Decoding infrared imprints of quantum origins of black holes

We analyze the emission spectrum of a (fundamentally quantum) black hole in the Kerr-Newman family by assuming a discretization of black hole geometry and the holographic entropy-area relation. We demonstrate that, given the above structure of black hole entropy, a macroscopic black hole always has non-continuously separated mass states and therefore they descend down in discrete manner. We evaluate the step size of the discrete spectrum, which vanishes in the extremal limit, leading to a continuum spectrum as expected from thermal nature of black holes. This further reveals an interesting relation, in each class, between the dynamic and kinematic length scales for all black holes belonging to the Kerr-Newman family, pointing towards a possible universal character across the class, dependent only on black hole mass. Further, we have presented the computation of maximum number of emitted quanta from the black hole as well as an estimation of its lifetime. We also argue the independence of these features from the presence of additional spacetime dimensions.


Introduction
Realization of the fact that black holes behave as thermodynamical objects [1][2][3][4][5][6][7][8][9][10] have boosted the hope to reconcile the quantum theory with the framework of gravity in a successful and consistent manner. In particular, macroscopic geometric constructions of black holes along with the microscopic quantum character as suggested by their thermodynamical nature, demanded a quantum theory of space-time geometry. Thus demystifying and understanding the black hole geometry has been one of the pioneering goals of modern physics, which in all likelihood, will provide an useful insight to the quest of arriving at a successful quantum theory of gravity.
Different approaches towards quantizing gravity [11][12][13][14], emerging out since the discovery of black hole thermodynamics, have been accordingly oriented, keeping the hope of unifying gravity and quantum theory in sight. Unfortunately, despite decades of efforts along numerous directions, which have resulted into many promising candidates for quantum gravity, we have not been able to come up with a theory that can actually provide a consistent description to unify quantum theory and gravity. Each of these candidates has offered its own different pathway of studying and understanding different constructions of gravity, more particularly the black holes. However with no black holes available to experiment with, and a plethora of quantum gravity theories with their own predictions, the black hole mysteries [15][16][17][18][19][20][21][22][23] have led this search to a cross road. The amount of tussle and conceptual discomfort one has to swallow, in order to come up with something as exotic an object as the black hole, from an as fundamental theory as the quantum mechanics, has led to fierce debates on the naturalness and existence of the black hole itself [24][25][26][27]. Fortunately, detections of the gravitational waves [28,29] from merger of two black holes have, for the moment, put the debates regarding the existence of black holes to oblivion. The spacetime ripples originating from the merger of two black holes following the general relativistic prediction has given a huge encouragement to the research of black holes in a more detailed fashion. There have been studies [30,31] to explore the imprints of some non-intuitive quantum features of the black holes, which appear in many frameworks of quantum gravity. This fundamental discovery has also paved the way for asking more fundamental questions regarding the black holes, e.g., is there any meaningful observational signatures of the quantum nature of black holes? Therefore, the dynamical evolution of black holes can really act as a lead for testing of our theoretical models claiming to quantize gravity [32][33][34].
Apart from the classical gravitational wave emissions when a black hole is perturbed from its equilibrium configuration, there is also a quantum channel by which a black hole can emit thermal radiation, known as the famous Hawking radiation [7]. There is a renewed interest in the quantum emission profile of black holes after the realization gained ground that quantum emission of black holes may well be hiding its intrinsic quantum face [35][36][37][38]. Studies of such quantum emissions hold their own ground as the tussle between non-compatibility of the quantum theory with principles of gravity become manifest here, thanks to the thermal nature of the radiation [20,39,40].
In this work, we will assume that the notion of black hole entropy as perceived in the context of thermodynamics of macroscopic black holes holds true through the microcanonical counting, in the quantum domain (see, for example [41][42][43]). Given the above assumption on the nature of black hole entropy at a microscopic level we would like to address the question, across these class of theories, is there any signature of the underlying quantum nature of black holes which gets fossilized in its emission? In a previous work [38,44], we have explicitly demonstrated that if black holes are indeed a macroscopic realization of a microscopic quantum geometry and the thermodynamic relation between entropy and area still holds in the microscopic level having holographic properties, then it is highly unlikely that the emission from them will have the suggested thermal behaviour. If the geometry (of black hole spacetime) gets discretized in any fashion, the emission lines from the macroscopic black hole become hugely discrete in the low frequency regime. Similar conclusions were reached in many other contexts and their far reaching implications are still being investigated [45][46][47][48].
However the scenario depicted in earlier works was for a Schwarzschild black hole, with mass as the only hair [38,41,44,[46][47][48][49][50][51]. However astrophysical black holes are most likely to inherit non-zero angular momentum as well. Thus in the process of decay, the black hole loses both mass and angular momentum and as we will show in this work, the inclusion of angular momentum modifies the quantum spectrum of a black hole drastically. In particular, as we will depict, there will be lot of interesting physics happening near the extremal limit, which was non-existent for Schwarzschild black hole. Further, since black holes are supposed to be thermodynamical objects in classical theory, they have to jump from one thermodynamical configuration to another, which are allowed by holography and the microscopic quantum description of the hole. Therefore, the emission lines must be consistent with both thermodynamics and the underlying quantum structure. As we will see, these two demands are strong enough to fix many characteristic features of black holes by not allowing them to emit at all wavelengths. In other words, the above requirements result into a discrete quantum spectrum for black holes [41,49,50]. This fact has been aptly demonstrated in the so called Bekenstein-Mukhanov effect, showing the existence of a minimum frequency and hence the largest wavelength that a black hole can emit. Note that the above result is based on the assumption that the entropy-area relation, obtained from the thermodynamics of macroscopic black holes, comes through in exact form from the underlying quantum counting of this macroscopic realization of the quantum discrete geometry. However, typically there can be subleading corrections to such a relation, which will come through the exact (say) von-Neumann counting. In Appendix B, we have addressed a class of such subleading correction which appear across many theories of quantum gravity [52][53][54][55]. Interestingly, for this class of corrections too, the results obtained in the paper survive. It will be interesting to analyze the most general structure of mass spectrum under most generic class of subleading correction in a closed form, which will come from a full quantum understanding of a non-extremal black hole. We leave this exercise for a future work.
For Schwarzschild-like black holes, it is easy to demonstrate that this largest wavelength is a multiple of the horizon scale [38,41,44], which we argue, is in fact related to the de-Broglie wavelength associated with its thermality. Further, by working out the corresponding situation in the context of Kerr-Newman family of black holes, we will derive an universal relation relating various wavelengths associated with this problem, which is independent of the number of hairs inherited by the black hole. This relation works as a quantum gravitational book keeping parameter throughout the emission. Therefore, this infrared character of the black hole emission will be able to make prediction of quantum gravity parameter of the black hole, serving as a infrared test for various theories. Thus, following the quantum rules, while keeping the thermodynamic properties in mind, in this work, we have been able to put forward some unique features of the emission spectra of a black hole. Furthermore, we have also demonstrated that, while in the earlier literature only integer shift scheme was advocated, it is possible to have a quantum structure which is Logarithmically discretized. Implications of such a new discretization scheme will also be discussed.
The paper is organized as follows: In Section 2 we analyze the relation of the macroscopic geometric aspects of a Kerr-Newmann black hole with the expectation of a fundamental quantum operator describing a black hole. This results into two ways a black hole can change its macroscopic configuration, either in integer steps or in a Logarithmic integer fashion. Section 3 discusses the mass gap between the nearest macroscopic configuration emerging out from such a quantum operator which is discretized in integer steps. We show how such a scheme gives a universal relation for all black holes in Kerr-Newman family. We also discuss the effect on the mass gap in various interesting scenarios where the macroscopic construction goes towards becoming an extremal black hole. In Section 4, we illustrate how the above results derived in the context of integer shift, translate into the physics of a Logarithmically discretized black hole. In Section 5, we estimate the maximum number of quanta that a Kerr-Newmann black hole can emit, through the smallest possible energy steps before it becomes Planck sized or extremal. Appealing to the emission time of release of a quanta we will calculate the time taken in the complete evaporation process of black hole as well. Furthermore, in Section 6 we have discussed how the extremal limit of a black hole can be arrived at and will connect the classical results with the discretization schemes presented here. We finally argue the robustness of our results for higher dimensional rotating as well as charged black holes in Section 7, before summarizing our main findings in Section 8. Some additional computations have also been presented in Appendix A and Appendix B respectively.

Holography and quantum spectrum of black holes
Throughout this work we will exclusively concentrate on Kerr-Newman family of black holes, characterized by the mass M , angular momentum J and charge Q. Such a black hole in the context of general relativity corresponds to the Kerr-Newman solution. The horizon length scale will turn out to be an important parameter in the above problem, which for the Kerr-Newman black hole is (classically) located at, Here a = J/M is the rotation parameter associated with the Kerr-Newman black hole. Since we are interested in black hole spacetime, in what follows we will assume that (a 2 + Q 2 ) < M 2 . Note that in the limit {(a/M ) 2 + (Q/M ) 2 } → 1 the term inside square root vanishes and is known as the extremal limit. We will discuss various properties of the emission spectrum as a black hole approaches the extremal limit extensively below.
In [44], it was demonstrated how the quantization of any geometrical attribute of a classical (like) black hole can be related to an averaged macroscopic quantity, resulting into the following counting of the number of microstates in micro-canonical ensemble picture as with f (j) marking the fundamental quantum spectra of the geometrical attribute and n j being the occupancy of the quantum level. Constant B contains information of the fundamental parameter of the correct quantum theory of gravity. Such a relation can be shown to exist even for the black holes in the Kerr-Newmann family []. The above relation provides the stringent constraints that the discretized levels must satisfy in order to maintain holography, i.e., the entropy-area relation. Clearly, as the number of microstates in the micro-canonical picture has to be an integer, for the right hand side of Eq.(2) to turn out to be an integer, one needs B 4 F γ = ln K for some variable integer K, corresponding to different area values of the horizon. This can happen in two possible ways -(a) B/4 = ln I 0 and F γ = I, for all possible {n j } (at least) for which j n j 1; or (b) Either B/4 = I 0 for some fixed integer I 0 , and F γ = ln I for variable integer I, i.e., in the macroscopic limit. The choice (a) essentially corresponds to a logarithmically discretized system [56] while choice (b) presents a integer shift [41,43]. A priori there seems to be no reason to prefer one above the other and thus we will discuss both of these scenarios. However it will turn out that the quantum spectrum of black holes heavily depend upon which of the above schemes have been chosen. We will first discuss the scenario depicted by case (a), i.e., in which As discussed, here B carries the imprint of the underlying quantum structure and the integer I marks the micro states {n j } in a collective manner. Such a integerly discretized scheme may be advocated on various grounds including quantum gravity as well [12,14,51,57,58]. As the black hole makes a transition from one macroscopic configuration characterized by {n j } to another macroscopic state denoted by {n j } the integer changes from I to I . We will be studying such transitions in the next section. As a final remark, we would like to re stress the fact that even though the above computation depends on the assumption of exact entropy-area relation at the macroscopic level. The accuracy of this relation, of course will be known as and when a full quantum theory of black holes gets available. However, we can try to repeat the analysis for (model-dependent) corrections to such relations, in anticipation of a closed form expression originating from true quantum counting (see appendix B, for example).

Minimum mass gap for black holes: Integer spectrum
In this section, we will assume the black hole to follow integer shift, i.e., its area is discretized, such that it follows Eq.(3). If large black holes are to behave as thermodynamical objects, exactly as predicted by the classical black hole thermodynamics, the change in mass and change in angular momentum should provide us a trail for a more general scenario, as prescribed by the thermodynamical laws. To demonstrate the same let us consider the Kerr black hole with the understanding that the associated results can be generalized to a Kerr-Newman scenario in a straightforward manner. The first law of thermodynamics in the context of Kerr black hole relates a change in mass δM to a change in area δA and a change in rotation parameter δJ, such that, In the above expression Ω H corresponds to the angular velocity of the black hole event horizon. It is advantageous to write down the change in angular momentum as a change in mass and a change in rotation parameter a, where J = aM , leading to, δJ = aδM + M δa. Note that it is possible to use the above expression in order to treat the rotation parameter a and the black hole mass M to be independent, while the angular momentum J is dependent on both a and M . Thus one can express the change in rotation parameter as For a macroscopic black hole the mass is supposed to be large, while J changes in integer steps if we were to invoke quantization of angular momentum [41][42][43]49] and hence the δJ/M term is always suppressed. Similarly, taking a cue from Schwarzschild black hole, δM itself scales as 1/M , making its presence having negligible effect for a macroscopic black hole, even when a/M ∼ 1. Thus one can safely ignore the effect coming from δa irrespective of whether the black hole is far from being extremal or not. However, it is important to note that in this process we are considering that black hole to be radiating away its mass as well as angular momentum (say in terms of gravitons). We discount the possibility of mergers/ collapse and processes which increase these parameters. Finally, using Eq.(5) to replace δJ in Eq.(4) we obtain the following relation connecting δM , δA and δa respectively, such that, In order to arrive at the above relation, the following identity, 1 − (a 2 /2M r h ) = r h /2M , has been used. Since we are interested in emission from the black hole as it jumps from one configuration to another, we will consider only those transitions for which both δA and δa are negative. Moreover we will not consider the transitions among degenerate levels, for which the black hole area (and hence entropy) does not change. Thus under the above set of assumptions imposed on the possible transitions among black hole micro-states it follows that the minimum mass gap demands δa = 0 and area changes minimally. Otherwise, change in rotation parameter will always add to the change in area and consequently the change in black hole mass will be higher. This provides the backdrop of the general scenario which we will discuss in the next section.

General Analysis
Having described a thermodynamical backstage, let us now concentrate on a general analysis. Moreover as the macroscopic black hole of the quantum theory, closely resembles the classical black hole configuration, the area and the mass are related through, Note that we have used the rotation parameter a and the black hole mass M as the black hole hairs, rather than the angular momentum J and black hole mass M . This is because, writing the geometrical objects associated with the black hole in terms of a facilitates simplification of the computation significantly. It is possible to invert the above relation in order to write down the mass in terms of the discretized area. This can be achieved by completing the square on the right hand and then solving an algebraic equation for M in terms of (a, Q, I). Such a relation can be written as, It is clear from the above expression that the macroscopic black hole's mass gets related to the integer I coming from the microscopic configuration counting. Different macroscopic configurations, identified with different I values, will therefore have different masses. We are interested in obtaining the smallest mass difference between two (mass-wise) nearest black holes. In principle, the nearest neighbour of a black hole of mass M (and hence integer I) can only be larger than the mass identified by the integer value I + 1.
With this preamble, let us compute the mass shift as the black hole makes the smallest possible transition from I + 1 to I. As we are considering the transitions for which both area and rotation parameter decrease and we are also excluding transitions among black hole micro-states having identical area, it follows that the mass change will be minimum when both Q and a are kept the same. Otherwise, the change in Q or a will add to the area change and hence the shift in black hole mass will be bigger. Thus the mass difference between two nearest micro-canonical configurations turns out to be In order to arrive at the above expression we have assumed the black hole to be macroscopic, which justifies expansion in inverse powers of I. This is because, as evident from Eq.(3) and the expression for F in terms of occupation number {n j }, large value of I implies that most of the micro-states are already occupied, resulting into a macroscopic description for black holes. However,the above expression is not very illuminating. To cast the above relation to a more useful form we have to provide an expression for (BI/4π) in terms of the hairs of the black hole. Such a relation is being supplied by Eq.(7), yielding, as well as, Therefore, the substitution of both these expressions, namely Eq.(10) and Eq.(11) in the mass gap expression Eq.(9) leads us to the desired relation, Here T H (η) corresponds to the Hawking temperature associated with the Kerr-Newman black hole and η 2 = (a 2 +Q 2 )/M 2 , a dimensionless parameter approaching unity as the black hole approaches the extremal limit. The above expression clearly demonstrates that the minimum mass gap identically vanishes in the extremal limit. This is because, the term in the denominator takes a value 2 in the η → 0 limit, while it becomes 1 in the extremal limit. On the other hand, the black hole temperature T H (η) is directly proportional to 1 − η 2 and hence identically vanishes in the extremal limit. Therefore the minimum mass gap also vanishes, implying continuous black hole spectrum in the extremal limit as Fig. 1 and Fig. 2 suggests. One can further verify that this minimum mass gap exactly coincides with the one derived from thermodynamical consideration, a crucial test for consistency of the scenario depicted here. It is sometimes beneficiary to write down the minimum mass gap in the following manner, This is because the right hand side of the above relation is solely dependent on the specific charge Q/M and specific angular momentum a/M , which makes the extremal limit apparent (see, for example Fig. 1). Thus, we have obtained an analytic expression for the (in principle) smallest mass gap and it gets related to the thermodynamic temperature of the black hole. If this mass gap gives rise to the emission of a quanta in the form of loss of energy, the frequency of the emitted quanta (ignoring the B/2 factor), will always be less compared to the thermal frequency associated with black hole temperature. This is because the term 1 + 1 − η 2 is always greater than unity. Notably as, a = Q = 0, then ∆M ∼ 1/M , as it should for a macroscopic Schwarzschild black hole [38,41,44]. We would also like to point out that as the black hole approaches the classical extremal limit, i.e., M → (a 2 + Q 2 ), ∆M ∼ 0, remarkably the mass gap vanishes, leading to continuous quantum spectrum of emission, though at extremely small temperature. It turns out that continuous emission from rotating black hole in the extremal limit can also be derived from calculation of characteristic time scale of the emission of a quanta from black hole, see [46].
The last property is very unique for Kerr-Newman class of black holes and is absent in Schwarzschild black holes. The result clearly shows that near the extremal limit the quantum spectrum of a geometry discretized hole is almost continuous, unlike the Schwarzschild scenario. Thus the initial discreteness in the quantum spectrum of a Kerr-Newman black hole gradually makes its way to a continuous spectrum as the black hole approaches extremality (see also [46]). The above provides a curious generalization of Bekenstein-Mukhanov effect for rotating black hole, where the quantum spectrum can be almost continuous. This scenario has been demonstrated in Fig. 2. It is also possible to estimate how fast does the mass gap between the nearest black holes in the configuration space, change as the rotation parameter a or the charge parameter Q is changed by simply differentiating Eq.(13) with respect to either a or Q. It turns out that ∂(M ∆M min )/∂a as well as ∂(M ∆M min )/∂Q attain a large negative value as one approaches the extremal limit. Therefore, we realize that the minimum mass gap ∆M min quickly falls down as one approaches the extremal limit by using either a or Q (see Fig. 1). Thus the minimum mass gap derived above, brings out the characteristic signature of discretizing the underlying geometry. For a Schwarzschild black hole this bound coincides with the maxima of the thermal Hawking spectrum suggesting possible non-thermal nature which may have potential implication for the black hole information loss paradox [16,20,36,38,46,47]. While for black holes in the Kerr-Newman class it turns out that in the near extremal limit this mass gap can become as small as one likes and hence the emission spectrum of black hole will become quasi-continuous. We will now see how this discontinuity in emission fares vizá viz the thermodynamic character of the black hole.
Before concluding this discussion, let us briefly mention what happens to the mass gap in a more general scenario, in which both the mass and rotation parameter a change. This is because we normally expect a black hole to emit its multi-pole moments which apart from carrying energy will carry away some angular momentum as well. In this situation both area and rotation parameter change once the black hole makes a jump to the nearest allowed configuration. That immediately leads to the following most general expression for the mass gap, Here we have used Eq.(6) with minimum change in area and expressing r h in terms of the black hole parameters. As evident from the above relation, by setting ∆a = 0 we get back the original relation for the minimum mass gap derived in Eq.(12) (with Q = 0). This explicitly shows that the mass gap due  Figure 2: The above figure schematically illustrates how the quantum spectrum gradually becomes continuous as the black hole approaches the extremal limit. Here both the axes are drawn in some arbitrary units. The blue, dot-dashed curve at the top illustrates the case of Schwarzschild black hole (i.e., with a = Q = 0), from which it is clear that the spectrum will be largely discrete, as the minimum mass gap appears around the maxima of the distribution. While the red, dashed curve depicts the corresponding situation for a = Q = 0.35 M and the lowermost green curve illustrates the case with a = Q = 0.6 M respectively. As evident from the red and green curves, the region spanned by the quantum spectrum gradually increases as the (a/M ) and (Q/M ) increases. Ultimately when the black hole becomes near-extremal it will turn out that the quantum spectrum will become continuous.
to quantum nature of underlying geometry does not betray black hole thermodynamics. Hence our initial assumption that black holes are macroscopic, behaving as a thermodynamic object holds good even when the geometry is discretized (see also [41,43,49]). Also note that for nonzero ∆a the mass gap will be higher from the minimum value. In what follows we will concentrate on this characteristic minimum energy that a quantized (yet, macroscopic) black hole may emit and whether using this distinctive signature originating from the interface of quantum theory and gravity one can make any concrete astrophysical prediction.

Connecting micro and macro length scales
In order to understand the behaviour of the mass gap, let us now introduce two more length scales in the problem -(a) the thermal de Broglie wavelength (for massless particles) λ T ∼ (1/T ), signifying the scale set by the thermality of the black hole [59][60][61] and (b) the Compton wavelength λ c ∼ r h [42,43,49,62], where r h is the location of the event horizon, marking the size of the black hole. The smallest mass difference ∆M min corresponds to a minimum emission frequency and hence to a maximum emission wavelength λ max . Thus given Eq.(12) the maximum wavelength takes the following form, Here we have again gone back to the Kerr-Newman black hole. On the other hand, the thermal wavelength λ T and the Compton wavelength λ c for Kerr-Newman black hole take the following form, It is possible to express the largest emission wavelength λ max in terms of both λ T and λ c individually, i.e., Hence, we readily obtain that in the near-extremal limit, a 2 + Q 2 → M 2 , such that λ max ∼ λ T , but not as λ c i.e., for nearly extremal black holes the maximum wavelength (or, minimum frequency) scales with the thermal wavelength but not the Compton wavelength. Thus even though potentially a black hole is capable of housing both the thermal and Compton wavelength, the minimum frequency gets related to the thermal wavelength. Since the minimum emission frequency is intimately connected to the quasi-normal mode frequency of a black hole [63][64][65], the above observation remarkably suggests that in the near extremal limit, the minimum quasi-normal mode frequency of a black hole, has nothing to do with its size (which is ∼ M ) but with the thermal scale (which scales as ∼ M 2 − (a 2 + Q 2 ))! Moreover, interestingly, it is possible to combine all the three length scales in a convenient manner to obtain, This expression relates the three distinct length scales of a black hole through its mass. Note that in the limit of vanishing charge and rotation, the thermal and the Compton wavelength both scale proportional to the black hole mass, such that the maximum wavelength for Schwarzschild black hole becomes λ max,sch = (8π/β)4M . Therefore, using the maximum wavelength for Schwarzschild, one readily arrives at the following dimensionless ratio, Since the maximum wavelength λ max dictates whether the quantum spectrum of a black hole is dense or not (see Fig. 2 for a related discussion), the above relation is a measure of the denseness of the emission spectral profile of a Kerr black hole viz-a-viz its Schwarzschild counterpart. One can actually write down a constant, independent of the nature of black hole hairs except its mass, out of the above length scales. This constant may be dubbed as universal among all the black holes having identical mass, i.e., it is independent of the charge and/or rotation parameter of the black hole. This can be achieved by noting that λ c,sch = 2M and λ T,sch = 8πM , thus we obtain, The universality of the ratio (λ max /λ T λ c ) stems from the fact that it is independent of whether the black hole is Kerr-Newman or Schwarzschild! The numerical value of this ratio depends on the underlying microstates of the black hole and hence independent of its other classical hairs apart from mass. Further, this tri-ratio of infrared physics cooks up an ultra-violet invariant of the theory of the black holes. Since this gives the parameter strength of the underlying quantum theory of black holes, it is an observationally realizable infrared test on the models of quantum gravity, which, from observational perspective, is much more feasible. When compared to using sub-leading corrections to the area-entropy relation [54,66,67], this tri-ratio is much more favorable to decide in favor of a theory observationally. This shows that there may exist an astrophysical avenue to test the discretization scheme of black holes. Measurement of λ c for a black hole will be feasible in near future as the Einstein telescope starts operating [68] as it can determine the photon radius of the black hole to sufficient accuracy. While measurement of λ T and λ max is not going to happen for a real black hole in foreseeable future, but it is indeed possible for analogue systems. There one can simply check whether such a relation exists and whether it can directly tell us something useful about the underlying quantum structure without having to probe "quantum gravity".

The situation with Logarithmic spectrum
As we have pointed out in Section 2, the area of the black hole besides being discretized in integer steps, can also change under a different shift scheme, while remaining faithful to the entropy-area relation. This corresponds to the Logarithmic discretization of the area, i.e., A =B ln I, where I is a variable integer constructed using occupancy of various quantum levels of the macroscopic black hole andB is a constant of discretization. Note that this constantB connects a macroscopic geometrical object with the underlying microscopic structure of spacetime and hence is related in some way to the fundamental Planck scale. Though this shift scheme becomes hugely disfavoured if the area and the discretized variable are not related in a straightforward manner as depicted in [44] (see Appendix A and Appendix B as well as Section 6), we discuss the implicit features of this scheme for completeness (see also [?]). In this case, the minimum mass shift corresponds to a shift in the quantum levels of a black hole such that (I + 1) → I. The area for a Kerr-Newman black hole has been already written in Eq. (7), in terms of the black hole mass M , the rotation parameter a and the charge Q. This relation can be inverted to obtain the mass of the black hole as a function of the area of the horizon and the charge and rotation parameter, leading to, The area being Logarithmically discretized, using the above relation one can write down the mass associated with the set of quantum numbers {n j } which collectively corresponds to the integer I as, In this case as well, the change in mass will be effected by a change in the horizon area, as well as a change in the rotation parameter and charge. However when degenerate transitions, keeping the black hole area unchanged, are not taken into account the minimum mass gap will correspond to ∆a = 0 = ∆Q. This is again due to the fact that both ∆a and ∆Q would add up to the change in area and hence the shift in black hole mass would be higher. Thus, as in the previous scenario here also the minimum change in the black hole mass will be due to a shift in the quantum levels as the black hole makes a transition from the state I + 1 to another state I, resulting into, In order to arrive at the above relation we have used the following result: ln(1 + I) = ln I + ln(1 + 1/I), which for large values of I can be written as ln I + (1/I), i.e., we have assumed the black hole to be macroscopic. We have denoted the minimum mass change by ∆M min,log , to distinguish with the minimum mass gap ∆M min appearing in the earlier calculation pertaining integer shift. Note that for Schwarzschild spacetime (i.e., with a = 0 = Q) the minimum mass change corresponds to ∼ (1/M ) exp(−M 2 ). Thus unlike the integer shift, in this case the black hole spectrum will be practically continuous. Broadly speaking, this is due to the following fact, one can write the minimum separation between two macroscopic areas as ∆A ∼B/I. Thus for large values of I, i.e., macroscopically the separation ∆A ∼ 0. Hence if one requires -(a) continuous Hawking spectrum, (b) entropy to be proportional to the area, then the black hole must be Logarithmically discretized. The same consideration applies to the rotating black holes as well. In this case as evident from Eq.(12), ∆M min,log ∝ ∆M min exp(−A). Since the area of the black hole is still a macroscopic quantity, it immediately follows that there is a huge suppression of exp(−A) in the Logarithmically discretized spectrum when compared to the integer shift.
Further, defining η 2 = (a 2 + Q 2 )/M , the above minimum mass change can be written in the following suggestive form, M ∆M min,log =B 8π It is interesting to see a comparison between the minimum mass gap ∆M min,log defined using Logarithmic discretization and the minimum mass gap ∆M min associated with integer shift. First of all, the magnitude of ∆M min,log will be much less compared to ∆M min , due to the exponential suppression. Moreover, note that the quantity M ∆M min has no dependence on black hole mass, it merely depends on the parameter η. While M ∆M min,log has explicit dependence on M . Further since the term within the exponential is always positive, the minimum mass gap for the Logarithmic discretization will always be smaller compared to integer shift. The other distinct feature corresponds to existence of a maxima for M ∆M min,log for a give choice of η (as evident from Fig. 3). One can determine the same by setting ∂(M ∆M min,log )/∂η = 0, leading to, One can check that for this value of η 2 the second derivative term, i.e., ∂ 2 (M ∆M min,log )/∂η 2 will turn out to be negative (see Fig. 3). For example, if one sets the black hole mass such that B /8πM 2 = 1, then one would obtain: . This refers to the cross over between the contrasting Schwarzschild or Kerr-Newman family of black holes viz-á-viz temperature evolution during evaporation. Thus as η increases the minimum frequency also increases at first, ultimately reaching a maximum value (still small compared to the minimum energy of the quanta in the integer shift scheme) and then goes to zero as η → 1. Thus in both the discretization schemes the discrete quantum spectrum of a black hole becomes continuous as the extremal limit is being approached. At this stage it is worthwhile to ask whether there exist any such relation connecting microscopic and macroscopic length scales pertaining to this discretization scheme as well. To our surprise, it turns out that there is indeed such a relation, again independent of whether it is a Schwarzschild black hole or a Kerr-Newman black hole, when the black hole is discretized in a Logarithmic manner. Expressions for the thermal wavelength and the Compton wavelength associated with the Kerr-Newman black hole has already been provided in the previous section, in particular see Eq. (16) and Eq.(17) respectively. These wavelengths are independent of the discretization scheme. On the other hand, the minimum frequency or, equivalently the maximum wavelength associated with ∆M min,log corresponds to Adopting the procedure employed in Section 3.2 it immediately follows that one can construct the following The upper curve (blue, thick) represents the minimum mass gap for integer shift, while the below one (red, dashed) provides the minimum mass gap when the Logarithmic discretization scheme is being used. It is clear that as the extremal limit is approached, i.e., η → 1 both of then vanishes, while for η = 1 the minimum mass gap of the Logarithmically discretized black hole is always smaller compared to the integer discretized scenario. This is in complete agreement with our analytical results as well. Further, the graph in the inset provides a zoomed in version of the M ∆M min,log illustrating the existence of a maxima in conformity with our results.
At this stage as well the difference with Eq. (21) in the context of integer shift should be immediate. As the minimum frequency is suppressed by the exponential factor, the maximum wavelength will be enhanced by the exponential factor in comparison to the integer shift scheme. Thus taking a cue from Section 3.2, in this case as well one can have the following relation independent of the nature of the black hole, Here also the left hand side corresponds to the expressions for wavelengths in the Kerr-Newman background, while those on the right hand side depicts identical quantities for a Schwarzschild background. This suggests, that the quantity on the left hand side depends only on the black hole mass, irrespective of its other hairs. Hence the above tri-ratio will act as a universal feature for all black holes having identical mass in this discretization scheme.
Having described both the discretization schemes in some detail and the relations among the relevant scales, as well as their compatibility with macroscopic thermodynamical laws, we will now concentrate on the possible implications and generalizations of these results. These will involve an estimate of the lifetime of a black hole as well as the maximum number of quanta that a black hole can emit. We will also explore the extremal limit of a black hole in light of the discretization scheme. Finally we will try to explore other alternative scenarios, e.g., the changes brought about by alternative gravity theories as well as by higher dimensions.

How long can a astrophysical black hole live?
We have obtained the expression for the smallest frequency to be emitted from a black hole as long as the mass of the black hole is large compared to the Planck mass. Therefore, in principle, we can follow the footprints of black hole emission obtained in earlier sections, till the time it turns into a Planck mass object, where the approximation of large mass, would no longer remain applicable. Therefore, through these calculations, we could in principle evaluate how many quanta a black hole could radiate before it settles into such a Planck mass system. Since we are interested in astrophysical black holes, the presence of the electric charge can be safely neglected and hence we will only concentrate on Kerr black hole in what follows.

Number of emitted quanta
In order to obtain an upper bound on the number of emitted quanta, we will assume that throughout its evaporation the mass change brought about in the hole is only through the minimum mass change at each step. Thus difference between the initial I i (which signifies the starting hole mass) and the final Introducing the final black hole mass a fraction of the initial mass M f = µM i , we obtain the (largest) number of emitted quanta for the process M i → M f , for the integer discretized black hole as, In general if a macroscopic black hole emits thermal radiation as prescribed by semiclassical physics [7,69,70], then the quanta emission for a Kerr black hole will go on till it either becomes extremal or Planck sized. At the extremal limit, the temperature of the black hole becomes vanishingly small and hence the minimum energy will also go to zero (see Fig. 1). In such a scenario, if the final state of the hole turns out to be extremal, then, η f = 1, hence the number of quanta emitted would be given by, With the end state being an extremal black hole, the maximum number of emitted quanta would correspond to the situation when besides being extremal the final black hole also turns out to be Planck sized, such that µ → 0, in which case the maximum number of quanta would be, While the minimum number of emitted quanta for the extremal case would correspond to the case µ ∼ 1, i.e., the black hole becomes extremal very quickly much before reaching the Planck size and hence, Further it will be useful to ensure that, the black hole, by emission of these quanta, is indeed going towards extremality. This can be done by imposing the condition that the rotation parameter and the black hole mass M are changing in such a manner that η f > η i . For example, if a 2 f > a 2 i , while M f < M i then the above criteria will be trivially satisfied. Also, note that for η f > η i , it follows that (1 − η 2 f ) < (1 − η 2 i ) and hence, we arrive at, the inequality, (I f /I i ) < µ 2 . Since the black hole is supposed to decay to lower and lower I values, the above condition ensures that it will certainly be driven to extremality.
The corresponding computation of the number of emitted quanta will follow an identical pathway in the context of Logarithmic discretization as well. The only change will be, in this case BI will be replaced byB ln I, such that the number of emitted quanta will be where N if corresponds to the number of emitted quanta in the case of integer shift. Thus it is clear that (and expected as well) in the case of a black hole discretized in a Logarithmic fashion the number of emitted The upper curve (red, thick) represents the behaviour of the time scale for integer shift, while the below one (blue, dashed) provides the associated time scale when the Logarithmic discretization scheme is being used. It is clear that as the extremal limit is approached, i.e., η → 1 both of them diverges, as anticipated. while for η = 1 the time scale associated with the Logarithmically discretized black hole is always smaller compared to the integer discretized scenario. This is in complete agreement with our analytical results. quanta will be exponentially large for a given change in its macroscopic parameters. Hence in contrast to the integer discretized scenario an exponentially large number of quanta will be emitted from a Logarithmically discretized black hole (see Fig. 4).
Thus given the initial mass and the final mass, along with the charge and rotation parameters it is possible to obtain the maximum number of quanta a black hole could emit. Depending upon the spin S of the emitted quanta, we can associate the dimensionality (2S + 1) N to the emitted radiation's Hilbert space quantum theoretically. Since the characteristic emission time for a quanta is related to the frequency it is emitted at, we can calculate, dynamically how long a black hole will last before it emits the quanta to turn into a Planck star or a extremal remnant. We can visualize the black hole emissions as jumps within macroscopically distinguishable configurations which must be microscopically orthogonal too. In that case we can also find out the fastest time in which a black hole can vanish [71]. This will also give the fastest route for a black hole to emit all its information. This exercise we leave for a future computation.

Lifetime of a black hole
Another very much observable aspect of this analysis is the computation the maximum possible lifetime of a black hole. As in the previous section, here also we will assume that all the emitted quanta have the minimum energy ω min , so that one can introduce a characteristic time scale τ , identical for all the emitted quanta. A natural way to determine the time scale τ is through the following equation for the mass loss rate [41] At this stage one assumes some form for the mass loss rate and hence determine τ . Subsequently this time scale is coupled to the number of emitted quanta, leading to an estimation of the lifetime of a black hole.
Since we are considering only the minimum energy quanta, it immediately follows that the time period computed here will provide an upper bound to the lifetime of a black hole.
To proceed further it is natural to assume that the above mass loss is due to thermal emission. From which it is possible to determine the mass loss rate in terms of the area and the temperature of a astrophysical Kerr black hole as where we have used expressions for the horizon area A and the black hole temperature T in term of the black hole parameters (M, a). Note that the minus sign in front of Eq.(39) ensures that the mass decreases with time. Finally equating Eq.(39) with the right hand side of Eq.(38), we obtain the characteristic time scale τ of emission of a minimum energy quanta to be, Note that in the case of Schwarzschild black hole (i.e., with a = 0) the time scale becomes proportional to M [41], while in the extremal limit, obtained by taking a 2 → M 2 , τ diverges. This suggests that in the near extremal region it takes a large time to emit the minimum energy quanta. This is expected as in the extremal limit the temperature of the black hole vanishes. The total lifetime of a black hole can be obtained by multiplying the average time scale τ with the total number of emitted quanta. In principle it should be obtained by integrating over the total time period of the black hole and since τ diverges, it takes an infinite time to reach the extremal limit. However as a crude estimate we can take the number of emitted quanta to be ∼ M 2 and if the black hole is far from being extremal, then total lifetime of a black hole will be ∼ M 3 , which is certainly large. The above computation was for a integerly evolving black hole, while for a Logarithmic discretization scheme the minimum frequency will be different and hence the associated time scale will become where τ is defined in Eq. (40). Note that the associated time scale is exponentially small compared to the corresponding integer discretized scenario. Still the divergence of τ log in the extremal limit remains due to the τ factor in its front (see Fig. 5). Thus in this case as well the total time to reach the extremal limit will be infinite. However if we consider the final state to be far from extremality, the it will follow that, the total lifetime of a black hole to be ∼ M 3 exp(M 2 ), which again is exponentially large compared to the integer spacing scenario. The feasibility of either of these schemes for an isolated primordial black hole survivability is an interesting point to dwell upon, which we defer for future.

Extremal limit of a black hole
In this section we would like to show that a few results derived in the context of extremal limit of a macroscopic black hole also hold even when the black hole is assumed to have discrete structure. In the context of macroscopic physics it has been demonstrated that, if one throws a charged particle inside a Kerr (or, Kerr-Newman) black hole then one can gradually arrive at an extremal black hole. We would like to show that the same is true for our case as well, i.e., if we consider the black hole characterized by mass M and angular momentum J ≡ M a to transform into another black hole with parameters (M ,J) by jumping down the discrete geometry spectrum, it actually evolves to an extremal configuration. We further assume that the rotating black hole emits an quanta with the minimum energy ω min with respect to the asymptotic observers, such that its mass change of the black hole corresponds to ∆M = ω min and the change in angular momentum being ∆J = (J/M )∆M , such that δa ≡ δ(J/M ) = 0. Then one can construct the following dimensionless quantity, where, T old corresponds to a rotating black hole with parameters (M, J), while T new relates to the black hole with (M − ∆M, J − ∆J). Assuming ∆M M , we have the dimensionless ratio Θ taking the following form, Thus note that for ξ ≥ 0.79 (which is a solution of the algebraic equation ξ 4 + ξ 2 − 1 = 0), the dimensionless quantity Θ < 0 [72]. Hence as the black hole emits the minimum quanta the ξ gradually increases and one will eventually end up with a higher negative value of the Θ parameter. This suggests that one arrives at the extremal limit as the black hole loses mass gradually by jumping into lower quantum states. An interesting fact with the above expression is that for ξ < 0.79, the temperature associated with the new black hole is actually greater than the temperature of the old black hole, which is the signal of domination of Schwarzschild character again. Let us now briefly mention how to connect this up with the physical process of lowering a charged particle gradually into a Kerr black hole. If one considers a situation where a particle of mass µ and charge q is being dropped slowly into a Kerr black hole. We will assume that this will make the black hole to absorb n quanta, such that it jumps from initial configuration I → I + n while the angular momentum J is unaltered. Thus the original black hole with mass M and rotation parameter a = J/M becomes another black hole with mass M + E and rotation parameter J/(M + E). Here E corresponds to the energy contributed by the charged object. Then from Eq.(7) we obtain, The above expression must remain finite in the extremal limit as well, since we would like to impose the condition that if the initial black hole is extremal then the final black hole should also remain extremal at most (i.e., does not become a naked singularity). Thus the requirement of divergent terms in the above expression in the extremal limit forces the energy associated with the charged particle to be Incidentally, this exactly coincides with the universal minimum energy that can be supplied by a charged body and is also the energy supplied in extremal situation by a charged particle [72]. This result was derived earlier using the classical black hole picture, while here we have merely computed the changed in the energy levels due to lowering of this charged particle in the black hole. Note that if the particle had no charge, i.e., if q = 0 then the above expression will require E to identically vanish. Further using the above expression for E, in the extremal limit we obtain, Thus remarkably, the area does not change when the energy carried by the charged body coincides with q 2 /4M , the universal minimum energy that a charged object can supply. Hence as the black hole approaches to the extremal limit, the classical universal features associated with them are borne out by our quantum schemes as well.

Generalization to Higher Dimensional Black Holes
So far, our discussion was concentrated on Kerr-Newman solution in four spacetime dimensions alone. In this setting we have observed that in the extremal limit the black hole emission spectra becomes almost continuous. We can similarly ask if the characteristic features of a rotating black hole are retained in higher dimensions. Let us start by considering Kerr black hole in higher dimensions, characterized with a single rotation parameter a [73][74][75][76][77]. In D spacetime dimensions the horizon of such a rotating black hole is located at r = r h , where r h is a solution of the following algebraic equation, In general obtaining a solution for r h for arbitrary D is difficult, so we will consider only the case D = 5. For a five dimensional spacetime the above algebraic equation can be readily solved resulting into the following horizon location: r h = √ M − a 2 . Note that in this case the extremal limit is defined as M → a 2 . The area of the event horizon for the rotating black hole in five dimensions can be easily computed, resulting into A = 2π 2 r h (r 2 h + a 2 ). Thus if the black hole is discretized in integer steps (see Section 2) then the following relation can be obtained, We can compute the minimum mass change for the higher dimensional solution following exactly an identical procedure as in Section 3. In particular we keep the rotation parameter a fixed and then consider the mass change as the integer I drops to (I − 1). This results into the following expression for minimum mass change, Thus again the mass change vanishes in the extremal limit, which corresponds to M = a 2 . Thus our conclusion remains unchanged as we consider a five dimensional rotating solution. One can trivially verify that the same conclusion holds for Logarithmic discretization scheme as well.
For completeness let us also consider a charged black hole in five dimensions. One can immediately compute the location of the event horizon by setting g rr = 0. This in turn implies that the horizon location can be obtained by solving the following algebraic equation, r 4 − 2M r 2 + Q 2 = 0. The corresponding solution for the event horizon becomes Here M → Q corresponds to the extremal limit. In this context the horizon area is being given by A = 2π 2 r 3 h , such that for integer discretization scheme, the above area-mass relation reduces to, One can in principle invert the above relation and obtain the mass in terms of the integer I. Then again considering the transition of the black hole from I to I − 1, the minimum mass separation can be immediately obtained as, In this case as well, note that in the extremal limit, i.e., as M → Q, one essentially obtains a vanishing contribution to ∆M . Thus in the case of a higher dimensional charged black hole, in the extremal limit, the quantum spectrum becomes dense again. An identical feature will hold for Logarithmically discretized black hole spectrum as well. Thus one can expect that in higher dimensions too, in the extremal limit, all the black holes in the Kerr-Newman class do produce continuous quantum spectrum, visually contrasting them with the Schwarzschild case.

Conclusion
In this work, we have studied the emission pattern of a classical black hole which is fundamentally a macroscopic realization of an underlying quantum gravity theory, where some geometric feature of spacetime gets discretized. We have obtained the spacing between distinguishable macroscopic configurations such a hole is allowed to have which will reveal the smallest energy jump between the allowed macro-configuration a black hole could undertake. We obtain the change in the mass of the hole through entropy and subsequent change in the horizon area, if the macroscopic black hole obeys the classical laws. This strategy revealed many interesting features when applied to Kerr-Newman class of black holes. The smallest frequency of emission gets related to the temperature of the hole through the de-Broglie thermal wavelength associated with the emission profile. This is unlike a normal blackbody where the minimum frequency of emission is determined by the size of the blackbody. Further, a tri-ratio compiled from infrared features was obtained for all black holes, which reveal the parameter of the quantum theory which will be dictating the microphysics. The computations have many significant implications involving black hole information release, life time, scrambling time etc. Below we provide the key findings of this work: • Smallest frequency of emission: The emission profile of the black hole admits a smallest mass (or frequency) gap depending upon two distinct ways in which the macroscopic area can change (by macroscopic we mean anything with a size appreciably large than set by the Planck scale). The area of such a large black hole can either change in integer steps or in logarithm of some integer. This comes just from the micro canonical counting of the entropy and is independent of any quantum modeling whatsoever. If the area changes by integer steps, the minimum frequency a black hole can emit, while jumping to nearest allowed configuration, scales inversely proportional to the hole's mass for the Schwarzschild case. Therefore, the collection of black hole radiation looks vastly discrete given the fact that the semi-classical radiation is expected to be thermal with the temperature inversely related to the mass. Similarly, for the Kerr-Newman black holes too, the minimum frequency gap between the nearest allowed configuration is set by the temperature the black hole is at.
On the other hand, if the black hole area scales as Logarithm of some integer, the minimum frequency becomes exponentially small and the spectrum appears rather dense. This will be the regime, like a classical system where number of allowed configurations are accumulated close by.
• Dense spectrum and extremal limit: For Kerr-Newman black hole it is possible to consider the extremal limit. In this case the minimum mass gap becomes vanishing. Thus unlike the Schwarzschild scenario in this case the quantum spectrum can become dense, as the rotation parameter and/or charge of the black hole increases. The situation has been explained in Fig. 2. For Logarithmic discretization it immediately follows that the spectrum becomes denser as the extremal limit is approached.
• Extremal is stable: In the case of a rotating (or charged) black hole the smallest quanta of emission gets related to the horizon temperature in a linear fashion. Thus, as the extremality parameter rises, the temperature drops down and the smallest frequency of emission also goes down, giving the emission spectra a dense profile. In fact, one can write down a relation between various length scales in the emission i.e, the thermal de Broglie wavelength, the Compton wavelength and the largest wavelength of emission, depending upon whether the horizon area changes as integer or the logarithm of an integer. However, once the extremality parameter becomes unity and the black hole turns extremal, the temperature vanishes and the black hole area can no longer change and hence it can only emit quanta through degenerate transitions. Of course, these studies are independent of any particular model and we can only estimate the "in principle" smallest frequency. An exact theory will presumably tell how large from this smallest value of mass gap the change in black hole's parameters will be. Further, in the extremal limit, it is possible to relate the discretization scheme of black holes to quantization of charge, as a charged particle is lowered into the black hole. This favors the integer discretization scenario.
• Connecting micro-physics to macro-physics: Once the black hole turns macroscopic, much of the information of its inherent quantum nature gets washed out. For example, its entropy, mass, maximum number of quanta it could emit would not change. However there are also quantities which know about the effect the quantum theory brought into the macroscopic parameters of the hole. We identified three such quantities: mass gap, time scale and a relation among the length scales. The last one among these is of significant interest. It relates (within a class of discretization) three length scales associated with a black hole in a universal fashion. Therefore observation of these associated quantities presumably will reveal the parameter (e.g. Immirizi parameter [55], string tension [78], non commutativity parameter of space-time [79] etc.) associated with the black holes' discretization.
• A road to Planck scale remnant: Based upon our estimates of the minimum frequency of emission, we can calculate maximum possible number of quanta a black hole can emit before it turns extremal and/or becomes Planck sized. One can determine the number of quanta starting from the initial configuration, upto the time where the classical/semiclassical approximation breaks down. Therefore, one can also estimate the maximum possible information content, available with the radiation, once the evaporation has ended. The total information content in emission will of course be based upon the dimensionality of the Hilbert space of total radiation, an exercise we leave for future. With such emission profile one can also obtain a characteristic time scale of the black hole, which tells us about the time phase corresponding to a particular configuration a black hole adopts in course of emitting radiation. These studies may potentially provide insights regarding at least two crucial issues -(a) a qualitative estimate of the amount of information a black hole is entitled to emit with its correspondence with information loss paradox and (b) an estimate for the lifetime of a black hole, i.e., survivability of a black hole.
• Correspondence with semi-classical physics: Whatever be the underlying quantum scenario, it presumably should reproduce the semi-classical physics in a natural way. We have shown that the minimum mass gap of a black hole will still be tallying with the thermodynamical laws of black hole, explicitly depicting how the semi-classical physics can be recovered. Similarly we have shown that the semiclassical bound ensuring extremality can be reached. Moreover we have also demonstrated that if a charge particle is thrown into a rotating extremal black hole, the minimum energy the charge particle must have to result into a final state extremal Kerr-Newman black hole coincides with semi-classical expectations. Thus our analysis is in close correspondence with semi-classical physics.
The results obtained here open up many frontiers of analysis, some of which are already discussed. We will like to pursue these aspects elsewhere. As the most useful insight obtained from this work, we would like to stress here is the infrared region of the black hole emission may be much more richer in terms of revealing the ultraviolet physics a microscopic theory is built up of. Further research in this direction is indeed warranted.
can be written as, where the results from [44] has been used. So far we have based our analysis on the leading order entropyarea relation. However in view of the above correction it is legitimate to ask the effect of the Logarithmic correction of area on the discrete quantum spectrum. We will show that such corrections terms will restrain the discretization scheme significantly and will also lead to more sparse emission spectrum. The number of microstates associated with the expression for entropy as presented in Eq.(57) can be expressed as Therefore, it immediately follows from the above expression that, where we have absorbed the constants σ and B/4l 2 p σ γ in the redefinition of F γ and in defining a new parameterB. Clearly for g(n) to be an integer with p, q ∈ Z + , one requiresB = ln I 0 and F γ = I to be the only possibility, where I and I 0 are integers. Therefore the discussion with the integer step area variation as presented in Section 3 will dictate the emission profile. For the case p < 0 one can obtain a superset of solution space from g(n) q = F −|p|γ expBF γ , again leading to the same result as above, though with the use of a further condition: I = nI 0 for some integer n. Thus quantum gravity corrections in micro-canonical counting scheme will also lead to a largely discretize the black hole emission spectra.