Sparsity of Hawking Radiation in D+1 Space-Time Dimensions Including Particle Masses
aa r X i v : . [ g r- q c ] O c t Sparsity of Hawking Radiation in D + 1 Space-Time Dimensions Including ParticleMasses ✩ Sebastian Schuster School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
Abstract
Hawking radiation from an evaporating black hole has often been compared to black body radiation. However, thiscomparison misses an important feature of Hawking radiation: Its low density of states. This can be captured in an easyto calculate, heuristic, and semi-analytic measure called ‘sparsity’. In this letter we shall present both the concept ofsparsities and its application to D + 1 -dimensional Tangherlini black holes and their evaporation. In particular, we shallalso publish for the first time sparsity expressions taking into account in closed form effects of non-zero particle mass.We will also see how this comparatively simple method reproduces results of (massless) Hawking radiation in higherdimensions and how different spins contribute to the total radiation in this context. Keywords:
Hawking Radiation, Black Holes, Higher Dimensions
1. Introduction to Sparsity
One of the core features of Hawking radiation [1, 2] isthe similarity of its emission spectrum to that of a greybody, hence that of a black body. While educational andedifying, this comparison omits one very important fea-ture: This radiation is sparse . Technically, this concept isencoded in a low density of states [3, 4]. Strictly speakingand as evidenced by Don Page’s work [5, 6, 7, 8], this fea-ture is known since the beginning, yet is often glossed over.A subsequent focus on high temperature regimes can beconsidered partly responsible for this [9, 10, 11, 12, 13, 14].Four years ago, a heuristic method to bring this forgottenfeature more to the forefront has been introduced [15, 16,17] — simply called sparsity η . This concept was originallyapplied to dimensional black holes (corresponding tothe solutions of Schwarzschild, Kerr, Reissner–Nordström,and ‘dirty black holes’), but soon found application alsoin higher dimensions [18], in phenomenological quantumgravity extensions [19], and in the context of the influenceof generalised uncertainty principles on Hawking evapora-tion [20, 21].Let us quickly introduce the concept: Sparsity η is ameasure to estimate the density of states of radiation. Forthis, one compares a localisation time scale τ loc of an emit-ted particle with a time scale τ gap characterising the timebetween subsequent emission events. It is worth emphas-ising the indefinite article here: Different choices can bemade for both time-scales, though their numerical values ✩ This document contains results of a PhD thesis funded by theVictoria University of Wellington PhD Scholarship.
Email address: [email protected] (SebastianSchuster) will not differ by much. The easiest way to choose the timescale τ gap is given by the inverse of the integrated numberflux density d Γ n of the radiation, and we will adhere onlyto this choice throughout the letter. Hence: τ gap = 1 Γ n . (1)For the localisation time scale τ loc , however, the identific-ation of a ‘simplest’ choice is less obvious. This is due tothe fact that for most spectra (in our context the Planckspectrum) peak and average frequencies do not agree, norare they the same when comparing number density spec-trum Γ n and energy density spectrum Γ E . We encode thisin the following way: τ loc = 1 ν c,q,s = 2 πω c,q,s , (2)where the index c ∈ { avg. , peak } indicates how the phys-ical quantity q (associated to a unique frequency by appro-priate multiplication with natural constants ~ , k B , c, G ) iscalculated, and s determines the spectrum we consider .As our τ gap is fixed, the sparsities η c,q,s := τ gap τ loc (3)inherit the freedom ( i.e. , the indices) of τ loc . The methodof calculating the sparsities is always the same: First, cal-culate the quantity q associated to the spectrum s . Second, Only in the definition of τ loc we employed the frequency, not theangular frequency as it makes for a more conservative estimate ofsparsity. This was suggested by an anonymous referee of [15]. Preprint submitted to Elsevier 17th October 2019 nd a corresponding angular frequency ω c,q,s . Here, thechoice of the quantity is of relevance — one should keep toquantities which can be related in a straightforward man-ner to a frequency. Third, and last, calculate the sparsity η c,E,s = ω c,E,s π Γ n . (4)Let us see this in action on two less trivial examples: Here,the frequency is gained from either the average wavelength λ or the average period τ of emitted particles for the num-ber spectrum. In order to compare the frequency corres-ponding to the average wavelength or of the average periodof emitted particles with / Γ n as it appears in the sparsitydefinition (3), one has to take their inverses before mul-tiplying with the appropriate factors of speed of light c ,and Planck’s constant ~ . In the resulting expression, Γ n cancels and one arrives at the convenient expressions η avg. ,τ,n = 1 R π ~ E d Γ n , η avg. ,λ,n = 1 R πck d Γ n , (5)where E is the energy of the emitted particle (equivalent toits angular frequency, as E = ~ ω ), and k its wave number.Note that both integrals are identical in the case of mass-less particles — in this case we simply call both the same, η binned . A convenient bonus of this result is the fact thatdifferent emission processes can be considered ‘happeningin parallel’, with the relevant notion borrowed from circuitanalysis, that is η tot = X channel i η i (6)for these two sparsities (and others reducing to a simple in-verse of a single integral). They also lend themselves nicelyto a interpretation as ‘binned’ or ‘bolometric’ sparsity meas-ures, as they can be understood as dividing up the emissionspectrum into infinitesimal bins.Much more common, however, is an expression notamenable to this property. For example, if one calculatesthe peak frequency ( i.e. , the peak energy) of the numberspectrum, the resulting sparsity η peak ,E,n = ω peak ,E,n π Γ n (7)involves finding the zeroes of the first derivative of Γ n w.r.t.energy/frequency. However, the peak frequencies will usu-ally not be given explicitly, as only in rare special casescan they be found analytically exactly.This is an opportune moment to describe the spectraunder consideration in more detail. We shall consider spec-tra of the form d Γ n = g (2 π ) D c ˆ k · ˆ n exp (cid:16) ~ √ m c + k c k B T − ˜ µ (cid:17) + s d D k d A (8)and d Γ E = g (2 π ) D c ~ √ m c + k c ˆ k · ˆ n exp (cid:16) ~ √ m c + k c k B T − ˜ µ (cid:17) + s d D k d A, (9) where g is the (possibly dimension-dependent) degener-acy factor of the emitted particles (more below in sec-tion 3.1), m their mass, D the number of space dimen-sions, T the temperature of the radiation, ˜ µ the chemicalpotential divided by k B T ( i.e. , the logarithm of the fu-gacity), ˆ n the surface normal to the emitting hypersurface A , and s ∈ {− , , +1 } a parameter distinguishing (re-spectively) between bosons, Maxwell–Boltzmann/classicalparticles, and fermions. The differential d D k takes on thefollowing form in spherical coordinates: d D k = k D − sin D − ϕ · · · sin ϕ D − sin ϕ D − d k d ϕ d ϕ D − , (10)where ϕ D − ∈ [0 , π ) , ϕ i ∈ [0 , π ) , if i ∈ { , . . . , D − } , and ϕ ∈ [0 , π ) (at least for our future integration steps). Inmany cases, the term ˆ k · ˆ n seems to be forgotten in higherdimensions (we will not mention the guilty parties) — eventhough without it, it will not be possible to correctly linkthese spectra to the dimensional case and its Stefan–Boltzmann law.Regarding our earlier mentioned peak frequencies in dimensions, the peak frequencies of the classicalmassive particle’s number and energy spectra can be foundin terms of cubics — but even these are neither useful norenlightening in most situations. For massless particles, onthe other hand, the result is in all dimensions expressiblein terms of the Lambert W-function: m = 0 : ω peak ,E,E = k B T ~ (cid:0) D + W ( sDe ˜ µ − D +2 ) (cid:1) , (11a) ω peak ,E,n = k B T ~ (cid:0) D − W ( s ( D − e ˜ µ − D +3 ) (cid:1) . (11b)In the following we will apply these methods to Tan-gherlini black holes. The sparsities we will calculate inthis letter are: η peak ,E,n , η peak ,E,E , η avg. ,E,n , η avg. ,τ,n , and η avg. ,λ,n . The latter two being the same in the masslesscase, they are relabelled as η binned in that case.
2. Preliminaries for Tangherlini Black Holes
The Tangherlini black hole [22, 23] is the higher di-mensional generalization of the Schwarzschild black hole;it is the D + 1 -dimensional, spherically symmetric vacuumsolution. The metric has the form d s = − (cid:18) − (cid:16) r H r (cid:17) D − (cid:19) d t + (cid:18) − (cid:16) r H r (cid:17) D − (cid:19) − d r + r dΩ D − , (12)where dΩ D − is the differential solid angle, and r H = D − s D ) GM/c ( D − π ( D − / (13)2s the D +1 -dimensional Schwarzschild radius, G the (dimension-dependent) gravitational constant, and M the mass of theblack hole. A Γ without the indices indicating numberor energy densities simply refers to the Γ -function. Inpassing, we note that the uniqueness theorems for blackholes hold (without further assumptions) only in di-mensions [24, 25, 26], related to a more complex notion ofangular momenta. This somewhat justifies our focus onhigher dimensional, non-rotating black holes even thoughsome solutions are explicitly known, like the Myers–Perrysolution [27].The surface area of the horizon becomes A H = 2 π D/ Γ( D/ r D − H , (14)while the corresponding Hawking temperature is D − πr H ~ ck B . (15)Due to the spherical symmetry, the angular and areaintegral required for the sparsity calculations can (a) beseparated from each other, and (b) the term ˆ k · ˆ n evalu-ates to a simple cos ϕ . This factor will prevent an integ-ration over the angular variables from being the area of ahypersphere. Rather, the result is (in all instances to beencountered in the following) Z π d ϕ D − Z π d ϕ D − sin ϕ D − × · · ·× Z π d ϕ sin D − ϕ Z π d ϕ cos ϕ sin D − ϕ = 2 πD − √ π D − Γ( ( D − . (16)
3. Sparsity Results and Comparison with the Lit-erature
The origin of the sparsity of Hawking evaporation in dimensions can be sought and found in the connectionbetween size of the horizon and the Hawking temperature.This feature is absent from black bodies — as long as theirtemperature can be maintained, they can be made of arbit-rary sizes. Put differently, in dimensions and for non-rotating black holes, the thermal wavelength λ thermal ful-fils λ thermal < A H . However, as we will see below, this doesnot translate to arbitrary dimensions as already shown byHod in [18]. Also, the inclusion of rotation would leadto the emergence of super-radiance further complicatingthe discussion. However, away from super-radiant regimesone can include easily the parameter ˜ µ (introduced above) Since, as mentioned before, rotating black hole solutions aremore subtle in higher dimensions we will limit the discussion tonon-rotating ones. As a shorthand, we will from now on assumeno rotation. to capture at least charges — allowing a spherically sym-metric solution —, or with less qualms and more bravadoabout deviating from spherical symmetry even very smallangular momenta.First, we will reproduce and improve Hod’s results onthe emission of massless particles, then we shall generaliseto massive particles. Due to the length of the results, thesewill be provided in tables 1 and 2. All results will be givenin terms of λ D − thermal / gA . Note that in this expression theTangherlini black hole mass drops out. Before starting, it is worth reminding ourselves that wewant to be as conservative as possible in our sparsity res-ults: A small sparsity would mean little phenomenologicaldeparture from the familiar black body radiation. Hence,we will not consider the area of the horizon to be the rel-evant area from which the Hawking radiation originates,but rather we will take the capture cross section σ capture for massless particles. This turns out to be [23] σ capture = 12 √ π Γ( D / )Γ( D +1 / ) (cid:18) DD − (cid:19) D − (cid:18) D (cid:19) D − D − | {z } =: c eff A H . (17)The factor c eff has been defined for future convenience.This change of area can be motivated and backed withnumerical studies highlighting that the renormalised stress-energy tensors of Hawking radiation do not have their max-imum at or very close to the horizon but rather a gooddistance away from it [28].It is relatively straightforward (though not necessarilynotationally easy-going) to manipulate standard integralexpressions [29, 30] into the required form. The Boltzmanncase ( i.e. , s = 0 ) often requires recognising removable sin-gularities, but apart from this is straightforward to includein these results. This is most apparent in the ubiquitousexpressions Li n ( − s ) / ( − s ) involving the polylogarithm oforder n . We have collected the results in table 1. In orderto emphasise the dependence of the degeneracy factor g on the dimension, it is written as g ( D ) in the table.These results correctly reproduce the earlier, -dimensional results found in [15, 31]. Note that the ex-act solution of the peak frequencies of equations (11) hasa different asymptotic behaviour for D → ∞ comparedto the approximation used in [18]. This does not influ-ence the general statement much: Sparsity is lost in highdimensions, Hawking radiation indeed becomes classicaland fully comparable to a black body spectrum. However,the exact dimension where the transition sparse to non-sparse happens changes. In figure 1 we compare the vari-ous sparsities and their dependence on D for massless grav-itons as done in [18], where η Hod ≈ e π (cid:0) πD (cid:1) D +1 . We cansee that the qualitative picture each measure of sparsitydraws is universal — and at least in the massless case this3 peak ,E,n = 12 π ( D − D − ) π ( D − / ( D − W (( D − se µ − D +3 )) Li D ( − se µ )( − s ) λ D − thermal g ( D ) c eff A H η peak ,E,E = 12 π ( D − D − ) π ( D − / ( D + W ( Dse µ − D +2 )) Li D ( − se µ )( − s ) λ D − thermal g ( D ) c eff A H η avg. ,E,n = D π ( D − D − ) π ( D − / Li D +1 ( − se µ )( − s ) (cid:16) Li D ( − se µ )( − s ) (cid:17) λ D − thermal g ( D ) c eff A H η binned = Γ( ( D − / )( D − π √ π D − ( D − Li D − ( − se µ )( − s ) λ D − thermal g ( D ) c eff A H Table 1: Sparsities for emission of massless particles in a D + 1 -dimensional Tangherlini space-time in terms of polylogarithms Li n ( x ) , andLambert-W functions W ( x ) . λ thermal is the thermal wavelength, c eff a correction factor to link capture cross-section with horizon area A H ,and g ( D ) the particles’ degeneracy factor. Figure 1: A comparison of various sparsities η for massless gravitonsin D space dimensions. The constant indicates the transition sparseto non-sparse. can be inferred from the way numerator and denominatorbehave in the definition (3).This is especially important once one takes into accountthe fact that we ignore grey body factors throughout ourcalculation: Their inclusion will push non-sparsity neces-sarily to even higher dimensions. That even their inclusionwill not change the qualitative result, is nonetheless shownby another comparison to the literature: In [32, 33, 34]numerical analysis was performed to take the effects ofgrey body factors into account. Even though these ana-lyses were performed without sparsity as such in mind, itis easy to compare how different particle types will be-have. As the different degeneracy factors g for masslessparticles with different spin depend characteristically onthe space dimension D , let us summarise these for the spinspresent in the standard model of particle physics plus grav-ity: g scalar = 1 , g spin / = 2 n − for D = 2 n or D = 2 n − and assuming Dirac fermions (and counting particles andanti-particles separately), g vector = D − , and g graviton =( D + 1)( D − / . These degeneracies depend, however, onthe specifics of the higher dimensional physics considered:In brane world models they are for all D the familiar, -dimensional ones for emission into the brane [34]. Suchbrane world models are already covered by the presentanalysis — up to a dimension- independent factor this cor- - Figure 2: The binned sparsity η binned for different (massless) particlespecies. responds to looking at the dimension-dependence of thescalar sparsities.The -dimensional case shows amply [15, 17] thatthe inclusion of grey body factors drastically changes thesparsity of, for example, gravitons. Even so, as shown infigure 2, the simplifications made while deriving our ex-pressions for sparsity still qualitatively reproduce the be-haviour of the earlier-mentioned, numerical studies. Whilethe order in which different particles change from η > to η < shows minor changes, the over-all behaviour is re-tained, as is the prediction that emitted gravitons becomeclassical radiation first. This then would correspond to athermal gravitational wave. Before starting the calculations for massive particles, itis a good idea to revisit the effective area A = c eff A H . Thecapture cross-section underlying this approach changes sig-nificantly for massive particles: They become dependenton the particle’s velocity β . While for any massless particle β = c , for massive particles this means that the effectivecapture cross-section diverges to ∞ for particles with velo-city β = 0 . The capture cross-section for massless particlesreappears as the limiting case for β → . Finding a cor-responding effective cross-section for any given β can stillbe done analytically in dimensions, but this fails inhigher dimensions. On top of this, in higher dimensions4table orbits do not exists [23, §7.10.2]; at least assumingthe dynamics of higher dimensional general relativity.To retain an ansatz for the following calculation weshall hence assume that the same effective cross-sectionalarea as for massless particles gives a good approximationfor the area from which Hawking radiation originates. Onthe one hand, in dimensions this seems a good startingpoint as we can expect the massless case to be a limitingcase for massive particles. An example of this approach isfound in [28]: The heuristic arguments based on an ana-logy to the Schwinger effect presented therein cover bothmassive and massless cases; the additionally studied renor-malised stress-energy tensor for massless particles consti-tutes such a limiting case. On the other hand, the assump-tion that this carries over in some way to higher dimensionsis also a good starting point and working hypothesis.These arguments in place, we can head straight for theintegrals involved, only this time with the relation E = k c + m c connecting the momentum k and the energy E of the particle emitted. The strategy here is alwayssimilar: First simplify the integration by rewriting it as theintegration of a geometric sum, then integrating by partsuntil one can make use of the substitution k = z cosh x .This allows employing the identity [29, 3.547.9]: Z ∞ exp ( − β cosh x ) sinh ν x d x =1 √ π (cid:18) β (cid:19) ν Γ (cid:18) ν + 12 (cid:19) K ν ( β ) , (18)valid for Re( β ) > , Re( ν ) > − / . The resulting sumsof modified Bessel functions of the second kind are theexpressions in table 2.At first glance, these seem to be rather unhelpful forfurther analysis. This is not quite the case: For example,remembering that for fixed νK ν ( β ) β →∞ ∼ r π β e − β , (19)tells that for high masses sparsity will be regained in any(fixed) dimension. From a phase-space point of view thisis what physical intuition would suggest. Likewise, asymp-totic expansions for z → will regain our earlier, masslessresults. Similar asymptotic analysis was employed in theservice of separating superradiant regimes from genuineHawking radiation in the analysis of the Kerr space-timein [31] and [15] (though it involved modified Bessel func-tions of the first kind and requires restricting oneself tosparsities fulfilling property (6)).
4. Conclusion
In this letter, we have provided a generalisation to D +1 dimensions of the exact, heuristic, semi-classical results fornon-rotating black holes found in [15] which introduced theconcept of sparsity. We have reproduced and improved on the results of [18], and shown agreement with previ-ous numerical studies [32, 33, 34]. This highlights twothings: First, it demonstrates the robustness of the heur-istic concept of ‘sparsity’. Second, this concept provides aquick, simple, and often pedagogical insight into radiationprocesses, here exhibited on the Hawking radiation from aTangherlini black hole in D + 1 space-time dimensions.Given the propensity of higher dimensional model build-ing encountered in the quest for quantum gravity, it seemsimportant to have an easy-to-calculate, but predictive phys-ical quantity like sparsity that helps to understand differ-ences between such models. This is particularly true forthe prime benchmark that is the Hawking effect: Tradition-ally, a focus for this differentiation between models relieson the connection between entropy and area, and how dif-ferent models vary this more or less severely compared tothe Bekenstein–Hawking result. Using instead a propertyof the emitted radiation (like sparsity) seems experiment-ally more readily accessible than entropy or horizon area.Here, we presented the results for models predicting a dy-namical situation as higher dimensional general relativitywould have. Sparsity is, however, more than just a toolof curved space-time quantum field theory and generalrelativity: Other phenomenological approaches involvinggeneralised uncertainty principles [20, 21], and attemptsto model backreaction [19], further illustrate the use ofthis tool also for other dynamics, as more particle physicsinspired extensions (like string theory) might imply.An obvious extension of the present letter is the ana-lysis of Myers–Perry black holes along the lines of the Kerranalysis in [31] and [15]; for sparsities amenable to thebinning property (6) even a combined superradiance-massanalysis could be performed based on the present results.Less straightforward would be an extension to other high-dimensional models not implying dynamics not akin tothose of general relativity. Acknowledgements
Part of the research presented here was funded by aVictoria University of Wellington PhD Scholarship. Theauthor would like to thank Finnian Gray, Alexander Van-Brunt, and Matt Visser for many helpful discussions.
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