Black Hole Evaporation as a Cosmic Censor ˙ Koray D¨ uzta¸s∗ and Ibrahim Semiz† Bo˘gazi¸ci University, Department of Physics ˙ Bebek 34342, Istanbul, Turkey In a recent work we have shown that it is possible to overspin a nearly extremal Kerr black hole by using integer spin test fields at a frequency slightly above the superradiance limit. In this work we incorporate the quantum effect of evaporation into the problem. We consider a nearly extremal evaporating black hole interacting with challenging test fields. Evaporation refers to either Hawking radiation or radiation from stationary Kerr black holes calculated by Starobinsky and Unruh in advance, which agree in the limit the surface gravity (temperature) tending to zero. We note that evaporation acts as a cosmic censor since it carries away the angular momentum of the black hole, proportionally more than its mass. The relevant amount of angular momentum carried away depends on the initial mass of the black hole and the period of interaction with the test field. We evaluate the efficiency of evaporation to prevent overspinning of black holes of different masses, against the maximum effect due to challenging test fields. We make an order of magnitude estimate to show that evaporation can prevent overspinning of black holes with an upper limit of mass M . 1017 g,, when we take the interaction period to be the age of the universe. Overspinning of black holes of higher masses by test fields remains possible, even if evaporation is taken into account. We also discuss the possibility to attribute a shorter interaction period for the problem which would reduce the effect of evaporation.
I.
INTRODUCTION
Development of the singularity theorems by Penrose [1] can be considered as the first genuine result in general relativity after Einstein. (See [2] for a review.) The original concepts such as geodesic completeness and trapped surfaces which were introduced in this work and also used in the following extensions of singularity theorems by Penrose and Hawking [3], became fundamental notions in black hole physics, cosmology, and mathematical and numerical relativity. Causal geodesic completeness requires that every timelike and null geodesic can be extended to arbitrarily large affine parameter value both into the future and into the past. Singularity theorems interpret the failure of causal geodesic completeness as the condition that a space-time containing a trapped surface possesses a singularity, provided that some generic conditions are also satisfied. In classical general relativity, trapped surfaces arise in the spherically symmetric gravitational collapse of a body, thus a singularity ensues. In the model developed by Penrose and Hawking the trapped surface is contained in the black hole (b.h.) region of the space-time, so it is surrounded by an event horizon. This singularity can be considered harmless as opposed to a naked one which intersects a Cauchy surface rendering the initial conditions undefined, thus disabling asymptotic predictability. The regions containing naked singularities also allow the evolution of closed time-like curves which violate causality [4]. To avoid these pathologies and preserve the deterministic nature of general relativity Penrose proposed the Cosmic Censorship Conjecture (CCC) [5]. In
∗ †
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its weak form, CCC asserts that the singularities that arise in gravitational collapse are always hidden behind event horizons. Distant observers do not encounter singularities or any effects propagating out of them. Conjecturing singularities to be inaccessible to distant observers assures the consistency of the theory of general relativity. It has not been possible to establish a concrete proof of CCC. Wald constructed an alternative problem to test the stability of event horizons when the black hole interacts with test particles or fields [6]. He considered a stationary Kerr-Newman space-time uniquely defined by three parameters (Mass M , charge Q, and angular momentum per unit mass a), satisfying M 2 ≥ Q2 + a2 .
(1)
(1) is valid for black holes surrounded by event horizons while it is violated by naked singularities. After the black hole absorbs some particles or fields coming from infinity the space-time is expected to settle to another stationary configuration with new values of M , Q, and a. If it is possible to reach a final configuration of the parameters which violates (1), the black hole can turn into a naked singularity and CCC is violated. This problem is formulated between two stationary states in the classical context, and ignores the flux out of the black hole due to evaporation. The most general form of black hole evaporation is the Hawking radiation [7], which follows the discovery that certain waves are amplified during reflection from rotating black holes [8–11]. On a quantum particle description, amplification of waves (superradiance) corresponds to a stimulated emission of particles. However, Bogoluibov transformations formalism for particle creation does not apply to stationary space-times such as Kerr family of solutions. Therefore, Hawking considered the time dependent phase of a gravitationally collapsing body in a context that treats matter
2 fields quantum mechanically on a classical curved spacetime background. This is a good approximation to a full quantum theory of gravity outside the regions of extremely high curvature which can be encountered near a singularity. Hawking found a steady flux of particles reaching future null infinity I + . The average number of particles with energy ω and angular momentum numbers l, m is Nωlm = Γlm (ω){exp[2πκ−1 (ω − mΩH )] ∓ 1}−1
We make an order of magnitude estimate to evaluate the efficiency of evaporation to prevent overspinning of black holes of different masses, against the maximum effect due to challenging test fields. In the calculations we take the interaction period to be the age of the universe. We also discuss the possibility to attribute a shorter interaction period for the problem which would reduce the effect of evaporation.
(2)
where the minus and plus signs apply to bosons and fermions respectively, κ is the surface gravity, and Γlm (ω) is the fraction of a purely outgoing wave-packet of frequency ω at I + , that would propagate through the collapsing body to I − when traced back in time. At sufficiently late times this equals the fraction of an ingoing wave-packet sent from I − , that would cross the horizon of the black hole which is the analytic extension of the collapsing space-time; i.e. the fraction that would be absorbed by the black hole. Note that when a bosonic wave 2 packet with frequency ω < mΩ (where Ω = a/(r+ +a2 ) is the rotational frequency of the black hole) is sent towards a Kerr black hole, the requirement that Nωlm is positive implies Γlm (ω) is negative so that the scattered part of the wave has a larger amplitude than the original incoming wave. This is the well-known effect of superradiance exhibited by bosonic fields [9]. For fermions Γlm (ω) remains positive for all frequencies which is in accord with the fact that they do not exhibit superradiant scattering. Hawking radiation as formulated in (2) refers to a higher rate of emission of particles with positive angular momentum m than negative angular momentum −m with the same frequency ω and quantum number l. In this way the emission of particles carries away the angular momentum of the black hole [7], working in favour of the inequality (1). The essential difference between Hawking radiation and spontaneous emission by stationary black holes calculated by Starobinsky [10, 11] and Unruh [12] in advance is that emission occurs in all modes; not only in superradiant modes. The temperature κ/2π tends to zero for massive black holes and even for tiny ones that are nearly extremal. In this limit, Hawking radiation (2) allows the emission of particles only in superradiant modes ω < mΩ with ∓Γlm . This flux of particles equals to those calculated by Starobinski and Unruh for a stationary Kerr black hole [7]. Superradiant modes carry higher angular momentum than energy, serving to reinforce (1). Therefore evaporation of black holes acts as a cosmic censor, considering both the general case of Hawking radiation and κ → 0 limit which is more relevant for problems testing the validity of CCC. In this work we consider a nearly extremal evaporating Kerr black hole, considering either the Hawking radiation or radiation from stationary Kerr black holes calculated by Starobinsky and Unruh in advance, which agree in the limit κ → 0. We check if it is possible to destroy the horizon of the black hole by sending in test fields as we did in [13], while evaporation acts as a cosmic censor.
II.
EVAPORATING BLACK HOLES AND CCC
Consider a nearly extremal Kerr black hole at early times with parameters M and a = J/M , satisfying M 2 ≥ a2 .
(3)
The black hole absorbs free fields incident from infinity where the space-time is asymptotically flat, and it keeps evaporating during the period. The interaction takes place in a period ∆t which is the time it takes the fields to reach the black hole from infinity. In Wald type problems it is sufficient to assume that the interaction period is long enough. When one takes evaporation into account one has to attribute a value to the interaction period ∆t to make an estimate of the amount of angular momentum radiated away. In this work we take ∆t to be the age of the universe. In other words we send in massless test fields from a distance of the size of the universe. Finally we shall also discuss the possibility to attribute a shorter interaction period for the problem. In fact the steady flux of particles reaching I + described in (2) means that the black hole will not be in a stationary state, thus can not be described by the Kerr metric. However the evaporation of the black hole is so slow that it can be described by a sequence of stationary solutions parametrized by M and a. This quasistationary approximation is valid until the mass of the black hole is reduced to the Planck mass 10−5 g [7]. So we formulate the problem as follows: Initially we have a nearly extremal Kerr b.h. satisfying (3) and we send in massless test fields from infinity, which we define as the asymptotically flat region. After a period ∆t the fields reach the b.h., which has been evaporating during the period. After the field interacts with the b.h. we have our final configuration of M and a. We check if the final configuration can violate (3), which is reinforced by evaporation and challenged by the incoming field. In being scattered by the b.h. the fields cause changes in the parameters dM and dJ, which satisfy [14]: dJ = (m/ω)dE
(4)
where dE = dM for the black hole. The condition for CCC violation in terms of angular momentum is Cfin ≡ (M + δE)2 − (J + δJ) < 0
(5)
where Cin ≡ M 2 − J. In the previous work [13] we have used a parametrization for closeness to extremality in the
3 form J/M 2 = a/M = 1 − 2ϵ2 ,
(6)
where ϵ ≪ 1 is implied. This turns (5) into δJ > 2ϵ2 M 2 + 2M δE + δE 2 . One can use (4) to convert this to ( ω0 ) 0 > 2ϵ2 M 2 + 2M 1 − δE + δE 2 ω
(7)
(8)
where ω0 ≡ m/2M . In that work we have shown that there exists a combination ω, and δE for any integerspin test field incident on a slightly subextremal Kerr black hole, that can overspin the black hole into a naked singularity. The frequency has to be in the range ωsl < ω < ω1 where √ ωsl is the superradiance limit mΩ be and ω1 ≡ ω0 /(1 + 2ϵ). The frequency interval can √ parametrized as ω = ω0 +(s−2)ϵω0 where 0 < s < 2− 2 to first order in ϵ. Then δE must be chosen in the range [ ] √ δE1,2 = (2 − s) ∓ (2 − s)2 − 2 ϵM, (9) to violate CCC. For extremal black holes (ϵ = 0) we have ωsl = ω1 so the interval vanishes. Therefore CCC can not be violated. The existence of a lower limit is merely due to superradiance. Had we used fermionic fields instead, the lower limit would have reduced to zero. In that case extremal black holes could also be destroyed, as long as we stay in the classical picture (see [15] for a general discussion). However the physical meaning of a classical fermionic field is not clear. We have mentioned that Cfin can be made negative. Before considering the effect of radiation let us check the highest negative value that can be attained by sending in the challenging fields ωsl < ω < ω1 . The highest contribution comes from the central value of δE in (9); that is δEc = (2 − s)M ϵ. This gives Cfin (δE = δEc ) = M 2 ϵ2 [2 − (2 − s)2 ] (10) √ where 0 < s < 2 − 2. The right hand side of (10) gives the highest negative value that can be attained by using each frequency in the range ωsl < ω < ω1 . We may come arbitrarily close to the lower limit ωsl (s = 0) to get the (absolute) maximum of these values; that is −2M 2 ϵ2 . So, for example let us start with a b.h. J/M 2 = a/M = 0.999 (Cin = 0.001M 2 ) and send in a test field with frequency slightly over ωsl , we end up with a value arbitrarily close to −0.001M 2 for Cfin and violate CCC. In this context an evaporating Kerr b.h. with initial values J/M 2 = a/M = 0.999, which has radiated away an amount of ∼ 0.001M 2 of its angular momentum in the period of interaction, can also be overspun by this test field as Cfin ∼ 0. Because the nearly extremal b.h. radiates mainly in the superradiant range, we can use the T = κ/2π → 0 approximation. Scalar particles and neutrinos are produced at a similar rate [12], photons and gravitons are
produced more copiously [10, 11].The rates at which the b.h. loses mass and angular momentum due to evaporation of scalar particles is given by the fluxes at infinity (see [16]): ∫ e−ζ 2 dM = lim dθdϕ⟨Trt ⟩vac ∼ − Ω r→∞ dt 4π ∫ dJ e−ζ = − lim dθdϕ⟨Trϕ ⟩vac ∼ − Ω r→∞ dt 2π
(11) (12)
m where ζ is a number of the order of unity. For a nearly extremal b.h. J ∼ M 2 e−ζ −1 M ∆t 4π e−ζ −2 ∆M ∼ − M ∆t 16π
∆J ∼ −
(13) (14)
With neutrino, photon and graviton contributions included the rate of emission is about two orders of magnitude higher than these values [16]. One can make an order of magnitude estimate for the amount of angular momentum radiated away in a period ∆t, where M is almost constant and J ∼ M 2 stays valid. Before testing the validity of CCC let us adopt absolute units, G = c = ~ = 1. In these units the mass of the sun is 1038 and the size, the age and the mass of universe are 1062 . Let us start with a nearly extremal b.h. J/M 2 = a/M = 0.999 (Cin = 0.001M 2 ). We send in a test field from infinity. We have shown that the effect of the test field for a stationary b.h. can lead to a final value of Cfin = −0.001M 2 , when evaporation is neglected. However, the b.h. keeps evaporating in the period. M is almost constant in the process and there is a loss of angular momentum. If the loss of angular momentum in the process satisfies |∆J| . 0.001M 2 , then Cfin at the end of this process is smaller than zero so the b.h. can be overspun by test fields. If there is a larger loss of angular momentum Cfin remains positive and evaporation prevents the b.h. from being overspun. Consider a nearly extremal b.h. with a solar mass M ∼ 1038 . Even if we take ∆t as the entire age of the universe ∆t = 1062 , angular momentum loss due to evaporation is |∆J| ∼ 4 × 1024 ≃ 10−51 M 2 . For evaporation to act as a cosmic censor we require at least |∆J| ∼ 10−3 M 2 starting with J/M 2 = a/M = 0.999. Thus, evaporation has no practical effect as a cosmic censor for a b.h. of solar mass against challenging test fields. In fact this is the case until M ∼ 1022 ∼ 1017 g. This is the critical mass that makes Cfin ∼ 0 if we take ∆t as the age of the universe. If the mass of the b.h. is lower than the critical mass evaporation can prevent over-spinning of the black hole. If it is larger than the critical mass the effect of evaporation as a cosmic censor is negligible.
4 A.
The effect of a lower interaction period
We have mentioned that the interaction period is the time it takes our massless fields to reach the black hole from infinity, which we define as the asymptotically flat region. If the object in consideration has a relatively small size one can argue that asymptotic flatness sets in at a distance much smaller than the size of the universe. For example let us consider a b.h. with M = 1020 ∼ 1015 g and J/M 2 = 0.999. If we send in a test field from a distance of the size of the universe we have shown that evaporation prevents the over-spinning of this b.h. However, if the test field is sent in from ∼ 10000 light years, then ∆t ∼ 1056 , |∆J| < 10−3 M 2 , the b.h. can be overspun.
III.
CONCLUSIONS
for the first time. We found that although evaporation of black holes acts as a cosmic censor, its effect is not strong enough to prevent overspinning of black holes of mass M > 1017 g by challenging test fields introduced in our previous work. For smaller masses, overspinning can be prevented by evaporation if the period of interaction is sufficiently long; that is, test fields are sent into the b.h. from a distance sufficiently far. Overspinning can still be achieved if test fields are sent in from a closer distance, as long as the space-time at that distance can be considered as asymptotically flat. The overall conclusion is that the effect of evaporation as a cosmic censor is rather weak against challenging test fields.
ACKNOWLEDGMENTS
In this work we have made a rough estimate of the cosmic censorship supporting effect of Hawking radiation
This work is partially supported by Bo˘gazi¸ci University Research Fund, by grant number 7981.
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