Hawking emission from quantum gravity black holes
aa r X i v : . [ h e p - ph ] N ov Prepared for submission to JHEP
Hawking emission from quantum gravity black holes
Piero Nicolini a,b
Elizabeth Winstanley c a Frankfurt Institute for Advanced Studies (FIAS), Frankfurt am Main, Germany b Institut f¨ur Theoretische Physik, J. W. Goethe-Universit¨at, Frankfurt am Main, Germany c School of Mathematics and Statistics, The University of Sheffield, Hicks Building, HounsfieldRoad, Sheffield. S3 7RH United Kingdom
E-mail: [email protected] , [email protected]
Abstract:
We address the issue of modelling quantum gravity effects in the evaporationof higher dimensional black holes in order to go beyond the usual semi-classical approxi-mation. After reviewing the existing six families of quantum gravity corrected black holegeometries, we focus our work on non-commutative geometry inspired black holes, whichencode model independent characteristics, are unaffected by the quantum back reactionand have an analytical form compact enough for numerical simulations. We consider thehigher dimensional, spherically symmetric case and we proceed with a complete analysisof the brane/bulk emission for scalar fields. The key feature which makes the evaporationof non-commutative black holes so peculiar is the possibility of having a maximum tem-perature. Contrary to what happens with classical Schwarzschild black holes, the emissionis dominated by low frequency field modes on the brane. This is a distinctive and poten-tially testable signature which might disclose further features about the nature of quantumgravity. ontents
The possibility for a black hole to emit thermal radiation like a black body, often calledblack hole evaporation, is the first and maybe one of the best-known results of the com-bination of quantum field theory with general relativity. Black hole evaporation is a topicof primary importance in fundamental physics, since it affects many research areas, span-ning thermodynamics, relativity and particle physics. In addition, black hole evaporationrepresents the first convincing insight into a possible theory of quantum gravity. However,despite the fact that the original derivation due to Hawking is dated back to 1975 [1] wedo not yet have direct evidence about the actual observation of this phenomenon. As-trophysical black holes behave like classical objects due to their large mass. On the otherhand, for microscopic black holes the evaporation is expected to be relevant. For black holemasses around M ∼ − kg, we have temperatures about T ∼ K and horizon radiiabout r h ∼ − m. These are typical parameters of primordial black holes, black holesthat might have formed due to the high density fluctuations of the early universe. Beingextremely bright, their detection is expected at the Fermi Gamma-ray Space Telescope [2].For even smaller sized black holes, we fully enter the regime of particle physics and we needan increased degree of compression of matter to create a mini black hole. According tothe hoop conjecture, a “particle black hole” would form if its Compton wavelength equalsthe corresponding horizon radius [3, 4] (see figure 1). This implies that mini black holesmust have masses of the order of the Planck mass, M ∼ M P , and radii of the order of thePlanck length, r h ∼ L P , a fact that creates formidable problems [5, 6]: on the experimentalside the Planck scale is about 15 orders of magnitude higher than the scale of current highenergy physics experiments, while on the theoretical side we do not have yet a full for-mulation of quantum gravity which is suitable for efficiently describing evaporating blackholes. This puzzling situation has no concrete ways out unless we make further hypothe-ses. If the space-time is endowed with additional spatial dimensions, it is possible to lower– 1 – igure 1 . Particle Compton wavelengths (dotted curve) and horizon radii (solid line) as a functionof the mass in Planck units. The intersection of the two curves corresponds to the formation of amini black hole. the fundamental scale of quantum gravity to an energy scale accessible to current particlephysics experiments, namely M ⋆ ∼ − fermi [10], i.e. the typical lengthscales under scrutiny at the LHC [11]. This fascinating opportunity has led to intensiveresearch activity whose main results can be found in various reviews, see, for example,[12–20].Despite these efforts and the large number of papers published in the field, the the-oretical scenario is still uncertain. For instance, the lower quantum-gravitational energyscale requires higher dimensional metrics that in the case of charged rotating black holesare not known analytically [21]. Even when we have analytic space-time geometries fordescribing some phases of the life of a microscopic black hole, the master equations for thepropagation of matter fields can be integrated only via accurate numerical methods (see,for example, [22–26]). Finally, we ignore the Planck phase, namely the fate of the blackhole in the terminal phase of the evaporation, when its temperature equals the fundamentalscale T ∼ M ∼ M ⋆ . We recall that evaporating black holes are conventionally described interms of semi-classical gravity, which is valid only if the black hole metric is not modifiedby the emitted particles, i.e. if T ≪ M .In the absence of a viable description of the Planck phase by some quantum theory ofgravity, there have been several attempts to incorporate one or more features we expectfrom quantum gravity in the formalism of the evaporation by means of effective theories.According to the formalism adopted, one has to deal with features like asymptotic freedom,non-commutative character, minimal area, minimal length, or non-locality, but in the end– 2 –he crucial aspect in all cases is the possibility for the space-time to undergo a transitionfrom a smooth differentiable manifold to a fractal surface, plagued by quantum uncertaintyand the loss of resolution [27–36]. As a result, the corresponding quantum gravity modifiedblack hole metrics have, in most cases, an equivalent qualitative behavior: the prevalentscenario is that of short-scale regularized metrics and the possibility of horizon extrem-ization even for the neutral, static case, with a consequent cooling phase towards a zerotemperature remnant as the mass approaches M ∼ M ⋆ [37–41]. There is an additionaladvantage: since these quantum gravity black holes are significantly colder than the cor-responding classical black holes, throughout the evaporation their metrics are not affectedby a significant back-reaction. The metric modifications are already taken into account bythe quantum-gravity corrections to the usual background geometries. As a consequence,one can safely use quantum field theory in curved space to study the evaporation of theseblack holes, without a breakdown of the formalism.Given this background, it is imperative to study the evaporation of quantum gravityblack holes by performing a detailed analysis of the brane/bulk emission, including thegrey-body factors. This would be the first step in the quest for consistent signatures ofevaporating black holes which are a requirement for starting any study of quantum gravityphenomenology.The outline of this paper is as follows. In section 2 we briefly review the existing sixfamilies of quantum gravity corrected black hole geometries, before focussing our attentionon neutral static, spherically symmetric non-commutative black holes. We outline the keyfeatures of the metric and temperature of these black holes, in particular showing thatthe back-reaction of quantum fields on these geometries is negligibly small, both for blackholes potentially created at the LHC and in cosmic rays. Hawking radiation of scalarparticles, both on the brane and in the bulk, is studied in detail in section 3. As well as theconventional fluxes of particles and energy, we also introduce an emission spectrum modifiedby non-commutative effects, although the latter does not give significantly different results.We present our conclusions in section 4. For sake of clarity, we shall classify the quantum gravity corrected metrics currently avail-able in the literature into six families, according to the mechanisms used to obtain themodifications with respect to classical space-times. They include non-local gravity blackholes [41, 42], non-commutative geometry inspired black holes (NCBHs) [43–56], general-ized uncertainty principle black holes [57–60], loop quantum black holes (LQBHs) [61–66],asymptotically safe gravity black holes (ASGBHs) [67–71] and a generic category of shortscale modified metrics [72–78] (for a review of earlier contributions see [79]).As a first paper in the area, we start our analysis from the most simple case, namelythe neutral, spherically symmetric static black hole, postponing the study of axisymmetricgeometries to the future. As a second point, we will not study all the existing geometriesmentioned above, but just the case of NCBHs. This choice is motivated by the followingreasons. NCBHs are the richest family of quantum gravity improved black hole space-times.– 3 –here exist higher-dimensional static [80], charged [81], rotating [52] and charged rotating[82] NCBHs, the latter only for low angular momenta as is the case for classical blackholes. Therefore NCBHs are the only ones that can currently provide a complete scenarioand it is worth starting from them in view of future investigations. In addition, NCBHshave been found to be a sub-class of non-local gravity modified space-times [41, 42]. Asa consequence, NCBHs encode features that are common to more than one formulationand might lead to model-independent results. As a side motivation, the analytic form ofNCBHs is compact enough to implement into numerical simulations, as has already beendone in previous contributions [83] (without studying the details of the Hawking emission),including in the development of a Monte Carlo event generator [84].We now proceed by recalling some basic facts about NCBHs. Non-commutative ge-ometry is an old idea, concerning the possibility that co-ordinate operators might fail tocommute in some extreme energy limit [85]. This opened up a vast research area, with ahigh degree of mathematical sophistication (for an incomplete list of reviews on the topicsee [86–89]). Despite the huge literature in the field, a formulation of the non-commutativeequivalent of general relativity is still missing. The best one can do is to consider theaverage effect of non-commutative fluctuations and study the consequences for the gravityfield equations. As a result, one can incorporate the presence of non-commutative effectsby a non-standard energy-momentum tensor, while keeping the Einstein tensor formallyunchanged. It turns out that for the specific case of a static spherically symmetric source,the usual point-like profile is no longer physically meaningful and must be replaced by aGaussian distribution T ( ~x ) = − M δ ( ~x ) → M (4 πθ ) n +32 e − ~x / θ , (2.1)where n is the number of extra dimensions and θ the non-commutative parameter withdimensions of a length squared, that encodes a minimal length in the manifold. This is akey result which has been derived both within non-commutative geometry [54, 90, 91] andnon-local gravity [41]. Covariant conservation and the additional condition g = − /g completely specify the energy momentum tensor, which then generates the solution ds = − h ( r ) dt + h ( r ) − dr + r d Ω n +2 (2.2)with h ( r ) = 1 − r n +1 (cid:18) M ⋆ √ π (cid:19) n +1 (cid:18) MM ⋆ (cid:19) γ (cid:16) n +32 , r θ (cid:17) n + 2 , (2.3)where d Ω n +2 is the metric of the ( n + 2) dimensional unit sphere d Ω n +2 = dϑ n +1 + sin ϑ n +1 (cid:0) dϑ n + sin ϑ n (cid:0) · · · + sin ϑ ( dϑ + sin ϑ dϕ ) . . . (cid:1)(cid:1) (2.4)and γ (cid:18) n + 32 , r θ (cid:19) = Z r / θ dt t n +12 e − t (2.5)– 4 –s the incomplete Euler gamma function. In the above, the angles are defined as 0 < ϕ < π and 0 < ϑ i < π , for i = 1 , . . . , n + 1, while the minimal length √ θ is not set a priori .However it is reasonable to have √ θ ∼ M − ⋆ ∼ − fermi, where the fundamental scale ofquantum gravity is M ⋆ ∼ (cid:18) L P R (cid:19) nn +2 M P ∼ , (2.6)with R the size of the extra dimensions.The above line element (2.2) approaches the usual higher dimensional Schwarzschildsolution for large radii, namely r ≫ √ θ , where we expect quantum gravity corrections tobe negligible. Conversely, for small radii r . √ θ , the line element (2.2) approaches a localde Sitter core h ( r ) ≈ − − n ( n + 2)( n + 3) (cid:18) M ⋆ √ π (cid:19) n +1 (cid:18) MM ⋆ (cid:19) (cid:18) √ θ (cid:19) n +3 r . (2.7)This is the signature of the regularity of the manifold at short scales. The de Sitter core isnothing but an effective geometry which accounts for the mean value of quantum gravityfluctuations and prevents the energy profile from collapsing into a Dirac delta, by meansof a locally repulsive gravitational effect. The gamma function (2.5) provides the smoothtransition between the classical geometry at large radii and the effective quantum geometryat small radii. Figure 2 . Metric function h ( r ) (2.3) for various eleven-dimensional NCBH solutions (solid curves),illustrating the possible horizon structures. The dashed curve shows the same function for a higher-dimensional Schwarzschild black hole for comparison. Further features of the line element (2.2) emerge by studying the horizon equation h ( r h ) = 0, a parametric equation depending on the mass parameter M which is the integral– 5 –f the energy density: M = 2 π n +32 Γ (cid:0) n +32 (cid:1) Z ∞ dr r n +2 T ( r ) . (2.8)There exists a threshold value M for M which lets us distinguish three cases:1. for M > M there exist two horizons, an inner Cauchy horizon r − and an outer eventhorizon r h ;2. for M < M there is no solution for h ( r h ) = 0 and no horizon occurs;3. for M = M the two horizons coalesce into a single degenerate event horizon r .These three possibilities are illustrated for eleven-dimensional black holes in figure 2. Thevalue of M depends on n and can be determined by numerical estimates [40, 80]. Theexistence of an inner Cauchy horizon for M > M opens the potential problem of theclassical instability of the solution. This is a feature that appears also in the case of LQBHs[92, 93] and has been investigated for NCBHs with controversial results [94, 95]. Even ifa Cauchy horizon is certainly a surface of infinite blue shift where classically unboundedcurvatures might develop, at a quantum level one may think that the same mechanism usedto cure the curvature singularity might be invoked to tame divergent frequency modes inthe vicinity of r − . In any case, we can for now circumvent this problem, as in the case ofclassical Reissner-Nordstr¨om or Kerr geometries, by saying that the potential instabilitywould not become manifest within typical evaporation time scales, which have been provento be extremely short [83].The no-horizon case corresponds to a manifold which is regular everywhere, an addi-tional gravitational object, within a plethora of non-perturbative gravitational objects, thatmight be produced in super-Planckian collisions [12]. In this class of no-horizon objectswe have to consider also the case of spherically symmetric solutions that can be obtainedby flipping the sign of the radial coordinate r → − r . Since the space-time is locally flatat the origin, the solution obtained by the r → − r map turns out to be geodesically com-plete. Therefore negative r solutions are not merely analytic continuations of positive r space-times, but genuinely new geometries [96]. The parity of the gamma function in (2.3)implies that only for even n we find distinct geometries by this procedure, which can beconsidered as geometries with positive r and negative mass parameter M (for more detailsabout these geometries see [97]). Finally, the last case, M = M , can be fully understoodby studying the thermodynamic properties of the solutions since it is intimately related tothe final configuration of the black hole at the end of the evaporation.The black hole temperature is given by T = n + 14 πr h − n + 1 (cid:18) r h √ θ (cid:19) n +3 e − r h / θ γ (cid:16) n +32 , r h θ (cid:17) . (2.9)We see that at large radii we recover the usual result T ∼ ( n + 1) / πr h . However, at r ∼ √ θ , quantum gravity corrections start to be dominant. As a consequence, in place of– 6 – igure 3 . The temperature T (2.9) of NCBHs as a function of event horizon radius r h , for variousvalues of n , the number of extra dimensions, in units in which M ⋆ = √ θ = 1. For n > n = 0 energies are measured in units of 10 TeV. The different units are used for the n = 0 case to facilitate comparison with the n > n for n >
0, while the cooling phase leads to a smallerremnant for higher n . the usual divergent behavior for the temperature at small radii, there is a value at whichthe temperature vanishes. If we consider the internal energy of the system, by defining M ≡ U ( r h ) as an implicit function of r h through the horizon equation h ( r h ) = 0, we canshow that it admits a minimum M = U ( r ) dU ( r h ) dr h = 18 ( n + 1)( n + 2) π n +12 r nh M n +2 ⋆ γ (cid:16) n +32 , r h θ (cid:17) − n + 1 (cid:18) r h √ θ (cid:19) n +3 e − r h / θ γ (cid:16) n +32 , r h θ (cid:17) (2.10)for the same value of r at which the temperature vanishes (for more details see [77]).This implies that the extremal black hole case M = M is actually a zero temperatureconfiguration. As the temperature is asymptotically vanishing, there should be a maximumtemperature for some r h > r , a fact that will have implications for the computation of theHawking emission. In conclusion, the temperature follows the usual curve at large radii,but as the black hole shrinks towards distances comparable with √ θ , it reaches a maximumtemperature before cooling down towards a zero temperature extremal black hole remnantconfiguration (see figure 3).At the maximum temperature, the system undergoes a phase transition from a locallyunstable configuration with negative heat capacity, C <
C >
0. The thermodynamic stability in the final phaseof the evaporation is a feature that appears also in LQBHs and in ASGBHs. It has beenargued that this is a general property of quantum gravity [98]. As a result, our analysis– 7 – igure 4 . The ratio
T /T S as a function of r h , in units in which M ⋆ = √ θ = 1, for various values of n . Here T is the temperature of an NCBH given by (2.9), and T S is the temperature of a classicalSchwarzschild black hole having the same mass. As n increases, the value of r h for which thetemperature is zero decreases, and the ratio T /T S for large r h also decreases. of Hawking emission could capture general features of the evaporation beyond the presentcase of NCBHs. From figure 4 we see that quantum gravity effects become important ina region within ∼ √ θ from the origin, but are negligible for larger black holes. It isalso clear from figure 4 that the temperature of an NCBH is considerably lower than thatof a Schwarzschild black hole having the same mass. This will turn out to be the mostimportant feature of the Hawking emission of NCBHs compared with higher-dimensionalSchwarzschild black holes. n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 M (TeV) 1 . × . . × . × . × . × r (10 − fm) 3 . × − .
68 2 .
51 2 .
41 2 .
34 2 .
29 2 .
26 2 . Table 1 . Minimum masses and minimum radii of NCBHs for different n . For n = 0 the units are M ⋆ ∼ √ θ − ∼ TeV. For n = 0 the units are M ⋆ ∼ √ θ − ∼ In view of particle physics experiments, we can now display potential values of theremnant size r and mass M . In table 1 we see that the minimum mass to have blackholes increases with n , while the corresponding radius slightly decreases. According to thelatest experimental constraints, the tightest and most pessimistic estimate for the size ofextra dimensions comes from the on-shell production of gravitons and sets R . − m– 8 – = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 T max (GeV) 15 ×
30 43 56 67 78 89 98 T max (10 K ) 0 . × Table 2 . NCBHs maximum temperatures for different values of n with units in which M ⋆ √ θ = 1. [99, 100]. This limit requires n ≥ M ⋆ at the terascale. This would imply thatthe NCBHs are too heavy to be produced at the LHC, at least as long as one assumes √ θ ∼ − fermi . Alternatively, the possibility of a smaller minimal length has beenconsidered, that is, M ⋆ √ θ <
1, in order to get into the LHC-accessible black hole massregion [80]. This possibility is based on the fact that in general we ignore the exact natureof the relation between √ θ and the mass scale Λ NC associated with the appearance ofnon-commutative effects. In other words, we cannot say anything more than √ θ ∝ / Λ NC and Λ NC ∼ M ⋆ . As a consequence, we can proceed by setting the value of M ⋆ √ θ in orderto have the minimum mass in the range 1 TeV . M .
10 TeV. From M = U ( r h ),we see that M ∼ r n +1 h M n +2 ⋆ . Once the radius r h is expressed in √ θ units we find that M ∝ ( M ⋆ √ θ ) n +1 M ⋆ . Thus we conclude that for M ⋆ √ θ ≈ .
27, the NCBH masses would beaccessible to current particle physics experiments for all n >
0. In addition, these thresholdmasses are compatible with recent limits established by experimental observations at theCMS detector [11].However, having M ⋆ √ θ < √ θ . In table 2 we have an estimate of the maximum temperatures n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 T/M < × − < × − < × − < × − < × − < × − < × − < × − Table 3 . Quantum back reaction estimates for different n with units in which M ⋆ √ θ = 1. in the case M ⋆ √ θ = 1. We see that the temperature can be at most ∼
100 GeV for n >
0. Consequently, the back reaction is negligible, since the ratio
T /M is alwaysquite small (see table 3). For M ⋆ √ θ ≈ .
27 maximum temperatures are in the range210 GeV . T max .
360 GeV for n ≥
3. However the ratio
T /M will never exceed In the present paper we have used the notation of Myers and Perry for the fundamental mass M ⋆ [101],which has also been adopted in [12, 13]. However, alternative definitions of the fundamental mass M ⋆ haveappeared in the literature. These lead to different values for the NCBH minimum masses. Our results areconsistent with all previous findings. For instance, Rizzo obtained minimum masses that correspond to8 πM , where M is given in table 1 [80]. Gingrich’s minimum masses correspond to 4(2 π ) − n M [84], whilethe notation of Spallucci and coworkers leads to masses n +32 )( n +2) π − n +12 M [40, 81]. Despite these differentdefinitions, the conclusion is unique in all cases: if M ⋆ √ θ = 1 minimum masses are not in the energy rangeaccessible at the LHC for n ≥ – 9 – max /M for all n . Since the latter ratio goes like T max /M ∝ ( M ⋆ √ θ ) − ( n +2) , we find that T /M < T max /M ≤ .
07 for n ≥
3. Thus we conclude that the back reaction can be stillconsidered to be negligible in this regime of parameters.Another potentially serious repercussion of having M ⋆ √ θ < σ NCBH ∼ πr . A smaller length scale gives smaller remnantradii r ∼ ( M ⋆ √ θ ) 10 − fermi and therefore a quadratically smaller cross-section. However,even for M ⋆ √ θ = 0 .
27 we still find a promising value for the cross-section, namely, σ ∼ − m ∼
100 pb. Given the latest peak LHC luminosity L ∼ . × m − s − [102],roughly a black hole every three seconds would be produced at the CERN laboratories, anastonishing number which does not differ significantly from that expected for conventionalblack hole metrics. Such a plentiful production of black holes might seem like a speculativeprediction to a skeptical reader. However, for our primary goal in this paper (which is tostudy the differences in Hawking emission from NCBHs compared to classical Schwarzschildblack holes), the condition M ⋆ √ θ < M ⋆ √ θ = 1. This implies that our phenomenological predictionswill have particular consequences for the physics of cosmic ray showers. Here the energyavailable can reach ∼ TeV, definitely much higher than that needed to produce NCBHs[12].
Hawking radiation from higher-dimensional black holes has been widely researched, see forexample [12–20] for some reviews. As well as being of intrinsic interest, a detailed quanti-tative understanding of the Hawking emission is essential for accurate simulations of miniblack hole events at the LHC [103, 104]. For spherically symmetric, higher-dimensionalSchwarzschild black holes the Hawking radiation both on the brane and in the bulk hasbeen extensively studied (some references are [26, 105–111]), including the graviton emis-sion. More recently, the emission from rotating higher-dimensional black holes has receivedattention (an incomplete list of references is [23–25, 112–123]). The emission of masslessparticles of spin-zero, spin-one-half and spin-one on the brane, and spin-zero in the bulk,has been computed in detail and implemented in simulations of black hole events at theLHC [103, 104]. However, only partial results are available for graviton emission [124, 125].The most recent work on Hawking radiation from higher-dimensional black holes has fo-cussed on the emission of particles of mass [126–129] or charge [128, 129], or studying morecomplicated black hole geometries. Of the latter, we mention only Gauss-Bonnet blackholes [130, 131] which have a lower temperature compared with the usual Schwarzschildblack holes, leading to a longer black hole lifetime [131].In this paper, we study the Hawking radiation of scalar fields from the black holes(2.2), both on the brane and in the bulk. We focus on a scalar field because this is thesimplest case and we anticipate that it will display many of the physical features of theemission common to all particle species. Of course, the emission of particles of higher spinis important for phenomenology and we plan to return to this in a future publication.– 10 –or the moment therefore, we restrict our attention to a massless, minimally coupledscalar field satisfying the Klein-Gordon equation ∇ µ ∇ µ Φ = 0 , (3.1)for comparison with previous results on the emission from Schwarzschild black holes [26].Since the black hole is non-rotating, and the scalar field uncharged, we are interestedin the fluxes of particles and energy. These fluxes are computed as expectation values ofparticle number operator and the component T rt of the stress-energy tensor respectively, theexpectation values being found in the Unruh vacuum [132] which models an evaporatingblack hole. Fortunately we can compute these expectation values without recourse tocurved-space renormalization (see for example [25]).We recall here that the expectation value of the stess-energy tensor for a quantizedscalar field in a general space-time of arbitrary dimension is given as the limit [133] h ψ | T µν ( x ) | ψ i = lim x → x ′ D µν ( x, x ′ ) [ − iG F ( x, x ′ )] (3.2)where G F ( x, x ′ ) is the Feynman propagator G F ( x, x ′ ) = i (cid:10) ψ (cid:12)(cid:12) T Φ( x )Φ( x ′ ) (cid:12)(cid:12) ψ (cid:11) (3.3)(here T denotes time ordering), | ψ i is a normalized quantum state of Hadamard type and D µν ( x, x ′ ) is a differential operator given, for a massless, minimally coupled scalar field, by D µν = g ν ′ ν ∇ µ ∇ ν ′ − g µν g ρσ ′ ∇ ρ ∇ σ ′ (3.4)where g µν ′ is the bivector of parallel transport from x to x ′ . The Feynman propagator is,by definition, a solution of ∇ µ ∇ µ G F ( x, x ′ ) = − [ − g ( x ) − / ] δ D ( x − x ′ ) (3.5)where D is the number of space-time dimensions. The issue is now to consider short-scalemodifications not only of the gravity sector but of the matter sector too. In other words,we consider the possibility that the field Φ is affected by the presence of a quantum-gravity-induced minimal length, i.e. a natural ultra-violet cut-off. Since this is the subject of muchresearch, we consider only the results given in [91] as a preliminary step and we reserve theanalysis of alternative modifications for forthcoming contributions.The introduction of space-time fluctuations in quantum field theory can be achievedby considering a modified form of the Green function equation (3.5). By analogy withwhat we have seen on the gravity side, we model all the relevant modifications by a non-standard source term in the Green function equation (3.5). For mathematical convenience,we temporarily switch to Euclidean signature and we find∆ G E ( x, x ′ ) = e θ ∆ √ g δ D ( x − x ′ ) , (3.6)– 11 –here ∆ = ∇ µ ∇ µ and the non-local operator e θ ∆ smears out any point-like object. Weintroduce the Euclidean Green function G ( x, x ′ ) corresponding to the usual case √ θ = 0,and then one can obtain the following relation between G ( x, x ′ ) and G E ( x, x ′ ), namely: G E ( x, x ′ ) = e θ ∆ G ( x, x ′ ) . (3.7)As a consequence, if we want to compute the stress-energy tensor corresponding to G E ( x, x ′ ),we need to consider terms emerging from the following non-trivial commutation relation h e θ ∆ , D µν i = 0 . (3.8)The above expression will depend on curvature terms. However, given the regularity of ourbackground metric we make the (brutal) approximation of neglecting these contributionsjust to have a flavour of the possible repercussions for the Hawking emission of the presenceof the non-local operator. As a result, we model the effects of an effective ultra-violet cut-off in the frequency ω of the emitted quanta, which modifies the expectation values of thestress-energy tensor in the following simplified way [83]: h T rt i ∝ X modes e − θω ω [exp ( ω/T ) −
1] (2 ℓ + 1) |A ωℓ | , (3.9)where ℓ is a quantum number labelling a scalar field mode, T is the Hawking temperatureand A ωℓ is a transmission coefficient which will be defined in the following subsections.This expression is phenomenologically motivated by the fact that all frequencies higherthan 1 / √ θ become largely suppressed. Setting θ = 0 in (3.9), we recover the standardHawking flux. We comment that the above approximation is reasonable: we have neglectedcurvature corrections that in the worst case are of order R ∼ /θ , corresponding to sub-leading disturbances of frequencies ω ∼ √ R ∼ / √ θ .The particle flux is not computed directly from the Feynman Green’s function, but wemodel the effects on the particle flux of the non-local operator described above in the sameway as for expectation values of the stress-energy tensor, namely by inserting a dampingfactor e − θω in the flux to give dNdt ∝ X modes e − θω [exp ( ω/T ) −
1] (2 ℓ + 1) |A ωℓ | , (3.10)which reduces to the usual Hawking flux when θ = 0. In the following sections, we shallcompare the usual Hawking emission quantities with those originating from the abovestress-energy tensor (3.9) and particle flux. From now on, throughout this section we useunits in which the length scale √ θ and M ⋆ are set equal to unity, so that energies are inunits of TeV for n > TeV for n = 0. We shall consider scalar field modesof frequency up to ω = 1 in these units (corresponding to frequencies up to 1 / √ θ ), sincefrequencies above this value are suppressed in our model.– 12 – .1 Emission on the brane To compute the emission of brane-localized modes, we consider a four-dimensional “slice”of the higher-dimensional black hole (2.2) obtained from fixing the co-ordinates ϑ i , i > ds = − h ( r ) dt + h ( r ) − dr + r (cid:0) dϑ + sin ϑ dϕ (cid:1) , (3.11)where we have set ϑ ≡ ϑ . We perform the usual frequency decomposition of the scalarfield Φ and consider field modes of the formΦ brane ωℓm ( t, r, ϑ, ϕ ) = e − iωt e imϕ R brane ωℓ ( r ) Y mℓ ( ϑ ) , (3.12)where ω is the frequency of the mode, m the azimuthal quantum number, and Y mℓ ( ϑ ) is ascalar spherical harmonic. The radial function R brane ωℓm ( r ) satisfies the equation0 = ddr (cid:20) r h ( r ) dR brane ωℓ dr (cid:21) + (cid:20) ω r h ( r ) − ℓ ( ℓ + 1) (cid:21) R brane ωℓ . (3.13)A suitable basis of linearly independent solutions of the radial equation (3.13) is given bythe “in” and “out” modes: R brane , in ωℓ = ( e − iωr ∗ r → r h r − h A brane , in ωℓ e − iωr ∗ + A brane , out ωℓ e iωr ∗ i r → ∞ (3.14) R brane , out ωℓ = ( B brane , in ωℓ e − iωr ∗ + B brane , out ωℓ e iωr ∗ r → r h r − e iωr ∗ r → ∞ (3.15)where we have defined the “tortoise” co-ordinate r ∗ by dr ∗ dr = 1 h ( r ) . (3.16)The conventional particle flux spectrum, the number of particles emitted per unit timeand unit frequency, is d N brane dt dω = 12 π ω/T ) − ∞ X ℓ =0 (2 ℓ + 1) (cid:12)(cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12)(cid:12) , (3.17)and the standard power spectrum, the energy emitted per unit time and unit frequency, is d E brane dt dω = 12 π ω exp ( ω/T ) − ∞ X ℓ =0 (2 ℓ + 1) (cid:12)(cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12)(cid:12) , (3.18)where T is the black hole temperature (2.9). In (3.17–3.18), the quantity (cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12) is thetransmission coefficient for the scalar field mode. If we consider a scalar wave which, nearthe event horizon, is out-going, the transmission coefficient is given by the proportion ofthe wave which tunnels through the gravitational potential surrounding the black hole and– 13 –scapes to infinity. We compute the transmission coefficients numerically from the “in”modes (3.14) as follows: (cid:12)(cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12)(cid:12) = 1 − (cid:12)(cid:12)(cid:12) A brane , out ωℓ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) A brane , in ωℓ (cid:12)(cid:12)(cid:12) . (3.19)As well as the usual particle and energy fluxes (3.17–3.18), we also study the fluxes discussedat the start of this section, where there is an additional damping term due to the non-commutativity (3.9–3.10) [83]: d N brane , NC dt dω = 12 π e − ω [exp ( ω/T ) − ∞ X ℓ =0 (2 ℓ + 1) (cid:12)(cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12)(cid:12) , (3.20) d E brane , NC dt dω = 12 π ω e − ω [exp ( ω/T ) − ∞ X ℓ =0 (2 ℓ + 1) (cid:12)(cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12)(cid:12) . (3.21)We also consider the absorption cross-section σ brane ( ω ), which has the form σ brane ( ω ) = πω ∞ X ℓ =0 (2 ℓ + 1) (cid:12)(cid:12)(cid:12) A brane ωℓ (cid:12)(cid:12)(cid:12) . (3.22)We begin the presentation of our numerical results by considering the transmissioncoefficient (3.19) (see figure 5 for the transmission coefficients for the first few modes for aneleven-dimensional black hole), and comparing them with those for a Schwarzschild blackhole having the same mass. For low frequencies, the transmission coefficient for the NCBH Figure 5 . Transmission coefficients (3.19) for a scalar field on the brane as a function of frequency ω for the first few modes. We consider an eleven-dimensional NCBH. Solid lines are the transmissioncoefficients for the NCBH, while dotted lines denote the transmission coefficients for a Schwarzschildblack hole with the same mass. The quantum number ℓ increases from ℓ = 0 to ℓ = 3 going fromleft to right. We use units in which M ⋆ and θ are set equal to unity, so that energies are in TeV. – 14 –s smaller than that for the Schwarzschild black hole, but it converges more rapidly to unityas the frequency increases than in the Schwarzschild case. These differences become moremarked as the quantum number ℓ increases. Figure 6 . Absorption cross-section (3.22) for a scalar field on the brane as a function of frequency.Five-dimensional NCBHs with varying masses are considered, together with a Schwarzschild blackhole having the same mass as the NCBH with maximum temperature. The dark grey dotted curve(having the largest low-frequency absorption cross-section) is for the Schwarzschild black hole, theother curves are for NCBHs, with the mass of the NCBH increasing as the value of the low-frequencyabsorption cross-section increases. The blue curve (third from the top in the low-frequency limit)is the curve for the NCBH having maximum temperature. We use units in which M ⋆ and θ are setequal to unity, corresponding to energies measured in TeV. This behaviour of the transmission coefficients is reflected in the absorption cross-section (3.22) as a function of frequency ω , see figures 6 and 7. The behaviour of theabsorption cross-section on the brane for NCBHs is qualitatively similar to that observed forSchwarzschild black holes [26]. As the frequency ω →
0, the absorption cross-section tendsto the area of the event horizon 4 πr h , as observed for brane emission from Schwarzschildblack holes [26]. For five-dimensional black holes (figure 6), the area of the event horizonincreases as the mass of the black hole increases, leading to the observed increase in thelow-energy absorption cross-section. The difference in low-energy absorption cross-sectionbetween the NCBH with the maximum temperature and the Schwarzschild black holewith the same mass is due to the latter having a considerably larger event horizon area.For NCBHs with maximal temperature (figure 7), the event horizon area decreases asthe number of space-time dimensions increases, which gives the observed decrease in thelow-energy absorption cross-section.As ω increases, the absorption cross-section oscillates about its asymptotic high-energy– 15 – igure 7 . Absorption cross-section (3.22) for a scalar field on the brane as a function of frequency.NCBHs with maximum temperatures are considered for varying numbers of space-time dimensions.The curves, from top to bottom, are for n = 0 up to n = 7. We use units in which M ⋆ and θ areset equal to unity. For n > n = 0 energiesare measured in units of 10 TeV. The different units are used for the n = 0 case to facilitatecomparison with the n > value (although the magnitude of the oscillations decreases significantly as the number ofspace-time dimensions increases [111]). The high-frequency limiting value (also known asthe geometric optics limit) of the total absorption cross-section corresponds to an effectiveabsorbing area of radius r c , where r c = σ brane ( ω → ∞ ) /π . From figure 6, it can be seenthat r c increases as the mass of the five-dimensional black holes increases, and, furthermore,that r c is always greater than the event horizon radius r h for these black holes. Thehigh-frequency absorption cross-sections for the NCBH with maximum temperature andthe Schwarzschild black hole with the same mass are identical, indicating that the high-frequency absorption cross-section depends only on the mass of the black hole and not thedetailed structure of the metric near the event horizon. Looking at figure 7, we observe thatfor four- and five-dimensional black holes, the effective high-frequency absorption area islarger than the black hole event horizon, whilst for black holes in six and more dimensions,the effective high-frequency absorption area is smaller than the black hole event horizonarea.For the brane emission of scalar particles from an ( n + 4)-dimensional black hole, ithas been found that, for Schwarzschild black holes, r c is related to the event horizon radius[134]: r c = (cid:18) n + 32 (cid:19) n +1) r n + 3 n + 1 r h . (3.23)For the black holes in figures 6 and 7, we find that σ ( ω → ∞ ) = πr c is a good approximationto the absorption cross-section when ω = 1 if r c is given by (3.23) with r h being the radius– 16 – igure 8 . Particle fluxes (3.17, 3.20) for scalar field emission on the brane, as a function offrequency, for five-dimensional NCBHs with varying masses and a Schwarzschild black hole havingthe same mass as the NCBH with maximum temperature. Solid lines indicate the standard particleflux (3.17), and dotted lines the particle flux with additional damping due to non-commutativeeffects (3.20). The top curve (dark grey) is the flux for the Schwarzschild black hole, with the othercurves being for NCBHs with increasing temperature from the bottom black curve to the top purplecurve. We use units in which M ⋆ and θ are set equal to unity, corresponding to energies measuredin TeV. of a Schwarzschild black hole having the same mass as the NCBH. In other words, as far ashigh-frequency modes are concerned, the NCBH is mimicking a Schwarzschild black holewith the same mass.Next we consider the scalar particle flux (3.17, 3.20), see figures 8 and 9, and scalarenergy flux (3.18, 3.21), see figures 10 and 11. The results for particle and energy emissionare very similar. Looking first at the emission from five-dimensional NCBHs with varyingmasses (figures 8 and 10), the flux is much smaller for NCBHs than from the Schwarzschildblack hole having the same mass as the NCBH with maximum temperature. This is dueto the considerably smaller temperature of the the NCBHs. The peak of the emission inall cases comes from low-frequency modes, which probe more fully the nature of the blackhole geometry near the horizon, for which there are differences in absorption cross-section(see figure 7) between NCBHs and Schwarzschild black holes. However, the dominanteffect across all frequency ranges is the much lower temperature of NCBHs compared withSchwarzschild black holes having the same mass. For five-dimensional black holes, theadditional damping due to the non-commutativity makes a negligible difference to theparticle and energy flux at high frequency (recall that ω = 1 in our units corresponds tothe extremely high frequency ω = √ θ , where θ is the minimal length scale).If we increase the number of space-time dimensions (figures 9 and 11), the maximumtemperature of the NCBHs increases (see figure 3) and we observe a corresponding increase– 17 – igure 9 . Particle fluxes (3.17, 3.20) for scalar field emission on the brane, as a function offrequency, for NCBHs with maximum temperature and varying numbers of space-time dimensions.The curves, from bottom to top, are for n = 0 up to n = 7. Solid lines indicate the standard particleflux (3.17), and dotted lines the particle flux with additional damping due to non-commutativeeffects (3.20). We use units in which M ⋆ and θ are set equal to unity. For n > n = 0 energies are measured in units of 10 TeV. The differentunits are used for the n = 0 case to facilitate comparison with the n > Figure 10 . Energy fluxes (3.18, 3.21) for scalar field emission on the brane, as a function offrequency. The same black holes are considered as in figure 8. As in figure 8, solid lines correspondto the standard energy flux (3.18) and dotted lines the energy flux (3.21) with additional dampingdue to non-commutative effects. The top curve (dark grey) is the flux for the Schwarzschild blackhole, with the other curves being for NCBHs with increasing temperature from the bottom blackcurve to the top purple curve. We use units in which M ⋆ and θ are set equal to unity, correspondingto energies measured in TeV. – 18 – igure 11 . Energy fluxes (3.18, 3.21) for scalar field emission on the brane, as a function offrequency. The same black holes are considered as in figure 9. The curves, from bottom to top, arefor n = 0 up to n = 7. As in figure 9, solid lines correspond to the standard energy flux (3.18) anddotted lines the energy flux (3.21) with additional damping due to non-commutative effects. Weuse units in which M ⋆ and θ are set equal to unity. For n > n = 0 energies are measured in units of 10 TeV. The different units are used forthe n = 0 case to facilitate comparison with the n > in the scalar field emission. The peak of the particle and power spectra increase, and theemission remains significant for larger frequencies. However, even for an 11-dimensionalNCBH, the emission is still only of the same order of magnitude as a five-dimensionalSchwarzschild black hole. The increase in brane emission as the number of space-timedimensions increases is much smaller than for Schwarzschild black holes [26]. The otherfeature in figures 9 and 11 is that the additional damping due to non-commutativity (3.21)becomes more important as the number of space-time dimensions increases and the tem-perature of the black holes increases. To study the bulk scalar modes, the Klein-Gordon equation (3.1) must be solved on thefull, higher-dimensional space-time (2.2). The scalar field modes now take the form [26]Φ bulk ωℓj ( t, r, θ i ) = e − iωt R bulk ωℓ ( r ) Y jℓ ( θ i , ϕ ) , (3.24)where Y jℓ ( θ i , ϕ ) is a scalar hyperspherical harmonic function of θ , . . . , θ n +1 , ϕ [135]. Foreach ℓ (which is the quantum number governing the constant arising in the separation ofthe Klein-Gordon equation), there are N ℓ hyperspherical harmonics, which we label by theindex j . The degeneracy factor N ℓ is N ℓ = (2 ℓ + n + 1) ( ℓ + n )! ℓ ! ( n + 1)! , (3.25)which reduces to the familiar 2 ℓ + 1 when the number of extra dimensions, n , is equal tozero. – 19 –he radial functions R bulk ωℓ ( r ) now satisfy the equation0 = 1 r n ddr (cid:20) h ( r ) r n +2 dR bulk ωℓ dr (cid:21) + (cid:20) ω r h ( r ) − ℓ ( ℓ + n + 1) (cid:21) R bulk ωℓ . (3.26)The “in” and “out” radial functions (3.14–3.15) are modified near infinity: R bulk , in ωℓ = ( e − iωr ∗ r → r h r − − n h A bulk , in ωℓ e − iωr ∗ + A bulk , out ωℓ e iωr ∗ i r → ∞ (3.27) R bulk , out ωℓ = ( B bulk , in ωℓ e − iωr ∗ + B bulk , out ωℓ e iωr ∗ r → r h r − − n e iωr ∗ r → ∞ (3.28)where the “tortoise” co-ordinate r ∗ is given in (3.16). The bulk particle and power spectraare most simply written as d N bulk dt dω = 12 π ω/T ) − ∞ X ℓ =0 N ℓ (cid:12)(cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12)(cid:12) , (3.29) d E bulk dt dω = 12 π ω exp ( ω/T ) − ∞ X ℓ =0 N ℓ (cid:12)(cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12)(cid:12) , (3.30)and we will also consider bulk particle and power spectra with additional damping termsdue to non-commutativity effects: d N bulk , NC dt dω = 12 π e − ω [exp ( ω/T ) − ∞ X ℓ =0 N ℓ (cid:12)(cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12)(cid:12) , (3.31) d E bulk , NC dt dω = 12 π ω e − ω [exp ( ω/T ) − ∞ X ℓ =0 N ℓ (cid:12)(cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12)(cid:12) , (3.32)In the above equations (3.29–3.32), the bulk transmission coefficient (cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12) appears, andthis is computed in the same way as for the brane emission: (cid:12)(cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12)(cid:12) = 1 − (cid:12)(cid:12)(cid:12) A bulk , out ωℓ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) A bulk , in ωℓ (cid:12)(cid:12)(cid:12) . (3.33)We can also define a bulk absorption cross-section σ bulk ( ω ) in terms of the transmissioncoefficient [26]: σ bulk ( ω ) = 2 n π ( n +1)2 ω n +2 ( n + 1) Γ (cid:18) n + 12 (cid:19) ∞ X ℓ =0 N ℓ (cid:12)(cid:12)(cid:12) A bulk ωℓ (cid:12)(cid:12)(cid:12) . (3.34)We begin our discussion of bulk emission by considering the absorption cross-section σ bulk ( ω ) (3.34). The bulk absorption cross-section shares many qualitative features with– 20 – igure 12 . Absorption cross-section (3.34) for a scalar field in the bulk as a function of fre-quency, for five-dimensional non-commutative black holes with varying masses, together with aSchwarzschild black hole having the same mass as the non-commutative black hole with maximumtemperature. The dark grey curve (having the largest low-frequency absorption cross-section) isfor the Schwarzschild black hole, the other curves are for NCBHs, with the mass of the NCBH in-creasing as the value of the low-frequency absorption cross-section increases. The blue curve (thirdfrom the top in the low-frequency limit) is the curve for the NCBH having maximum temperature.We use units in which M ⋆ and θ are set equal to unity, corresponding to energies measured in TeV. that for brane emission. For low-frequency waves, the absorption cross-section tends to thearea of the event horizon, which for higher-dimensional black holes is given by A h = 2 πr n +2 h π ( n +1)2 Γ (cid:0) n +32 (cid:1) . (3.35)For five-dimensional black holes (figure 12), the event horizon radius increases as the massof the black hole increases (but is always smaller for NCBHs than for Schwarzschild blackholes with the same mass). In figure 13, the area of the event horizon increases dramaticallyas the number of extra dimensions increases, and this leads to the large increase in theabsorption cross-section.As the frequency increases, the absorption cross-section oscillates about its final, high-frequency limit, although the oscillations are of small amplitude for more than two extradimensions. In the high-frequency limit, the absorption cross-section tends towards theprojected area of an absorptive body of effective radius r c (3.23). The effective radius r c isthe same for both bulk and brane modes [134], but care is needed in computing the relevantprojected area (for a detailed discussion, see [26]). The expected limiting behaviour of theabsorption cross-section is [26]: σ ( ω → ∞ ) = 2 π n ( n + 2) Γ (cid:0) n +22 (cid:1) (cid:18) n + 32 (cid:19) n +2 n +1 (cid:18) n + 3 n + 1 (cid:19) n +22 r n +2 h . (3.36)– 21 – igure 13 . Absorption cross-section (3.34) for a scalar field in the bulk as a function of frequency,for non-commutative black holes with maximum temperatures in varying numbers of space-timedimensions. The curves, from bottom to top, are for n = 0 up to n = 7. We use units in which M ⋆ and θ are set equal to unity. For n > n = 0 energies are measured in units of 10 TeV. The different units are used for the n = 0 caseto facilitate comparison with the n > Comparing this formula with our numerical results in figures 12 and 13, we find excellentagreement by taking r h in (3.36) to be the event horizon radius of a Schwarzschild blackhole having the same mass as the NCBH, except for larger numbers of extra dimensions,where the absorption cross-sections in figure 13 have not yet converged to their asymptoticlimit. Therefore, in the bulk, as on the brane, for high frequency waves the NCBHs aremimicking Schwarzschild black holes of the same mass.We now consider the particle and energy fluxes (3.29–3.32), see figures 14–17. Theresults for particle and energy emission are similar. For five-dimensional black holes (fig-ures 14 and 16), as with the emission on the brane, the emission in the bulk is much smallerfor the NCBHs than it is for Schwarzschild black holes. This dominant effect is due to thesmaller temperature of the NCBHs. The additional damping term present in (3.31–3.32)once again has a negligible effect on the emission.As we increase the number of extra dimensions (figures 15 and 17), the results are verydifferent from those obtained for Schwarzschild black holes [26]. In the latter case the bulkemission increases greatly as the number of space-time dimensions increases, due mostlyto the linear increase of the black hole temperature with n for fixed horizon radius. In ourcase, the maximum temperature of the black holes does increase with n , but not so quicklyas for Schwarzschild black holes (see figure 3), and the temperature of the NCBHs is so lowthat the peak of the emission of both particles and energy decreases as n increases. Theemission spectrum broadens as n increases, with emission at higher frequencies making amore significant contribution to the total. We also observe that the additional damping– 22 – igure 14 . Particle fluxes (3.29, 3.31) for scalar field emission in the bulk, as a function offrequency, for five-dimensional NCBHs with varying masses and a Schwarzschild black hole havingthe same mass as the NCBH with maximum temperature. Solid lines indicate the standard particleflux (3.29), and dotted lines the particle flux with additional damping due to non-commutativeeffects (3.31). The top curve (dark grey) is the flux for the Schwarzschild black hole, with the othercurves being for NCBHs with increasing temperature from the bottom black curve to the top purplecurve. We use units in which M ⋆ and θ are set equal to unity, corresponding to energies measuredin TeV. Figure 15 . Particle fluxes (3.29, 3.31) for scalar field emission in the bulk, as a function of frequency,for NCBHs with maximum temperature and varying numbers of space-time dimensions. Solid linesindicate the standard particle flux (3.29), and dotted lines the particle flux with additional dampingdue to non-commutative effects (3.31). The curves, from top to bottom, are for n = 0 up to n = 7.We use units in which M ⋆ and θ are set equal to unity. For n > n = 0 energies are measured in units of 10 TeV. The different units areused for the n = 0 case to facilitate comparison with the n > – 23 – igure 16 . Energy fluxes (3.30, 3.32) for scalar field emission in the bulk, as a function of frequency.The same black holes are considered as in figure 14. As in figure 14, solid lines correspond to thestandard energy flux (3.30) and dotted lines the energy flux (3.32) with additional damping due tonon-commutative effects. The top curve (dark grey) is the flux for the Schwarzschild black hole,with the other curves being for NCBHs with increasing temperature from the bottom black curveto the top purple curve. We use units in which M ⋆ and θ are set equal to unity, corresponding toenergies measured in TeV. Figure 17 . Energy fluxes (3.30, 3.32) for scalar field emission in the bulk, as a function of frequency.The same black holes are considered as in figure 15. As in figure 15, solid lines correspond to thestandard energy flux (3.30) and dotted lines the energy flux (3.32) with additional damping due tonon-commutative effects. The curves, from top to bottom, are for n = 0 up to n = 7. We use unitsin which M ⋆ and θ are set equal to unity. For n > n = 0 energies are measured in units of 10 TeV. The different units are used for the n = 0case to facilitate comparison with the n > – 24 –erm in the spectrum due to non-commutativity (3.31–3.32) becomes more important as n increases, although, even for n = 11 the difference between the spectra with the additionaldamping and the spectra without the additional damping is not great, because of the lowtemperature of the black holes. We now consider the total emission from the NCBHs and the proportion of this emissionwhich is in the bulk space-time. We begin by comparing the total emission of particlesand energy, both on the brane and in the bulk, from NCBHs with maximum temperature.The totals for emission frequencies up to ω = 1 are presented in Tables 4 and 5, where thefluxes have been rescaled so that the flux from a four-dimensional NCBH with maximumtemperature is equal to unity. n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7Particles (undamped) 1 3.3 6.0 8.5 10.7 12.4 13.8 15.0Particles (damped) 1 3.3 6.0 8.4 10.6 12.3 13.7 14.8Power (undamped) 1 5.5 13.1 21.8 30.2 37.6 44.0 49.3Power (damped) 1 5.5 13.0 21.5 30.0 36.8 42.9 48.0 Table 4 . Total fluxes of particles and energy on the brane for frequencies up to ω = 1, fornon-commutative black holes with maximum temperature, compared with the emission from afour-dimensional non-commutative black hole. n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7Particles (undamped) 1 0.56 0.24 0.091 0.032 0.011 0.0033 0.0010Particles (damped) 1 0.56 0.24 0.088 0.030 0.0095 0.0029 0.00083Power (undamped) 1 1.46 1.08 0.59 0.27 0.11 0.042 0.014Power (damped) 1 1.45 1.05 0.56 0.25 0.098 0.035 0.012 Table 5 . Total fluxes of particles and energy in the bulk for frequencies up to ω = 1, for non-commutative black holes with maximum temperature, compared with the emission from a four-dimensional non-commutative black hole. From table 4, it is clear that the total emission of particles and energy on the branesteadily increases as n increases, due to the increased temperature of the black holes withincreasing n . This is also evident from the plots of the brane emission as a function offrequency in figures 9 and 11. For emission in the bulk, the results are rather different, inaccordance with our earlier discussion of figures 15 and 17. The fluxes of particles decreaseas n increases, however, the flux of energy is larger for n = 1 and n = 2 than it is for n = 0,but for n ≥ n increases. In both table 4 and 5, it can be seen thatthe additional damping due to non-commutativity effects makes only a small difference tothe total emission. – 25 – igure 18 . Ratio of bulk/brane emission as a function of frequency, for NCBHs with maximumtemperature. The curves, from top to bottom, are for n = 1 to n = 7 (the ratio for n = 0 is unityfor all ω ). The ratios are the same for particle and energy fluxes, and independent of whether thereare additional damping terms in the spectra due to non-commutativity. The ratio of bulk/brane emission is shown as a function of frequency ω in figure 18. Itcan be seen that the bulk emission is greatly suppressed compared to the brane emissionfor low frequencies and large n . The bulk/brane ratio increases with frequency for all n ,and for large n becomes greater than unity for large frequencies. The shape of the curvesin figure 18 are qualitatively similar to those in [26] for Schwarzschild black holes, bearingin mind the different units we are using. However, by comparing the values in table 4 and5, the ratio of total bulk emission to total brane emission can be seen to decrease rapidlyas n increases, from about 20% bulk emission compared with brane emission for n = 1down to about 0 .
02% bulk emission compared with brane emission for n = 7. This is inmarked contrast to the results for Schwarzschild black holes [26], where the bulk/braneratio decreases down to about 22% for n = 3 but then increases as n increases until itis about 93% for n = 7. The reason why the bulk emission is so suppressed in our caseis that the NCBHs have a very much smaller temperature compared with Schwarzschildblack holes with the same mass. Thus, while the bulk/brane ratio increases quickly withincreasing frequency ω (see figure 18), the low temperature means that there is negligibleemission in high frequencies either on the brane or in the bulk, so that low-frequencyemission dominates and this is mostly on the brane. In the present paper we have addressed the problem of the Hawking emission from quantumgravity corrected black hole space-times. One of the motivations for this study is recentLHC bounds on the fundamental scale of quantum gravity M ⋆ , namely M ⋆ & − .
02% of the braneemission as one increased the number n of extra dimensions, while it increases with n forthe Schwarzschild case. In other words, we find that the emission is dominated by lowfrequency modes, mostly on the brane. This is the most phenomenologically interestingeffect. The amount of energy lost in the bulk can be measured as missing energy by anobserver on the brane. Such a missing energy determines the remaining available energyfor emission on the brane in terms of easily detectable standard model particles. We stressthat the observation of such peculiarities would not only confirm our predictions but mightdisclose further features about the nature of quantum gravity itself.The work initiated in this paper is far from being concluded. The present analysisconcerns just the black hole direct emission of scalar particles, while little is known aboutthe subsequent evolution of matter and radiation. For four-dimensional black holes, anincrease of the spin of matter fields is responsible for a suppression of the emission sinceparticles have to traverse a higher angular momentum barrier. The number of extra di-mensions also affects the black hole emission, but in a specific manner for each value of thespin. For example, for ten-dimensional, spherically symmetric, black holes, the emissionof scalars, fermions and gauge bosons is comparable for each field degree of freedom [26].We cannot infer that the same pattern emerges when considering QGBHs. The study ofhigher spin fields in QGBH backgrounds will be of primary importance, since it is directlyconnected to observations in particle detectors. The emission of higher spin fields is also– 27 –onnected to the onset of the photosphere and the chromosphere, regions around the blackhole where an electron-positron-photon plasma and a quark-gluon plasma might develop bymeans of particle production and bremsstrahlung mechanisms. To date quantum gravityeffects have been neglected in studies of both the photosphere and chromosphere.Another open direction of investigation is the evolution of black holes in phases pre-ceding the spherically symmetric neutral configuration. For now we ignore how quantumgravity effects could affect black hole formation and the eventual loss of charge and/orangular momentum. We expect a variety of new effects from the combination of higherspin fields, number of extra dimensions and non-spherical QGBH emission. In particular,it would be interesting to study the role of super-radiance in QGBH decay. We plan toaddress these issues in forthcoming contributions. Acknowledgments
This work is supported by the Helmholtz International Center for FAIR within the frame-work of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich- ¨Okonom-ischer Exzellenz) launched by the State of Hesse. This work is supported in part by theEuropean Cooperation in Science and Technology (COST) action MP0905 “Black Holes ina Violent Universe”. P.N. would like to thank the School of Mathematics and Statistics,University of Sheffield for the kind hospitality during a visit supported by COST actionMP0905 at the initial stages of this work. E.W. would like to thank the Frankfurt Insti-tute for Advanced Studies, Goethe University Frankfurt for the kind hospitality during avisit during the final stage of this work. E.W. also thanks Perimeter Institute, Waterloo,Canada and the School of Mathematical Sciences, Dublin City University for hospitalitywhile this work was in progress; Sam Dolan for helpful discussions and Carl Kent for helpwith numerical computations.
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