Hawking radiation may violate the Penrose cosmic censorship conjecture
aa r X i v : . [ g r- q c ] F e b This essay received an Honorable Mention from the Gravity Research Foundation 2019
Hawking radiation may violate the Penrose cosmic censorship conjecture
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: February 11, 2021)
Abstract
We analyze the Hawking evaporation process of Reissner-Nordstr¨om black holes. It is shown thatthe characteristic radiation quanta emitted by the charged black holes may turn near-extremal black-holespacetimes into horizonless naked singularities. The present analysis therefore reveals the intriguingpossibility that the semi-classical Hawking evaporation process of black holes may violate the fundamentalPenrose cosmic censorship conjecture.Email: [email protected] ntroduction . — The seminal work of Hawking [1, 2] has revealed the intriguing fact thatsemi-classical black-hole spacetimes are characterized by filtered black-body emission spectra withwell defined thermodynamic properties [1–3]. Soon after his groundbreaking discovery, Hawkingnoted that the thermally distributed black-hole radiation spectrum may contradict the fundamentalquantum principle of a unitary time evolution [1, 2]. The incompatibility of general relativity andquantum mechanics, as reflected by the Hawking black-hole radiation phenomenon, is certainlyone of the most important open problems in modern physics.In the present essay we would like to discuss another disturbing feature of the Hawking evap-oration mechanism of black holes. In particular, we shall explicitly prove that the Hawking semi-classical radiation process may turn a near-extremal Reissner-Nordstr¨om (RN) black-hole space-time into an horizonless naked singularity which violates the black-hole condition Q ≤ M [4, 5].Thus, our analysis, to be presented below, suggests that the Hawking radiation of black holes mayviolate the fundamental Penrose cosmic censorship conjecture [6, 7] which asserts that spacetimesingularities are always hidden behind event horizons inside black holes. The Hawking evaporation process of near-extremal Reissner-Nordstr¨om black holes . — Weconsider the semi-classical Hawking evaporation process of RN black holes in the near-extremalregime [8] 0 ≤ ∆ ≡ M − Q ≪ M . (1)Note that, for a given value of the electric charge Q , a minimal mass (extremal) black-hole spacetimeis characterized by the simple relation ∆ = 0 [that is, M min ( Q ) = Q ].Our analysis is based on the following two well-known facts [9, 10]:(1) For near-extremal RN black holes in the large-mass regime M ≫ e ~ πm e , (2)the quantum emission of massive charged fields (here m e and e are respectively the proper massand the electric charge of the elementary positron field) is exponentially suppressed as comparedto the Hawking quantum emission of massless neutral fields.(2) In addition, due to the partial back-scattering of the emitted field quanta by the centrifugalbarrier which surrounds the black holes, the neutral sector of the Hawking radiation spectra ofspherically symmetric black holes is dominated by electromagnetic field quanta with unit angularmomentum [11].As we shall explicitly show below, these two facts may allow a near-extremal charged RN black2ole in the regime (2) to jump over extremality by emitting a characteristic neutral Hawkingquantum which reduces the mass of the black hole without reducing its electric charge.For one bosonic degree of freedom, the Hawking radiation power of non-rotating black holes isgiven by the simple integral relation [1] P = ~ π X l,m Z ∞ Γ ωe ~ ω/T BH − dω . (3)Here T BH = ~ ( M − Q ) / π [ M + ( M − Q ) / ] (4)is the Bekenstein-Hawking temperature of the RN black hole which, in the near-extremal regime(1), is characterized by the strong inequality M T BH / ~ ≪
1. The integer parameters l and m arerespectively the spheroidal and axial angular harmonic indices of the emitted field mode and thefrequency-dependent greybody factors Γ = Γ lm ( ω ) in (3) quantify the partial back-scattering ofthe field modes by the curved spacetime outside the black-hole horizon [1].The familiar black-body (thermal) factor ω/ ( e ~ ω/T BH −
1) which appears in the Hawking ex-pression (3) for the black-hole bosonic radiation power implies that the corresponding emissionspectrum has a characteristic peak at the dimensionless emission frequency
M ω peak ∼ M T BH ~ ≪ , (5)in which case the frequency dependent greybody factors Γ lm ( ω ) are given by the simple low-frequency analytical expression [9, 10]Γ m = 19 ǫ ν (1 + ν )(1 + 4 ν ) · [1 + O ( M ω )] , (6)where ǫ ≡ M − Q ) / M + ( M − Q ) / and ν ≡ ~ ω πT BH . (7)Substituting (6) into the semi-classical Hawking relation (3), one finds the compact expression [12] P = ~ ǫ πGM Z ∞ F ( ν ) dν with F ( ν ) ≡ ν + 5 ν + ν e πν − F ( ν ) one learns that the Hawking emission spectra ofthe near-extremal RN black holes have a peak at the characteristic dimensionless frequency [10] ν = ν peak ≃ . . (9)3he corresponding energies of the emitted black-hole quanta are characterized by the simple near-extremal relation [see Eqs. (1), (4), and (7)] E = ~ ω = ~ ν peak r M . (10)The quantum emission of the characteristic neutral Hawking field mode (10) would produce a newspacetime configuration whose mass and electric charge are given by [see Eq. (1)] M new = M − E = Q + ∆ − E and Q new = Q. (11)Intriguingly, one learns from Eq. (11) that the black-hole condition Q new ≤ M new (and withit the Penrose cosmic censorship conjecture [6, 7]) would be violated due to the emission of thecharacteristic Hawking quanta (10) from near-extremal RN black holes in the dimensionless regime∆ < ~ ν peak ) M . (12) Summary . — In the present compact essay we have analyzed the Hawking emission spectraof charged Reissner-Nordstr¨om black holes in the dimensionless near-extremal regime (1). Inter-estingly, it has been shown that the semi-classical radiation spectra of these near-extremal blackholes can be studied analytically in the large-mass regime (2).We have explicitly proved that the characteristic Hawking emission of quantum fields from blackholes may turn an initially near-extremal RN black hole with [see Eqs. (1), (4), and (12)] [13] T BH < ν peak ~ πM (13)into an horizonless naked singularity which is characterized by the inequality Q new > M new [10].Our analysis has therefore revealed the intriguing fact that the Hawking evaporation process ofblack holes may violate the fundamental Penrose cosmic censorship conjecture.4 CKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I thank Don Page for interestingcorrespondence. I would also like to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and AlonaB. Tea for stimulating discussions. [1] S. W. Hawking, Commun. Math. Phys. , 199 (1975).[2] S. W. Hawking, Phys. Rev. D , 2460 (1976).[3] J. D. Bekenstein, Phys. Rev. D , 2333 (1973).[4] Here M and Q are respectively the gravitational mass and the electric charge of the black hole. Weshall assume Q > G = c = k B = 1.[6] R. Penrose, Riv. Nuovo Cimento I , 252 (1969).[7] R. Penrose in General Relativity, an Einstein Centenary Survey , eds. S.W. Hawking and W. Israel(Cambridge University Press, 1979).[8] Note that charged RN black-hole spacetimes are characterized by the inequality M ≥ Q (∆ ≥ Q > M .[9] D. Page, arXiv:hep-th/0012020 .[10] S. Hod, The Euro. Phys. Jour. C (Letter) , 634 (2018).[11] In particular, due to the presence of the centrifugal barrier which surrounds the emitting black holes,the Hawking radiation power of massless (electromagnetic and gravitational) fields with larger valuesof the angular momentum parameter ( l >
1) is suppressed by several factors of the large ratio M/ ∆ ascompared to the Hawking radiation power of unit angular momentum ( l = 1) photons [9, 10].[12] Here we have used the relations [see Eqs. (1) and (7)] ǫ = p /M · [1 + O ( ǫ )] and ω = ν p /M · [1 + O ( ǫ )] for the near-extremal ( ǫ ≪
1) black holes.[13] Here we have used the relation T BH = ~ p ∆ / π M · [1 + O ( ǫ )] [see Eqs. (1), (4), and (7)].)] [see Eqs. (1), (4), and (7)].