Micron-sized black holes do not necessarily have a constant horizon
temperature distribution. The black hole remote-sensing problem means to find
out the `surface' temperature distribution of a small black hole from the
spectral measurement of its (Hawking) grey pulse. This problem has been
previously considered by Rosu, who used Chen's modified Moebius inverse
transform. Here, we hint on a Ramanujan generalization of Chen's modified
Moebius inverse transform that may be considered as a special wavelet
processing of the remote-sensed grey signal coming from a black hole or any
other distant grey sourceComment: 5 pages, published versio

Chen N. X.

HARET C. ROSU

MICHEL PLANAT

Page D. N.

Visser M.

English

arXiv

Rosu, Haret

Planat, Michel

HAL - Université de Franche-Comté

On the arithmetic detection of grey pulses with application to Hawking radiation

HAL Descartes

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ON ARITHMETIC DETECTION OF GREY PULSES WITH APPLICATION TO HAWKING RADIATION

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"Micron-sized black holes do not necessarily have a constant horizon temperature distribution. The black hole remote-sensing problem means to find out the "surface" temperature distribution of a small black hole from the spectral measurement of its (Hawking) grey pulse. This problem has been previously considered by Rosu, who used Chen's modified Möbius transform inverse transform. Here, we hint on a Ramanujan generalization of Chen's modified Möbius transform inverse transform that may be considered as a special wavelet processing of the remote-sensed grey signal coming from a black hole or any other distant grey source.

Rosu Barbus, Haret-Codratian

Michel Planat

Repositorio IPICYT

This is the Author's Pre-print version of the following article: HARET C. ROSU and MICHEL PLANAT, Mod. Phys. Lett. A 17, 1377 (2002).  Electronic version of an article published as https://doi.org/10.1142/S0217732302007430  © World Scientific Publishing Company http://www.worldscientific.com/worldscinet/mpla       ON ARITHMETIC DETECTION OF GREY PULSESWITH APPLICATION TO HAWKING RADIATIONHARET C. ROSUhcr@ipicyt.edu.mxDept. of Applied Mathematics, IPICyT, Apdo Postal 3-74 Tangamanga, San Luis Potośı, MEXICOMICHEL PLANATplanat@lpmo.eduLaboratoire de Physique et Métrologie des Oscillateurs du CNRS, 25044 Besançon Cedex, FRANCEMicron-sized black holes do not necessarily have a constant horizon temperature distribu-tion. The black hole remote-sensing problem means to find out the ‘surface’ temperaturedistribution of a small black hole from the spectral measurement of its (Hawking) greypulse. This problem has been previously considered by Rosu, who used Chen’s modifiedMoebius inverse transform. Here, we hint on a Ramanujan generalization of Chen’s mod-ified Moebius inverse transform that may be considered as a special wavelet processing ofthe remote-sensed grey signal coming from a black hole or any other distant grey source.Keywords: Hawking radiation, Moebius transform, Ramanujan sums, grey-body, blackhole, wavelets1. INTRODUCTIONThere may exist black holes in the micron size range carrying on some external dis-tribution of matter. Theoretical examples are (i) down-scaled Weyl black holes,w forwhich the metric potentials are solutions of polar Laplace equations (ii) black holeswith Einstein shells,e (iii) primordial black holes (PBH) and/or any mini black holehovering through the universe and carrying on some matter distributions, (iv) hairyblack holes,bow with additional conserved quantum numbers beyond those allowedby the classical no hair theorems and dirty black holes in the sense of Visser,v i.e.,black holes in interaction with various classical fields, for which the Hawking tem-perature appears to be supressed relative to the vacuum black holes of equal area.In some cases, for small enough black holes, the external distribution of matter canbe of such a kind as to disturb only slightly the pure horizon Hawking radiation andconsequently from the praxis standpoint we have a grey-body radiation problem.Hawking radiation by itself is distorted with respect to a pure black-body spec-trum, especially in the low frequency regime due to a grey-body factor usuallyidentified with the square of the absorption amplitude for the mode.be A usefulwork on the nature of the grey body problem for black holes has been written bySchiffer.sIn this letter, we first review the grey body inverse problem and the modifiedMoebius inverse transform (Chen’s transform) in sections 2 and 3, respectively.roIn section 4, we hint on a possible Ramanujan extension of Chen’s transform withpossible application to small black holes and the cosmological background radiation.2. INVERSE GREY-BODY PROBLEMPlanck’s law provides the analytical formula for the emitted power spectrum fromblack body sources. In laboratory physics the emitted power spectrum is also calledspectral brightness, or spectral radiance of the black body radiation. The latternotion is used in radiometry to characterize the spectral properties of the source asa function of position and direction from the source. For point, i.e., far away, greysources the total radiated power spectrum, also called radiant spectral intensity isW (ν) ∼∫ ∞0A(T )B(ν, T )dT , (1)whereA(T ) is the area temperature distribution of the grey body,B(ν, T ) is the Boltzmann-Planck occupation factor.Finding out A(T ) at given W (ν) is known as the inverse grey-body problem.l W (ν)may be known either experimentally or within some theoretical model. This in-verse problem was solved in principle by Bojarski,bo by means of a thermodynamiccoldness parameter u = h/kT , and an area coldness distribution a(u), as moreconvenient variables than T and A(T ) to get an inverse Laplace transform of thetotal radiated power. The coldness distribution is obtained as an expansion in thisLaplace transform. Explicitly, the total grey power spectrum is rewritten as anintegral over the coldness variableW (ν) =2hν3c2∫ ∞0a(u)exp(uν) − 1du (2)and furthermore asW (ν) =2hν3c2∫ ∞0exp(−uν)[ ∞∑n=1(1/n)a(u/n)]du . (3)Therefore the sum under the integral that we shall denote by f(u) is the Laplacetransform of g(ν) = c22hν3 W (ν) and the inverse Laplace transform of g will providethe sought coldness distribution.Despite the formal mathematical solution the inverse grey body problem is un-stable for most numerical implementations, i.e., it belongs to the broad class ofill-posed inverse problems.x23. MODIFIED MOEBIUS TRANSFORM (MMT)Chen,c obtained a(u) by means of the so-called modified Moebius transform ( MMT)of f(u)a(u) =∞∑n=1μ(n)nf(u/n) . (4)To understand Eq. (4) we recall a few basic results from the theory of numbers.baThe Moebius expansion refers to special sums over prime factor divisors, (d-sums)running over all the prime factors of n, 1 and n included, of any function f(n)defined on the positive integersSf (n) =n∑d|nf(d) . (5)The remarkable fact in this case is that the last term of the sum can be writtenin turn as a sum over the Sf arithmetical functions. The latter sum is called theinverse Moebius transform (or the Moebius d-sum) of ff(n) =n∑d|nμ(d)Sf (n/d) , (6)in which the d-sum Sf (n) becomes the first term of the Moebius d-sum, and whereμ(d) is the famous Moebius function. Since at the left hand side of (6) one has onlya term of a d-sum whereas on the right hand side there is a sum of d-sums thereis clear overcounting, unless the Moebius function is sometimes either naught ornegative. The partition of the prime factors of n implied by the Moebius functionis such that, by definition, μ(1) is 1, μ(n) is (−1)r if n includes r distinct primefactors, and μ(n) is naught in all the other cases. In particular, all the squareshave no contribution to the inverse Moebius transforms. That is why the integersselected by the Moebius function are also called square-free integers.Chen’s MMT means to apply such an inversion of finite sums to infinite summa-tions, and to ordinary functions of real continuous variable(s). MMT means thatify1(x) =∞∑n=1y2(x/n) (7)theny2(x) =∞∑n=1μ(n)y1(x/n) (8)For the inverse grey-body problem, y1(u) = uf(u) and y2(u) = ua(u). So, onecan get the coldness distribution by multiplying the Laplace transform of the totalpower spectrum by the coldness parameter, and then applying the MMT.34. RAMANUJAN GENERALIZATION OF CHEN’S TRANSFORMRamanujan sums are well known in number theory but only recently some physicalapplications have been suggested.p They are of the formcq(m) =q∑p=1cos(2πmp/q) (9)with irreducible fractions p/q. Due to the orthogonality property cq(m)cs(m) =δqs and obvious periodicities they may be considered as periodic wavelets. Sincecq(1) = μ(q) they may be seen as a generalization of the Moebius function. TheMMT for black holes in the Ramanujan notation will be[ua(u)](1) =∞∑q=1cq(1)uf(u/q) , (10)but the point is that many other Ramanujan inverse transforms can be introducedthrough[ua(u)](i) =∞∑q=1cq(i)uf(u/q) , (11)for integers i > 1. One can see that a sequence of two-dimensional analyses (i, q)of the signal are available in this approach similarly to the time-frequency analysisthat is so characteristic to wavelets.As an example, for micron-sized Schwarzschild black holes (M ∼ 1024 g), no knownmassive particles are thermally emitted, and according to the calculations of Page,paabout 16% of the Hawking flux goes into photons, the rest being neutrino emission.Let us consider these black holes as grey objects of the following two classes (i) bytheir own,be and (ii) of the Weyl type. The coldness parameter will be in the firstcase uS = 1ν ln(1 + eβhh̄ω−1Γ(ω)), where βh is the horizon inverse temperature param-eter, and Γ(ω) is the penetration factor of the curvature and angular momentumbarrier around the black hole, as can be infered from the work of Bekenstein,bewhereas in the latter case uS = h/kTd, where Td can be considered as an effectivehorizon temperature of the distorted black holes Td = (8πM)−1 exp(2U), where Uis given in the work of Geroch and Hartle. With the coldness parameter at handone can apply the aforementioned number methods for the black hole emissivityproblem.Before closing, we mention that an alternative viewpoint on MMT developed byHughes et al,hu in terms of Mellin transform and Riemann’s ζ function is alsoextremely interesting if one takes into account that the Bekenstein-Mukhanov spec-trum, bm could be considered as the eigenvalue problem of relativistic Schroedingerequations in finite differences.berThus, it appears that a richer information content of the spectrum of importantastrophysical signals that may reveal hidden discrete features can be obtained by4extensive use of number theoretical techniques. The cosmological background radi-ation signal can be studied by the same approach and with the same aim.REFERENCES[w] H. Weyl, Ann. Phys. 54 (1917) 117; for recent works, see R. Geroch and J.Hartle, J. Math. Phys. 23 (1982) 680; J.P.S. Lemos and P.S. Letelier, Phys. Rev.D49 (1994) 5135, and references therein.[e] A. Einstein, Ann. Math. 40 (1939) 922. For recent papers see P.R. Brady, J.Louko, and E. Poisson, Phys. Rev. D44 (1992) 1891; G.L. Comer, D. Langlois, andP. Peter, Class. Quantum Grav. 10 (1993) L127; G.L. Comer and J. Katz, Class.Quantum Grav. 10 (1993) 1751.[bow] M.J. Bowick et al, Phys. Rev. Lett. 61 (1988) 2823.[v] M. Visser, Phys. Rev. D46 (1992) 2445.[be] J.D. Bekenstein, Phys. Rev. Lett. 70 (1993) 3680; C.F.E. Holzhey and F.Wilczek, Nucl. Phys. B380 (1992) 447; V. Balasubramanian and F. Larsen, Nucl.Phys. B495 (1997) 206; J.M. Maldacena and A. Strominger, Phys. Rev. D55 (1997)861.[s] M. Schiffer, Gen. Rel. Grav. 27 (1995) 1, and references therein.[ro] H.C. Rosu, Mod. Phys. Lett. A13 (1998) 695.[l] For a concise review, see A. Lakhtakia, Mod. Phys. Lett. B5 (1991) 491.[bo] N.N. Bojarski, IEEE Trans. Antennas Propag. 30 (1982) 778.[x] Xin Tan et al, J. Opt. Soc. Am. A11 (1994) 1068.[c] N.-x. Chen, Phys. Rev. Lett. 64 (1990) 1193 [E: ibidem 64 (1990) 3202]; J.Maddox, Nature 344 (1990) 377.[ba] A. Baker, A Concise Introduction to the Theory of Numbers, (Cambridge Uni-versity Press, Cambridge 1984).[p] M. Planat, Ramanujan-Fourier transform for low frequency noise spectroscopy,contribution for the Noise Conference, Florida, (May 2002); M. Planat, H.C. Rosu,S. Perrine, Arithmetical chaology and the signatures of 1/f noise, contribution forTH-2002, Paris, (July 2002).[pa] D.N. Page, Phys. Rev. D13 (1976) 198.[hu] B.D. Hughes, N.E. Frankel, B.W. Ninham, Phys. Rev. A42 (1990) 3643.[bm] J.D. Bekenstein, V.F. Mukhanov, Phys. Lett. B360 (1995) 7.[ber] V.A. Berezin, A.M. Boyarsky, A. Yu. Neronov, On the spectrum of relativisticSchroedinger equation in finite differences, gr-qc/9902028.5

On arithmetic detection of grey pulses with application to Hawking radiation

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On arithmetic detection of grey pulses with application to Hawking
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On arithmetic detection of grey pulses with application to Hawking radiation

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