Hawking-Radiation Recoil of Microscopic Black Holes
HHawking-Radiation Recoil of Microscopic Black Holes
Samuel Kov´aˇcik Faculty of Mathematics, Physics and Informatics, Comenius University of Bratislava, Bratislava, Slovakia Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Brno, CzechRepublic
Abstract
The Hawking radiation would make microscopicblack holes evaporate rapidly which excludes themfrom many astrophysical considerations. However,it has been argued that the quantum nature ofspace would alter this behaviour: the temperatureof a Planck-size black hole vanishes and what isleft behind is a Planck-mass remnant with a cross-section on the order of 10 − m which makes directdetection nearly impossible. Such black hole rem-nants have been identified as possible dark mattercandidates. Here we argue that the final stage ofthe evaporation has a recoil effect which would givethe microscopic black hole velocity on the order of10 − c which is in disagreement with the cold darkmatter cosmological model. Quantum theory of gravity is not known at the mo-ment, yet we can assume some of its properties.One of them is that space has a structure that be-comes evident at the Planck scale. This idea isnot new and has been explored from various angles[1, 2, 3, 4, 5, 6, 7].A general feature of theories of quantum space isthe impossibility of distinguishing points with sep-aration on the order of the Planck length. The pho-ton or other particle used to perform the measure-ment would be hidden under its own Schwarzschildradius and would form a microscopic black hole,preventing extraction of any information. Infinitelyshort distances are expected to be absent and dualto it, infinitely large energies.In physics, we often rely on the notion of point-particle matter density (Dirac delta function), often as a mathematical tool or as an approximation. Yetthere is one prominent example where it is takento describe physical reality: the matter distribu-tion of a black hole is zero everywhere but at onepoint. Theories of quantum space often lack thenotion of exact point localisation and any matterdistribution is rendered nonsingular; regular blackholes have been studied before, for example in [8].This affects behaviour of black holes size of which iscomparable to the fundamental length scale. Oneof the striking difference compared to the ordinaryblack hole theory is that the Hawking temperature[9] defined to be proportional to the surface grav-ity at the horizon does not grow indefinitely butinstead drops to zero at small but positive mass,resulting in a microscopic black hole remnant.Black holes remnants have been considered aspossible dark matter constituents [10]. In the or-dinary space, small black holes evaporate rapidly.In quantum space, they can be eternal and arevery difficult to detect due to their miniscule cross-section. If they contributed significantly to theoverall dark matter density, proving it would bedifficult as direct detection seems to be impossible.Gravitational collapse leading to black hole for-mation or merger of two black holes can producegravitational waves carrying a significant momen-tum in a single direction, recoiling the resultingblack hole the opposite way [11, 12]. Here we pro-pose that the Hawking radiation modified by thequantum structure of space leads to a similar ef-fect: thermal radiation recoil.1 a r X i v : . [ g r- q c ] F e b Planck-size black holes
Any effort to distinguish two points of space with aseparation comparable to the Planck-length scaleresults in the production of a microscopic blackhole. Therefore, in a similar consideration thatleads to quantisation of phase space in quantummechanics, one can suspect a quantised structureof the ordinary space. Infinitely short distancesshould be prohibited and dual to it also infiniteenergies.Both of those vices, infinitely short distances andinfinitely large energies, are present in the ordinarydescription of black holes. Their mass is locatedin a single zero-size point and therefore the densitythere has to be infinite. Also, the temperature ofthe Hawking radiation grows to infinity during thefinal moment of the evaporation. Quantum treat-ment of the matter distribution (singularity) makesit regular and prevent the temperature from reach-ing infinitely large values.There were various attempts to investigate theconsequences for the quantum nature of space onthe behaviour of black holes, for example, [8, 13,14]. Various models of space modified on the lengthscale λ may provide different forms of the matterdensity ρ λ ( r ) that reproduces the singular delta-function in the λ → λ tobe the Planck length and express all other dimen-sionful quantities in terms of Planck units.One can take such matter distribution ρ λ ( r )and plug it into the Einstein field equations, look-ing for a solution in the Schwarzschild-like form g = diag (cid:0) f ( r ) , − f − ( r ) , r , r sin θ (cid:1) (with coordi-nates ( t, r, θ, ϕ ) and signature ( − , + , + , +)). Thisapproach has been done for matter densities of theform ρ αλ ( r ) = ρ e − ( r/λ ) α , where the values α = 1 , α = 3 case from considerationsof vacuum polarisations. The value of ρ is fixed ineach of those cases separately. The correspondingEinstein field equation is1 + f ( r ) + rf (cid:48) ( r ) r = 8 πρ αλ ( r ) , (1)we are using units G = c = k B = 1. Solutionsthat satisfies f (0) = − f ( r ) ≈ − Mr for r (cid:29) λ have been be found for α = 1 , , m ∼ λ for which the solution has only one horizon, defined by f ( r ) = 0. For any larger mass m > m there aretwo horizons r = r ± for which f ( r − ) = f ( r + ) = 0.For any smaller mass m < m , there is no solu-tion to f ( r ) = 0 and therefore no horizon. Formasses m (cid:29) m , the larger horizon is close tothe Schwarzchild value r + ≈ m and the smalleris close to the origin r − ≈
0. The function f ( r )for the three considered cases is shown in the figure1, generalisation of those result is discussed in theappendix. r / λ - - f ( r ) f ( r ) for α = ( solid ) , 2 ( dashed ) , 3 ( dotted ) for m = m m m m m m r / λ - - f ( r ) Figure 1:
The function f ( r ) for the three different choices ofmatter density ρ αλ ( r ) = ρ e − ( r/λ ) α evaluated at m = m , where m is the minimal black hole mass. The zero points f ( r ± ) = 0mark the event horizons. For m (cid:29) m one of them is close to theorigin r − ≈ r + ≈ m . As the mass is decreased, they move toward eachother and meet for m = m . At this point, the derivative f (cid:48) ( r )vanishes at the horizon and so does the Hawking temperature.For m < m , f ( r ) = 0 has no solution so there is no horizon.The small plot shows f ( r ) for α = 1 and various values of mass. As expected due to the removal of arbitrarily smalldistances and therefore arbitrary short wavelengthsthe infinite temperatures are avoided, instead ofgrowing indefinitely the temperature drops to zeroas m approaches the minimal mass m and theblack hole becomes frozen. In the figure 2 we showthe temperature dependence for the three consid-ered cases rescaled with respect to the minimummass m . = (2 . / . / . λ and the maximaltemperature T max . = (0 . / . / . λ − where λ − is the Planck temperature.Another shared feature of the considered modelsof regular black holes is that the temperature as a2 m / m T ( m )/ T max T ( m )/ T max for α = ( solid ) , 2 ( dashed ) , 3 ( dotted ) Figure 2:
The hawking temperature of the black hole withmatter density ρ αλ ( r ) = ρ e − ( r/λ ) α expressed in terms ofthe minimal mass m for which the Hawking temperaturevanishes and the maximal temperature. For the consideredcases with α = 1 , , m . = (2 . / . / . λ and T max . = (0 . / . / . λ − where λ − is the Planck temper-ature. Note that when expressed in dimensionless units, all threecases have very similar temperature profiles. function of mass grows very rapidly in the vicinityof m . This means that when the black hole has themass m Tmax for which it reaches the maximal tem-perature, the radiation is so energetic that the massdifference ∆ m = m Tmax − m is radiated in a rela-tively small amount of quanta, N q ≈ ∆ mT max ≤ .Each quantum carries momentum on the order of p ≈ ∆ mN q and due to the conservation law the blackhole receives the opposite momentum. As the ra-diation is random so are the momentum impulsesthe black hole receives. As a result, it performsa random walk in the momentum space, and afterradiating N q quanta will carry momentum of mag-nitude p r ≈ ∆ m √ N q . As a result, its final recoiledvelocity will be on the order of v r ≈ ∆ mm (cid:112) N q (2)This is the recoil effect due to thermal Hawkingradiation of Planck-size black holes. For the con-sider cases of ρ αλ we have ∆ mm . = 0 . / . / .
22 and N αq . = 88 / . / .
8; therefore v αr . = 0 . / . / . c = 1). The primordial black hole scenario builds on the as-sumption that an ensemble of black holes was cre-ated in the earliest stages of the Universe [9, 15, 16].It has been assumed that small black holes wouldhave evaporated by now, however, further researchrevealed that the quantum structure of space wouldprevent complete evaporation. The resulting frozenPlanck-size black holes would be impossible to de-tect due to their minuscule cross-section σ ∼ λ . =10 − m . The only possible evidence could be fromvery short gamma-ray bursts as discussed in [17],unless the radiation process took place before therecombination era. The recoil effect on ordinarymicroscopic black holes in the context of extra spa-tial dimensions has been discussed in [18, 19].The recoil effect due to the Hawking radiationmodified by the quantum structure of space dis-cussed here makes the Planck-size black holes im-probable dark matter candidates as during the lastmoments of radiation they would obtain veloci-ties large enough to be incompatible as cold darkmatter. Velocities of the Planck-size black holeswould also exceed escape velocities from most as-tronomical objects. This considers only the Planck-size black holes, expected behaviour of larger blackholes, such as those analysed in [20], are not af-fected by the present discussion.Our discussion has not been very detailed as wedo not have a detailed description of the quantumgravity and the behaviour of the Hawking radia-tion on this scale. However, at least under currentassumptions the recoil effect due to thermal radia-tion of microscopic black holes should be taken intoconsideration. Acknowledgement
This research was supported by VEGA 1/0703/20and the MUNI Award for Science and Humanitiesfunded by the Grant Agency of Masaryk University.Valuable comments from V. Balek, J. Tekel and N.Werner were greatly appreciated.
Appendix
Similar behaviour as shown for the three consideredcases of matter density ρ λ ( r ) will appear for other3atter densities that are quickly decreasing, mono-tonic and continuous functions of r . The differen-tial equation (1) is of the form rf (cid:48) ( r ) = R ( r ) − F ( r )where R ( r ) = ρ λ ( r ) r and F ( r ) = 1 + f ( r ). Fromthe regularity of the mass density and finitenessof the mass we have R (0) = F (0) = R ( ∞ ) = 0.Quickly decreasing matter density yields depen-dency at least as R ( r ) ≈ /r q with q > F ( r ) ≈ r − and therefore there holds F ( r ) > R ( r ).On the other hand, close to the origin the otherholds, R ( r ) > F ( r ). Therefore, from the continu-ity of the solution we know that there is a point r such that R ( r ) − F ( r ) = 0 and therefore f ( r ) isextremal there. As f (0) = f ( ∞ ) = −
1, if f ( r ) >
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