PreprinttypesetinJHEPstyle-HYPERVERSION Hawking-like radiation from evolving black 1 1 holes and compact horizonless objects 0 2 n a J 9 1 Carlos Barcelo´ ] c Instituto de Astrof´ısica de Andaluc´ıa, IAA–CSIC, Glorieta de Astronom´ıa, q 18008 Granada, Spain - r g E-mail: [email protected] [ 2 Stefano Liberati v 1 SISSA/ International School for Advanced Studies, Via Bonomea 265, 1 34136 Trieste, Italy and INFN, Sezione di Trieste 9 5 E-mail: [email protected] . 1 1 Sebastiano Sonego 0 1 Universit`a di Udine, Dipartimento di Fisica, Via delle Scienze 208, : v 33100 Udine, Italy i X E-mail: [email protected] r a Matt Visser School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, New Zealand E-mail: [email protected] Abstract: Usually, Hawking radiationisderived assuming (i)thatafutureeternal event horizon forms, and (ii) that the subsequent exterior geometry is static. However, one may be interested in either considering quasi-black holes (objects in an ever-lasting state of approach to horizon formation, but never quite forming one), where (i) fails, or, following the evolution of a black hole during evaporation, where (ii) fails. We shall verify that as long as one has an approximately exponential relation between the affine parameters on the null generators of past and future null infinity, then subject to a suitable adiabatic condition being satisfied, a Planck-distributed flux of Hawking-like radiation will occur. This happens both for the case of an evaporating black hole, as well as for the more dramatic case of a collapsing object for which no horizon has yet formed (or even will ever form). In this article we shall cast the previousstatementinamorepreciseandquantitativeform,andsubsequently provide several explicit calculations to show how the time-dependent Bogoliubov coefficients can be calculated. Keywords: Hawking radiation; adiabatic; quantum gravity; black holes; spacetime singularities; horizons; null infinity; affine parameter; peeling; surface gravity. 27 November 2010; 18 January 2011; LATEX-ed January 20, 2011. Contents 1. Introduction 2 2. Motivating the adiabatic condition 4 3. The exponential approximation 5 3.1 Definitions and exact results 5 3.2 Introducing the approximation 6 3.3 Range of validity of the approximation 7 3.4 Nonlocal normalization 10 3.5 Logarithmic approximation 11 4. Peeling properties versus inaffinities 12 4.1 Metric asymptotics 12 4.2 Inaffinity estimation in the bulk 12 4.3 The surface gravity of + 13 H 4.4 Summary 14 5. Bogoliubov coefficients: Basic framework 14 5.1 Bogoliubov coefficients in terms of the Klein–Gordon inner product 14 5.2 Approximation scheme 16 6. Stationary phase evaluation of the Bogoliubov coefficients 17 6.1 Boltzmann spectrum using stationary phase 18 6.2 Planck spectrum using stationary phase 19 6.3 Mathematical range of validity for stationary phase 20 7. Gamma function evaluation of the Bogoliubov coefficients 21 8. Physical particle detection: Wave-packets 23 9. Physical necessity of the adiabatic condition 25 10. Discussion 25 A. Surface gravity: peeling versus inaffinity 27 A.1 The peeling notion of surface gravity 27 A.2 The inaffinity notion of surface gravity 28 – 1 – 1. Introduction Ever since Hawking’s original 1974 derivation that black holes emit a steady Planck- distributed flux of quanta [1, 2], there has been a steady and continual stream of articles that re-derive the Hawking flux in various different ways [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] — the justification being that in doing so one might strip the calculation to its essence and so discover what aspects ofblack holephysics aretruly importantfor thephenomenon, andwhat aspects cansafelybeput aside. Several important points arebynow well established, though often not well appreciated: Hawking radiation is more ubiquitous than Bekenstein entropy, and Hawking • radiation will still occur in situations where the notion of black hole entropy has no meaning [10, 11, 12]. Eternal black holes, and their associated bifurcate Killing horizons, are useful • mathematical models [3], and the source for many useful heuristics [26], but physically they do not accurately reflect the formation and evolution of real astrophysical black holes. The subtle differences between various forms of horizon present in general rel- • ativity (event, apparent, isolated, dynamical, trapping, etc...) are physically important [8,12, 27,28,29,30, 31, 32]—andprecisely which (ifany)ofthese is essential for Hawking radiation has direct impact, for instance on the question of “information loss” [33, 34, 35, 36, 37]. In the current article we will address two specific questions (throughout this pa- per we will adhere to strictly general relativistic analyses, not considering modified dispersion relations): (i) How can we derive the Hawking flux emitted by a slowly evolving (as opposed to static) black hole-like object? (ii) What type of horizon (if any) is required to generate a Hawking-like Planck- distributed flux of quanta? We shall do so by adapting, modifying, and extending the by now reasonably well established result that if (in spherical symmetry) you have an (approximately) ex- ponential relation between the affine parameters U and u on the null generators of past and future null infinity, I− and I+, then Hawking radiation will occur. The standard form of this argument (as per Hawking’s original calculation [1, 2]) is to say that if asymptotically U U A e−κHu, (1.1) H ≈ − – 2 – as u + for some arbitrary positive constants A and κ , then Hawking radiation H → ∞ happens, with fixed non-evolving temperature ~ κ H k T = . (1.2) B H 2π Here U and u can be viewed as different labels that we attach to a null curve con- necting I− with I+. We can equivalently write 1 U U H u ln − . (1.3) ≈ −κ A H (cid:18) (cid:19) Of course the original Hawking calculation [1, 2] explicitly assumes the existence of an event horizon at U , and that the resulting black hole quickly settles down to a H static configuration. In counterpoint, in 1987 Hajicek [8] demonstrated that a strict event horizon was not necessary, and that a (suitably long-lived) apparent horizon was quite sufficient to generate a Hawking flux. (See also [12, 33, 34].) More recently (within the con- text of“analoguespacetimes” [10, 11, 38, 39])thepresent authorshave demonstrated that apparent/ trapping horizons can also be dispensed with, or their appearance postponed indefinitely into the future [28, 29]. We shall now make these results more general, precise, and quantitative [40]. Inspired in particular by the work of Hu [41], we focus on the existence of an (in our case, approximate) exponential relation be- tween the affine parameters on past and future null infinities as the necessary and sufficient condition for generating a Hawking flux. Here is a summary of our key result: Consider null curves starting from I− and arriving on I+. There will be some relation between the affine parameters U on I− and u on I+: U = p(u); u = p−1(U). (1.4) Now pick a particular null curve, labelled by U on I− and u on I+. We can ∗ ∗ without loss of generality write u u¯ U = U +C exp κ(u˜) du˜ du¯, (1.5) ∗ ∗ − Zu∗ (cid:20) Zu∗ (cid:21) for some constant C and the function κ(u) = p¨(u)/p˙(u). Assume (and this is ∗ − where the physics comes in) that κ(u) satisfies an “adiabatic condition” κ˙(u ) κ(u )2. (1.6) ∗ ∗ | | ≪ Then we shall show that this is sufficient to guarantee (under mild technical assump- tions) the existence of a Hawking-like Planck-spectrum of outgoing particles reaching I+ at u , now with a time-dependent Hawking temperature ∗ ~ κ(u ) ∗ k T (u ) = . (1.7) B H ∗ 2π – 3 – As we move along I+ (that is, as u increases, possibly even with u + ), this ∗ ∗ → ∞ relationcontinues tohold, withtheHawking temperaturecontrolledbyκ(u ), as long ∗ as the adiabatic condition continues to hold. We do not need to assume a horizon — of any sort —ever forms. Therest ofthisarticlewillbedevoted toadetailedproofof this result. Along the way we shall revisit (and hopefully clarify) the salient features that go into calculating the relevant Bogoliubov coefficients — specifically we shall very carefully look at the issue of defining and accurately estimating appropriate time-dependent Bogoliubov coefficients. 2. Motivating the adiabatic condition Why might weeven expect something like the“adiabaticcondition”(1.6)toeither be true or relevant? We can physically interpret the “adiabaticity condition” as equiv- alent to the statement that a photon emitted near the peak of the Planck spectrum, with ~ω k T , that is ω κ, should not see a large fractional change in the ∞ B H ∞ ≈ ∼ peak energy of the spectrum over one oscillation of the electromagnetic field. (That is, the change in spacetime geometry is adiabatically slow as seen by a photon near the peak of the Hawking spectrum.) It is this slow change in the spacetime geometry that ultimately permits us to apply a variant of Hawking’s original calculation. To then verify that this adiabatic condition holds for macroscopic black holes let us (for example) think of the standard Hawking calculation for a Schwarzschild black hole. The Hawking temperature is [1, 2]: M2 k T ~ κ Planck, (2.1) B H ∼ ∼ M and consequently (assuming self-consistent back-reaction and that the Hawking for- mula continues to be true for a slowly evolving almost-Schwarzschild black hole) the mass loss rate for a black hole evaporating into vacuum is given by the standard result M4 M˙ Planck. (2.2) | | ∼ M2 In particular, as the black hole evolves its temperature changes. This brings up and reinforces an important point: Any truly fundamental derivation of the Hawk- ing effect should be able to deal with a time-dependent Hawking temperature. If your favourite derivation is intrinsically incapable of dealing with time dependent situations, then such a derivation is missing fundamental parts of the physics. Looking at the surface gravity of the Schwarzschild black hole we can estimate κ˙ M2 Planck. (2.3) κ2 ∼ M2 So the standard Hawking process for standard Schwarzschild black holes does satisfy the “adiabaticity condition” we have enunciated above, at least as long as the black hole is heavier than a few Planck masses. – 4 – Overall this now provides us with a coherent physical picture all the way down to the Planck mass, where we see that adiabaticity breaks down, and “quantum gravity” (inthe sense of“that quantum theory that approximately reduces to general relativity in an appropriate limit”) takes over. 3. The exponential approximation 3.1 Definitions and exact results Consider anasymptotically flatspherically symmetric spacetime with aMinkowskian structure in the asymptotic past. (The discussion that follows applies equally well to any number of spatial dimensions and can easily be generalized to deal with acoustic spacetimes in 1+1 dimensions having two asymptotic regions [42].) In the t,r { } sector of the geometry we define an affine parameter W on I−, and use it to label the null curves travelling towards the centre of the body. Similarly, u is taken to be an affine parameter on I+, used to label the null rays travelling away from the central body. The independent coordinates W,u provide a double-null cover of the { } relevant parts of spacetime (the domain of outer communication). As is standard, one can define a canonical functional relationship connecting I− with I+ by using null curves that reflect off the centre at r = 0. This relation can be expressed as U = p(u), u = p−1(U), (3.1) wherethelabels U,u arenownolongertobethoughtofasindependent coordinates { } but, since we have explicitly linked them via the function p( ), as different ways of · labelling the same null curve once it is reflected through the origin. It is to be understood that p−1( ) need not be defined on all of I− if a true event horizon · indeed forms; however this function will certainly be well defined on those parts of I− that lie in the domain of outer communication. We shall soon see that the function p( ), or equivalently itsinverse, is sufficient to encodeall the relevant physics · of Hawking radiation. Specifically, let us choose a reference null curve completely traversing the body. It is labelled by u on its way out of the body, and by U on its ∗ ∗ way in. We want to use “local” information from the vicinity of this reference null curve to study Hawking-like radiation that reaches I+ in the vicinity of u . ∗ Let us now start the technical computation by defining a quantity κ(u) via the relation d2U/du2 p¨(u) κ(u) = = , (3.2) − dU/du −p˙(u) so that κ(u) is simply a functional parameterization of the relationship between the affine parameters U and u. When this function happens to be almost constant it controls the e-folding relationship between u and U, and so provides a notion of “surface gravity” in terms of the “peeling” properties of null geodesics. (And so is – 5 – intimately related to κ as defined in [28, 29]. See also [43, 44, 45] for comments effective on the importance of these “peeling” properties.) Now pick some generic null curve labelled by u , then through integration one can express any U = p(u) as a function ∗ of its corresponding κ(u): u u¯ U = U +C exp κ(u˜) du˜ du¯, (3.3) ∗ ∗ − Zu∗ (cid:20) Zu∗ (cid:21) forsomeconstant C . Notethatitisimpossible, eveninprinciple, touselocal physics ∗ to specify a unique normalization for C . This is ultimately due to the fact that any ∗ constant multiple of an affine null parameter is still an affine null parameter. Note also that C is a constant in the sense that it is the same for all null curves u in the ∗ vicinity of the null curve specified by u . However C does depend on the choice of ∗ ∗ the specific null curve u one is working around. This formalism continues to make ∗ perfectly good sense even for κ = 0, where it simply implies a linear relation between u and U. Let us now write κ(u) = κ(u )+δκ(u) = κ +δκ(u); (3.4) ∗ ∗ then in particular we have u¯ u¯ κ(u˜) du˜ = κ (u¯ u )+ δκ(u˜) du˜, (3.5) ∗ ∗ − Zu∗ Zu∗ so that (still an exact result) u u¯ u u¯ exp κ(u˜) du˜ du¯ = exp[ κ (u¯ u )] exp δκ(u˜) du˜ du¯. (3.6) ∗ ∗ − − − − Zu∗ (cid:20) Zu∗ (cid:21) Zu∗ (cid:20) Zu∗ (cid:21) Here we are interested in situations in which the second exponential on the RHS is in some suitable sense (to be more carefully defined below) “close to unity”. 3.2 Introducing the approximation We now begin the approximation procedure: Let us now suppose that u δκ(u˜) du˜ ǫ2 1, (3.7) ≤ ≪ (cid:12)Zu∗ (cid:12) (cid:12) (cid:12) where we shall soon check th(cid:12)e conditions(cid:12)under which this happens. Under this (cid:12) (cid:12) hypothesis we can re-write the exact result (3.6) as u u¯ u exp κ(u˜) du˜ du¯ = exp[ κ (u¯ u )] 1+O(ǫ2) du¯, (3.8) ∗ ∗ − − − Zu∗ (cid:20) Zu∗ (cid:21) Zu∗ (cid:8) (cid:9) which we can integrate to yield u u¯ 1 exp[ κ (u u )] exp κ(u˜) du˜ du¯ = − − ∗ − ∗ +O(ǫ2). (3.9) − κ Zu∗ (cid:20) Zu∗ (cid:21) (cid:26) ∗ (cid:27) – 6 – The analysis here is somewhat delicate, because one is integrating a small quantity over what could be a very long time. Note that the way we have set things up, this approximation will always be valid over some interval — the only real question is how long this validity interval will be. (See section 3.3 below.) The net result ofthe discussion upto thispoint is that, if weaccept the condition (3.7), then for u sufficiently close to u we can effectively replace κ(u) by κ , and so ∗ ∗ write u U U +C exp[ κ (u¯ u )]du¯ ∗ ∗ ∗ ∗ ≈ − − Zu∗ C ∗ = U exp[ κ (u u )] 1 ∗ ∗ ∗ − κ { − − − } ∗ C C = U + ∗ ∗ eκ∗u∗ exp( κ u) ∗ ∗ κ − κ − (cid:26) ∗(cid:27) (cid:26) ∗ (cid:27) = U∗ A exp( κ u), (3.10) H − ∗ − ∗ where we have defined C C U∗ = u + ∗ and A = ∗ eκ∗u∗. (3.11) H ∗ κ ∗ κ ∗ ∗ In spite of the similarity with Hawking’s approximation, it is vitally important to note that U∗ is not the location of the horizon (extrapolated back to I−) — it is H instead the best estimate (based onwhat you can see locally at u ) of where a horizon ∗ might be likely to form if the relation between U and u keeps e-folding in the way it is at u . There is no actual implication that a strict event horizon (or indeed any ∗ sort of horizon) ever forms, only that it “looks like” a horizon might form in the “not too distant future”. Once we have this approximate relation, U = p(u) U∗ A exp( κ u), (3.12) ≈ H − ∗ − ∗ which we shall refer to as the “exponential approximation”, then the rest of the calculation simply drops out (in the quite usual manner). The only tricky point lies in estimating the range of validity of this “exponential approximation”. 3.3 Range of validity of the approximation The exponential approximation condition (3.7), can always be satisfied for small- enough integration intervals (u u ). But how small is small-enough? As a mild ∗ − technical assumption, let us consider only functions κ such that we can define a constant 1/(n+1) (n) 1 κ ∗ D := sup | | , D < + . (3.13) (n+1)! κn+1 ∞ n>0" ∗ # Physically this amounts to the assumption that the only two relevant (reciprocal) timescales in the problem are κ and Dκ . (Any other scale is assumed to be smaller, ∗ ∗ – 7 – which is simply another way of saying that κ(u) is slowly varying over the region of interest.) This condition is rather mild, covering even functions with poles at specific values of u. So κ(u) is even allowed to exhibit “sudden singularities”, with this terminology being borrowed from cosmology [46, 47, 48]. Under this hypothesis we have u +∞ 1 δκ(u˜) du˜ = κ(n) (u u )n+1 (n+1)! ∗ − ∗ (cid:12)(cid:12)Zu∗ (cid:12)(cid:12) (cid:12)(cid:12)Xn=1 (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12)+∞ 1 (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) κ(n) u u n+1(cid:12) ≤ (n+1)!| ∗ | | − ∗| n=1 X +∞ Dn+1κn+1 u u n+1. (3.14) ≤ ∗ | − ∗| n=1 X Let us temporarily set x = Dκ u u , then ∗ ∗ | − | u +∞ 1 x2 δκ(u˜) du˜ xn = 1 x = . (3.15) ≤ 1 x − − 1 x (cid:12)Zu∗ (cid:12) n=2 − − (cid:12) (cid:12) X Now, as long as (cid:12) (cid:12) (cid:12) (cid:12) x2 ǫ2, (3.16) 1 x ≤ − we are sure that condition (3.7) is satisfied. Taking into account that ǫ 1, the ≪ previous condition is certainly guaranteed to hold as long as x2 ǫ2/2. (This is ≤ not the optimal condition, but it is simple and quite good enough for our purposes.) Then 2D2κ2 (u u )2 ǫ2 1. (3.17) ∗ − ∗ ≤ ≪ Thus the range of validity of the exponential approximation condition (3.7) is cer- tainly at least as large as ǫ 1 u u . (3.18) ∗ | − | ≤ √2Dκ ≪ √2Dκ ∗ ∗ Inthemostsimplesituationsthefirstterminthedefinition(3.13)dominates, thereby yielding κ˙ 2D2 = | ∗|. (3.19) κ2 ∗ For instance, for an evaporating Schwarzschild black hole one can estimate M4n+2 κ(n) 1 4 7 (3n 2) Planck, (3.20) | | ∼ × × ×···× − M3n+1 and so 1/(n+1) 1 κ(n) 1 M2n 1/(n+1) M2 M 2/(n+1) | ∗ | Planck Planck (3.21) (n+1)! κn+1 ∼ n+1 M2n ∼ M2 M " ∗ # (cid:20) (cid:21) (cid:20) Planck(cid:21). – 8 –