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Total Spectral Distributions from Hawking Radiation Eur. Phys. J. C (2017) 77:756 https://doi.org/10.1140/epjc/s10052-017-5336-1 Regular Article - Theoretical Physics Total spectral distributions from Hawking radiation Bogusław Brodaa Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Łód´z, Pomorska 149/153, 90-236 Lodz, Poland Received: 23 June 2017 / Accepted: 29 October 2017 / Published online: 10 November 2017 © The Author(s) 2017. This article is an open access publication Abstract Taking into account the time dependence of the and especially in [10]. The finiteness of N is not quite unex- Hawking temperature and finite evaporation time of the black pected, but it is not obvious for gapless particles, e.g. for hole, the total spectral distributions of the radiant energy and photons. Moreover, this fact (i.e., the finiteness of N) could of the number of particles have been explicitly calculated and possibly also influence discussion around the BH information compared to their temporary (initial) blackbody counterparts paradox [5]. Since we deal with a finite process (te < +∞), (spectral exitances). and the Hawking temperature TH is a (non-constant) function of time, what could be physically more representative for the BH evaporation process than the initial or temporary black- 1 Introduction body spectrum expressed by the spectral radiant exitance eω (see (2)) is the total spectral distribution of energy Eω coming One of the most famous features of the Hawking black hole from the total particle production from the BH. Therefore, (BH) radiation is its blackbody thermal (Planckian) spectrum the aim of the present work is to calculate the total spectral [1–5]. However, it is also well known that the BH blackbody distribution of energy, Eω, and its actinometric counterpart, thermality is modified in a number of ways [6,7]. Actually, Nω (the total spectral distribution of the number of parti- the shape of the BH spectrum is only approximately Planck- cles), both emitted during the whole BH evaporation time te. ian, with certain key modifications, especially for small BHs. In other words, we assume complete evaporation (the final A direct consequence of the Hawking radiant flux from the mass Mf = 0). This assumption implicitly ignores possible BH is its evaporation [1,2], i.e., the mass M of the BH mono- modifications of the evaporation process for the temporary tonically decreases in time. Moreover, since the BH black- mass of the order of the Planck mass, when unknown quan- body temperature TH (the Hawking temperature) depends on tum gravity effects should probably be taken into account. the reciprocal of M (see (11)), TH is a growing function of For simplicity, we also assume that at any instant of time time [1,2]. The author of [6] has presented an elegant and the radiation is given by the Hawking blackbody formula for exhaustive discussion of various modifications and limita- the Schwarzschild BH. Moreover, we limit our discussion to tions of the thermality of the Hawking radiation. According one species of massless particles, i.e., to photons. Thanks to to [6] the Planckian shape of the Hawking spectrum will be these simplifications, we are able to express both formulas modified by at least three distinct physical effects: graybody in closed analytical forms as polylogarithm functions. The factors, adiabaticity constraints, and available phase space. both calculated distributions are not only interesting in itself, From our perspective, somewhat arbitrarily, we could add to but also they give rise to the approximate notion of the (time) the list the following (not quite independent, nor new) mod- average(d) temperature T (see (30) and (31)), which we aim ifications of the thermality: the BH radiation lasts for a finite to estimate, as a quantity alternative to the (temporary) Hawk- period of time te (the evaporation time)[1,2], and the total ing temperature TH. number N of the particles emitted by the BH is finite. Inter- According to standard terminology, the notion radiomet- estingly, it appears that the particles are being emitted from ric refers to the quantities corresponding to radiant energy, the BH (in a sense) rather rarely [8]. The mentioned (finite) whereas the notion actinometric(photometric or photonic) total number N (strictly speaking, its average) of particles refers to the quantities corresponding to the number of radi- (photons) emitted has been calculated quite recently in [9], ant particles (photons) [11]. In the case of spectral (i.e., ω- dependent) quantities, both possibilities are multiplicatively a e-mail: [email protected] related with the Planck multiplier h¯ ω. The simplest relation 123 756 Page 2 of 5 Eur. Phys. J. C (2017) 77 :756 of this type is where A denotes the area of the surface of the source. Analo- gously, the total spectral distribution of energy, we are most eω = h¯ ωNω, (1) interested in, is defined by where eω is the spectral radiant exitance, in the case of the Eω = Aeωdt, (8) blackbody, given by the well-known Planck law [12](cf. [13,14]) whereas the total spectral distribution of the number of par- ticles defined by h¯ ω3 1 h¯ ω3 − h¯ ω = · ≡ kBT , eω ¯ ω/ Li0 e (2) 4π 2c2 eh kBT − 1 4π 2c2 Nω = ANωdt (9) and Nω is the corresponding spectral particle (photon) exi- tance assuming the standard form is directly related to (8) by the relation (cf. (1)) ω = ¯ ω ω. ω2 1 ω2 − h¯ ω E h N (10) N = · ≡ kBT . ω ¯ ω/ Li0 e (3) 4π 2c2 eh kBT − 1 4π 2c2 Here, h¯ is the reduced Planck constant (h/2π), c the speed 2 Total number of particles of light, kB the Boltzmann constant, ω the angular frequency, T the blackbody temperature, and Li is the polylogarithm For the Schwarzschild BH, the spectral radiant exitance eω, 0 N function Li of the zeroth order (see the appendix). The and the spectral particle (photon) exitance ω isgivenbythe notation assumed in the paper is not quite standard—see standard thermodynamic formulas (2), and (3), respectively, Chapts. 34 and 36 in [15] for the standard notation. with T equal the Hawking BH temperature TH, i.e., Except where necessary, for clarity, we confine ourselves hc¯ 3 only to radiometric quantities (actinometric quantities can T = T ≡ , H π (11) easily be reproduced according to (10)). Nevertheless, we 8 kBGM aim to treat both types of quantities equally and comple- where G is the Newton gravitational constant. menting each other. Analogously, the total radiant exitance e, and the total The total radiant exitance, denoted in our paper by e,is particle (photon) exitance N is given by another couple of given (for the blackbody) by the famous Stefan–Boltzmann standard thermodynamic formulas, (4) and (5), respectively, law, with T = TH. The total radiant energy (see (6)) is directly given by the ∞ π 2 4 kB 4 4 e = eωdω = · T ≡ σ T , (4) Einstein mass–energy equivalence formula 2 3 0 60 c h¯ E = Mc2, (12) where σ is the Stefan–Boltzmann constant, whereas the cor- responding total particle (photon) exitance, whereas the total number N of particles (photons) emitted by the BH has been determined quite recently in [9,10]. ∞ ζ( ) 3 Since, in the next section, for technical reasons, we will 3 kB 3 N = Nωdω = · T , (5) 2π 2 c2h¯ 3 need in our calculations the formula expressing the mass flow 0 rate dM/dt in terms of the mass M, involved in the derivation of the evaporation time te, we rederive it in the present sec- with ζ(3) ≈ 1.20206. For processes being considered in a tion. As a byproduct of our derivation, we compute the total finite time interval, we can introduce the notion of the total number N of particles (photons) emitted during the whole energy, expressed by integration within this time interval, BH evaporation process. By virtue of the equivalence formula (12), and of the Stefan–Boltzmann law (4), the mass flow rate E = Ae dt, (6) dM 1 dE 1 1 =− =− Ae =− Aσ T 4, (13) dt c2 dt c2 c2 and the notion of the total number of particles, where dE/dt can be interpreted as the velocity of evapora- tion. Inserting to (13) the standard BH horizon surface area = N , N A dt (7) formula, 123 Eur. Phys. J. C (2017) 77 :756 Page 3 of 5 756 16πG2 or, equivalently, using the chain rule as a technical trick, A = M2, (14) c2 −1 dEω dM and the Hawking BH temperature, Eq. (11), we obtain the dif- = Aeω. (21) dM dt ferential equation expressing the mass flow rate in terms of M 4 Making use of the mass flow rate equation (15), the area dM hc¯ − =− M 2. (15) equation (14) and the Planck law (2), we obtain from (21) dt 15 · 210πG2 the (differential) element of the total spectral distribution of This equation yields the evaporation time energy 12 4 π · 10π 2 15 · 2 G − 8 G ωM 5 2 G =− ω3 4 c3 , t = M3, (16) dEω M Li0 e dM (22) e hc¯ 4 0 c10 where M0 denotes the initial mass of the BH. or in terms of an (auxiliary) dimensionless BH mass As a byproduct of our derivation (15), we can now calcu- 8πG late the total number N of particles (photons) emitted during x ≡ ωM, (23) c3 the whole BH evaporation process. To this end, we will apply 5 15c −2 4 −x a technical trick which consists in using the chain rule to the dEω =− ω x Li (e )dx. (24) π 5 0 number flow rate (given by differential form of (7)), and next 8 G performing some further formal manipulations (insertions).
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