JHEP04(2018)056 Springer April 8, 2018 April 11, 2018 : : January 17, 2018 mportant role in : Accepted Published trapping horizon of the zon and is a dynamical hich depends on the soft Received nsider supertranslation of rg soft theorems. However as ic dynamical spacetime with zation property of the black t work has emphasized that the h involves an angular dependent s, National Tsing-Hua University, Published for SISSA by https://doi.org/10.1007/JHEP04(2018)056 [email protected] a , . 3 and Yoji Koyama 1801.03658 a,b The Authors. c Black Holes, Space-Time Symmetries

Soft hair of black hole has been proposed recently to play an i , [email protected] Hsinchu, 30013, Taiwan Department of Physics, NationalHsinchu Tsing-Hua 30013, University, Taiwan E-mail: Physics Division, National Center for Theoretical Science b a Soft hair of dynamicalradiation black hole and Hawking Open Access Article funded by SCOAP Abstract: Chong-Sun Chu hole S-matrix. Keywords: ArXiv ePrint: the Vaidya black hole andsoft construct hair. a non-spherical symmetr Weblack show hole. that We this finddynamical spacetime that black hole. Hawking admits radiation a Thehair is Hawking trapping of emitted radiation hori the from has black the a hole spectrum and w this is consistent with the factori the resolution of the black holesoft modes information cannot paradox. affect Recen the blacksoft hole S-matrix hair due is to generated Weinbe by supertranslationshift of of geometry time, whic it must have non-trivial quantum effects. We co JHEP04(2018)056 3 5 7 9 1 3 7 11 14 11 17 totically Hawking and superrotation es are characterized by, cture of gravity and its diation is described by a rge and angular momen- ] for a review of the black r. 7 , coordinate in the asymptotic ld be non-unitary which is in ed to black hole in Minkowski 6 he black hole. Therefore if the ciency in our understanding of ], black holes obey the first law of ]. Furthermore, Hawking, Perry 3 , 13 2 – 8 ] advocated a new approach to the black hole – 1 – 16 ], one would expect that the Hawking radiation to 5 ] and Hawking [ 1 ] is the symmetry of asymptotically flat spacetime generated 18 , ] and Strominger [ 17 15 , ], with a temperature that arises from the quantum process of 4 14 ]. The existence of an infinite number of new charges for asymp 21 , 20 ], also an infinite number of BMS charges of supertranslation 19 Recently, Strominger initiated a study of the infrared stru BMS symmetry [ 4.1 Hamilton-Jacobi method4.2 Supertranslated Vaidya black hole 2.1 Supertranslated2.2 Vaidya spacetime Implantation of supertranslation hair 3.1 Trapping horizon 3.2 Surface gravity connection with the asymptotic BMS symmetry [ region. With thein addition BMS to symmetry, the asymptoticallytum standard [ flat ADM spacetim charges of mass, electricflat cha spacetime is a very interesting observation. When appli and Strominger [ information paradox based on a new kind of blackby hole supertranslation, soft an hai angular dependent shift of the time be completely independent of thematters state were of in matters a enteringmixed pure t thermal quantum state, state, the and blackcontradiction since hole with the evaporation Hawking quantum process ra mechanics. wou hole See, information for problem. example, [ thermodynamics [ radiation. From the no-hair theorem [ Black hole model in generalgravity. As relativity explained identifies by deep Bekenstein insuffi [ 5 Conclusion and discussion 1 Introduction 4 Hawking radiation from supertranslated Vaidya black hole 3 Properties of supertranslated Vaidya black hole Contents 1 Introduction 2 Supertranslation of Vaidya spacetime charges [ JHEP04(2018)056 ckwave of radia- ory, the process of r studying the evap- ss) is factorized from f Schwarzschild black trum of the Hawking arges. Nevertheless soft radiation would have a hair configuration. Our sider a dynamical black ut how and what aspects wever this is not entirely e quantum effects on the quantum gravity, at least of time is translated to a s recently. As emphasized r could help to resolve the cess can be modeled with g method to compute the rix could be constrained by soft modes. If one use the ing radiation would develop rom having physical effects r. is not a contradiction to the non-trivially on the classical . On the other hand, if one ate how the soft hair would ack hole evaporation due to uld backreact on the metric. ation process. It was further n the dressed state approach. lated black hole is diffeomor- ssically with the gravitational find that the Hawking radia- r suggests that soft hair could ned by the infinite symmetries ir is implanted on the eternal ies an infinite amount of soft hair S-matrix: the final state of black – 2 – ]. 25 ]. – 30 22 ] that, if viewed as a scattering amplitude in the quantum the ], the scattering of the hard modes (like the black hole proce ] showed that one could grow soft hair on it by throwing in a sho 14 29 15 – In this regard, we consider the effects of soft hair on the spec Although the S-matrix turns out not to be a good observable fo Soft hair can be generated via physical process. In the case o 26 hole evaporation consists ofdressed Hawking states radiation as plus aspectrum a which basis sea is for of independent theuse of the soft undressed the modes, states soft as thena hair in dependence configuration our the computation, on Hawking then the the soft Hawk hair. hole. We thereforeHawking consider radiation the of more thisaffect realistic dynamical the process black Hawking of hole radiation.a bl and supertranslated investig The Vaidya black spacetime.Hawking hole radiation evaporation We and pro employ findresult the that is it tunnelin consistent has with a the dependence factorization property on of the the soft radiation. For thetion soft-hairy is Schwarzschild insensitive black to hole,Schwarzschild the black we hole soft with hair. anconsistent as In energy Hawking flux this radiation of process,One carries shock soft should away wave. ha energy allow Ho and for sho the time dependence of the metric and con on the black holechange physics. of the Quantum quantum mechanically,black vacuum, the hole so change physics. soft The hairof main is the motivation quantum expected of physics our to of work leav black is hole to is find affected o by its soft hai oration process of black hole, it does not exclude soft hair f hole, [ the scattering of the softThe modes. decoupling The has decoupling can also beperturbatively been seen [ argued i to be a generic feature of tion. This relation ofencode the information soft about hair the withargued black the [ hole collapsed creation matte and evapor memory effect [ characterized by the BMS charges.standard The no-hair existence theorem of of softphic black hair to hole the since untranslated the one,hair supertrans and modifies so the they definition carry ofphase the time space. at same It null ADM is infinity ch physical and and it its acts effects can be observed cla spacetime, this means that supertranslated black hole carr the black hole formation andof evaporation BMS should and be hence constrai theinformation existence paradox. of The an problem infinite ofthe amount how soft of the soft modes black hai has hole been S-mat by further [ analyzed by a number of author JHEP04(2018)056 ] d es 45 , (2.2) (2.1) 44 , ], 18 A 15 , [ , we discuss Θ 3 od developed 17 − I dvd  ) BD B standard form and the C Θ adiation. We discuss the d he supertranslated Vaidya BD (Θ). eneral one wants to choose ymptotically flat metric in , we introduce the Vaidya Next we define the horizon es. In section A ty of spherically symmetric f 2 C and the angular momentum enerated by the vector field: ) are the angular coordinates. t null infinity ( Θ e briefly review the definition supertranslated Vaidya black , ¯ time can be obtained from the z ituation one wants to describe, = d A te of observation. We conclude nd Sachs (BMS) [ e energy) are ruled out. The as gravitational radiations are A . Supertranslations are charac- ∂ AB z, f 1 8 CD f∂ C AB A C = ( γ − D A m CD r 1 A C ∂ + v r , we apply the Hamilton-Jacobi tunneling AB 3 4 4 γ B f∂ 1 4 2 − Θ – 3 – d + D A A 1 2 N B Θ 3 4 Θ − d d  v A r 1 AB , the traceless tensor f∂ γ Θ 2 d − m r = dvdr, A f AB + ζ Θ AB ). BMS supertranslation is the diffeomorphism which preserv rC ], to compute the Hawking radiation from the supertranslate C A dvd + 32 Θ dvdr AB 2 AB v, C dv 2 + 2 C r 2 B 1 m r 2 16 D dv − − + − = ], which has been used as frequently as the null geodesic meth 2 is the metric on the unit two sphere and Θ is the covariant derivative with respect to depend on ( 31 ds A A AB N ) the Bondi mass aspect D γ The organization of the paper is as follow. In section 2.1 the falloff conditions suchincluded, that but interesting unphysical solutionschoice solutions of such falloff (e.g. conditions of those Bondi,considers with van der metric infinit Burg, with Metzner the a asymptotic expansion near the pas 2.1 Supertranslated Vaidya spacetime Let us start withfour dimensions. a brief At the review infinity,one depending on on may supertranslation the impose physical for different s falloff an conditions as on the metric. In g dependence on soft hairresults is with respect encoded to in dressingthe of the paper soft temperature modes with in of some the further the final discussion. r sta 2 Supertranslation of Vaidya spacetime surface gravity associated withhole the by trapping extending horizon the ofblack the definition holes using of Kodama themethod vector. horizon [ surface In gravi section by Parikh and WilczekVaidya [ black hole. The Hawking radiation spectrum takes the spacetime and its supertranslation. Wecollapse show of that a this space certainsome energy properties momentum of tensor the with supertranslatedof soft Vaidya trapping charg black horizon hole. for W spacetime dynamical is black a holes dynamical and black show hole that with t a trapping horizon. where terized by an arbitrary function of the angular variables, the Bondi gauge and the asymptotic falloff conditions. It is g where In ( aspect JHEP04(2018)056 - ) 2.2 (2.3) (2.5) (2.4) (2.6) 0, the A of ( > Θ ′ f ζ . M ), it is clear ) v dvd 0, the metric ( ) ) is given by 2.6 r . The ingoing f A < v 2 M ′ Θ 2 D M − ring the stage the black + v, r, 1 cation where tunneling ≡ V f (2 er to utilize the tunneling scribes the formation of a . 0 as a model to discuss the rom the Vaidya black hole 1 y flat spacetime. An exam- versed, he outgoing Vaidya metric the quantization of gravita- 0. A the advanced (ingoing) null ime, one written in terms of B nates ( erior region of a white hole. y flux (for Newton constant n. . < D < ck hole by an outgoing energy idya metric which covers the Θ ) . For example, with momentum flux to the Vaidya ′ ′ ′ d v − l. This may also be seen physi- rom the form of ( ,V ( A M M M B Θ M Θ d v oss . ) d ∂ dvdr f A µν 2 g Θ := + 2 f D d ′ ζ 2 L AB AB , which means that the back reaction caused dv 0 which corresponds to ingoing energy flux γ + f rγ 2  ) of the energy loss from Hawking radiation with the is obtained by acting the vector field r ≥ µν – 4 – f − 2 g ) ,M + µν 2 f 4.22 v ) g r ( 2 ) can be obtained, as we will show in the next sub- = v B ′ ( ′ MD D πr µν M 2.6 ) is a function of the advanced time dvdr 4 A g M v − . In this paper, we restrict ourselves to linearized theory ( r ′ rD + 2 = M 2 r vv fM + 2 ¯ = T 2 V dv ), ) can be extended to finite distance and is a solution to the lin − AB M − γ 2 V 2.3 2.6 = r  ν − + ( dx µ = ) give the influence of the soft hair on the Hawking radiation du ν dx 4.21 dx µν µ g ), ( dx = 4.20 2 µν g ds The supertranslated Vaidya metric We are interested in dynamical black holes in asymptoticall = 1) Most part of our analysis actually holds true for any sign of 1 N evaporation of dynamical black hole due to Hawking radiatio method, one must employoccurs, a i.e. metric the which horizon. isboth smooth the This across interior is the andsince suitable lo exterior it for of covers the the the ingoingTherefore black exterior Va we hole; region will of employ but the a not ingoing black for Vaidya metric hole t with and the int can also be taken as aflux. model describing In the this evaporation of paper,with bla soft we hair will using determine the the tunneling Hawking method. radiation f We note that in ord results, ( The Bondi mass aspect The result is cally from the fact thatsection, the by metric throwing ( in a non-spherically symmetric energy hole was formed.change However of the mass of identification the ( black hole holds only in the case of mass l earized Einstein equations for all that the supertranslated Vaidya spacetime is non-spherica on the Vaidya metric ( ple is the Vaidya metric.the There retarded are (outgoing) two kinds null ofcoordinates. coordinates Vaidya The spacet and Vaidya one metric in in terms the of advanced Bondi coordi Vaidya metric satisfies theG Einstein equation with the energ Null energy condition implies that by the energy momentum tensor oftional the fluctuations gravitational will waves not and be considered in our analysis. F in the metric perturbation, or equivalently in Note that the metric ( being absorbed by theblack black hole hole. by the This collapse geometry of naturally matter. de However with the sign re JHEP04(2018)056 ,  (2.7) (2.9) (2.8) ) (2.10) (2.14) (2.11) (2.12) 0 v ed on it. − v ( , θ ) vA (Θ) T 2 (1) r ˆ t ( ground Vaidya metric = 0 imposes that ) is the first example l black holes and show ) + , →∞ µν lim 0 r 2.6 ˆ g the relation T v first demonstrate that the π µ , (Θ) o be generalized. 8 − space. Our claim is that one ): ergy momentum tensor, ropriate choice of the energy ¯ ) (2.13) ∇ . y throwing in an energy flux upertranslated Schwarzschild  2 ave an event horizon. In fact 0 v (1) ) g horizon and describes a non- − − ( ˆ t v µ r ) δ (Θ) ( − ˆ + T O BC ′ v ) (Θ) ( v C ) in the Bondi gauge, the Einstein δ ( . (Θ) = (1) C ˆ A ˆ CM µν T 2.1 M ˆ t D , ˆ (Θ) T 6  )( A A  ˆ 0 T , 3 − ) + v D ) 0 − 1 v πr vv − − µν 4 = ,D T ¯ − T v 2 – 5 – ( r + v AC θ (Θ) = = ( ( AB (Θ) C   θ ) can be obtained by throwing in a shockwave-like ) C (1) v C µν (1) 0 v ˆ →∞ ∂ T lim v ˆ T ∂ D r T 2.6 (Θ) B B π A − ˆ t D D v + 2) ( ( A Θ) = 2 + 4 δ B ) + D (Θ) = ) can be solved by taking D v, 0 D ( ( 1 4 A v AB ˆ (Θ) 4 1 ˆ µ T C ˆ − T A 2.10 v + v ∂ D ( B m δ v D Θ) + ∂ ) describes a black hole with supertranslation hair implant A A (Θ) v, D ( 2.6 A give the following constraints at order ˆ ˆ 4 1 µ T vD ) is step function. The covariant conservation −   0 I = = v 2 2 A 1 1 − stands for the covariant derivative with respect to the back m πr πr v N 4 4 v µ ∂ v ( ¯ ∂ θ ∇ = = In the next section, we will review the definition of dynamica Now for general asymptotically flat metric ( vv . In addition, we consider energy momentum tensor satisfyin vA ˆ ˆ T T µν equations at Physically, the configuration ofof soft a hair particular can form. be Consider implanted an b ansatz for the perturbed en of a non-spherical dynamicalsupertranslated black hair configuration hole. ( energy Before momentum flux that, to let the us Vaidya black hole. 2.2 Implantation ofThe supertranslation hair spacetime ( where and g spacetime, the supertranslated Vaidyafor spacetime a does dynamical not black h hole, the concept of eventthat horizon the has supertranslated t Vaidya spacetimespherical admits a dynamical trappin black hole. As far as we are aware of, ( spacetime. We also remark that unlike the static case of the s To start, the constraint ( where we neglected quadratic perturbationcan terms reproduce off the from flat supertranslated Vaidyamomentum metric tensor with an app where JHEP04(2018)056 (2.20) (2.18) (2.15) (2.16) (2.21) (2.19) (2.23) (2.22) (2.17) . Note that the µν ˆ T , supertranslation of the  ˆ C. . . ˆ . m flux ) ˆ C A C, we can also implant a con- 2  ˆ 2  C D ˆ v ν 2 C , . Also it shifts the mass pa- D ˆ δ . 2 ′ C, ˆ ) D 2 v M µ C can be written as a perturbation  MD A δ D 3 µ M ˆ 2 C 1 D AB µ ˆ r µν = r C, 2 + ) 2 γ AB g ′ γ D f M − + 1 2 ) + 3 − ′ + 2) µν ˆ (Θ) = (Θ) = ˆ C CM µν 2 + h ˆ A − C ): B g ˆ ′ )( (1) T ˆ D v ˆ B ˆ ˆ C t + 0 CM C D ( ( v 2 = D V A + f µν vM M A D  − D g L µ )( – 6 – 4 1 D v ( A is nonzero. Physically it means that during the 0  ( 2 ˆ = v r )(2 C, 2 − ˆ ) D θ C,  0 2 is shockwave-like, with a delta function and a step µ ) 0  v ) A − µν 0 v + D 0 ) ). g v v µν D 0 v − ′ ( ˆ − T v 2 M − M θ v 2.8 ), that − v (Θ) = 3 ( M (Θ) is an arbitrary function of angles. Then ( v − ˆ θ v = = T ( ˆ θ C ( v 2.9 θ ( A = m = − θ rθ = N ˆ (Θ) = (Θ) = = = = C µν AB ], the perturbation can be written as A h ), let us consider C ˆ t (1) ) and ( vv vA 15 AB ˆ h T h h 2.8 2.13 ), we obtain 2.11 ) can be solved and gives is obtained by solving ( is a constant and (1) 2.13 ˆ t µ The energy momentum tensor process of the formationfiguration of of the soft dynamical hair Vaidya on black it hole, with the aid of the energy momentu over the background Vaidya metric of mass rameter by a constant amount in case where function component. And the corresponding spacetime In turn from ( which implies, from ( As for the equation ( with the perturbations: and Therefore our shockwave-likeVaidya energy metric momentum with flux the creates supertranslation a parameter Then ( where Like in the static case [ JHEP04(2018)056 = 0. (3.1) ] (see ν − 49 l – µ ∇ 46 µ − l can be written as terms of trapped and thus describes = 0, − l r on BMS supertrans- ν − . l eneral dynamical case ll 4 µ − res a knowledge of the l and nition here. Consider a + µν rizon supertranslations on lack hole thermodynamics ir implanted Vaidya black e the theorem of Hawking l ted to have dependence on its effect on the physics at g have been spent on looking ranslated Vaidya black hole nown as horizon supertrans- hole and show that Hawking ection ally flat spacetime is a Killing possible unless the spacetime uences. Below we will extend . slations (and superrotations) as lack hole spacetime [ the results of Hawking radiation We will show that this is indeed ansion of the outgoing null rays ]. Recent work on gravitational = 0 and a relative normalization µ associate with the Killing horizon + 40 ν + l l – ν µ + − l 33 l , , especially in the neighborhood of the µν + r and satisfies orthogonal to g 14 ν S µ − + l l µ but also in the interior of spacetime. − l – 7 – − + I such that ]. µν µ + g 43 l – ) holds for all = 41 µν 2.21 q ) actually satisfies the linearized Einstein equations for a 1. Then the metric in the two-space ] for reviews). These local horizons are typically defined in − 54 2.21 – = µ 50 However the situation becomes much more complicated in the g We end this section by noting that while in this paper we focus The fact that the metric ( + l µ − l Let us pick another null vector field asymptotic symmetries defined atlations, black which hole has horizons, been also studied k for example in [ lations as asymptotic symmetriesthe defined black hole at horizon, null there infinity is a and different sort of supertran bundle of null geodesics with tangent vector also [ surfaces, which are space-like 2-surfacesnormal for to which the the exp surface vanishes. Let us review the precise defi is stationary and nothingwhich changes. says that the In event thishorizon. horizon of case In addition, a one using stationary the can asymptotic nulla Killing invok surface vector, one gravity can whichand plays Hawking the radiation role effect. of temperature in the b 3.1 Trapping horizon The event horizonentire of causal a structure spacetime of is spacetime. a global This concept is physically and im requi 3 Properties of supertranslated VaidyaIn black this hole section wewhich will investigate be helpful some for properties interpretingfrom and the of understanding supertranslated of the Vaidya black supert hole obtained in the s memory and quantum mechanical effectsRindler associated horizon with can the be ho found in [ linear perturbations not only at around hole. Naturally the Hawkingthe radiation soft charges spectrum (superrotation can charges)the be of case. expec the black hole. black hole trapping horizon,the has consideration important of physical Hawking conseq radiation radiation are for dynamical created black at the dynamical horizon of the soft ha metric ( where Killing horizon generallyfor does appropriate not local definitions exist. of horizon Much for efforts dynamical b JHEP04(2018)056 . (3.2) (3.3) (3.4) 0. A > = 0, i.e. + + θ θ .  rrounding radial 0 , 0 and 0 and 0 , with null normals 0 (outer type) , 0 (gravity is not so < R < < a spacetime. Let us In this paper, we find  ∈ − + f − θ t θ 2 θ θ > ee-dimensional submani- 2 µ , cal spacetime, the horizon t r if ∂ till exhibit infinite redshift. S µ − surfaces are interesting since MD lack hole with a future outer ust be an event horizon, with and considered. Among them rather than a white hole. The r behaviour to characterize the llel. The second condition says rresponding to the ingoing and − rresponding tangent vectors are on (FOTH) introduced by Hay- izon. This is a highly non-trivial , l to be the tangent vectors of the ′ d to the presence of singularities. ity is the expansion scalar which faces µ + r l fM 2 . ν − and l µ µ V − l = 0. In general given a null flow character- ∇  0, i.e. the outgoing null rays are contracting ν ± µν 2 1 makes it trapped, hence it is outer rather than marginally trapped l < q , – 8 – − 1 0 (future type) µν l + q := θ  < : θ = S − µ + along t 0 and . A normal surface would have , θ S 0. Now a space-like closed and orientable two surface in S < , l , one can characterize the flow with a shear, rotation and an − µ θ ) specifies the location of marginally trapped surfaces where θ < l 0) , 0 3.3 , 1 − is orthogonal to , ± ]: the bundle of null geodesics is expanding if l is one with = (0 55 µ − measures the expansion rate of the infinitesimally nearby su l θ ], where later refinements and generalizations are based on. 46 = 0 (marginally trapped) instead of expanding. The surface is Now let us apply these concepts to the supertranslated Vaidy A future outer (marginally) trapping horizon is a smooth thr + S constructed by ingoing and outgoing rays such that: θ ± null geodesics [ Various definitions of black holeof horizon have particular been importance proposed is theward [ future outer trapping horiz under certain physically reasonable assumptions,The they cosmic lea censorship hypothesis thenthe suggests trapped that surface there located m insideproblem of in certain the black dynamical hole case hor sinceis when not embedded expected in a to dynami be a null hypersurface although it should s the outgoing null rays momentarily stop expanding. Trapped fold of spacetime which is foliated by closed space-like sur expansion part. Of particularis interests defined to by general the relativ divergence of the flow: at inner kind. trapped surface strong) and contracting if Physically, bundles of ingoing and outgoinggravitational null field geodesics, and surrounding use thei nearby surrounding radial outgoing nullthat geodesics the trapping are horizon para isthird of condition says future a type, motion i.e. of a black hole that the superstranslated Vaidya spacetimetrapping is horizon a of Hayward. dynamical b four-dimensions has two independent normaloutgoing directions, null co rays. It is thus natural to take ized by the tangent vector The first condition in ( l By construction, consider radial ingoing and outgoing null geodesics. The co JHEP04(2018)056 (3.5) (3.8) (3.6) (3.7) , and so | f 2 D | , | f ′ ) is actually equiv- ) and fixing angular M 3.7 , dependent terms give angular coordinates is 2.6 ) is of future and outer FOTH is in general not ≫ |  corresponds to a marginally f 3.6 f f s of the null Killing vector . 2 2 M tion ( 2 2 the trapping horizon of the r D ) into ( fine a notion of surface gravity 2 1 dv e will consider the case where MD e is generally no asymptotically cuss in the next section. There- − f. mbiguously. Instead, depending faces (  3.6 , 2 te. As a result, quantities such as f 0 − = 2 D ′ g ( r 2 1 < D ′ ) and corresponds to a marginally trapped r  + fM ′ M 3.6 ′ 2 1 4 M M − f + fM V ′′ −  + 2 1 – 9 – r 1 fM  . For the stage of evaporation the black hole mass M ). For the supertranslated Vaidya black hole, the 0, whose foliation gives a future inner trapping horizon if f 2 = + 2 M ′ f for which the null energy condition will be violated > 1 + h ( ) is trivially satisfied. The third condition yields = 2 D r ′ + M f O h 2 θ 2 r  3.3 µ M ∂ D − 1 4 ′ µ = l , θ = 4 − r M r 2 2 ′′ 1 4 2 − ), we get ds − 0 due to ingoing negative energy flux and the FOTH would be fM = ′′ 3.4 < − is taken as a small perturbation and the − ′ ) yields θ > fM f ′ M 3.3 − = 0 gives M ) and ( < A = 0 has two solutions. One is given by ( ′ 3.1 Θ 0. + d M θ ), ( f > 2 2.6 D 2 1 0. The second condition in ( For stationary black hole, surface gravity is defined in term − Actually > 2 = h four-acceleration and surface gravity cannot beon defined the una local definition of horizons that is adopted, one may de of the Killing horizon. Intime-like Killing a vector time-dependent to spacetime, define ther a preferred time coordina Next we discuss thesupertranslated horizon Vaidya black surface hole. gravity associated with time-like. 3.2 Surface gravity will be decreasing and space-like if first condition in ( where we have discarded terms of correction to the Schwarzschildthe radius. corrections are For consistency, small w such that the first term dominates, surface of outer type as discussed above. The other solution Hence the FOTH oftime-like supertranslated if Vaidya black hole at a fixed alent to a positive horizonfore, surface a gravity trapping which we horizon will foliatedtype dis by and marginally trapped is sur anull FOTH but of it the still supertranslated has infinitecoordinates Vadiya red black shift. hole. In A fact, substitutin In our calculation, r trapped surface of inner type with which is satisfied for the same assumption as above. The condi Using ( r JHEP04(2018)056 ) d κ (3.9) 3.10 (3.12) (3.10) (3.11) and the µ v δ = for dynamical µ κ . We would like ), although it is ¯ direction for dy- ) K v ( 2.6 1 ( M 4 dama vector coincides µν g = ctor is rizon. The coefficient κ ) coincides with the time- acetime physics. in thermal equilibrium. In horizon corresponds to the ric f it exists, we can use ( face gravity ally flat spacetime, with an nt on the FOTH. Physically ence of a unique vector field amical black holes, it is not librium system. Nevertheless lated Vaidya black hole case = 0. It is interesting to think . ) is ¯ 3.11 ing relation holds for Kodama y solutions, and the defining ystem is in equilibrium locally. th the time translation Killing κ ically symmetric case. In fact, , = 0 zschild in static limit. In the su- rtranslated Vaidya black hole has ν 3.10 µ , K ν K  µν ′ µ g κK M ∇ M ] = − f 58 µ = − ] , 1 µ  K – 10 – ). The inner trapping horizon would correspond ν [ ,K 1 M ) = 0 µ v 4 ∇ µ δ µ ) term. Evaluated on the FOTH, the surface gravity 3.12 K = 2 ]. For the FOTH of Hayward, there is a quite explicit K = f µν κ ( µ ) reduces to that of Killing surface gravity in static limit 57 G , O ( K ν 56 3.10 ∇ , as a local measure of the temperature of the superstranslate given by ( 54 κ κ ) up to . Note that the Kodama-like vector ( f 3.9 is the Einstein tensor. Kodama vector gives a preferred time ], ) is found to be 59 µν 0 and the degenerate (extremal) one corresponds to G For a spherically symmetric Vaidya spacetime, the Kodama ve 3.10 , called the Kodama vector, which satisfies [ κ < µ horizon surface gravity of Vaidya black hole defined by ( about what these thermodynamical notions mean in terms of sp black holes with sphericalknown in symmetry. general whether For the non-spherical Kodama dyn vector exists. However i in ( at the linear order in to extend the definitionusing of the surface Kodama gravity vector. to For the the supertranslated supertrans Vaidya met We propose to interpret to translation Killing vector forpertranslated the case, supertranslated the Schwar horizon surfacea gravity of dependence the on supe the supertranslationthis means hair that and the is black notgeneral hole a one does cannot consta not define a correspondone specific to may temperature a still for system define a a nonequi temperature locally provided that the s not spherically symmetric, we find that to define the surface gravity. by virtue of Killing equation and thus can be used to define sur and satisfactory definition offor surface any gravity spherically for symmetricK the metric, spher one can show the exist where the equality holdsdefines when the it horizon is surfacewith evaluated gravity the on of time the aequation translation FOTH trapping for Killing of ho surface Hayward. vector gravity ( field Ko for stationar Vaidya black hole. As wepositive mentioned surface above, gravity the outer trapping correspondingly, see e.g. [ solves the equation ( namical spacetimes in sphericalappropriate symmetry. normalization, the In Kodama anvector vector asymptotic at coincides spatial wi infinity. Haywardvector has [ shown that the follow Here JHEP04(2018)056 . (4.1) (4.2) ation ) = 0. x ( ] so that φ ). At the 63 ~  ( – O 61 + ~ iS/ e ) x radiation of black hole. ( lar particles, nary part of the particle he two methods provide milton-Jacobi method to A king radiation observed at ted Vaidya black hole. As k hole geometry. That this null geodesic, the Hamilton- = ion of a particle getting out al black holes [ g solution). The motion of on. Moreover, the Hamilton- ated Vaidya black hole. d the imaginary contribution wavelength near the horizon, g solution along a trajectory ry is because of the huge red orrespond to the two different eading order of the energy of φ a point particle approximation und. One can then reconstruct d particles. Also covariance of ation of the whole spacetime, the o a black hole along a classically . ] for a review on the tunneling methods S, µ 57 = 0 ∂ µ S ν dx P at the location of horizon. To leading order in S∂ – 11 – Z µ S ∂ µ = µν ∂ S g ], the Hawking radiation can be computed as a tunneling is real in general, but can become complex if the trajectory 32 to possibly become imaginary only for the outgoing solution S , the Klein-Gordon (KG) equation reduces to the Hamilton- ~ S ]. While the null geodesic method is based on the self-gravit 60 is the null geodesic. P Consider now, for example, a minimally coupled massless sca In the case ofcrossing static the black event horizon holes, whicharises this we from occurs call the a for tunneling residue the path, of outgoin an the pole in is a classically forbiddenpermitted one. path, Since we expect particles can fall int leading non-trivial order of the particle action by a line integral Generally there are two solutionssolutions to this of equation the and KG they(outgoing c equation. solution) Physically, and theyparticle falling describe is in given mot by to a null the geodesic black in the hole black hole (ingoin backgro where Jacobi equation We look for a solution to this equation by a WKB ansatz compute the Hawking radiation spectrum for the supertransl Hamilton-Jacobi method would beaction simple without to reference compute tothe the method the imagi is self-gravitation well of understoodJacobi method in emitte can the be Hamilton-Jacobi applied equati it to is either stationary suitable or dynamic for our purpose. We will therefore adopt the Ha and Hawking radiation. In additionJacobi to method the has original also methodIt been of has developed to been computethe shown the same for Hawking result stationary forthe black the emitted semi-classical hole particles [ emission spacetimes(backreaction) rate of that at emitted t particles the and l the energy conserv is possible without usingshift the factor full-fledged at quantum the fieldasymptotic horizon theo infinity of arises the from blackand emitted therefore, hole. wave with as As far vanishing as anear the the result, horizon tunneling Haw is process good. is of We refer concern, the readers to [ process based on null geodesic motion of particle in the blac 4.1 Hamilton-Jacobi method Now let us considerargued Hawking by Parikh radiation and from Wilczek [ the supertransla 4 Hawking radiation from supertranslated Vaidya black hole JHEP04(2018)056 (4.4) (4.6) (4.3) (4.5) (4.8) (4.7) to the putation = 1. For in S ~ e a thermal ], the metric has poles at, ) are 15 ) 4.5 f out , and 2 S M D have set 2 + ωdv. r > , V f tions to ( Z emiclassical emission rate B nary part since absorption is the semiclassical absorp- (2 , , − es would be suitable for the ] Θ A anslated Schwarzschild black instein equations. We can see d ab = . As shown by [ classical emission rate satisfies 15 holes. We consider radial null or regions of a black hole and D A = 0 is the 4-momentum of particle. = 0 S. 0) for f A in 2 v Θ µ , S in Θ ∂ d r p  D S
− 1 2 r f S > ω∂ out dvd where Γ 2 r 2 = + S and ]. The integrand for ∂ ,S − D ab S − v 65 M f µ Γ ∂ 2 2 AB ∂ 2 ) , ∂ 2 Im r µ which is the location of the (shifted) event βω = S f = 2 rγ dvdr ξ − 2 r MD − ξ 2 f ωdr r e ∂ − r  2 − ( 2 − – 12 – + 2 ω MD = f D =  2 2 1 2 V B exp µ f − is given by the WKB formula em 2 p dv D + being the (Hawking) temperature of the system. 2 Z µ ∝ V A r  ξ 1 ]Γ em + f M − − = em MD . 2 rD 64 β Γ 2 f , = r 2 − out ωdv + 2 := 2 31 ω S D MD V r 1 2 ), the corresponding action is h Z  ∂ r AB − − − γ 4.2 2 = V r = :=  r f r − + out . This coordinate system has a nice feature for tunneling com ) plane for which the Hamilton-Jacobi equation is S = = r r M is the energy of the emitted particles. This allows one to hav 2 r 2 − terms, ω ds v − f corresponds to the outgoing solution ( denotes the particle action for the outgoing solution and we = 1 out out S V S Let us consider the semiclassical emission rate for supertr is the energy of particle and is a constant in motion. The solu is the time-translation Killing vector field , the semiclassical emission rate Γ ingoing solution. Using ( static black hole spacetimes, itthe is detailed easy balance to relation show [ that the semi where ω up to order since it is non-singular at the event horizon horizon, and at ξ geodesics in ( can be extended to thethat interior this and is coordinate still system ahence covers solution it of both is the interior E an andfor appropriate exteri black holes. coordinate system Oncomputation the to of other compute the hand, the semiclassical retarded s absorption Bondi rate coordinat for white where As mentioned above, the ingoingis action complete does for not black have holes an in imagi the classical limit [ where hole. In the advanced Bondi coordinates, the metric reads [ tion rate and ~ where interpretation of the result with JHEP04(2018)056 iǫ em as Γ (4.12) (4.10) (4.11) πωM motion. 8 − e = ab ]. 66 prescription should cor- is introduced with the op- iǫ shed line represents the ]. iǫ prescription is consistent with ath. After tunneling out 64 moving backward in time is πωM, . ) iǫ ]. If a type-II path and would be al function decays and the relation ab −→ er with positive energy escapes to out 31 = 4 S vanishes on a type-II path. For ) πωM . f ncy solutions is usually implicit in the 8 r ) (type-II) (4.9) (type-I) out f way. A pair of virtual particles is created − r S − lton-Jacobi equation [ c ce of vacuum while in the tunneling method r xtension as in [ − , )( dr dr hat for the time reversed one Γ r 2 2 iǫ  )( h f ωr ωr − ) = exp( r 2 2 2 2 h − r r out b r S ], it is discussed that the – 13 – − MD ( I r 57 ( − Z 2Im , the integral is divergent. Adopting a Feynman’s which contains a path b h the imaginary part of V − = r → 4  a a −→ abc = Z 1 2 It is easy to see that out , exp( r S − 3 ∝ = 0 = = Im or higher. The semiclassical emission rate is thus obtained em 2 dv dr dv Γ out f S Im , we need to perform the integration for along a trajectory of out for black hole semiclassical emission rate is obtained [ S ab Γ , we show a trajectory is introduced in such a way that the positive energy exponenti 1 ]. . A trajectory from inside to outside of the black hole. The da πωM 8 iǫ 64 − e . One with negative energy falls into the black hole and the oth b To obtain = This tunneling process can also be interpreted in a different Here 3 4 em respond to boundary conditionswith on the the wave WKB function approximation andcoordination choi identification and of in making the ansatz positive for freque the solution to the Hami the analytic continuation when spacetime allows the Kruskal e near As we consider s-wave motion, there are two types of path: infinity [ prescription to deform the integral, Γ a type-I path, we have In figure Figure 1 posite sign, one obtains a result which would be regarded as t As our path crosses the horizon dropping terms of order and crossing the horizon.from the This horizon, part theobserved emitted is as particle described Hawking escapes radiation. by to a infinity type-I on p for white hole semiclassical absorption rate. In [ event horizon. JHEP04(2018)056 ) of v , the ( (4.13) (4.16) (4.15) (4.14) out M S . r = 0 S B ωdv. ) with the metric . S∂ Z A 4.1 ∂ − sibility is to consider = 0
reted in the tunneling iation. The black hole = f in 2 i method can be applied S in r this dynamical black hole; D l particle pair created near ck hole. However there are dya spacetime with a time- across the trapping horizon e to consider the process of e to the features of soft hair uation ( orate this back reaction effect f the black hole mass AB ion in this case is independent S. v rγ ∂ ,S ), we can read off the Hawking , ∂ + − 1 f f 2 − = B ω/T  D S − f MD 2 µ A in favor of the Kodama-like vector which 2 ∂ r − µ ω rD dr r 2 ′ MD 2 K − − − ωr – 14 – 2 2 fM ′ the surface gravity of Schwarzschild black hole. = ) is real for most part of the path except possibly along the geodesics. In this case, the Hamilton- 2 AB S µ γ . r r 1 M − 2 p 0 during the evaporation process. fM 4 out ∂ r µ 2 2 ( S < K 4  = ′ − 1 V r r f s 2 V M is the advanced null coordinate. The decrease of the black + ) to = const κ Z 2 ≡ −  r v S A + ω A ω MD 4.16 with S∂ − = 2 r ′ ωdv s ∂ π ) κ 2 Z r ) where out f fM v 2 S − ( 2 = r D ∂ − with the Boltzmann factor, exp( = M T + has no imaginary contribution since the energy is real. As fo V out em  ). in V f S S + (2 2.6 S A v D 2 S∂ 1 r ) is rather complicated to solve in general. The simplest pos r For the supertranslated Vaidya metric, the Hamilton-Jacob ∂ + 2 4.13 dependent mass In the above, we findof that a the stationary Hawking Schwarzschild radiationblack black is hole hole. insensitiv evaporation and Next,and compute we the investigate would Hawking how lik radiation theevolution of process soft can hair be would modeled affect with the a Hawking supertranslated rad Vai Note that in thiscontributes computation, to the only result. the Noteof infinitesimal also the region that soft the hair Hawking features radiat of the4.2 black hole. Supertranslated Vaidya black hole radial null geodesics with Θ given by ( line integral contribution ( Here we define the invariant energy of particle As before, And the particle action is reconstructed as hole mass duemethod to as the one ingoing havingthe negative the FOTH falls negative energy into energy flux the particle blackof can hole. of ingoing Hawking be Here a radiation we virtua interp effectively by incorp the diminishing behaviour o temperature as Identifying Γ generalizes the particle energy in the static case. the background geometry with ( in a similar wayimportant to differences. the previous We start example with of the Schwarzschild bla Hamilton-Jacobi eq Jacobi equation has simple solutions JHEP04(2018)056 = 0 r (4.19) (4.17) (4.18) crossing ) and it is clear that . , 3.6 ck hole at a fixed value of dynamical, it is no longer rtual pair is created inside path corresponds to a pair n how the path crosses the scribes a backward null ray ccurs at the location of the ime, while type-II tunneling espectively: ch comes out from the future y forward in time. Note that e interior point in the trapped le. pe-I tunneling path crosses the (type-I) (type-II) y escapes to infinity. The main , given by ( h κ , vanishes and no tunneling occurs for πω r  = f out 2 f S 2 h 2 r r , the wavy line represents the singularity ) with − h . The imaginary contribution due to the pole MD r πωr h h 2 r r 4.10 − = – 15 – ′ = r = r r fM out 2 S − , Im . It is clear that V . On the segment of the geodesic which crosses the trapping  out 2 S 1 2 = = const. v dr dv prescription as before and yields iǫ = 0, crosses the trapping horizon and escapes to infinity. The r . Typical Penrose diagram of the supertranslated Vaidya bla Let us now compute Im . The dashed line represents the FOTH A type-II paths. For type-I paths, we have ( is evaluated by an the integrand has a pole on the FOTH horizon, they are described by the type-I and type-II paths r difference between the two types ofFOTH tunneling paths along is a that ty classicallypath forbidden crosses trajectory the backward FOTH inthe along t crossing a with classically type-II allowed path trajector is absent for a static black ho with type-II path correspondswhich to comes a out from pair the forming futureregion, inside singularity, and and and then reaches de som crosses the trapping horizon and eventuall Figure 2 forming outside and describes asingularity backward at radial null ray whi The two types of tunnelingor paths outside correspond of to the whether trapping the horizon. vi The crossing with type-I trapping horizon. Howevernull. since now This the leads trapping totrapping horizon. two horizon types is See of figure tunneling paths, depending o and the thick curve represents a collapsing matter. at around the location where the integrand diverges, which o Θ JHEP04(2018)056 vap- ), we (4.23) (4.20) (4.21) (4.22) we get ′ 4.19 M , κ ) for Θ 2 e undressed hard 4.22 γd √ ]. It should be clear 4 2 h r β 29 ck hole. A proposal for atrix does not imply that ervable. th soft gravitons, then the Z discussion of gravitational f soft hair on gravitational ]. For Einstein gravity with 2 al black hole is given by the . . Note that non-trivial angle- . 30 π 68 ated Vaidya black hole can be ). Solving (  v  M . ′ ( 1 κ . In the last equality of ( πω M Θ = − M M 2 A 2 f  = − ] is caused by the Hawking radiation. 0 γd −  P 1 do not contribute to the mass loss in 67 √ M 1 πM 2 00 h , and Θ ≥  r a 3 √ l v is the standard power loss due to Hawking  − 1 M = exp 4 1 1 ]. The non-trivial dependence on the soft hair dω − – 16 –  out = 1 ) 29 S 0 2 of the FOTH. – f P , 3 − r 0 κ 2Im ω 26 − is thus formally written in the same form as that in 2 h P πM (Θ) dependence comes into the spectrum of Hawking − βω − r e e = f  h em ′ ′ r ∝ ∞ 0 M M ) of the emission rate is consistent with the factorization M = Z em π 00 κ  Γ a free and determined the Hawking radiation in terms of it. In a √ := (7680 4.21 ′ Z 0 − 2 P M 1 1 π , we obtain a consistent model of dynamical black hole whose e 2 ′  = = M and ′ ) to write the result in terms of the horizon surface gravity M π/κ − 3.12 := 2 is evaluated on the FOTH for a fixed ) is due to the fact that we are performing an observation of th β ω Let us discuss the entropy of the supertranslated Vaidya bla Note that our result ( So far we have left 4.21 oration is driven by the power loss due to Hawking radiation. With this value of radiation in the leading order approximation of a constant the static case. dependent supertranslations corresponding to The semiclassical emission rate Γ the case of supertranslated Vadiya black hole with consistent model of evaporation, the mass loss [ radiation through the surface gravity We find that the supertranslation where matter minimally coupled to gravity, the entropy of dynamic The leading contribution to theestimated mass from loss the of Hawking the radiation supertransl spectrum as: memory where one maythat also the factorization remove of thethe soft memory soft modes with hair and dressing hard hasmemory [ modes no and in physical Hawking the radiation implication S-m spectrum at is all. physical and The is effects obs o property of the blackin hole ( S-matrix [ Hawking radiation quanta. Ifsoft one is hair to dependence dress would the hard disappear. modes wi This is similar to the have used ( dynamical black hole entropy is presented by Iyer and Wald [ where JHEP04(2018)056 (5.1) (4.24) (4.25) (4.27) (4.26) defined by κ ]. Furthermore, one 57 lack hole mass changes , awking quanta, then the ) and it is consistent with aidya spacetime with soft 32 . pear in the mass function. π y sight of the soft hair. ) rface gravity at is being observed. If one the black hole. Our results 4 d the semiclassical emission 2 s clear observable effects on we show that soft hairs also hole. The Hawking radiation ω √ e of the black hole evaporation ( / . ) as ft hairs are physical and can be . In particular we computed the ′ O holes [ 00 t is consistent that a local change a , 4.21 + i ) + . MM ′ v α ( 00 em S 2 M i| em Θ 00 πa α 2 ln Γ πM √ ) at different angles. As Hawking radiation H πa | γd π Ω α √ 4 d √ c + 4 = 4 = ln Γ 4.27 2 h 2 r – 17 – Z α + 4 S X dyn Ω Z ∆ = d S πM d = 4 1 S πM i ∆ = 4 = (8 Ψ | ω due to the energy conservation, and the entropy changes − ) as saying that the Hawking radiation carries away from with dyn S ω = S ] has emphasized the presence of soft modes in the final state 4.26 − S 16 4 as for stationary black holes. We find ∆ + ∆ / M h A dyn → S = ) can be expressed as M → , dyn ω S 4.24 dyn S Recently Strominger [ When the black hole emits a massless Hawking radiation, the b The entropy ( the Kodama-like vector of thespectrum supertranslated has Vaidya a black dependencemake on it the clear soft that hairthe soft distribution Hawking hair radiation over of spectrum. black Ofwishes, hole course one is this physical may depends and also onobserved it wh choose spectrum ha to would make become detection the canonical of one the without dressed an H This can be related to the semiclassical emission rate ( and interpret the relation ( of the black hole evaporation andprocess proposed to that the be final a stat pure state of the form observed through the classicalhave memory non-trivial effects effect. on the In quantum physics thisHawking of radiation paper, black hole for a dynamicalhair. black We hole find modeled that tunneling byrate occurs the of at V the Hawking trapping radiation horizon an is characterized by the horizon su of entropy of the black hole occurs. 5 Conclusion and discussion Supertranslation of black hole adds soft hair to it. These so the black hole different amountoriginates of entropy locally ( at the surface of the trapping horizon, i may define a differential entropy change: This generalizes the standard relation for spherical black same formula by amount of the fact that only the zero mode part of the shift of time can ap accordingly JHEP04(2018)056 ommons gravity. ]. (1975) 199 awking radiation SPIRE 43 , e (1993) R3427 IN [ onservation. In other ]. ]. , oto, Hiroyuki Kitamoto ]. ]. op on Fields and Strings 1-15, 2017. This work is D 48 , Pei-Ming Ho, Yoshinori ved, the resulting reduced graviton modes. However S symmetries and soft hair nce (NCTS) and the grant SPIRE ute for Theoretical Physics, and for valuable comments ically. Indeed, similar deco- cussions and comments. We nology of Taiwan. redited. IN SPIRE [ SPIRE SPIRE of unobservable soft modes has IN ]. It is important to understand (1973) 2333 IN IN rence when “environmental vari- ][ 70 ][ ][ , Phys. Rev. D 7 69 , (1974) 30 Commun. Math. Phys. , Stationary Black Holes: Uniqueness and 248 Phys. Rev. gives the thermal density matrix of Hawking ra- , – 18 – arXiv:0909.1038 ]. [ arXiv:1205.6112 Nature [ arXiv:1703.02143 , [ thermal SPIRE ρ IN = ), which permits any use, distribution and reproduction in (2012) 7 | Ψ ]. 15 ih (2009) 224001 Ψ | (2017) 092001 26 SPIRE soft (1976) 206] [ 80 IN CC-BY 4.0 Black holes and entropy ][ Black hole explosions Particle Creation by Black Holes 46 This article is distributed under the terms of the Creative C = tr The Information paradox: A Pedagogical introduction r Black hole entropy is the Noether charge ) is related to the BMS symmetry and the infrared structure of ρ The Black Hole information problem: past, present and futur 5.1 Living Rev. Rel. describes the state of radiation in the thermal ensemble of H , i α describes the cloud of soft modes that is required by charge c i H | α gr-qc/9307038 Erratum ibid. S Beyond Class. Quant. Grav. Rept. Prog. Phys. [ [ | [1] J.D. 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We benefited much fromat the KEK, discussions Japan, on BM during2017” the held conference in November “East 13-17, AsiaJapan, 2017, Joint and during Worksh also the at conference Yukawasupported Instit “CosPA2017” in held part in by December the 1 National Center of Theoretical Scie Acknowledgments We would like toHonma, thank Yu-tin Carlos Huang, Cardona, JanetChris Dimitrios Hung, Lau, Giataganas Misao Satoshi Sasaki, Iso, Jirothank Soda Shoichi and in Kawam Wei particular Song for Misao useful Sasaki dis for reading the manuscript Attribution License ( JHEP04(2018)056 ] ]. lativity SPIRE IN , ]. , ][ , ]. ry , , (2014) 152 arXiv:1106.0213 [ ]. SPIRE , , ptotically flat 07 SPIRE IN [ IN ]. ][ SPIRE ]. ]. ]. IN JHEP ][ , ]. gr-qc/9911095 (2011) 105 SPIRE [ ]. IN SPIRE SPIRE SPIRE 12 ]. ][ IN IN IN SPIRE ][ ][ , SPIRE IN Gravitational waves in general [ Semiclassical Virasoro symmetry of the IN SPIRE JHEP ]. ][ , IN arXiv:1706.07143 arXiv:1406.3312 ]. 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