A black hole with quantum core

Accepted Manuscript

A Black Hole with Quantum Core Biplab Paik PII: DOI: Reference:

S0577-9073(18)30827-X https://doi.org/10.1016/j.cjph.2018.08.015 CJPH 613

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

15 June 2018 29 July 2018 20 August 2018

Please cite this article as: Biplab Paik, A Black Hole with Quantum Core, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.08.015

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Highlights • A proposal of black hole geometry that conforms loop quantum cosmology. • Quantum improved black hole holds a characteristic quantum core. • Hawking radiation process gets a predictable final phase.

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• Quantum core is regular, of radius (M lP 2)1/3 , packed with Planck-order-dense matter.

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• Improved black hole entropy-area law carries a logarithmic quantum correction.

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A Black Hole with Quantum Core Biplab Paik∗

Abstract

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Rautara MNM High School Rautara, Habra, (N) 24 Parganas, West Bengal-743234, India.

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A model black hole, holding a ‘quantum core’ characterized by the Planck order matter density, is revisited here. Based on the quantum improved Newton’s potential drawn out of the loop quantum cosmology we propose a Schwarzschild class, quantum improved black hole line-element that upholds the existence of Planck-dense quantum matter core. Causality is kept preserved in this proposal. Quite in a natural way the quantum core emerges closely homogeneous in its interior matter distribution. The radius of the quantum core turns out to be necessarily proportional to one-third power of the black hole mass. Hawking process of black hole evaporation leads to a shrinking quantum core, and as the mass of black hole approaches near about the Planck mass, the rate of evaporation diminishes rapidly and eventually leaves a cold remnant having a Planck order mass. Proposed model supports the standard quantum geometrical logarithmic correction to black hole entropy-area law.

Introduction

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Keywords: Quantum black hole; Quantum core; Hawking radiation and evaporation. PACS Numbers: 04.60.-m; 04.70.Dy; 04.70.-s

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In absence of any complete quantum theory of gravity, trying to build and further explore a nonsingular model-spacetime of black hole is one of the few ways for anticipating quantum gravity. For one to work upon this interest, a starting point would be to note the simple fact that any nonsingular black hole must hold a quantum matter-core, which in turn in general demands a necessary existence of Cauchy class horizon inside the event horizon. Nonsingularity of black hole was first proposed by Bardeen [1], an interpretation of which is found in [2]. Follow-up models can be found in [3, 4, 5, 6, 7, 8, 9, 10]; in a minute qualitative sense these are the “Bardeen-class” [11]. These models share a basic causal geometry that associates an inner Cauchy class horizon along with the outer event horizon. No certain clarification is however yet established on the origin of nonsingularity of black hole core. One inspiring point of tackling this issue comes out in the following manner. Physical Hilbert√space admits a definite upper bound on the matter-energy density, ρcrit ∼ 0.4ρP , where ρcrit = 3c5 /(32π 2 γ 3 G2 ~) with γ being ∼ 0.24 [12, 13] (- a quantum theoretical fact). In loop quantum cosmology (LQC) one finds that as the matter-density and with it the curvature enter the Planck scale, quantum geometrical effects indeed become dominant creating an effective repulsive force which rises very quickly, overwhelms the classical gravitational ∗

[email protected]

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attraction, and causes a bounce thereby resolving both the big-bang and big-crunch singularities. This new repulsive force associates the quantum nature of geometry that originates in effect of the Heisenberg uncertainty principle and is strong enough to counter the classical gravitational attraction irrespective of how large the mass is [13, 14]. Thus what emerges is that the ultraviolet quantum correction to gravity must be repulsive. While this is negligible in normal conditions, dominates when curvature approaches the Planck regime to ensure the prevention of an indefinite gravitational matter collapse (that would have classically led to singularity) and the formation of a regular “quantum core”. In this respect there is a curious similarity with the situation in the stellar collapse where a new repulsive force comes into play when the core approaches the critical density, halting further collapse and leading to stable white dwarfs and neutron stars. Let us now make an important observation. A series of proposals of nonsingular smeared source-mass black hole model piles up in literature in recent times ([15]- [23])1 . However with a model (and/or an interpretation) of this class there appears a serious conceptual concern : black holes modeled in this way seem to violate the fundamental causality principle of relativistic physics. In simple words this happens because this class model permits matter-smearing with matter spreading across horizons in two ways- a phenomenon which is strictly forbidden by the causal spirit of spacetime physics. So criticizing this class model seems obvious. A basic point, which we think must be kept in consideration, is that “any form of mass-energy, being attributed to spacetime fabric, cannot stay at a trapping zone of a black hole spacetime. Once matter gets gravitationally squeezed inside the Cauchy class horizon boundary (of a quantum black hole), it must remain confined within that boundary, because a horizon is strictly a ‘one way 3-surface’. Moreover, we consider another crucial aspect, which is to say whether the quantum gravity arises at a Planck length-scale or else it arises as matter gets compacted at a Planck level density. In both the senses, however, the spacetime physics demands that there must be a quantum core, even though the cores distinctly differ in the respective cases. If the quantum core-radius is assumed to be of a Planck scale, independent of the mass, M , of a black hole, then it must grant an ever increasing density, viz. ρQ ∼ (M/mP )ρP , 2 with the increasing mass of the black hole. In the present paper we rely on the alternative paradigm according to which serious quantum modification in gravity arises around the Planck-scale matter density - the maximal matter density, quantum theoretically admissible in physical Hilbert space. Gravity must change its notion and become repulsive as the matter density approaches the Planck limit- an idea which indeed is qualitatively essential in order to perceive a black hole with regular quantum core. In the usual general relativistic picture, black hole shrinks due to the quantum field theoretical phenomenon of Hawking radiation, until the central singularity is reached [24]. However, this situation would be avoided if the singularity does not exist, instead, quantum core is present there. Such a spacetime geometry would also lead to have a predictable final phase of the phenomenon of black hole evaporation [15]. For a black hole, quantum core is a result of repulsive nature of gravity at the ultraviolet completion. A picture of this kind happens to be quite an old appraisal, first put forward in [25]. This first proposal [25] by Frolov and Vilkovisky has been recently rediscovered in nonlocal gravity [26]. Then there appears a follow-up paper where the spacetime structure of 1

Hayward model is not explicitly classified as a smeared matter-source model. Nevertheless in its Einsteinian interpretation, the effective Einstein stress-energy tensor implies a spread of source mass-energy across horizons (in this respect one may also note ref.[16]). 2 This evaluation may in only way alter if one assumes presence of an effective negative vacuum energy density for the space occupied by the extremely dense (classical, nonvacuum) matter, at the expense of a positive vacuum energy distribution for the space devoid of the classical matter compactification. While the net black hole massenergy would now be M , the net vacuum energy would be zero - a curious proposition by which ([15] - [23]) could obey the causality principle. Note that vacuum energy would effectively act as an absolute space property.

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the gravitational collapse had been constructed and extensively studied in nonlocal gravitational theories [27]. In a recent paper [28], qualitatively, a similar class causal geometry of a black hole (triggered by the Planck energy-density) is conjectured by Rovelli and Vidotto. In all these constructions, viz. ([25] - [28]), LQC has been a great influence. One also finds the formal picture of a LQC black hole emerging through [29, 30], while [31] enabled the first model of gravitational collapse inspired by LQC. Few other approaches to understand the LQC black holes and analyses over an estimation of the quantum gravity effect on the Hawking radiation-evaporation process build up in ([32, 33, 34]). In the present paper we shall revisit the metric proposed in [28] (we call it as Rovelli-Vidotto metric), and modify in accordance with the arguments justifying this revisit. The content of this paper can be divided into following four principal segments : 1. A model of a Schwarzschild class black hole holding “quantum core” (that revises RovelliVidotto’s model [28]) is proposed based on the quantum improved Newton’s gravitational potential inferred out of the Friedman equation of LQC - section (3). 2. In accordance with the proposed model, the quantum geometrical effect is analyzed on the key aspects of black hole spacetime and thermodynamics - section (4). 3. Next we make an overall comparison of newly proposed model and Rovelli-Vidotto model section (5). 4. Finally we conclude - section (6). As we proceed let us make a specification of the physical unit system to be adopted. We shall use the p geometrized units in which G = c =√1 , and also let kB = 1 where it applies. Having lP = ~G/c3 would thereby imply that lP ≡ ~ .

Assumptions in brief

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The model of quantum black hole to be proposed here, is inspired by [28]. Nevertheless it will be clarifying to give an explicit mention of the assumptions behind our proposal : 1. Quantum gravity does not lead to violation of the fundamental relativistic principle of causality. 2. Quantum gravity gets its repulsive UV completion only for a regime governed by a Planckdensity state of matter. 3. Loop quantum cosmological understanding of quantum gravity must correlate with quantum black hole gravity. 4. There is no local quantum suppression of matter density (because of an effect like local matter annihilation) inside black hole core. Under these assumptions classical Einstein equations of gravity would require quantum geometrical modification for solving a regular model of black hole gravity.

LQC and a black hole with quantum core

Quantum mechanics puts a definite upper bound on the density of matter-energy that can prevail in physical space. In order for supporting this quantum fact, general relativity must get quantummodified in order for allowing the minimal space for matter confinement. Indeed this picture emerges real in a thorough theoretical construction of Loop Quantum Cosmology [13, 14, 37, 38]. It is very well known that black hole poses singularity-problem analogous to what arises in the classical big bang cosmology. Being inspired by the loop quantum cosmological understanding of the regular evolving Universe, we expect that quantum gravity would be in the exact nature to 4

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support a regular quantum core for a black hole; core being characterized by the maximal physical energy density. Note that the matter-energy density in space is constrained by ρ . ρQ ,

(1)

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with ρQ being the maximal physical energy density, and the core of a black hole will be assumed as that regime where matter density reaches its highest possible limit. In order that such a quantum core is compatible to spacetime causality, it has to be that a Cauchy class inner horizon in spacetime does exist inside the usual Schwarzschild event horizon. This Cauchy horizon would act like a boundary to quantum core. It is to point out that the quantum core is bound to be closely homogeneous with maximal physical matter density. An approximate core homogeneousness is physically reasonable because the Planck order density has been considered as the fundamental repulsive quantum gravity generating state of matter, and evidently one point of the quantum core cannot be more dense at the expense of some other points. Let us now fix the issue of locating Cauchy class inner horizon. By letting rcore to be the radius of quantum core, we expect to have 3 M . (2) ρQ = 3 4π rcore It is to remember that rcore is also to denote the radius of the Cauchy class inner horizon. Let us write ρQ = βρP . This leads us to write 1 M l2 . β P

(3)

M

3 rcore =

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Hence, with respect to our prescription of quantum core, quite in a straightforward manner, it comes out to be rcore ∝ M 1/3 . This property was in fact also upheld in [28], however, was not fully trusted there. But, demand of an approximate core-homogeneousness can fix the issue. Indeed, Planck density of matter is critical for restraining the classical gravitational collapse, and hence a closely homogeneous status of quantum core is expected. Thus rcore ∝ M 1/3 appears as a plausible 3 2 functional dependence of rcore on M . Now casting rcore = M lQ , yields the entity β as β=



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Typically, it is β ∼ 0.1 [12]. Thus, so far in this section, the readers are aimed at providing with an useful preliminary basis. Now we shall go through a few subsections to eventually propose a model spacetime for a “black hole with quantum core”.

3.1

Rovelli-Vidotto model

In [28], Rovelli and Vidotto proposed a model for quantum improved black hole geometry that allows a black hole to carry quantum matter core. They called the core a “Planck star” which typically has a Planck order matter density. The key point for us in this subsection is to assess the Rovelli-Vidotto model, and indicate (/ mention) the exact areas of interest and concern. So let us make a brief review of [28]. The black hole line element of the Rovelli-Vidotto model (that describes the exterior of quantum core) is Schwarzschild class, being given by ds2 = −F (r) dt2 + 5

dr2 + r2 dΩ2 F (r)

(5)

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where dΩ2 = dθ2 + r2 sin2 θdφ2 represents a unit 2-sphere, and  −1 2M 2r02 M F (r) = 1 − 1+ ; r r3 (  5/3  4/3 )−1 2M M lQ = 1− 1 + 22 , r r r

(6) 2/3

for r0 = lQ M 1/3

(7)

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with r0 being a mass dependent parameter and  is a real number of the order of unity. The function, (6) with r0 ∝ M α (α = 1/3 being one of the proposed possibilities), stands for a red shift modified Hayward function (of ref.[15]). Let us cast the Rovelli-Vidotto line-element in the form :   1 2 2 dr2 + r2 dΩ2 , (8) ds = − (1 + 2Φ(r)) dt + 1 + 2Φ(r)

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by which the Rovelli-Vidotto model appears to correspond to a quantum improved Newton’s gravitational potential, given by  −1 M 2r02 M Φ(r) = − 1+ ; (9) r r3 (  5/3  4/3 )−1 M lQ M 2/3 1 + 22 , for r0 = lQ M 1/3 . (10) = − r r r

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The model spacetime, [28], was presented by Rovelli and Vidotto to be a red-shift correction over the Hayward metric model. However this idea is not essential. Note that, once Einstein field equations are employed, Hayward black hole gets prescribed as a smeared gravitating source of matter in space while that matter-spreading is independent of whether the space is trapped or not. Eventually this Hayward-Einstein way of proposing a nonsingular black hole seems to violate the fundamental causal spirit of Einstein relativity. On the contrary, the Rovelli-Vidotto model proposes spacetime outside of a quantum matter-core, but does not infer the spacetime of the coreinterior. Hence the causal spirit of spacetime physics is kept intact in the Rovelli-Vidotto model by making the model to comply with the simple assessment that for a black hole the Cauchy class inner horizon acts as an impenetrable core-boundary as perceived from the core-interior. Thus the black hole metric proposed in [28] happens to be a choice qualitatively independent of the Hayward model. The basic motivation for the Rovelli-Vidotto metric model was to allow the existence of p 3 2 quantum core (viz. “Planck star”,) enclosed by the inner horizon of radius rcore ∼ r0 ∼ M lP , such that the chosen metric causes repulsive quantum gravitational effect to causally allow the formation of a region packed with matter at Planck-density. Thus Rovelli-Vidotto ansatz conveys a regular quantum black hole geometry to be a solution of a yet unknown quantum modified Einstein field equations. It is crucial to note that the function (6) does not represent the geometry (5) for r < rcore . This is so because while the net physical mass confined within a hypersphere of radius r ≥ rcore is always M (- equals the ADM mass at spatial infinity), inside the core the physical mass has a smeared picture leading to M −→ M(r). Note that interpreting RovelliVidotto geometry in classical Einstein framework would only lead to conflicting concepts. Nevertheless, Rovelli-Vidotto geometry has its own points of concern. One of the inconveniences arise with the following general choice of r0 in [28]:  α M r0 =  lQ (11) lQ 6

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3.2

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for 1/3 ≤ α ≤ 1. This convention on r0 breaks the central paradigm that gravity gets its quantum UV completion only around a regime characterized by a Planck-density state of matter. For example, a particular case α = 1, which equivalently implies r0 ∼ M , was originally attempted in [28] to propose a possibility of tackling the issue of black hole information loss paradox. However in accordance with a quantum core of this radius, viz. rcore ∼ r0 ∼ M , the critical, repulsive, quantum gravitational effect arises while matter density is much lower than the Planck order density. Therefore, clearly, the basic idea of expecting repulsive quantum geometrical effect arising around a region packed with Planck matter density, is never met. Again, even though with a particular choice r0 ∝ M 1/3 , the Rovelli-Vidotto model metric seems to be compatible with the loop quantum cosmological fact that the repulsive quantum gravity effect arises around a Planck order matter density, a simple point raised in the following subsection assesses that this particular choice for model metric may not be proper.

Friedman equation in LQC and the effective Newton potential

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This subsection works in two ways : first, it infers quantum improved Newton’s gravitational potential (and paves way for anticipating a quantum black hole model); second, it shows how LQC disagrees with Rovelli-Vidotto black hole metric. So, we shall now proceed to have the quantum improved form of Newton’s gravitational potential by interpreting the quantum improved, general relativistic Friedman equation of motion of the evolving fluid of the Universe in the framework of the Newtonian physics. Indeed, to a certain extent, general relativistic cosmology can be recovered in view of the Newtonian gravitational physics [35, 36]. We will explore this analogy. Loop quantum cosmological quantum improved Friedman equation for the evolving Universe in ‘k = 0’ (k = 0 implies spatial flatness) Friedman-Robertson-walker (FRW) cosmology model reads as [13]:    2 a˙ ρ 8π 2 ρ 1− = H = (12) a 3 ρcrit

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where a is the standard cosmological scale factor, H is the Hubble constant, and ρcrit ≡ ρQ ≡ −2 (3/4π) lQ = βρP . Let us now denote r = a(t)R, (with R being a fixed radial coordinate assigned to some expanding spherical coordinate grid) as the instantaneous radius of an expanding spherical shell of the expanding Universe. Due to the condition of homogeneity and isotropy, Universe will have a spherically symmetric matter distribution with any arbitrarily chosen space point as a center of symmetry. On invoking a Newtonian cosmological sense [36] to the above mentioned LQC improved Friedman equation, we may retrieve the energy conservation equation of a comoving particle sitting on the surface of a particular expanding spherical shell of an instantaneous radius r (enclosing a composite static physical mass-energy M inside) :   2  M lQ M 1 2 r˙ + − 1− 3 = E = 0, (13) 2 r r where E represents a fixed (and time-independent) net energy attributed to the expanding element of cosmological fluid. One may note that the spatially curved (i.e., k = ±1) Universe (- a model which is not observationally favored) would correspond to nonzero E. In view of Eq.(13) we may have the following classic Newtonian interpretation - a resultant of kinetic energy of expansion (K.E.) and potential energy of gravitational attraction (P.E.), being experienced by any element of cosmological fluid, is always balanced (conserved); viz. K.E. + P.E. = 0. Hence we infer the

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quantum corrected Newton’s gravitational potential to be  2  M lQ M Φ(r) = − 1− 3 r r

(14)

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such that this potential can recover the FRW equation in LQC in the Newtonian framework 3 . Now once we have a derived form of Newton’s potential at a radius r = r˜ = a(t˜)R, we observe that this potential is completely determined in terms of the parameter of a net enclosed massenergy, M , within the sphere of radius r˜, and the details of distribution of matter for points r < r˜ is irrelevant. In cosmological conditions as r varies for one fixed R, the same enclosed mass M must affect variable distances r in accordance with the force FN ∝ −dΦ/dr, and hence the notion of uniformity of evolving Universe makes way to transpire the derived functional form of potential necessarily as a fundamental identity of quantum improved Newton’s gravitational potential field at vacuum spaces outside an isolated astrophysical body of mass-energy M (for more clarification readers need to review the known correspondence [35] between classical Newton’s potential and classical FRW cosmology). Indeed in Newtonian mechanics gravitational potential would be treated as fundamental, and FRW cosmology can simply follow by the assumptions of homogeneity and isotropy (uniformity) of expanding matter distribution of Universe. Regarding a spherically symmetric black hole geometry, presumably holding a quantum core, any point on or outside the core should necessarily correspond to the inferred quantum improved Newton’s potential in Schwarzschild analogy since a sphere of radius r ≥ rcore always encloses the same physical mass-energy M (for there being vacuum anyway for r > rcore ). Only at a radius inside the quantum black hole core, in order for Eq.(14) to exactly hold for the substitution M −→ M(r), conditions analogous to FRW cosmology needs to be met so as to ensure that gravitational potential field due to all matter exterior to a radius r cancels. It is explicit that the Eq.(9), that comes out of the Rovelli-Vidotto model, does not represent the potential (14) (despite of the fact that the model by Rovelli and Vidotto could have easily led to the arousal of repulsive UV quantum gravitational effect for Planck matter density). On the contrary, once we simply use the LQC inspired Newton’s potential given by Eq.(14) for constituting a modified red-shift factor, the resultant spacetime successfully ensures a repulsive character of quantum gravity at the UV-completion around some matter compacted at Planck density. Indeed, a repulsive nature of UV quantum gravity is essential to support the necessary formation of a Cauchy class inner horizon (to have a q quantum core, thereby) for a regular black hole. It is amusing to note that, in the limit r >>

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2 M lQ , the resultant quantum improved spacetime would

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agree with the “Hayward geometry” (viz. [15]). However, the present model of spacetime differs from the Hayward model in terms of predicting the locus of Cauchy horizon.

3.3

LQC inspired quantum black hole metric

We are finally in state to propose a revised form of Rovelli-Vidotto metric. Our proposal straightforwardly follows. Taking the line-element describing the spacetime geometry to be of the form
ds2 = − (1 + 2Φ(r)) dt2 + (1 + 2Φ(r))−1 dr2 + r2 dθ2 + sin2 θdφ2 , (15) 3

In appendix we have an observation that the inferred quantum improved Newton potential is in an encouraging conformity also with loop quantum gravity (LQG).

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we replace the classical Φ(r) by what is given by Eq.(14). Thus, the nonzero, diagonal, quantum improved metric components are now given by  2  M lQ 2M 1 = 1− 1− 3 ; gθθ = r2 ; gφφ = r2 sin2 θ (16) −gtt = grr r r

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for r ≥ rcore , while for r < rcore we have a UV quantum gravity masked zone of quantum core 4 . The central motivation behind this choice of metric has been that the Newton’s gravitational potential drawn out of this choice reproduces the LQC improved Friedman equation in the Newtonian cosmological framework. Hence, apart from including this inspiration behind Eq.(14), the proposed model of black hole metric (viz. Eq.(16)) relies on the sole assumption that the line-element-form given by Eq.(15) holds for all region of spacetime on and above the Cauchy horizon (away from the black hole center). Indeed if we assume the red-shift factor to be completely determined by the Newton’s gravitational potential acting in space, we may expect the quantum improved line element to be of the prescribed form given by Eq.(15). The following note is critical. One must not interpret the proposed geometry in view of the classical Einstein equations. Employing the Einstein general relativity means to interpret the last equation as a result of a presumed phenomenon of matter-spread across horizons - a phenomenon that relativistic principle itself forbids. This condition essentially reflects an inevitable nature of classical Einstein gravity leading to singularity. Evidently for us it appears that a quantum improved geometry a black hole must solve some kind of quantum modified Einstein’s equations around a region packed with the Planck-dense state of matter.

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Exploring the properties of newly modeled black hole

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In the last section we developed a LQC inspired model geometry of black hole. In the current section we shall first explore the causal structure of quantum black hole gravity and the quantum core. Next we shall determine the loci of quantum black hole horizons and hence analyze the effect of quantum gravity on Hawking radiation phenomenon and black hole thermodynamics. While these analyses should make way to anticipate and recognize some key, characteristic properties of a quantum black hole, also provides standing for one to evaluate the current model in view of other existing regular black hole models in literature.

Causal structure of spacetime and quantum core

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Here we explore the causal structure of quantum black hole geometry and the status of quantum core. We rewrite the proposed model line element as ds2 = −F (r) dt2 +

dr2 + r2 dΩ2 , F (r)

(17)

where dΩ2 = dθ2 + r2 sin2 θ dφ2 . A region of spacetime is trapped if F (r) < 0, and not trapped if F (r) > 0. The trapping horizons bounding such a region are found at the zeros of F (r). It is again useful for us to have this metric in the Eddington-Finkelstein coordinates: ds2 = −F (r)dv 2 + 2dv dr + r2 dΩ2 . 4

Later in subsection(4.1) we shall provide a discussion on the spacetime for r < rcore .

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Figure 1: In this figure the well known Eddington-Frinkelstein coordinates (t¯, r) are used to describe the basic causal structure of a nonsingular black hole spacetime associating an event horizon, viz. r = r+ , and a Cauchy class inner horizon, viz. r = r− ≡ rcore .

(19)

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Here the Schwarzschild t-coordinate gets transformed as Z Z dr 2Φ(r) dr = v − t = v−r+ 1 + 2Φ(r) F (r) where

1 (F (r) − 1) 2   M 2 lP2 M M lP2 M + . = − 1− = − r βr3 r βr4

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Φ(r) =

(20)

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Some ingoing radial light-rays will follow the curve: v = Const.; else, the radial light rays must follow the curves satisfying the equation dr 1 = F (r), dv 2

(21)

and therefore are outgoing for F (r) > 0 and ingoing for F (r) < 0. There we see an extra term in Φ(r) (see Eq.(20)) modifying the classical case; this term indicates the short-range repulsive nature of quantum gravity. Its effect is to turn back a light cone vertical at the surface of inner horizon rcore . On crossing the event horizon, a time-like geodesic must hit the rcore hypersurface, but essentially not the r = 0 coordinate-point. After entering into the quantum core region, that resides inside the Cauchy class inner horizon, a light ray will move upward in the coordinate t¯ = v − r (observe, Fig.(1)) (note that t¯ is not the Schwarzschild time coordinate). Quantum core of a black hole would be supported by an effective metric that is going to solve some kind of modified Einstein’s equations outside a region packed with the Planck-dense matter. 10

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4.2

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But, the spacetime for the core-interior gets hard to predict because of quantum dominated gravity and lack of any definite knowledge over ordinary Planck dense matter. Nevertheless, the already drawn property of spacetime upto the core-boundary may be qualitatively extended to include the core interior. We have noted that in the Eddington-Finkelstein coordinates, a light cone, which bend inwards at the event horizon, returns smoothly pointing upward when entering the Planckdensity-regime. That is, there is a second trapping horizon inside the classical Schwarzschild event horizon, at a length-scale rcore representing the size of the quantum core of a black hole. Now the core-interior spacetime can in principle always be interpolated to the exterior likewise to the normal astrophysical stars, even though there could be many possibilities for the interior spacetime apart from the constraint that the spacetime must not lead to trapped surfaces. For example, there could be series of nearly-lightlike surfaces right from the core-surface to the interior (until the Planck length is reached)- a projection that could effectively allow relativistic fluid elements to remain fixedly assigned to space-points inside the black hole core. In any way, the interior of the quantum core of a black hole would be under great repulsive quantum pressure, and the core-stability analysis is not plausible in absence of an adequate, smooth description for the core spacetime and/or the Planck-dense fluid residing inside. Yet, in an analogy with the normal astrophysical stars we may anticipate this stability issue to deal with the preciseness of the upper cutoff state of matter density, which implies that it is not possible to have any further enhancement as well as diminution in density, once the matter density becomes Planckian. For completeness it would be satisfying to find approximate, yet a direct way to perceive the curvature regularity of quantum core. Indeed such an idea exists there that could defend that spacetime singularity inside black hole would not exist quite in an obvious way as long as there is a quantum core. It is only to note that a simple dimensional analysis leads to a naive expression of the Ricci scalar curvature : R ∼ M/r3 . Hence, as M → M(r) ∝ r3 (for quantum black hole core), the Ricci scalar curvature turns out finite for r → 0. Note that for the matter confined within a quantum core of radius rcore = r− , if we for the moment ignore the presence of a faint 2 (with ρQ being the maximal physical density of inhomogeneity, it would be ρ(r) ∼ ρQ ∼ 1/lQ 3 2 matter). Hence we get R|core ∼ M(r)/r ∼ ρQ ∼ 1/lQ . Thus a crude estimation of Ricci scalar curvature does anticipate the core-spacetime to be regular.

Loci of spacetime horizons

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It is essential to determine the spacetime horizons in order to explore the physics of black hole. Horizons act to trap regions of spacetime. These are null 3-surfaces - at each point of which there exists one null tangent direction (that points along a light-ray), which is orthogonal to two independent spacelike directions. Loci of the spacetime horizons are found by the condition −gtt (r) = 0 .

(22)

Use of this condition with respect to the LQC inspired present model of black hole (16) yields the event horizon, viz. r+ , as the largest (real) root of the following transcendental equation:  2  M lQ (23) r± = 2M 1 − 3 r± where the finite (real) solutions representing horizons are denoted by ‘r± ’. The Cauchy class inner horizon, viz. r = r− is determined as the smallest (real, positive) root of the above equation (viz. Eq.(23)). Since the Eq.(23) is transcendental in nature, it may be solved graphically in order to 11

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have the exact answers for the horizon-radii, r± . For an approximate determination r− , it is very useful to recast the last equation in the following form n r− o−1 3 2 r− = M l Q 1 − . (24) 2M Hence for M >> lQ , the radius of the inner (Cauchy) horizon may easily be approximated as =

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3 r−

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Thus it indeed appears that r− ∝ M 1/3 is a suitable approximation. This property seems essential as the way for admitting a proper quantum core (closely homogeneously packed with matter) for black hole. As is well understood in literature, the Hawking radiation coming of a black hole would result in an evaporation of mass-energy of a black hole. Hence it also follows that the radius of the Cauchy class inner horizon must decrease as the evaporation process progresses. We would now like to address the locus of even horizon at a weak quantum gravity affected regime. Such a regime happens to be fact for a black hole with mass, M , lying in the scale M >> lQ . Hence, upto the leading order quantum correction term (lQ /M )4 (in the dimensionless ratio: lQ /M ), the event horizon radius comes out to be (  2  4 ) 3 lQ 1 lQ − . (26) r+ ≈ 2M 1 − 8 M 64 M

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Therefore the effect of having a quantum gravitational improvement on the classical black hole geometry can be observed as a cause for a reduction in the radius of the event horizon as compared to the classical Schwarzschild radius (- a qualitative aspect which gets also apparent in view of the original equation given as (23)). The last expressed equation is closed in nature. However, the actual equation, viz. Eq.(23), giving the exact answer for r+ , needs to be solved graphically. For a quantum masked black hole, horizon-solutions, in general, are found by graphically plotting the entity, ‘gtt (r)’, and analyzing the intersections of the resultant curve with the r-axis. No horizons will then be observed to exist for quantum gravitational spacetime prevailing around any mass smaller than a critical mass (of value, ME ≈ 1.089 lQ ) ; that is, a real black hole solution can exist only for the masses greater than a finite minimal mass, and there are only particles for masses lower than that minimal mass. A LQC inspired understanding on quantum gravity is thus observed distinguishing a “black hole mass-regime” from the “particle mass-regime”.

4.3

Hawking radiation and thermodynamics

Hawking radiation is a key property to probe a quantum black hole model. The said radiation is explained plainly as a quantum field theoretical effect in which vacuum-field-fluctuations (and particle-antiparticle pair-creation) occur around the event horizon of a black hole. The law which is obeyed in this process, is the conservation of energy (as referred to infinity), viz. ξ.p + ξ.¯ p=0

(27)

where ξ = (1, 0, 0, 0) is the temporal Killing vector and p, p¯ are the 4-momenta of the pairparticles. In order for fulfilling this criterion if we imagine one particle, viz. p as appearing inside 12

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4.3.1

Hawking temperature and specific heat capacity

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a timelike spacetime region, the other one (i.e., the anti-particle), viz. p¯, must be found inside the spacelike trapped region of the black hole spactime. Since the energy that reaches infinity must be +ve, as referred to infinity the in-flowing energy into the black hole must be −ve. Evidently so the mass-energy of black hole must diminish and in effect both the Cauchy class inner horizon and the event horizon boundaries are to shrink as the black hole mass evaporates. It is to note that the rate of shrinking of the inner horizon is much slower compared to that of the event horizon. Hence the event horizon would eventually catch the inner horizon and the black hole evaporation would cease as rH → rE ∼ lQ (rE being the extremal horizon) and a cold relic is left thereby.

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Thermal nature of Hawking radiation attributes a characteristic finite temperature to a black hole of a given mass. Classical black holes corresponds to ever rising Hawking temperature with the decreasing black hole mass until the central singularity of classical spacetime is reached - a scenario which is an indicator of incompleteness in perceiving a regular black hole property with respect to Hawking radiation. A quantum improved black hole geometry can stabilize the thermal status of an evaporating black hole. Let us observe how it happens as referred to our proposed model. The Bekenstein-Hawking temperature for any Schwarzschild class black hole of the form (17) can be found by (in kB = 1 units) ~κ+ (28) TH = 2π 0

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where

 2 −2  2  M lQ 4M lQ 1− 3 1− 3 r+ r+ ( )  2  2 −1 2  2 3M lQ M lQ 6M 2 lQ lP lP2 = 1− 3 1− 3 = 1− 4 4πr+ r+ r+ 4πr+ r+

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lP2 = 8πM

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TH

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M

with κ+ = [F (r)/2]r=r+ being the surface gravity of black hole (event horizon). Since the Hawking radiation is a quantum field theoretical effect on the background spacetime, while the relation TH ∝ κ+ still holds, the effect of quantum modified geometry on Hawking temperature remains only implicit in κ+ . Let us now explicate this implicit quantum effect. On utilizing the expression 0 κ+ = [F (r)/2]r=r+ , temperature of LQC inspired quantum improved black hole comes out (in c = G = 1 units) as

r+ M= 2



2 M lQ 1− 3 r+

−1

.

(29) (30)

(31)

√ In context of last few equations let us stress that in the chosen unit system we imply ~ ≡ lP . As long as black hole has got a size well above the Planck scale, that is, r+ >> lP (note that lP ∼ lQ ), Hawking temperature increases as its mass is decreasing. However, the characteristic of the phenomenon of Hawking radiation dramatically changes as the mass of black hole approaches the Planck scale; the Hawking temperature, now, starts diminishing with a decreasing mass of black hole. A similar qualitative picture had also emerged in several other models, viz., in [32, 33, 39]. This picture contradicts the traditional semi-classical one within which the Hawking temperature rises indefinitely as the mass of black hole decreases. Up to terms of order (1/r+ )5 , quantum

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Figure 2: In this figure Hawking temperature of the proposed LQC inspired quantum improved black hole is plotted in units of βlQ (observe the solid magenta curve). It describes the fate of a radiating quantum black hole. Unlike the ordinary classical case (being represented by a dashed cyan curve), one now gets a cold remnant at the termination of evaporation process.

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modified Hawking temperature (due to the present black hole model) can be given by (  2  4 ) 3 lQ 3 lQ lP2 1− − . TH = 4πr+ 2 r+ 2 r+

(32)

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It is worth to note that the net effect of quantum gravity is to reduce the traditional semi-classical Hawking temperature of real physical black holes. This picture also reflects that still at the event horizon quantum improved gravity is less attractive as compared to the classical gravity.

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Now we shall analyze for the specific heat, CBH , of the LQC inspired current model of quantum black hole. Specific heat acts to provide a picture of energy exchange of a thermodynamic system with its ambient surroundings. It happens to be a fact that a radiating classical black hole always corresponds to negative specific heat capacity which reflects no possibility of thermal equilibrium of classical black hole with its environment. However a black hole at a quantum masked regime may oppose the classical black hole. Indeed it is evident from the definition, CBH = ∂M/∂TH , that CBH can be positive once a quantum black hole enters a Planck mass regime where the quantum improved Hawking temperature (due to the event horizon) starts to fall with the decreasing black hole mass. That is, it is likely to have a thermodynamic change of phase as CBH changes from its prior negative value to a positive value. Hence it could be possible for a Planck scale black hole relic to be in thermal equilibrium with its immediate surroundings. Let us now observe explicitly. The specific heat, viz. CBH of our LQC inspired quantum black hole is estimated to be  2   ∂M ∂r+ r+ C1 1 ∂M = = −2π (33) CBH = ∂TH ∂r+ ∂TH lP C2 C3 14

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where 2 6M 2 lQ 4 r+ 2 −1 2  M lQ 3M lQ 1− 3 1− 3 r+ r+  2  2M lQ 1− 3 r+   2 2 30M 2 lQ 6M lQ C1 1− + 3 4 r+ r+ C2 ( ) 2 −1 2 2  M lQ 6M lQ 15M lQ C1 1− 3 + 3 1− . 3 r+ r+ r+ C2

(34)

=

(35)

C2 = C3 = =

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C1 = 1 −

(36) (37) (38)

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Note that for a massive black hole C1 → C2 → C3 → 1, and one recovers the classical value for 2 /lP2 . Let us also introduce a new variable: specific heat, viz. CBH = −2πr+ dQ =

3 r+ . 2 M lQ

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Use of this entity simplifies the mathematical expressions. Hence we have  −1 3 1 C1 = 1 − 1− dQ dQ   2 C2 = 1− dQ ) (  −1 15 1 6 C1 . C3 = 1− 1− + dQ dQ dQ C 2

(39)

(40) (41) (42)

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In accordance with our present model there are physical black holes only for dQ ≥ 4 (one may note Fig.(2)). It is to note that in this range C2 6= 0, C3 6= 0, and therefore the zero of the specific heat is determined by the zero of C1 ; and the result p is as it should be, viz. dQ = 4. There occurs a change of phase during black hole evaporation for 3 dQ ≈ 2.243 (which is equivalent to a black hole of radius r+ = r˜+ ≈ 2.50 lQ ; in order to estimate this equivalence one may use a forthcoming equation, viz. Eq.(43)). Fig.(3) clearly depicts this fact. Theoretically, the change of phase occurs −1 at that particular point, viz. r+ = r˜+ , for which CBH |r+ =˜r+ ≡ (∂TH /∂M )|r+ =˜r+ = 0, and this event leaves its mark in terms of a change of sign in the value of CBH . Let us hence also mention that there exists a certain regime, specified by 1.633 lQ < r+ < 2.50 lQ , for which the specific heat remains positive and one can get a stable thermal phase for a quantum masked black hole. It is very useful to recast some of the previous formulas as a function of the variable ‘dQ ’. On using Eq.(31) we may cast the event horizon r+ as  2  −1 r+ dQ 1 = 1− , (43) lQ 2 dQ and the Hawking temperature given by Eq.(30) may be cast as s  (  −1 ) lP2 1 1 3 1 TH = √ 1− 1− 1− . dQ dQ dQ 2 2πlQ dQ 15

(44)

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Figure 3: This figure depicts the β-multiplied black hole specific heat capacity βCBH as a function q 2 3 of r+ . Here r+ is scaled in units of M lQ . There is point of discontinuity which shows that there q 2 happens to be a phase change once a radiating black hole reaches the size r+ ≈ 2.243 3 M lQ ; hence CBH turns +ve from a prior −ve value.

(45)

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Again, using Eq.(43), we recast Eq.(33) as #  2 " (  −1 )   lQ dQ 1 C1 1 1− CBH = −2π lP 2 dQ C2 C3

4.3.2

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with C1 , C2 , C3 being the known functions of dQ (note Eqs.(40), (41), (42)). The last two equations are used in the graphical representations of TH and CBH respectively in figures (2 , 3) . Cold black hole remnant

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As we have observed, a quantum black hole suffers a change of phase, cools as it approaches a quantum masked Planck scale and henceforth attains a positive heat capacity. Evidently the end state of an evaporating black hole is an absolute cold relic. Such a black hole remnant, while appears physically appreciable, is interesting in quite a few respects. For example it could lead to a resolution to the Hawking information loss paradox [40]; apart from that a thermodynamically stable Planck black hole relic also emerges as a natural candidate for being a dark matter (WIMPs) particle [41]. So we shall now proceed to have the exact figure of the cold black hole remnant. There is a critical mass ME such that for M < ME there are no black hole solutions, instead, there are particles. A black hole relic with the critical mass M = ME , is extremal in nature carrying a single degenerate horizon. This critical mass ME will also represent the cold black remnant left after a quantum-affected stopping of the Hawking radiation process. For M > ME , there are real physical black holes, each with two distinct horizons. It is possible to find a solution for ME in an analytic method. This solution for ‘ME ’, is to be found by solving F (rE ) = κ(rE ) = 0 16

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with rE being the radius of the single degenerate horizon (viz. r+ = r− = rE ) of the extremal black hole of mass ME . Using the condition F (rE ) = 0, first we get 2 rE4 − 2M rE3 + 2M 2 lQ =0.

(46)

4.3.3

Entropy-area law

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Further, utilization of the condition κ|r± =rE = 0 (which is equivalent to the condition F 0 (r)|r=r+ =rE = 0) along with the Eq.(46) provides us 3 (47) rE = M . 2 Next use of Eq.(47) in Eq.(46) yields √ 4 2 ME = √ lQ . (48) 3 3 Therefore, the “black hole remnant” left at the termination of Hawking radiation and energyevaporation process (in other words the minimal mass black hole) would be of mass ME ≈ 1.089 lQ , or, equivalently of radius rE = 1.633 lQ . This remnant is thermodynamically stable in nature. Eventually, as an important observation we find that the remnant of black hole is universal in size, being of the order of the Planck mass - a result that resembles what emerges out of the other quantum black hole models, viz. [15, 39, 42, 43]. We had pointed out that a Planck mass black hole remnant could be a candidate for the WIMPs-class dark matter particle because of its barely detectable interaction with the ordinary matter. Planck mass black hole relics might have a realistic origin in the primordial black holes (PBHs) because of the matter fluctuations in the primordial Universe. In this regard one promotes the possibility of cold Planck mass PBHs. However it should be noted that a Planck mass PBH is unlikely to be absolutely cold, rather it would have a finite temperature (with a positive heat capacity) in order to stay stable in its ambient thermal surroundings. Yet it is also true that the mass of such a practical relic would be very close to the mass of the absolute cold remnant.

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Entropy, being determined in terms of the horizon area, is one intriguing aspect of black hole thermodynamics. The classical Bekenstein-Hawking relation, viz. SBH = AH /(4lP2 ) (while AH = 16πM 2 ), still leaves scope for quantum improvement. Indeed such a possibility has gained people’s interest over the last few decades [39, 44, 45]. In view of our proposed model of quantum black hole, we shall here try to find out the quantum geometrical refinement in the black hole entropyarea law with respect to event horizon. So we use the 1st law of black hole thermodynamics, viz. dM = TH dS, which upholds entropy, S, as a state function. Let us hence cast

where

1 ∂M dr+ TH ∂r+  −1 2πr+ 2 = 1− dr+ lP2 dQ

dS =

2 lQ 1 = 2 dQ 2r+



1 1− dQ

−1

.

(49) (50)

(51)

Let us explicate: the expression (50) is found by substituting Eq.(30) (while also appropriately using Eq.(31)) in Eq.(49). Now, since dQ ≥ 4 for any physical black hole (as has been earlier 17

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clarified), it is suitable to expand the the above identity (viz. Eq.(51)) as  2 2  lQ lQ 1 4 1 + 2 + O(1/r+ ) + ... . = 2 dQ 2r+ 2r+

(52)

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2 Noting that the event horizon area is given by A+ = 4πr+ , we find the differential entropy-area law of thermodynamics: −1  1 2 dS = 2 1 − dA+ (53) 4lP dQ

with dQ being a function of A+ . On using Eq.(52) in Eq.(53), we get (by setting the arbitrary constant to zero):  2     πlQ A+ A+ 1 S≈ 2 + ln −O . (54) 4lP lP2 lP2 A+

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This quantum improved entropy-area law reproduces the well known leading order logarithmic quantum correction to the original semi-classical Bekenstein-Hawking law [45]. It could be worth to note that the same entropy-area relation is also checked to hold with respect to the classical black hole area Aclass = 16πM 2 . Further it would be clarifying to determine an approximate change of black hole entropy, ∆S, over the completion of the Hawking evaporation process :  2   πlQ A+ A+ − AE + ln . (55) ∆S ≈ 4lP2 lP2 AE

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where AE is the event horizon area of the final cold black hole remnant, whereas A+ corresponds to an arbitrary initial state of the black hole.

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Where does the current model stand?

5.1

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We shall here briefly compare the outcomes of Rovelli-Vidotto model with the results obtained due to the currently proposed black hole model. So we shall look into the major aspects. In the LHS of the following equations, symbols with no subscripts will denote entities due to the current model, while the symbols with the subscript “RV ” will represent the Rovelli-Vidotto model.

Loci of horizons and size of quantum core

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To have the exact loci of the horizon(s), for both of the models under concern, one has to solve the corresponding transcendental equations. However, in the limit M >> rcore , we can have the following simple analytic formulas: ( (  2 )    2−2α ) 1  r0 2 2 lQ 1 lQ ; r+ |RV ≈ 2M 1 − = 2M 1 − . (56) r+ ≈ 2M 1 − 8 M 4 M 4 M Evidently the result due to the currently proposed model differs from that of the Rovelli-Vidotto model. Next we note the loci of the inner Cauchy class horizon for the concerned models: r− ∝ M 1/3 ;

r− |RV ∝ M α while 1/3 ≤ α ≤ 1 .

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In particular for a special case α = 1, or, equivalently for r0 = M , the Rovelli-Vidotto model (6) leads to following ranges for the loci of horizons

5.2

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r+ |RV = ηM, 4/3 ≤ η ≤ 2 ; (58) r− |RV = ζM,  ≤ ζ ≤ 4/3 ; (59) √ while  is restricted to be  < 4/ 27 for a valid black hole solution. It may also be noted that in any case with respect to α, it would be r− |RV ∼ r0 and r+ |RV ∼ M [28]. With this understanding, it becomes manifest that the sizes of the black hole core (rcore ≡ r− ) of the two models can coincide provided one chooses α = 1/3 in the Rovelli-Vidotto model-metric.

Hawking temperature

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Next let us look into the issue of Hawking temperature for a comparison. Hawking temperatures for the concerned model black holes run as (note: Eq.(30) and Eqs.(6), (11), (28)) (   2α  2−2α ) 2  6M 2 lQ lP2 lP2 lQ M TH = 1− ; TH |RV = 1 − 32 . (60) 4 4πr+ r+ 4πr+ r+ r+

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So we observe that the results for the two models differ from each other. In the limit rcore << M , difference between two models comes out as ( (  2 )  2α  2−2α ) 3 lQ lP2 lQ lP2 1 1− ; TH |RV ≈ 1 − 32 . (61) TH ≈ 4πr+ 2 r+ 4πr+ 2 r+

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For a special case α = 1 of Rovelli-Vidotto model one can have   lP2 32 TH |RV = 1− 2 . 4πr+ η

(62)

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Here, for every  there will be a specific η. But, in order for evaporation to cease for a prior evaporating Rovelli-Vidotto black hole of α = 1 class, the ratio /η must vary along with M - this requirement would turn to explicate a fallacy, since,  and thereby η are theoretically fixed-valued.

Size of remnant

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It is essential for this subsection to put forward the required basis for the Rovelli-Vidotto model, before we can move on to compare the said model with the new LQC inspired black hole model of the current paper. So let us note that for the case α = 1, clod state extremal Rovelli-Vidotto black hole corresponds to √ 4 (63) rE |RV = 3M ;  = √ ; 3 3 which can therefore be valid for any mass-scale. In trying to alleviate black hole information loss paradox, √ Rovelli√ and Vidotto demanded an ad-hoc final cold black hole state defined by Mf ∼ Mi / 2 ≡ M/ 2 [28]. Thereby, with respect to the size of black hole remnant, the ‘α = 1’ class Rovelli-Vidotto model largely differs from the current model of this paper: r tevp rE ∼ 3 lP ; rE |RV ∼ 0.1lP 3 . (64) tP 19

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5.4

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Here tevp represents the duration of time over which a black hole evaporated since its formation to reach the extremal cold remnant state. In case of the primordial black holes surviving in the present-day Universe, tevp ∼ 1010 years, and hence in accordance with the Rovelli-Vidotto model, the size of the present-day primordial relics comes out to be as large as ∼ 10−14 cm. However for a universal convention rcore ∝ M 1/3 , the mass-scales of the remnants of black holes for the concerned models agree. It is to emphasize that indeed the deal of Rovelli and Vidotto with the issue of remnant was precisely triggered by the choice α = 1,[28] (- a choice which in principle emerges to be a misconstruction since it does not go with the central notion of having repulsive quantum gravity around a regime governed by the Planck-dense source-matter).

Entropy-area law

SRV SRV

Z 2 r0 A+ 6π + 2 dr+ , ≈ 2 4lP lP r+  −1 1 32 A+ , ' 1− 2 η η 2lP2

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and

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As a final point we would like to bring in comparison between the two models (under discussion) with respect to the issue of entropy-area law. Hence it follows. On setting the arbitrary constants to zero, the entropy-area law for the concerned model black holes take the form :  2   πlQ A+ A+ + ln ; (65) S ≈ 4lP2 lP2 lP2 for r0 << M ;

(66)

for r0 = M.

(67)

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Thus, with rcore ∝ M α while 1/3 ≤ α ≤ 1, no logarithmic correction is recovered for the quantum improved entropy area law in the Rovelli-Vidotto model for the entire range of value of α. In contrast, the currently proposed model of this paper does produce a well desired, stable logarithmic correction to the (semi)-classical entropy-area law.

6

Concluding remarks

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Regular black hole must be associated with a “quantum core”. We assess quantum core to be characterized fundamentally by the maximal physical matter-density, viz. by the Planck-order density. In this paradigm the core size is not universally fixed, rather, is variable; in fact, must increase with an increasing black hole mass. This idea, which is actually quite old in its qualitative sense [25], recently came into attention due to Rovelli and Vidotto [28]. In our paper we argued specifically against the choice of the spacetime metric of Rovelli and Vidotto. The loop quantum cosmological quantum improved Friedman equation of the evolving Universe is utilized in predicting the quantum black hole spacetime. It emerges that this LQC inspired quantum improved geometry p 3 for a black hole of mass M agrees with the Hayward geometry [15] for r >> M lP2 , while differs in predicting the core-size. As an another point, the newly anticipated geometry (of the present paper) does successfully infer the regularity of quantum core to be characterized by the critical Planck order spacetime curvature while judging the same core to be packed approximately with the Planck dense matter. The Hawking process of black hole evaporation goes through a phase change as the size of quantum core approaches the Planck size 5 . Eventually a cold remnant of black hole 5

More minutely this size is expected to be ∼ 5 lP .

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is left, which bears a Planck order mass. We also observe the proposed LQC inspired black hole to reproduce a standard leading order logarithmic quantum correction in the entropy-area law. Even though the basic motivation behind the presently proposed black hole model and the basis of the proposal made by Rovelli and Vidotto in [28] are the same, viz. “black hole holds quantum core characterized by Planck order matter-density”, results obtained in our present model differ from the Rovelli-Vidotto model in some crucial aspects. In our case the quantum improved entropy-area law goes with what is quite a general consensus, whereas, this law gets never satisfied with the entire range of parameter-choice in the Rovelli-Vidotto metric model. Again, unlike the Rovelli-Vidotto model, wherein the black hole remnant left at the end of the Hawking radiation process was speculated to be determined by the mass of a black hole at the time of its formation and thereby was claimed possibly to be often much larger than the Planck size, we find the size of the remnant black hole to be universal, being quite of the order of the Planck size. We may now summarize : On revisiting the Rovelli-Vidotto model, we realize that it is necessary to have a new choice for the spacetime metric, and hence on exploring the resultant quantum improved black hole spacetime we find a new set of results. These results imply some robust characteristic properties of quantim black hole. We hope that the present analysis and its outcomes would be helpful in understanding the quantum improved black hole physics.

Appendix: Does LQC-Newton potential conform LQG ?

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Here we shall show that a generalized uncertainty principle (GUP) due to loop quantum gravity (LQG) leads to quantum corrected properties of a black hole that fairly conforms the properties of the presently proposed LQC inspired black hole holding a quantum core. Hence first we note the GUP formula due to LQG [46]:

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∆x &

~ − α∆p . 2∆p

(68)

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for α being strictly positive number, since LQG Rspacetime lattice p p structure causes to suppress 3 2 |ds | ∼ r+ /2M [39] (based on classical uncertainty in position. On identifying ∆x ≡ geometry), and TH ∼ ∆p , one gets for proper scaling of the uncertainty parameters   s s ( )  2  3/2  r+ lP ηM lP2 ∆x TH = − 1 − 1 + 2α '− √ 1− 1+ 3 (69) 2α  ∆x  r+ πη 2M (  2  4 ) lP2 η lP lP ≈ 1− −O .... for r+ >> lP (70) 4πr+ 8 r+ r+ in Adler method [47] (here, η is treated as an unspecified constant of the order of one, also note 5 the forthcoming equation (71) for understanding 1/r+ order correction in (70)). Hence the result is in fair qualitative agreement with (32) that corresponds to the constructed model LQC inspired black hole. Note that one cannot anyway expect to uphold an exact agreement since the available generalized uncertainty principle is itself non-exact. Further Adler method is not self-complete, since it requires a known background general relativistic geometry to assess ∆x ≡ ∆x(r+ , M ). Nevertheless it may in principle be possible that an improved GUP due to LQG could more well conform the LQC inspired geometry of ‘black hole holding a quantum core’.

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Let us check another property. Through a black hole uncertainty principle (BHUP) correspondence [48], locus of a quantum improved generalized event horizon due to LQG-GUP will follow from Eq.(68) (by one of the suitable physical substitutions ∆p = ±M in this respect) : (  2 ) 1 lP ; (71) r+ ≈ 2M 1 − 4 M

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for α being set to 2. Hence again we can observe an anticipated qualitative conformity between LQG and the LQC-Newton potential by comparing Eq.(71) with Eq.(26).

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[2] E. Ay´on Beato and A. Garci´a, “Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics,” Phys. Rev. Lett. 80, 5056 (1998). [3] V. P. Frolov, M. A. Markov and V. F. Mukhanov, “Through A Black Hole Into A New Universe,” Phys. Lett. B216: 272-276 (1989). [4] V. P. Frolov, M. A. Markov and V. F. Mukhanov, “Black Holes as Possible Sources of Closed and Semiclosed Worlds,” Phys. Rev. D41, 383 (1990).

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[10] R. Balbinot and E. Poisson, “Stability of the Schwarzschild-de Sitter model,” Phys. Rev. D41, 395 (1990). [11] A. Borde, “Regular black holes and topology change,” Phys. Rev. D55, 7615-7617 (1997). [12] A. Ashtekar, A. Corichi and P. Singh, “Robustness of predictions of loop quantum cosmology,” Phys. Rev. D77, 024046 (2008). [13] A. Ashtekar “Singularity Resolution in Loop Quantum Cosmology: A Brief Overview,” J. Phys. Conf. Ser. 189, 012003 (2009) [arXiv:gr-qc/0812.4703]. [14] A. Ashtekar, T. Pawlowski, P. Singh and K. Vandersloot, “Loop quantum cosmology of k = 1 FRW models,” Phys. Rev. D75, 0240035 (2006). 22

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[15] S. A. Hayward, “Formation and evaporation of non-singular black holes,” Phys. Rev. Lett. 96, 031103 (2006). [16] T. De Lorenzo, C. Pacilioy, C. Rovelli and S. Speziale, “On the effective metric of a Planck Star,” Gen. Rel. Grav. 47: 41 (2015) [arXiv:gr-qc/1412.6015]. [17] J. C. S. Neves and A. Saa, “Regular rotating black holes and the weak energy condition,” Phys. Lett. B734, 44-48 (2014).

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