),assuming now r < r div . This selection breaks the symmetry of the construction and is moreakin to a quantum version of the extremal RN solution. Consistency with the idea thatnow the redshift function slope should be positive makes us to select again the λ + valueinside the horizon. Thus, the final local form of the metric is (6.8), which is now valid fora sufficiently small open interval containing r = r div .One first implication of this solution is that, in crossing the horizon, one changes fromthe unconcealed branch to the concealed branch of the semiclassical corrections. Recallthat the concealed branch has no well-defined classical limit. Therefore, this entire solutioncannot be found perturbatively starting from the extremal RN solution. This will be moreclearly seen when analyzing the next order expansion of the metric around r = r div (see rightbelow). Thus, our self-consistent analyses reveal that horizons enforce non-perturbativebackreaction effects, typically eliminating the horizons whatsoever, or at most maintainingan extremal-like horizon with the special characteristics we are describing.Another notable difference with the classical extremal black-hole resides in the value ofthe exponent in the redshift function, 1 / < /λ + < α → r div . Despite this unusual behaviour of the metric, curvature scalars calculated from (6.8)that are quadratic in the Ricci and Riemann tensors are finite and analytic at r div (althougha different type of singularity is indeed present, as seen below). This somewhat surprisingresult comes from the fact that any appearance of the function ψ in the Kretschmann scalar(2.9) is accompanied by C factors which complete eliminate potential divergences.The finitude and analyticity of curvature invariants at leading order comes from theparticular form of (6.8), which describes the leading-order contributions in an expansionaround r = r div . The next order in the expansion introduces additional terms in ψ whichdo not coincide with the classical solution in the l P → ψ acquires an additional contribution: ψ = 1 λ + ( r − r div ) + Ψ( r ) , (6.9)where Ψ( r ) is some function that diverges more slowly than ( r − r div ) − , or does not divergeat all. Inserting this ansatz in (3.3), expanding in r − r div , and taking into account the– 27 –ancellation of the leading order contributions, we obtain the expressionΨ (cid:48) (cid:39) (cid:18) A + 2 A Ψ + 3 B Ψ λ + (cid:19) (cid:20) λ + ( r − r div ) + Ψ (cid:21) − Ψ [ A − B ( r − r div ) Ψ] + A , (6.10)where all coefficients are evaluated at r = r div .Let us assume first that Ψ diverges slower than 1 / ( r − r div ). This assumption impliesthat the leading form of the previous equation isΨ (cid:48) = (cid:18) A + 3 Bλ + (cid:19) λ − Ψ( r − r div ) , (6.11)which after solving implies Ψ ∝ r − r div ) , (6.12)but this is against our starting assumption. Therefore Ψ cannot diverge at r = r div .Hence, the only remaining possibility for the consistency of (6.10) is that Ψ equals aconstant that causes the vanishing of all divergent contributions. An extra term linear in r − r div can be added to (6.9) so that the remaining constant terms vanish as well. Thesolution ψ takes the form ψ = 1 λ + ( r − r div ) + Ψ + Ψ ( r − r div ) + O (cid:2) ( r − r div ) (cid:3) , Ψ = − A λ + B + 2 A λ + , Ψ = 3 B Ψ + λ + [ A + ( A + A Ψ ) Ψ ] − B + λ + ( λ + − A ) λ + . (6.13)The coefficients Ψ and Ψ fail to return the extremal black hole solution in the classicallimit. The main reason behind this is linked to how semiclassical modifications expand thespace of solutions. Indeed, the quasi-extremal geometry emerges as a consequence of theterms cubic in ψ the RSET introduces in (3.3) and, as a consequence, it does not connectwith the extremal black hole solution when these cubic terms are suppressed. From adifferent perspective, as we already mentioned, part of the support of the solution lieswithin the concealed branch, which is intrinsically quantum and has no classical limit.Replacing expression (6.13) in Eq. (3.2) we obtain the following approximate expres-sion for the compactness1 − C = κ (cid:18) r − r div r div (cid:19) (cid:104) κ ( r − r div ) + O ( r − r div ) (cid:105) , (6.14)with κ = 2 λ + l r div (cid:26) λ (cid:2) r + l ( α + r div Ψ ) (cid:3) − l Q r − Q + αl r + αl (cid:27) > . (6.15)The Kretschmann scalar (2.9) takes the form K = 4 (cid:0) λ + κ (cid:1) ( λ + r div ) (cid:40) κ r div (2 + λ + ) ( κ + 2 λ Ψ ) − λ r div (cid:0) λ + κ (cid:1) ( r − r div ) + O (cid:2) ( r − r div ) (cid:3)(cid:41) . (6.16)– 28 –he constant term corresponds to the “bulk” contribution coming from the leading-ordercontributions in the line element (6.8). Additional vanishing terms appear in curvatureinvariants when subdominant contributions are taken into account. The quasi-extremalmetric is non-analytic at the horizon but, due to its particular form, this behavior does notreflect on curvature invariants. Similar tendencies were found in [23, 51] in the context ofextremal black holes in dilatonic gravity coupled to a quantum scalar field, in the sense thatquantum-corrected extremal geometries develop non-analyticities at the corrected horizon.We can appeal to another notion of curvature singularity that does not rely on cur-vature scalars. This is the definition given in [52] of non-scalar singularities: those wherethe components of curvature tensors, when evaluated for a suitable tetrad at the singularregion, show divergences. By suitable we mean a particular tetrad field which is paralleltransported along a physical curve that approaches the singular point. For this particularcase, we choose a tetrad field associated to an ingoing timelike geodesic path on the metric(6.8). In terms of a spherically symmetric spacetime of the form (2.1) we have drdτ = − √ − C √ − e φ e φ (6.17)for ingoing geodesics, where τ is a proper time which equals the coordinate time t asymp-totically. Replacing the components of the local metric (6.8) in (6.17) we obtain drdτ = − (cid:112) κ/f (cid:12)(cid:12)(cid:12)(cid:12) r − r div r div (cid:12)(cid:12)(cid:12)(cid:12) − /λ + . (6.18)For λ + >
1, the velocity of the observer when crossing the quasi-extremal horizon vanishes.Moreover, Eq. (6.18) reveals that subsequent derivatives of the velocity (acceleration, jerk,and so on) are divergent for an increasing range of values of λ + . The tangent vector to theingoing radial geodesic (6.18) can be used as one of the vectors of the desired tetrad field e µ (0) = f − (cid:40)(cid:12)(cid:12)(cid:12)(cid:12) r div r − r div (cid:12)(cid:12)(cid:12)(cid:12) /λ + , (cid:112) κf (cid:12)(cid:12)(cid:12)(cid:12) r − r div r div (cid:12)(cid:12)(cid:12)(cid:12) − /λ + , , (cid:41) . (6.19)To prove there is a non-scalar curvature singularity at r = r div , it is sufficient to obtain asingle divergent component of the Riemann tensor contracted with the vector field (6.19)of the tetrad. For simplicity, we select the component R (0) θθ (0) = R tθθt e t (0) e t (0) + R rθθr e r (0) e r (0) = κf (cid:18) λ + − λ + (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) r − r div r div (cid:12)(cid:12)(cid:12)(cid:12) − /λ + . (6.20)Note that, for 1 < /λ + <
2, this physical component of the curvature is singular. Weonly recover a well-defined geometry in the classical limit λ + →
1. Note that the curvaturesingularity at the quasi-extremal horizon follows from the non-analyticity of the metric for λ + >
1. If the exponent of the redshift function takes values below 1, divergences willappear at higher-order derivatives of the components of the Riemann curvature tensor. Wehave thus proved that the quasi-extremal black hole has a non-scalar curvature singularityat the horizon. – 29 –he extremal black hole is more stable against quantum perturbations than black holeswith an outer horizon. As we have seen, vacuum polarization eliminates these horizonsreplacing them with a wormhole neck. The particular case of the extremal black holeresults in a different kind of modification, in the sense that the horizon itself is preservedand becomes a curvature singularity. An important difference between these two cases canbe understood by looking at the geometry beyond the quasi-extremal horizon. The shapeof the solution can be again univocally determined by arguments concerning the roots andexact solutions. Figure 7 contains a plot of one of these solutions, showing details of theinner part of the geometry. In the inner region r < r div , the solution ψ , now living in theconcealed branch, approaches −∞ from below the root R . Then, it crosses r = Q in asimilar fashion as it occurred in the over-charged regime (see Sec. 5), with the differencethat the exact solution intersected is now ψ + . After intersecting the exact solution, itmeets the condition for a branch jump given by the vanishing of G (3.8) and the solutionswitches to the unconcealed branch back again. More explicitly, the branch jump requiresthat the solution ψ takes the value ψ ( r jump ) = − r + αl l r jump . (6.21)Once the transition to the unconcealed branch has occurred, the solution stays withinthis branch, growing until it reaches a maximum and remaining trapped between R and ψ − . The behaviour of ψ close to the radial origin is the same as in the over-charged regime.The main difference with the over charged regime is that the central singularity is no longernaked, but covered by a likewise singular horizon at the end of an infinite wormhole neck.It is interesting to recall that this is the separatrix between two solutions both harboringsingularities not covered by event horizons of any sort. These can be located either at radialinfinity in the wormhole regime, or at the radial origin for the over-charged family. Beyondits neck, the wormhole solution stays within the concealed branch, extending towardsan asymptotically singular region at radial infinity. However, since the quasi-extremalsolution does not have its radial coordinate reverted but just elongated through an infinitetube, it extends towards r = 0. Therefore, below the (singular) quasi-extremal horizonthere is just a narrow region where the solution is non-perturbative, and the perturbativeregime is recovered once we go sufficiently deep beyond the horizon. Nevertheless, thisnarrow band where quantum corrections become non-perturbative, as well as the non-scalar singularity, persist under changes of the regulator, only disappearing in the limit ofinfinite charge. Thus, we also expects that for soluctions with M = Q crit >> l P , the maincharacteristics of the quasi-extremal solution described in here will be preserved in morerefined approximations to the RSET. In this work we have obtained the complete set of self-consistent electro-vacuum solutions inthe semiclassical approximation taking the RP-RSET as describing the quantum materialcontent of the spacetime. For the Boulware vacuum state, the only state compatible with– 30 – .2 0.4 0.6 0.8 - - - - - - - R ψ R R ψ + ψ − r d i v Q r d i v ψ r Figure 7 . Numerical plot of a quasi-extremal solution (black curve), with Q = 0 . M (cid:39) .
59 and r div (cid:39) .
62. The roots and exact solutions appear represented, together with the vertical dashedand dotted lines denoting r = r div and r = Q , respectively. The solution has a branch jump atthe quasi-extremal horizon. In the region r < r div , ψ intersects the exact solution ψ + and theroot R exactly at r = Q (zoomed figure). Then, the solution jumps to the unconcealed branchat r jump (cid:39) .
58 and grows towards smaller r until crossing R , reaching a maximum, and startsdecreasing confined between R and ψ − . The asymptotic behaviour near r (cid:39) staticity and asymptotic flatness, three families of solutions have been obtained dependingon the charge-to-mass ratio.Under-charged solutions ( Q < Q crit ) are wormhole geometries with essentially thesame features as the Schwarzschild geometry counterpart from [13]. An increase of chargeexerts a repulsive effect that makes the wormhole neck shrink, which nevertheless alwayssits above the classical gravitational radius r + (2.5). On the other side of the neck, there isa null naked singularity at finite affine distance for all geodesics. The regulator α allows toextend the space of solutions to wormholes of Planckian size, something forbidden for thePolyakov RSET due to its unphysical singularity at r = l P . In any case, in physical termssolutions with M ∼ l P should not be very trustable as the semiclassical approximationshould break down in that regime.Over-charged solutions ( Q > Q crit ) describe naked singularities at r = 0. This can bethought of as composed by a cloud of infinite negative mass coming from vacuum polar-ization, originated from the backreaction of vacuum energy caused by the infinite chargedensity of the electromagnetic field. Depending on the characteristics of the regulator, thecontribution of this cloud of vacuum polarization can be suppressed so that the dominant– 31 –ingular contribution at short distances comes from the electromagnetic SET. This geome-try exemplifies a situation where slight modifications in the regulator parameter α stronglychange the features of the solution close to r = 0.Lastly, the appeareance of a “quasi-extremal” geometry ( Q = Q crit ) partially respondsto a question raised in our previous work [13], where we wondered whether semiclassicalgravity would allow for the existence of horizons where the roots of the redshift functionhave a greater multiplicity. It can be arguably expected that the Boulware state wouldbe able to coexist with horizons of the extremal kind, as these represent zero-temperatureconfigurations. The quasi-extremal case here studied exemplifies this situation. The back-reacted geometry that we find is more alike to its classical counterpart than geometriesof the wormhole kind, with the caveat that non-perturbative corrections still occurr in anarrow region behind the singular horizon. We find that indeed quantum backreactionretains the horizon structure; however, it transforms it from having a parabolic shape tohaving a cusp-like shape, its sharpness being modulated by the exponent λ + . Then, thiscusp translates into a non-scalar curvature singularity of the kind defined in [52].The incompatibility between Schwarzschild-like horizons (at nonzero temperature) andthe Boulware state can be interpreted as an indication that trapping horizons must be dy-namical and be subjected to an evaporation process (see [14] for a discussion on the possibleoutcomes of dynamical evaporation scenarios). The case of the extremal horizon is excep-tional since, due to having zero temperature, in principle they do not need entering into anevaporation regime. It is reasonable to interpret our non-scalar curvature singularity resultas pointing out that extremal configurations have additional elements of non-physicalitybeyond those already present at the classical level.In view of our analyses, we conclude that semiclassical electro-vacuum static geometriesin the Boulware vacuum state show the unavoidable appearance of curvature singularities,whose spacetime location depends on a balance between charge and mass. Semiclassicalself-consistency ensures that these singular geometries are devoid of event horizons of anykind. The sole exception is the “quasi-extremal” solution, for which the horizon itselfconstitutes a singularity. These results indicate the incompatibility between the presenceof quantized fields in the Boulware state and regular event horizons. Acknowledgments
Financial support was provided by the Spanish Government through the projects FIS2017-86497-C2-1-P, FIS2017-86497-C2-2-P (with FEDER contribution), FIS2016-78859-P(AEI/FEDER,UE), and by the Junta de Andaluc´ıa through the project FQM219. Au-thors JA and CB acknowledge financial support from the State Agency for Research of theSpanish MCIU through the “Center of Excellence Severo Ochoa” award to the Institutode Astrof´ısica de Andaluc´ıa (SEV-2017-0709).
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