Trapped surfaces, energy conditions, and horizon avoidance in spherically-symmetric collapse
aa r X i v : . [ g r- q c ] J un Trapped surfaces, energy conditions, and horizon avoidance in spherically-symmetric collapse
Valentina Baccetti
School of Science, RMIT University,Melbourne, VIC 3000, Australia
Robert B. Mann
Department of Physics and Astronomy, University of Waterloo,Waterloo, ON, N2L 3G1, Canada andPerimeter Institute for Theoretical Physics,Waterloo, ON, N2L 6B9, Canada
Daniel R. Terno ∗ Department of Physics and Astronomy, Macquarie University,Sydney, NSW 2109, AustraliaE-mail: [email protected]
We consider spherically-symmetric black holes in semiclassical gravity. For a collapsing radiating thin shellwe derive a sufficient condition on the exterior geometry that ensures that a black hole is not formed. This isalso a sufficient condition for an infalling test particle to avoid the apparent horizon of an existing black holeand approach it only within a certain minimal distance. Taking the presence of a trapped region and its outerboundary — the apparent horizon— as the defining feature of black holes, we explore the consequences oftheir finite time of formation according to a distant observer. Assuming regularity of the apparent horizon weobtain the limiting form of the metric and the energy-momentum tensor in its vicinity that violates the nullenergy condition (NEC). The metric does not satisfy the sufficient condition for horizon avoidance: a thin shellcollapses to form a black hole and test particles (unless too slow) cross into it in finite time. However, there maybe difficulty in maintaining the expected range of the NEC violation, and stability against perturbations is notassured.Keywords: black holes; null energy condition; thin shells
I. INTRODUCTION
Event horizon — the null surface that bounds the spacetimeregion from which signals cannot escape — is the definingfeature of black holes in general relativity [1–3]. This classi-cal concept plays an important role in their quantum behaviour[4–6]. Emission of the Hawking radiation completes a ther-modynamic picture of black holes, and its most straightfor-ward derivation relies on the existence of an horizon [4]. Thisradiation is also one of the ingredients of the black hole infor-mation loss paradox [7], perhaps the longest-running contro-versy in theoretical physics [5–7].Event horizons are global teleological entities that are inprinciple unobservable [8, 9]. Theoretical, numerical and ob-servational studies therefore focus on other characteristic fea-tures of black holes [3, 10]. A local expression of the idea ofabsence of communications with the outside world is providedby the notion of a trapped region. It is a domain where bothoutgoing and ingoing future-directed null geodesics emanat-ing from a spacelike two-dimensional surface with sphericaltopology have negative expansion [1, 3, 10, 11]. The apparenthorizon is the outer boundary of the trapped region [1, 10].According to classical general relativity the apparent horizonis located inside the event horizons if the matter satisfies en-ergy conditions [1, 12].Quantum states can violate energy conditions [12]. Blackhole evaporation proceeds precisely because T µν = h ˆ T µν i vi-olates the null energy condition (NEC): there is a null vector k µ such that T µν k µ k ν < . In this case the apparent horizonis outside the event horizon. In fact, the very existence of thelatter is uncertain [13, 14]. While existence of spacetime sin-gularities is no longer prescribed, their appearance without thehorizon cover (“naked”) is not excluded either. This situationmotivated introduction of many models of the ultra-compactobjects [9].According to a distant observer formation of a classicalblack holes takes an infinite amount of time t , even if effectiveblackening out happens very fast. Similarly, the plunge of atest particle into an existing black holes takes infinite amountof time t , but finite proper time of a comoving observer. Onthe other hand, if quantum effects responsible for finite-timeblack hole evaporation allow for the formation of an appar-ent horizon, it happens in finite t . The question is then if it ispossible to fall into such a black hole.Working in the framework of semiclassical gravity withspherical symmetry [16], we consider formation of a trappedregion in the finite time of a distant observer as a definitionof existence of a black hole. Assuming that it exists, we de-rive the condition that allows an infalling observer to avoidhorizon crossing. We then show that if the apparent horizonis regular, the energy-momentum tensor and the metric in itsneighbourhood are uniquely defined by the Schwarzschild ra-dius r g and its rate of change. The resulting metric does notsatisfy the sufficient condition for horizon avoidance. Finally,we discuss intriguing implications of these results. II. SPHERICAL SYMMETRY. SUFFICIENT CONDITIONFOR HORIZON AVOIDANCE
A general spherically symmetric metric in Schwarzschildcoordinates is given by ds = − e h ( t,r ) f ( t, r ) dt + f ( t, r ) − dr + r d Ω (1)The function f ( t, r ) = 1 − C ( t, r ) /r is coordinate-independent, where the function C ( t, r ) is the Misner-Sharpmass [3, 15]. In an asymptotically flat spacetime t is the phys-ical time of a distant observer.Trapped regions exist only if the equation f ( t, r ) = 0 hasa root [11]. This root (or, if there are several, the largest one)is the Schwarzschild radius r g ( t ) . Apparent horizons are ingeneral observer-dependent entities. However they are un-ambiguously defined in the spherically symmetric case for allspherical-symmetry preserving foliations[15]. In this case theapparent horizon is located at r g . In the Schwarzschild space-time C ( t, r ) = 2 M and h = 0 , hence r g = 2 M .In thin shell collapse models [2] the geometry inside theshell is given by the flat Minkowski metric. The matter con-tent of the shell is given by the surface energy-momentum ten-sor. The trajectory of a massive shell is parameterized by itsproper time τ and expresses as (cid:0) T ( τ ) , R ( τ ) (cid:1) in the exteriorSchwarzschild coordinates. Initially the shell is located out-side its gravitational radius, R (0) > r g . Its dynamics is ob-tained by using the so-called junction conditions [2, 18].The first junction condition is the statement that the inducedmetric h ab on the shell Σ is the same on both sides Σ ± , ds = h ab dy a dy b = − dτ + R d Ω . Since for massive particles thefour-velocity u µ satisfies u µ u µ = − , the shell’s trajectoryobeys ˙ T = p F + ˙ R e H F , (2)where ˙ A = dA/dτ , H = h ( T, R ) , F = f ( T, R ) . This con-dition is used to identify the radial coordinate of the shell ininterior and exterior coordinates, R − ≡ R .Discontinuity of the extrinsic curvature K ab is described bythe second junction condition [2, 18] that relates it to the sur-face energy-momentum tensor. Given the exterior metric thejunction conditions result in the equations of motion for theshell. For a classical collapse in vacuum the exterior geom-etry is given by the Schwarzschild metric, and the resultingequation for R ( τ ) is simple enough to have an analytic solu-tion τ ( R ) , leading to the finite proper time τ ( r g ) and infinitetime T ( r g ) .This equation of motion is modified for a general exteriormetric and its solution has some remarkable features [19, 20].Here we focus on the possibility of crossing the Schwarzschildsphere of an evaporating black hole ( r ′ g ( t ) < ) in finiteproper time. For a finite evaporation time t E the finite propercrossing time is equivalent to having a finite time t g of a dis-tant observer. By monitoring the gap between the shell andthe Schwarzschild radius [16, 21], X ( τ ) := R ( τ ) − r g (cid:0) T ( τ ) (cid:1) , (3) we discover the sufficient condition for a thin shell to nevercross its Schwarzschild radius. The same condition appliesto the study of an infalling test particle into an existing blackhole. The analysis is generalized to null shells and test parti-cles.The rate of approach to the Schwarzschild radius behavesas ˙ X = ˙ R − r ′ g ( T ) ˙ T . (4)Close to the Schwarzschild radius we have ˙ T ≈ − ˙ Re − H /F ,and hence ˙ X ≈ ˙ R (1 − | r ′ g | e − H /F ) . (5)If for a fixed t the function exp( h ) f goes to zero as x := r − r g → , then there is a stopping scale ǫ ∗ ( τ ) . If the shellcomes to the Schwarzschild radius closer than ǫ ∗ the gap hasto increase, ˙ X > , evidently indicating in this case that theshell never collapses to a black hole. It is so if, e.g., h ( t, r ) ≤ . In particular, this is the case when the exterior geome-try is given by the outgoing Vaidya metric. Then ǫ ∗ =2 C | dC/dU | , where U ( τ ) is the retarded null coordinate ofthe shell [16, 21]. However, it is not a priori clear that in ageneral evaporating case this criterion is satisfied. III. METRIC OUTSIDE AN APPARENT HORIZON
Using only one additional assumption it is possible to ob-tain the explicit form of the metric near r g . In fact this metricsatisfies the sufficient condition for the horizon avoidance. Weconsider an evaporating black hole that is formed at some dis-tant observer’s finite time, i.e. its apparent horizon radius r g ( t ) is a decreasing function of time. In addition we assume thatthe horizon is regular (the standard “no drama at the horizon”postulate[6], where the established regularity of the classicalresults is assumed to hold in the quantum-dominated regime).The regularity is expressed by finite values of the curvaturescalars that can be directly expressed in terms of the energy-momentum tensor[17], T := T µµ and T := T µν T µν .The existence of an apparent horizon and regularity as-sumptions strongly constrain the energy-momentum tensor,and consistency with the known results on the background ofan eternal black hole[4, 22] specify its limiting form uniquely.The leading terms in the ( tr ) block of the energy-momentumtensor turns out to be the functions of a := r ′ g r g , and thefunctions C ( t ) and h ( t ) take the following form [17], C = r g ( t ) − a ( t ) √ x + 13 x . . . . (6)and h = − ln √ xξ ( t ) + 43 a √ x + . . . , (7)where x = r − r g . The function of time ξ ( t ) is deter-mined by the choice of the time variable. In asymptoticallyflat spacetimes with t being a physical time of a stationaryBob, lim r →∞ h ( t, r ) = 0 and the exact solution of the Ein-stein equations allows to determine ξ . The constant in thefunction h is set (using the freedom in re-defining the timevariable) in such a way that h ≈ for a macroscopic BHwhen x → r g , i.e. far relative to the scale of quantum effects α . The metric takes a particularly simple form in ingoingVaidya coordinates[17].The energy-momentum tensor that corresponds to this met-ric violates the null energy condition in the vicinity of the ap-parent horizon[17]. The comoving density and pressure at theapparent horizon of an evaporating black hole are negative, ρ = p = − r ′ g π ˙ r r g , (8)where r is the radial coordinate of the comoving observer[23].We focus on the question of horizon avoidance. Return firstto a collapsing thin shell problem where the exterior metric isgiven now by Eq. (1) with the metric functions that are givenabove. Expanding Eqs. (6) and (7) for X → the rate ofapproach ˙ X = − ( ˙ R − πr g Υ )2 | ˙ R |√ πr / g Υ √ X + . . . , (9)which is independent of the function ξ . Hence if a test parti-cle is in the vicinity of the apparent horizon, X ≪ a , it willcross the horizon unless | ˙ R | < √ πr g Υ ∼ √ κ ∼ . .For the evaporating case we match the r ′ g that is obtained asa consistency requirement on the functions of C and h fromthe Enstein equations [17], with the known results [24] forthe quasi-static mass loss r ′ g = − κ/r g . In Planck units κ ∼ − − − , and we obtain ξ ≈ r κ r g ≈ q πr g Υ = a . (10)We have seen that the violation of the NEC is necessaryfor existence of a black hole. Such violations, however, arebounded by quantum energy inequalities (QEI). Adapting the QEI of Ref. 25 we obtain that the region where the NEC isviolated is bounded by x max < π | r ′ g | r g ∼ , (11)that is obtained by ignoring the sub-Planckian features of thebound[16]. Even this estimate fails short of the conventionalestimate x max ∼ r g . Our results indicate either that the re-quired negative energy density for having a Schwarzschildsphere at finite time t S cannot be maintained or the trappedregions forms at an much later stage of the collapse. Alter-natively, the domain of validity of our metric (that has theform[16] of the approximate metric that is obtained by takingthe backreaction into account[4] has a much narrower domainof validity. Both possibilities may indicate that the semiclas-sical approximation and its associated classical notions aremodified already at the horizon or larger scales. A rigourousanalysis of this situation is in order.Another question results from two properties we discussedabove. On the one hand, existence of the apparent horizonrequires violation of the NEC. Test particles cross it in finitetime unless moving too slow. It is not clear how the apparenthorizon that requires NEC violation for its existence reacts toa perturbation by infalling normal matter. Given that collaps-ing thin shells are known to cross the apparent horizon andform a black hole with nearly all their rest mass intact[24], itis possible that the answer will involve considerable adjust-ment of our concept of black holes. ACKNOWLEDGEMENTS
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