Temperature and entropy-area relation of quantum matter near spherically symmetric outer trapping horizons
aa r X i v : . [ g r- q c ] F e b Temperature and entropy-area relation of quantummatter near spherically symmetric outer trappinghorizons
Fiona Kurpicz ,a , Nicola Pinamonti , ,b , Rainer Verch ,c Institute for Theoretical Physics, University of Leipzig, D-04009 Leipzig, Germany. Dipartimento di Matematica, Universit`a di Genova - Via Dodecaneso, 35, I-16146 Genova, Italy. Istituto Nazionale di Fisica Nucleare - Sezione di Genova, Via Dodecaneso, 33 I-16146 Genova, Italy.E-mail: a [email protected] , b [email protected], c [email protected]
Abstract
We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are gen-eralizations of spherically symmetric black hole spacetimes where the central mass can vary with time,like in black hole collapse or black hole evaporation. While these spacetimes possess in general no time-like Killing vector field, they admit a Kodama vector field which in some ways provides a replacement.The Kodama vector field allows the definition of a surface gravity of the outer trapping horizon. Spher-ically symmetric spacelike cross-sections of the outer trapping horizon define in- and outgoing lightlikecongruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on thespacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal formand a thermal spectrum with respect to the time-parameter of the Kodama flow, where the inverse temper-ature β = 2 π/κ is related to the surface gravity κ of the horizon cross-section in the same way as in theHawking effect for an asymptotically static black hole. Similarly, the tunneling probability that can beobtained in the scaling limit between in- and outgoing Fourier modes with respect to the time-parameterof the Kodama flow shows a thermal distribution with the same inverse temperature, determined by thesurface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon inthe scaling limit. Moreover, the scaling limit 2-point function allows it to define a scaling-limit-theory,a quantum field theory on the ingoing lightlike congruence emanating from a horizon cross-section. Thescaling limit 2-point function as well as the 2-point functions of coherent states of the scaling-limit-theory are correlation-free with respect to separation along the horizon-cross section, therefore, theirrelative entropies behave proportional to the cross-sectional area. We thus obtain a proportionality of therelative entropy of coherent states of the scaling-limit-theory and the area of the horizon cross-sectionwith respect to which the scaling limit is defined. Thereby, we establish a local counterpart, and mi-croscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamicallaws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black holedynamics originally shown for stationary black holes by Bardeen, Carter and Hawking. Introduction
The famous four laws of black hole mechanics and their analogy with the laws of thermody-namics have been derived and developed in [BCH73] assuming stationarity. The temperaturethereby assigned to a black hole is related to the horizon’s surface gravity and can physicallybe interpreted in terms of Hawking radiation [Ha75] in the framework of quantum field theoryin curved spacetime, see also [FH90, KW91, Se82, Wa75]. Similarly, the area of the black holehorizon surface is analogous to an entropy. Discussions of black hole entropy and its physicalnature have been given in a variety of contexts ([Be73, Wa01, FNS05, So11, Pe17, Ma15] andliterature cited therein is just a small sample of references on the topic) but it has been difficultto find a simple, direct counterpart of the entropy-area relation for black holes in the setting ofquantum field theory in curved spacetime (see however [HI19], and further discussion below).Although Hawking radiation is derived neglecting backreaction, assuming that the space-time geometry is stationary (or asymptotically stationary), the emission rate of Hawking ra-diation is usually associated to the rate of black hole mass loss due to evaporation, see e.g.[Ha75, Ca80, FNS05]. However, black hole evaporation is a dynamical process and should bedescribed locally. A local theory for the geometry of non-stationary black holes using conceptsof dynamical horizons and trapped horizons has been developed, see e.g. [Ha97, AK04]. Inparticular, in [Ha97] it is shown that the first law holds as an energy balance along the trappedhorizon. In contrast to the (asymptotically) stationary case, Hawking radiation and a relationbetween temperature and local geometrical quantities of dynamical or trapped horizons has sofar not been derived for quantum fields in the background of non-stationary (or dynamical) black holes.An essentially local derivation of the Hawking effect has been proposed by Parikh andWilczeck [PW00]. In that approach, an estimate is given for the tunneling probability of quan-tum particles across the horizon, showing that this probability has a thermal distribution. Thisidea has been generalized to the case of dynamical black holes in [DNVZZ07, HDNVZ09,DHNVZ09], hence furnishing a connection between the surface gravity and a thermal distri-bution of the tunneling probability. These considerations didn’t use quantum field theoreticalmethods as in the original derivation of the Hawking effect but relied on single-particle quan-tum mechanics in a WKB-type approximation. In order to overcome the limitations of sucha quantum mechanical treatment, it has been shown in [MP12] that for a scalar quantum fieldon a stationary black hole spacetime — more generally, any spacetime with a bifurcate Killinghorizon — a thermal distribution in the tunneling probability is obtained in a certain scalinglimit located on the horizon whenever the quantum field is in a Hadamard state. The asso-ciated temperature is the Hawking temperature and is independent of the chosen Hadamardstate. A similar result can also be obtained in the case of self-interacting fields, see [CMP14].For some related results, focusing on the thermal nature of field theories restricted on null sur-faces (horizons) and thus not focussing on the local aspect related to tunneling processes, see2Se82, KW91, GLRV01, FH90, HNS84, SV96].In this paper we aim at generalizing the result of [MP12] to the case of spherically symmet-ric, dynamical black holes. For generic spherically symmetric black holes there is no Killingvector field which generates an horizon. Nevertheless, there are generalizations available thatserve a similar purpose in the context of black hole thermodynamics. In particular, we shall usethe concept of outer trapping horizons [AK04] and the
Kodama vector field [Ko80]. A spheri-cally symmetric (non-stationary) black hole spacetime is a warped product of a 2-dimensionalLorentzian space and a 2-dimensional Euclidean sphere. The future-directed light rays in thetwo-dimensional Lorentzian space determine, at each spacetime point, two geodesic congru-ences of null type; one is called outgoing and the other ingoing . The outer trapping horizon H is the 3-dimensional hypersurface which divides the inside region where the expansion param-eters θ ± of the ingoing ( − ) and outgoing (+) null geodesic congruences are both negative fromthe outside region where θ + > and θ − < . The outside region usually reaches out to spatialinfinity. If the expansion parameter of the null geodesic congruence is positive (negative), thearea of a congruence-orthogonal spatial sphere grows (decreases) towards the future along thecongruence. Hence, in the region where both θ ± are negative all light rays tend to fall into theblack hole while in the region where θ + > the outgoing lightrays tend to reach points whichare far away (measured by the radius of the orthogonal spatial sphere) from the center of theblack hole. Thus, an outer trapping horizon H is the surface from which nothing can escapeinstantaneously. It is worth noting that in a dynamical spherically symmetric spacetime, H neednot be lightlike but can have timelike or spacelike parts.In [Ko80], Kodama has shown that in the case of spherically symmetric spacetimes, it ispossible to find a vector field K a which can be used as a replacement of the timelike Killingvector field of a stationary black hole (the full definition will be given in Sec. 2.2). This Kodamavector field K a is a conserved current, and also W ab K b is a conserved current whenever W ab is a symmetric tensor field that is invariant under the spherical symmetries of the spacetime.Furthermore, K a is timelike outside of, spacelike inside of, and lightlike on an outer trappinghorizon H , respectively. On H , one has K a ( ∇ a K b − ∇ b K a ) = κK b (1)where the function κ is the surface gravity along H .Outer trapping horizons and the conservation of currents generated by the Kodama vectorfield have been used by Hayward [Ha97] to derive a first thermodynamical law for dynamicalblack holes. In particular, it holds that M ′ = κ π A ′ + w V ′ We shall mostly employ the abstract index notation for vector and tensor fields as in [Wa84] f ′ = z a ∇ a f where z a is any (nowhere vanishing) vector field having zeroangular components tangent to the outer trapping horizon. Furthermore, M is the Hawking massof the black hole, A = 4 πr is the area of the surface, V = πr is the surface-enclosed volume,and κ is the surface gravity associated to the Kodama vector field. The term w = − G UV g UV is related to the trace of the Einstein tensor taken with respect to the lightlike coordinates ofthe horizon, symbolized by indices U and V ; see Section 2 for full details. As usual, M isinterpreted as the black hole’s internal energy, see (7) below, and w V ′ is the work done on thesystem. Interpreting κ/ (2 π ) as a temperature, A ′ / represents the variation of entropy.We will consider a (for simplicity, scalar) quantum field φ ( x ) propagating on a sphericallysymmetric spacetime with an outer trapping horizon and a Kodama vector field. Here, wefollow common practice to write symbolically φ ( x ) where x is a spacetime point as if φ ( x ) was an operator-valued function, while actually it is an operator-valued distribution. We willtake due care of this circumstance whenever required in the main body of the text. To furthersimplify matters, we assume that φ ( x ) is a quantized Klein-Gordon field fulfilling the fieldequation ( ∇ a ∇ a − M ( x )) φ ( x ) = 0 where ∇ is the covariant derivative of the spacetime metric g ab and M is a smooth, real-valued function on spacetime. (This assumption could, in fact, begeneralized.)States (and in particular, quasifree states) of the quantized Klein-Gordon field on curvedspacetimes admitting a physical interpretation consistent with the principles that apply for quan-tum field theory on Minkowski spacetime are Hadamard states . These states are defined ashaving a 2-point function of
Hadamard form , meaning that w (2) ( x , x ) = 18 π ∆ / ( x , x ) σ ǫ ( x , x ) + W ǫ ( x , x ) , (2)where ∆ is the van Vleck - Morette determinant of the spacetime metric and σ ( x , x ) is itsSynge function, i.e. the squared geodesic distance divided by 2. Both quantities are determinedby the spacetime metric; the subscript ǫ denotes a regularisation that is used to properly definethe quantity on the right hand side as a distribution (after integration with test functions) inthe limit ǫ → (see [KW91] and Sec. 3 for further details). Similarly, in the limit ǫ → , W ǫ ( x , x ) is a distribution which diverges at most logarithmically in σ for σ → and containsthe state-dependence as a smooth contribution. For a discussion as to why Hadamard states areof particular significance, see e.g. [FV13, KM15, Wa94] and references cited there.We will show that close to an outer trapping horizon of a spherically symmetric spacetime,the universal leading short-distance singularity behaviour of any Hadamard state results, ina scaling limit, in an interpretation of the surface gravity κ as a temperature parameter, inclose analogy to previous considerations for the case of quantum fields on stationary blackholes [KW91, Ho00, MP12, SV96, GLRV01]. Our approach follows the spirit of [MP12] veryclosely and thus makes contact with the tunneling interpretation of Hawking radiation. Fora spherically symmetric spacetime with outer trapping horizon H , one introduces Eddington-4 C ∗ U T ∗ H S ∗ Figure 1: The picture represents the
U, V plane in adapted coordinates ( U, V, ϑ, ϕ ) . The thickline corresponds to the outer trapping horizon H , S ∗ ⊂ H is a sphere which is used to identifythe null congruence C ∗ towards which we compute the scaling limit of the quantum states. Thescaling limit state is then restricted onto the null congruence T ∗ .Finkelstein coordinates v, r, ϑ, ϕ . Some point on H will be determined by certain coordinatevalues ( v ∗ , r ∗ , ϑ ∗ , ϕ ∗ ) and, by spherical symmetry, it determines the associated spatial sphericalcross-section S ∗ = S ( v ∗ , r ∗ ) of H . The outgoing null geodesic congruence emanating from S ∗ defines a null hypersurface denoted by C ∗ , and similarly the ingoing null geodesic congruenceemanating from S ∗ defines a null hypersurface denoted by T ∗ . As will be discussed in the mainbody of this paper, given S ∗ , there exists a natural choice of an affine parameter V along thegeodesic generators of C ∗ and of an affine parameter U along the geodesic generators of T ∗ sothat local coordinates ( U, V, ϑ, ϕ ) near S ∗ can be introduced, with the following properties:(1) U = 0 and V = 0 exactly for the points on S ∗ ,(2) U = 0 exactly for the points on C ∗ ,(3) V = 0 exactly for the points on T ∗ ,(4) ds = − A ( U, V ) dU dV + r ( U, V ) d Ω is the metric line element where d Ω denotesthe line element of the two-dimensional Euclidean sphere, and A = 1 on C ∗ ∪ T ∗ ,(5) dU a K a ( U, V = 0 , ϑ, ϕ ) = − κ ∗ U + O ( U ) on T ∗ near U = 0 , with κ ∗ = κ | S ∗ .We call ( U, V, ϑ, ϕ ) with the properties stated above adapted coordinates with respect to S ∗ (See Fig. 1 for an illustration.)To analyze the short distance behavior of the 2-point function of Hadamard states when both x and x are very close to H we proceed as follows. Once a sphere S ∗ (having radius r ∗ )of the outer trapping horizon is chosen and the null surface C ∗ of outgoing null geodesics isdetermined, we take a suitable scaling limit of the 2-point function towards C ∗ . As we shall5rove in Theorem 4.1, the 2-point function (distribution) Λ thus obtained is universal, and itcan be tested with compactly supported smooth functions on T ∗ . Using adapted coordinates, itsregularized integral kernel has the form Λ ε ( U, ν ; U ′ , ν ′ ) = − π r ∗ ( U − U ′ + iǫ ) δ ( ν , ν ′ ) , (3)where ( U, ν ) denotes a point on T ∗ , U is the null coordinate and ν = ( ϑ, ϕ ) denotes standardangular coordinates on the sphere S ∗ . Furthermore ε > is a regulator (to be taken to 0after integrating against test functions) and δ ( ν , ν ′ ) is the Dirac delta function supported oncoinciding angles. As we shall see in Sec. 5.2, the thermal properties are manifest when Λ is tested with respect to the flow Φ τ ( τ ∈ R ) generated by K a . Applied to Λ the flow actsas Φ τ ( U ) = e κ ∗ τ U where κ ∗ is the surface gravity on S ∗ , and the Fourier frequencies, orenergies, with respect to the flow-parameter τ in the spectrum of Λ are distributed according tothe spectral density ρ ( E ) = E − e − πκ ∗ E . (4)The presence of a Bose factor with inverse temperature π/κ ∗ in the spectral density distributionmakes the thermal interpretation manifest, analogously as in [MP12]. Making use of this fact,in following [MP12] we show that the tunneling probability, or transition probability, betweena one-particle state inside the outer trapping horizon H , i.e. for U > and another one particlestate outside of H , i.e. for U < , takes in the scaling limit the high-energy asymptotic form e − βE , when the the one-particle states have a Fourier distribution peaked at E . This is theform of a transition probability for a thermal energy level occupation at inverse temperature β = 2 π/κ ∗ .The 2-point function Λ obtained in our scaling limit is very similar to the restriction of2-point functions to Killing horizons considered in [Se82, HNS84, KW91, SV96, GLRV01].In these articles, the restrictions or scaling limits of 2-point functions to the analogues of C ∗ exhibit a thermal spectrum with respect to the Killing flow. In contrast, in the case of dynamicalblack holes the relevant part of the state is the transversal component of the 2-point function(the component supported on T ∗ ), showing thermal properties with respect to the Kodama flowin the scaling limit. The C ∗ -part of the 2-point function depends on the details of the quantummatter entering the horizon, blurring an exact thermal spectrum. On the other hand, at least inthe case of static black holes, the T ∗ -part is related to the radiation emitted by the black holeand is the source of Hawking radiation, see e.g. [FH90].The 2-point function Λ can be used to define a quantum field theory – the “scaling-limit-theory” – on the lightlight hypersurface T ∗ ; Λ also induces a quasifree state ω Λ on the algebra Formally, this means that f ( ν ) = R S δ ( ν , ν ′ ) f ( ν ′ ) d Ω ( ν ′ ) for any continuous function f on the unit sphere.
6f observables W of the scaling-limit-theory, where W is a CCR-Weyl algebra. This state turnsout to be a KMS-state [HHW67] at inverse temperature β = 2 π/κ ∗ with respect to the Kodamaflow. It is then possible to define and calculate the relative entropy S ( ω Λ | ω ϕ ) in the sense ofAraki [Ar76] between ω Λ and coherent states ω ϕ on W analogously as in [Ho19, Lo19] wherethe fucntion ϕ describes a coherent excitation of the scalar field over the state ω Λ . We findthat S ( ω Λ | ω ϕ ) coincides with the classical energy of the coherent excitation, measured by anobserver moving along the Kodama flow, multiplied with the inverse temperature β = 2 π/κ ∗ (see equation (74) below). Furthermore, we observe that S ( ω Λ | ω ϕ ) is proportional to r ∗ , thegeometrical area of the outer trapping horizon’s cross section S ∗ with respect to which thescaling-limit-theory is constructed. We argue that this is not accidental but a consequence ofthe fact that ω Λ and ω ϕ are correlation-free product states with respect to separation in theangular coordinate ν of S ∗ , together with the additivity of the relative entropy for correlation-free product states. Thus we arrive at S ( ω Λ | ω ϕ ) ∼ r ∗ , analogous to the entropy-area relationsuggested in the classic article of Bardeen, Carter and Hawking [BCH73].The idea to relate the relative entropy of quantum field states on a spacetime containing ahorizon to a form of black hole entropy goes back to Longo [Lo97] (in the setting of quantumfield theory on Minkowski spacetime, where the lightlike boundaries of a wedge-region playthe role of a horizon). These ideas have been extended in [KL05] to a relation between relativeentropy of quantum field states and non-commutative geometrical quantities for area. In aseries of articles by Schroer [Sc03, Sc06] it was mentioned that – in the setting of quantumfield theory in Minkowski spacetime – quantum field theories restricted to lightlike hyperplanestypically show no correlations in the transversal spacelike directions of the hyperplane, whichwould result in an additive behaviour of entropy quantities for the restricted quantum fields andan area proportionality. In our present article, we see that the earlier ideas of Longo and ofSchroer can indeed be combined to result, in our scaling limit, in a proportionality betweenthe relative entropy of quantum field states and the horizon area of a black hole spacetime,and more generally, the cross-section area of an outer trapping horizon. In a recent work,Hollands and Ishibashi [HI19] consider linearized perturbations of the spacetime metric aroundSchwarzschild spacetime which are quantized similarly like a linear scalar field. Using preferredstates for the characteristic data of the perturbations (which on the black hole horizon take aform as our scaling limit state) they define relative entropies in a similar way, and taking intoaccount the backreaction of the coherent excitation on the background geometry, they showthat the combined variation of the relative entropy and a cross-sectional area of the black-holehorizon along Schwarzschild time equals the future out- and ingoing flux of radiation. Relatedcontent, in the context of spherically symmetric dynamical black holes, appears also in [Da20].This article is organized as follows. In Section 2 we discuss the geometric setup of spher-ically symmetric spacetimes in which a Kodama vector field and outer trapping horizons canbe defined. Section 3 contains the specification of the quantum field theory on the spacetimeswe consider, together with a discussion about Hadamard 2-point function. Section 4 begins7y introducting a conformal transformation of the spacetimes considered which is useful forderiving the scaling limit of Hadamard 2-point functions, presented thereafter in Theorem 4.1.The behaviour of the scaling limit 2-point function Λ under the Kodama flow is also discussed.In Section 5 we show how the scaling-limit-theory is constructed from the scaling limit 2-pointfunction Λ , and derive and discuss several of its properties, like the thermal spectrum and ther-mal tunneling probability with respect to the Kodama time. We also consider the coherent statesin the scaling-limit-theory and their relative entropy which we find to be proportional to the areaof the horizon cross-section S ∗ with respect to which the scaling-limit-theory is defined. A con-clusion is given in Section 6. Section 7 is a technical appendix containing the proof of Theorem4.1. We consider a spacetime ( M, g ab ) , where M is the 4-dimensional spacetime manifold and g ab is the spacetime metric, with signature ( − + + +) . It will be assumed that the spacetime is(spatially) spherically symmetric, i.e. its set of isometries contains the group SO (3) , and allthe orbits of the SO (3) action are spacelike. We also assume that the spacetime has an outertrapping horizon H and a Kodama vector field . Thus, we assume that M contains an opensubset N , diffeomorphic to L × S with an open, connected subset L of R , on which advancedEddington-Finkelstein coordinates ( v, r, ϑ, ϕ ) can be introduced, where ( v, r ) are coordinateson L and ( ϑ, ϕ ) are angular coordinates for the sphere. The spherical symmetry group then actson the S part of N . Using such coordinates, the metric g ab on N assumes the line-element ds = − e v,r ) C ( v, r ) dv + 2 e Ψ( v,r ) dvdr + r d Ω (5)where d Ω is the normalized spherically symmetric Riemannian metric on S . With respect toangular coordinates ( ϑ, ϕ ) , one has d Ω = dϑ + (sin ϑ ) dϕ . (6)The coordinate v takes values in a real interval and r in a positive real interval; the precise formof the intervals depends on the smooth coordinate functions C ≥ and Ψ . Furthermore, thefunction C can be written in terms of the Hawking mass M ( v, r ) = r (cid:0) − g ab ( v, r ) ∇ a r ∇ b r (cid:1) (7)as C ( v, r ) = 1 − M ( v, r ) r . (8)8s a side remark, we notice that, if Ψ( v, r ) = 0 and M ( v, r ) = M ( v ) , the metric g ab reducesto the Vaidya metric, which is one of the simpler models of dynamical black holes (see [GP12]and references cited there).Consider the following null vector fields, ℓ a = 2 ∂∂v a + e Ψ C ∂∂r a , ℓ a = − e − Ψ ∂∂r a . (9)Then ℓ a is a future pointing outgoing null vector field and ℓ a a future pointing ingoing nullvector field. Furthermore, in terms of these vector fields we have that g ab = − (cid:0) ℓ a ℓ b + ℓ a ℓ b (cid:1) + h ab (10)where h ab = r d Ω and it also holds that g ab ℓ a ℓ b = − . (11)The expansion parameters of the congruences of outgoing and ingoing null geodesics tan-gent to ℓ a and ℓ a are given by θ + = h ab ∇ a ℓ b = 2 e Ψ Cr , θ − = h ab ∇ a ℓ b = − e − Ψ r . (12)On N , θ − is always negative, while θ + has the same sign as C , and vanishes if C = 0 . In the caseof a black hole we have that C is positive far from the center, thus far from r = 0 , the transversalarea of the congruence tangent to ℓ a increases towards the future while the transversal area ofthe congruence tangent to ℓ a decreases. The set of points where C = 0 is the union of trapped surfaces. The possibility arises that thisset has several disjoint connected components. Thus, we define the outer trapping horizon H as the outermost connected component, in the following sense: We assume that there is a timefunction T on N so that all the r -coordinate values of H on hypersurfaces of constant T arelarger than the respective r -coordinate values of the other connected components. If there isonly one connected component then, writing C = 0 in terms of the Hawking mass, H = (cid:26) ( v, r, ν ) ∈ N : 2 M ( v, r ) r = 1 (cid:27) . (13)The hypersurface H is spacelike for black holes which are growing in a collapse, it is lightlikefor stationary black holes and it is timelike for black holes which evaporate.9n contrast to the case of Schwarzschild spacetime, there is in general no timelike or causalKilling vector field near H that could be used to define and test black hole thermodynamicalquantities. Hayward [Ha97] proposed to use the Kodama vector field as a replacement (see also[He15] for a review). The Kodama vector field [Ko80] can be defined in terms of ℓ , ℓ and r as K a := 12 ( ℓ [ r ] ℓ a − ℓ [ r ] ℓ a ) = e − Ψ ∂∂v a , (14)with ℓ [ r ] = ℓ a ∇ a r , and similarly for ℓ [ r ] . The Kodama vector field is conserved and can beused to build other conserved quantities: It holds that ∇ a K a = 0 , ∇ a ( W ab K b ) = 0 , (15)for any symmetric tensor field W ab that is invariant under the spherical symmetries of the space-time.Notice that K a is timelike in the region where θ + > and is lightlike on H . Furthermore,the surface gravity κ associated to the Kodama vector field is a function on H defined by K a ( ∇ a K b − ∇ b K a ) = κK b on H . (16)With respect to the metric component function C ( v, r ) of (5), one obtains κ = 12 ∂C ( v, r ) ∂r if C ( v, r ) = 0 , (17)and in terms of the Hawking mass it is κ = (cid:18) M ( v, r ) r − ∂ r M ( v, r ) r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r =2 M ( v,r ) . (18)This definition generalizes the concept of surface gravity known for stationary black hole hori-zons, or for bifurcate Killing horizons. In the case of a stationary black hole, it is known thatthe surface gravity is proportional to the Hawking temperature. We have already indicated in the Introduction that there are lightlike congruences emanatingfrom the outer trapping horizon H . They are determined once one chooses any point ( v ∗ , r ∗ , ν ∗ ) on H . Any such point then determines its orbit S ∗ = { ( v ∗ , r ∗ ) } × S under the sphericalsymmetry group of the spacetime. Clearly, S ∗ is a subset of H . The lightlike vector fields ℓ a and10 a restricted to S ∗ are tangent to two lightlike congruences C ∗ (“outgoing”) and T ∗ (“ingoing”),respectively. Owing to the spherical symmetry, these lightlike conguences are 2-dimensionallightlike hypersurfaces. It holds that S ∗ = C ∗ ∩ T ∗ .One can introduce local coordinates ( U, V ) covering an open neighbourhood of ( v ∗ , r ∗ ) inthe L -part of N . U and V are null (or lightlike) coordinates, so that the metric line element (5)takes the form ds = − A ( U, V ) dU dV + r ( U, V ) d Ω , (19)where the radial coordinate is now a function of U and V , r = r ( U, V ) . This can actuallyalways be achieved for a spherically symmetric spacetime metric; in the case at hand, there isan integrating factor α = α ( v, r ) so that the required coordinates can be defined on an openneighbourhood of S ∗ by α · dU a = 12 e v,r ) C ( v, r ) dv a − e Ψ( v,r ) dr a , dV a = dv a . (20)One can further re-define the coordinates U and V so that they have additional properties. First,we have the freedom to choose the U and V coordinates such that U = 0 and V = 0 exactly forthe points in S ∗ . Furthermore, we can choose the U and V coordinates so that U = 0 exactlyfor the points on C ∗ and V = 0 exactly for the points on T ∗ ; this freedom of choice is related tothe fact that we have ℓ a = β ( U ) · dU a on T ∗ and ℓ a = β ( V ) · dV a on C ∗ , with smooth, non-zerofunctions β ( U ) and β ( V ) . Re-defining U and V again so that β = 2 and β = 1 , one obtainsfrom (11) that A ( U,
0) = 1 and A (0 , V ) = 1 . (21)Thus, given S ∗ , we can choose coordinates ( U, V, ϑ, ϕ ) = (
U, V, ν ) in an open neighbourhoodof S ∗ with the properties (1) to (4) stated in the introduction. In the next section we shall seethat also property (5) is satisfied. S ∗ As discussed above, once a sphere S ∗ contained in the outer trapping horizon H is fixed we candetermine the cone C ∗ formed by outgoing radial null geodesics passing through S ∗ , and thetransversal cone T ∗ , formed by ingoing radial null geodesics passing through S ∗ . Later we shallanalyze the scaling towards C ∗ of the 2-point function of any Hadamard state. The resultingdistribution can be restricted to T ∗ and it will be tested with respect to an observer movingalong the integral line of the Kodama field. Hence, we need to analyze the form of the actionof K a on T ∗ near S ∗ . We denote by Φ τ ( τ ∈ R ) the flow generated by K a . We recall that thismeans that, whenever p ∈ N and τ ∈ R so that Φ τ ( p ) ∈ N for all τ in an open interval around τ , it holds that K a ∇ a f | Φ τ ( p ) = ddτ | τ = τ f (Φ τ ( p )) for all smooth, real-valued functions f on N .11e use adapted coordinates ( U, V, ν ) with respect to S ∗ as described in the previous section.Then we write Φ τ ( U, V, ν ) = (u τ ( U, V, ν ) , v τ ( U, V, ν ) , ν ) (22)i.e. u t ( U, V, ν ) is the U -coordinate of Φ t ( U, V, ν ) and v t ( U, V, ν ) is the V -coordinate. (Notethat Φ τ doesn’t act on ν , and therefore u τ and v τ actually do not depend on ν .) Lemma 2.1.
With κ ∗ the value of κ on S ∗ (corresponding to U = 0 and V = 0 ), there is anopen interval of U coordinate values around 0 so that (a) dU a K a ( U, V = 0 , ν ) = − κ ∗ U + O ( U ) , (b) u τ ( U, V = 0 , ν ) = e − κ ∗ τ U + O ( τ, U ) Proof.
On using that ℓ a = ∂∂U a and the definition of K a , one obtains that dU a K a = − ℓ [ r ] = ∂ τ u τ . On the other hand, ℓ [ r ] vanishes on S ∗ , and it holds that − ℓ [ ℓ [ r ]] (cid:12)(cid:12)(cid:12)(cid:12) S ∗ = 12 ∂C∂r (cid:12)(cid:12)(cid:12)(cid:12) S ∗ = κ ∗ . (23)This yields dU a K a ( U, V = 0 , ν ) = − κ ∗ U + O ( U ) , having used ℓ a = ∂∂U a once more and thefact that S ∗ is the locus of U = 0 and V = 0 . This proves (a).Furthermore, we have ∂ τ u τ ( U, V = 0 , ν ) = dU a K a ( U, V = 0 , ν ) = − κ ∗ U + O ( U ) ,yielding on integration u τ ( U, V = 0 , ν ) = e − κ ∗ τ U + O ( τ, U ) on fixing the constants of integration such that u ( U, , ν ) = ( U, , ν ) to be consistent with Φ τ =0 ( p ) = p . This proves (b). ✷ This shows that the property (5) stated for adapted coordinates in the Introduction is also ful-filled.
The main point of our article is an investigation of quantized fields on a spherically symmetricspacetime ( M, g ab ) with an outer trapping horizon H and Kodama vector field K a . To this end,our investigation starts with the free quantized scalar field φ ( x ) . We assume that the underlyingspacetime ( M, g ab ) is globally hyperbolic. Actually, global hyperbolicity of the spacetime atlarge distances from H is not required for our considerations; what we need is a sphericallysymmetric, globally hyperbolic open neighbourhood of the outer trapping horizon H containedin the open set N ≃ L × S on which the Eddington-Finkelstein coordinates discussed before12an be introduced. For notational convenience, we assume in the following that this sphericallysymmetric globally hyperbolic open neighbourhood just coincides with M .The quantized real free scalar field on ( M, g ab ) is then defined in the standard manner whichwe will briefly sketch. For a fuller discussion, the reader may consult [Wa94, HW15, KM15].As ( M, g ab ) is globally hyperbolic by assumption, there are uniquely determined advancedand retarded fundamental solutions G adv / ret (“Green’s operators”) for the 2nd order hyperbolicKlein-Gordon operator ∇ a ∇ a − M defined on smooth scalar test-functions on M . Here, ∇ a denotes the covariant derivative of the spacetime metric g ab , and M ≡ M ( x ) is a smooth, real-valued function on M . Then one can define the causal Green’s function G ( F, F ′ ) = Z M (cid:0) F ( x )( G adv F ′ )( x ) − F ( x )( G ret F ′ )( x ) (cid:1) d vol g ( x ) , F, F ′ ∈ C ∞ ( M, R ) , where d vol g denotes the volume form of the spacetime metric g ab . Hence there is a ∗ -algebra A = A ( M, g ab , M ) which is generated by a family of elements φ ( F ) , F ∈ C ∞ ( M, R ) , and aunit element , subject to the relations: ( i ) F φ ( F ) is R -linear ( ii ) φ (( ∇ a ∇ a + M ) F ) = 0( iii ) φ ( F ) ∗ = φ ( F ) ( iv ) [ φ ( F ) , φ ( F ′ )] = i G ( F, F ′ ) · , F, F ′ ∈ C ∞ ( M, R ) . Here, [ φ ( F ) , φ ( F ′ )] = φ ( F ) φ ( F ′ ) − φ ( F ′ ) φ ( F ) denotes the algebraic commutator. The φ ( F ) are abstract field operators, at this level without a Hilbert space representation. One can symbol-ically write φ ( x ) to mean that φ ( F ) = R M φ ( x ) F ( x ) d vol g ( x ) which can best be made rigorouswhen the φ ( F ) are given in some Hilbert space representation.We recall that w (2) is a 2-point function for the Klein-Gordon field operators φ ( F ) if w (2) : C ∞ ( M, R ) × C ∞ ( M, R ) → C , F, F ′ w (2) ( F, F ′ ) is real-bilinear, extends to a distribution in D ′ ( M × M ) , and moreover fulfills w (2) ( F, F ) ≥ , w (2) ( F ′ , F ) = w (2) ( F, F ′ ) , Im w (2) ( F, F ′ ) = 12 G ( F, F ′ ) , (24) w (2) (( ∇ a ∇ a − M ) F, F ′ ) = 0 = w (2) ( F, ( ∇ a ∇ a − M ) F ′ ) ( F, F ′ ∈ C ∞ ( M, R ) . (25)There is a one-to-one correspondence between states on A and Hilbert space representa-tions of A which is given by the Gelfand-Naimark-Segal (GNS) representation. At this point,however, we don’t make use of this, but we come back to a more operator-algebraic point ofview in Sec. 5. Rather, we are interested in quasifree Hadamard states on A ; as these are com-pletely determined by their 2-point function w (2) , it is the behaviour of these 2-point functionsnear the outer trapping horizon that will be in the focus of our investigation.At this point it is useful, for later purpose, to look at the Hadamard form of the 2-pointfunction in more detail, following mainly [KW91, Ra96, SV01] (see also [SV00] for the relationof the Hadamard condition with equilibrium states).13aving chosen some S ∗ , in the adapted coordinates we can define the time-function T ( x ) = T ( U, V, ϑ, ϕ ) = ( U + V ) / . We can consider the hypersurface Σ = { x = ( U, V, ϑ, ϕ ) : T ( x ) =0 } . Then there is an open neighbourhood B in M of S ∗ so that Σ B = Σ ∩ B is a spacelike,acausal hypersurface containing S ∗ , and the open interior of the domain of dependence D (Σ B ) is a globally hyperbolic open neighbourhood of S ∗ having Σ B as a Cauchy-surface. Then anopen neighbourhood N B of Σ B is called a causal normal neighbourhood if, given x and x ′ in N B , with x ′ ∈ J + ( x ) , there is a convex normal neighbourhood (with respect to the metric g ab )containing J − ( x ′ ) ∩ J + ( x ) . It has been shown in [KW91] that causal normal neighbourhoodsalways exist.Then a 2-point function w (2) is said to be of Hadamard form near S ∗ if, for all F, F ′ ∈ N B , itholds that w (2) ( F, F ′ ) = lim ε → Z w ǫ ( x, x ′ ) F ( x ) F ′ ( x ′ ) d vol g ( x ) d vol g ( x ′ ) ( F, F ′ ∈ C ∞ ( N B , R )) , (26)where w ε ( x, x ′ ) = ψ ( x, x ′ ) (cid:18) π ∆ / ( x, x ′ ) σ ε ( x, x ′ ) + ln( σ ε ( x, x ′ )) Y ( x, x ′ ) (cid:19) + Z ( x, x ′ ) (27)with σ ε ( x, x ′ ) = σ ( x, x ′ ) − iεt ( x, x ′ ) + ε and t ( x, x ′ ) = T ( x ) − T ( x ′ ) . (28)Here, σ ( x, x ′ ) is the half of the squared geodesic distance between x and x ′ , and ∆( x, x ′ ) denotes the van Vleck-Morette determinant. Both functions are well-defined and jointly smoothin x and x ′ on an open neighbourhood U of all pairs of points ( x, x ′ ) ∈ N B × N B for which x and x ′ lie within a convex normal neighbourhood. The function ψ is in C ∞ ( N B × N B , [0 , ,vanishes outside of U , and is equal to 1 on U ∗ , the set of pairs of points ( x, x ′ ) ∈ N B × N B for which J ± ( x ) ∩ J ∓ ( x ′ ) , if non-empty, is contained in a convex normal neighbourhood. Inparticular, the factor ψ ( x, x ′ ) guarantees well-definedness and C ∞ property of σ ( x, x ′ ) and ∆( x, x ′ ) on the support of w ε . Moreover, Y ( x, x ′ ) and Z ( x, x ′ ) are in C ∞− ( N B × N B ) , i.e.they can be chosen to be in C k ( N B × N B ) for any k ∈ N . For further details, which aren’tneeded in our later discussion, we refer to the references [KW91, Ra96, SV01]. (We note thatour function ψ has been denoted by χ in these references; later, in the proof of Thm. 4.1 we use χ for a different function.) It is also worth remarking that ∆( x, x ′ ) is determined by the metric g ab while Y ( x, x ′ ) is determined from the metric and the Klein-Gordon operator ∇ a ∇ a − M bythe Hadamard recursion relations. Changing ψ and T leads to a change of Z , but that preservesthe Hadamard form. Again, we refer to the references for a full discussion; see also [KM15] fora review. 14 Scaling limit of Hadamard 2-point functions near S ∗ andrestriction to T ∗ In the adapted coordinates ( U, V, ϑ, ϕ ) discussed in Sec 2.3, the line element of the spacetime ( M, g ab ) under consideration assumes the form ds = − A ( U, V ) dU dV + r ( U, V ) d Ω . Our investigation of the scaling limit of the quantized linear scalar field near points of an outertrapping horizon H that we will consider below will be facilitated by using a conformally trans-formed metric. This applies in particular to the proof of Thm. 4.1. To simplify notation, we willwrite again ν for ( ϑ, ϕ ) noting that the angular variables ( ϑ, ϕ ) really represent an element ν ofthe unit sphere.The conformal transformation is defined with respect to an arbitrarily chosen point ( v ∗ , r ∗ ) on H defining S ∗ and consequently C ∗ and T ∗ , as explained in the Introduction. Given ( v ∗ , r ∗ ) (or equivalently, the corresponding S ∗ ), we introduce the conformally transformed metric ˜ g ab on M by ˜ g ab = η g ab with the conformal factor η ( U, V ) = r ∗ r ( U, V ) ; (29)the associated line element is d ˜ s = − A ( U, V ) r ∗ r ( U, V ) dU dV + r ∗ d Ω . (30)One feature of ˜ g ab is the splitting of the squared geodesic distance between points ( U, V, ν ) and ( U ′ , V ′ , ν ′ ) according to the Pythagorean theorem: ˜ σ ( U, V, ν ; U ′ , V ′ , ν ′ ) = ˜ σ ( L ) ( U, V ; U ′ , V ′ ) + s ( ν ; ν ′ ) (31)where ˜ σ ( L ) ( U, V ; U ′ , V ′ ) denotes the squared geodesic distance between the points ( U, V ) and ( U ′ , V ′ ) on the two-dimensional “Lorentzian” part of the spacetime with metric line element − A ( U,V ) r ∗ r ( U,V ) dU dV , and where s ( ν ; ν ′ ) is the squared geodesic distance between points ν and ν ′ on the two-dimensional sphere with radius r ∗ .It is worth noting that on S ∗ where r = r ∗ , the conformal factor is equal to 1: η | S ∗ = 1 . At the level of 2-point functions, the conformal transformation has the following effect.Suppose that w (2) ( F, F ′ ) = lim ε → Z w ε ( x, x ′ ) F ( x ) F ′ ( x ′ ) d vol g ( x ) d vol g ( x ′ ) ( F, F ′ ∈ C ∞ ( N B , R ))
15s a 2-point function of Hadamard form, near S ∗ , for the quantized scalar field that we consideron ( M, g ab ) . Then, defining ˜ w ε ( x, x ′ ) = η − ( x ) w ε ( x, x ′ ) η − ( x ′ ) , the distribution ˜ w (2) ( F, F ′ ) = lim ε → Z ˜ w ǫ ( x, x ′ ) F ( x ) F ′ ( x ′ ) d vol ˜ g ( x ) d vol ˜ g ( x ′ ) ( F, F ′ ∈ C ∞ ( N ˜ B , R )) , is a 2-point function of Hadamard form near S ∗ on the conformally related spacetime ( M, ˜ g ab ) ,with a suitably small neighbourhood ˜ B of S ∗ , and an associated causal normal neighbourhood N ˜ B defined with respect to ˜ g ab . This has been shown in [Pi09].The scaling limit which we will consider in the next section gives the same results on w (2) or on ˜ w (2) on account of η | S ∗ = 1 , but it is easier to study the scaling limit using ˜ w (2) becauseof (31). To this end, we put on record the following observations for later use.The volume form d vol g of the original metric and the volume form d vol ˜ g of the conformallytransformed metric are related according to d vol ˜ g ( x ) = η ( x ) d vol g ( x ) and therefore one has w (2) ( F, F ′ ) = ˜ w (2) ( ˜ F , ˜ F ′ ) with ˜ F = η − F , ˜ F ′ = η − F (32)for all F, F ′ ∈ C ∞ ( M, R ) . The statement that ˜ w (2) is of Hadamard form near S ∗ on ( M, ˜ g ab ) means that ˜ w ε ( x, x ′ ) = ˜ ψ ( x, x ′ ) π ˜∆ / ( x, x ′ )˜ σ ε ( x, x ′ ) + ln(˜ σ ε ( x, x ′ )) ˜ Y ( x, x ′ ) ! + ˜ Z ( x, x ′ ) ( x, x ′ ∈ N ˜ B ) (33)where ˜∆ and ˜ σ refer to ˜ g ab , ˜ ψ has properties analogous to ψ , and ˜ Y and ˜ Z can be chosen as C k function for any k ∈ N . We select a sphere S ∗ of radius r ∗ lying in the outer trapping horizon, and a patch of adaptedcoordinates ( U, V, ν ) relative to S ∗ . Moreover, we assume that N B is a causal normal neigh-bourhood of a partial Cauchy surface Σ B so that S ∗ ⊂ Σ B , as described in Sec. 3. Then if l > is small enough, the open set O = { ( U, V, ν ) : | U | < l , | V | < l , ν ∈ S } (34)16s a subset of N B and an open neighbourhood of S ∗ . We assume that l is chosen small enoughso that O is also contained in a causal normal neighbourhood N ˜ B of S ∗ with respect to theconformally transformed metric ˜ g ab described in Sec. 4.1.Note that, if < λ ≤ and < µ ≤ then ( U, V, ν ) ∈ O ⇒ ( λU, µV, ν ) ∈ O .Consequently, when defining the scaling transformations ( u λ F )( U, V, ν ) = 1 λ F ( U/λ, V, ν ) , (35)for < λ ≤ , one can see that the u λ map the space of test functions C ∞ ( O , R ) into itself, and supp( u λ F ) = { ( λU, V, ν ) : ( U, V, ν ) ∈ supp( F ) } (0 < λ ≤ . (36)We stress that the scaling transformations are defined with respect to the chosen S ∗ , and thecorresponding adapted coordinates.We also define another type of transformations which serve, in a limit, to restricting dis-tributions to T ∗ by effectively acting like a δ -distribution concentrated at V = 0 . Let ζ ∈ C ∞ (( − ℓ, ℓ ) , R ) with ζ ( V ) ≥ and R ζ ( V ) dV = 1 . Then we define, for any F ∈ C ∞ ( O , R ) , ( v µ F )( U, V, ν ) = 1 µ ζ ( V /µ ) F ( U, V, ν ) (0 < µ ≤ . (37)Clearly, also every v µ maps C ∞ ( O , R ) into itself.Adopting this notation, we now present the result on scaling limits of Hadamard 2-point func-tions near S ∗ and subsequent restriction to T ∗ . Theorem 4.1.
Let w (2) be any 2-point function of Hadamard form for the scalar field φ on thespacetime ( M, g ab ) as in Sec. 3. (I) For all f, f ′ ∈ C ∞ ( O , R ) it holds that lim λ → w (2) ( u λ (2 ∂ U f ) , u λ (2 ∂ U ′ f ′ )) = L ( f, f ′ ) , where (38) L ( f, f ′ ) = lim ε → − r ∗ π Z f ( U, V, ν ) f ′ ( U ′ , V ′ , ν )( U − U ′ + iε ) Q ( V, V ′ , ν ) dU dU ′ dV dV ′ d Ω ( ν ) with Q ( V, V ′ , ν ) = ˜∆ / (0 , V, ν , , V ′ , ν ) P (0 , V, , V ′ ) , (39) P ( U, V, U ′ , V ′ ) = r ∗ η − ( U, V ) A ( U, V ) η − ( U ′ , V ′ ) A ( U ′ , V ′ ) . (40)(II) For all f, f ′ ∈ C ∞ ( O , R ) it holds that lim µ → lim λ → w (2) ( v µ u λ (2 ∂ U f ) , v µ u λ (2 ∂ U ′ f ′ )) = lim µ → L ( v µ f, v µ f ′ ) = Λ( f, f ′ ) (41)17 here Λ( f, f ′ ) = lim ε → − r ∗ π Z f ( U, , ν ) f ′ ( U ′ , , ν )( U − U ′ + iε ) dU dU ′ d Ω ( ν ) (42)The proof of this Theorem will be given the Appendix (Sec. 7). Remarks (i)
The more difficult step is proving Part (I) of the Theorem, Part (II) then is merely a corol-lary. Actually, the statement follows easily when inserting the scaled test functions u λ (2 ∂ U f ) and u λ (2 ∂ U ′ f ′ ) into the ε -regulated integral expression of the Hadamard form and exchangingthe λ → and ε → limits. The more involved part of the proof consists in showing that thiscan be justified. We have opted to give a full, self-consistent proof in this article, despite somesimilarities of our proof with a related argument in [MP12] (that relied in parts also on resultsfrom [KW91]) which applies to the case of the quantized Klein-Gordon field on spacetimeswith bifurcate Killing horizons. (ii) As is familiar from the quantization of the massless free quantum field in 2-dimensionalMinkowski spacetime, respectively its chiral components on lightrays, the 2-point functionis well-defined for test-functions which are first derivatives of compactly supported smoothfunctions. Without derivatives, an infrared divergence occurs, see e.g. [BLTO90], Sec. on the“Schwinger model”. This is the reason why the test-functions used for the scaling limit consid-erations are U -derivatives of compactly supported smooth functions. (iii) One may choose U or V -coordinates so that the Van Vleck - Morette determinant is equalto 1; this simplifies the form of the function Q in the first part of Theorem 4.1, however we neednot make use of this possibility here. (iv) The factor 2 doubly inserted in the scaling limits (38) and (41) has been introduced tomatch Λ with the convention for 2-point functions on lightlike hyperplanes used in the litera-ture, see e.g. [DMP17]. See also the remark towards the end of Sec. 5.5. T ∗ and its action in the scaling limit Under the same assumptions as for the previous theorem, we can establish that the projectedaction T τ on T ∗ of the flow of the Kodama vector field K a acts like the dilation group in thescaling limit. To make this more precise, we define T τ ( U, V, ν ) = (u τ ( U, , ν ) , V, ν ) ( τ ∈ R ) (43)for all ( U, V, ν ) ∈ O , with the convention that the definition applies whenever (u τ ( U, , ν ) , V, ν ) is again in O . Recall that (cf. Lemma 2.1) u τ ( U, V = 0 , ν ) = e − κ ∗ τ U + O ( τ, U ) so that theprojected action of the Kodama flow on T ∗ takes the form T τ ( U, V, ν ) = (e − κ ∗ τ U + O ( τ, U ) , V, ν ) (44)18nd there is some τ > and an open neighbourhood O of S ∗ with O ⊂ O so that T τ ( U, V, ν ) ∈ O for all ( U, V, ν ) in O and all | τ | < τ . We also define: ( T τ F )( U, V, ν ) = F (T − τ ( U, V, ν )) ( F ∈ C ∞ ( O , R ) , | τ | < τ ) , (45) S τ ( U, ν ) = (e − κ ∗ τ U, ν ) , (( U, ν ) ∈ T ∗ , τ ∈ R ) , (46) ( S τ ϕ )( U, ν ) = ϕ (S − τ ( U, ν )) ( ϕ ∈ C ∞ ( T ∗ , R ) , τ ∈ R ) (47) Proposition 4.1.
For any 2-point function w (2) of φ that is of Hadamard form, lim µ → lim λ → w (2) ( T τ v µ u λ ( ∂ U f ) , T τ ′ v µ u λ ( ∂ U ′ f ′ )) = Λ( S τ f, S τ ′ f ′ ) (48) holds for all f, f ′ ∈ C ∞ ( O , R ) and | τ | , | τ ′ | < τ .Proof. For any F ∈ C ∞ ( O , R ) and | τ | < τ , one obtains ( T τ u λ F )( U, V, ν ) = ( u λ F )(T − τ ( U, V, ν ))= ( u λ F )(e κ ∗ τ U + O ( τ, U ) , V, ν )= 1 λ F ( λ − (e κ ∗ τ U + O ( τ, U )) , V, ν )= 1 λ F (e κ ∗ τ ( U/λ ) + O ( λ ) · O ( τ, ( U/λ ) ) , V, ν ) (49)for small enough λ > . One can now see that in the proof of Thm. 4.1, all estimates involving F λ ( x ) = ( u λ F )( U, V, ν ) (and similarly, the primed counterparts) are preserved when replacing ( u λ F )( U, V, ν ) by ( T τ u λ F )( U, V, ν ) (and similarly for the primed counterparts). Moreover,the limit considerations in the proof of Thm. 4.1 where F ( x ) = F ( U, V, ν ) appears (and theprimed counterpart) render the analogous results upon replacing F ( U, V, ν ) by F (e κ ∗ τ U + O ( λ ) · O ( τ, U ) , V, ν ) (analogously for the primed counterpart) as λ → , except that F ( U, V, ν ) isin the limit to be replaced by F (e κ ∗ τ U, V, ν ) and F ′ ( U ′ , V ′ , ν ′ ) by F ′ (e κ ∗ τ ′ U ′ , V ′ , ν ′ ) . Thatfollows from the fact that O ( λ ) · O ( τ, U ) → as λ → uniformly as τ and U vary overcompact sets. Observing this and carrying out the steps of the proof of Thm. 4.1 thus yields theclaimed result. ✷ T ∗ T ∗ (and its extension) The 2-point function Λ defines a quantum field theory on T ∗ which naturally extends to a (chiral,conformal) quantum field theory on R × S ∗ ≃ R × S . We will refer to this as the “scaling-limit-theory” induced by the scaling limit 2-point function Λ .19o discuss this, fix again S ∗ ⊂ H , which is a copy of the sphere S with radius r ∗ . Then onecan introduce on the (real-linear) function space D S ∗ = C ∞ ( R × S ∗ , R ) the symplectic form ς ( ϕ, ϕ ′ ) = 2Im Λ( ϕ, ϕ ′ ) = r ∗ Z ( ∂ U ϕ ( U, ν ) ϕ ′ ( U, ν ) − ϕ ( U, ν ) ∂ U ϕ ′ ( U, ν )) dU d Ω ( ν ) . (50)Note the dependence of ς on r ∗ . Given this symplectic form, one can form the Weyl-algebra W ( D S ∗ , ς ) (the “exponentiated CCR algebra”) over the symplectic space ( D S ∗ , ς ) ; by definition,it is a C ∗ algebra with unit element , generated by unitary elements W ( ϕ ) , ϕ ∈ D S ∗ , fulfillingthe Weyl-relations W (0) = , W ( ϕ ) ∗ = W ( − ϕ ) , W ( ϕ ) W ( ϕ ′ ) = e − i ς ( ϕ,ϕ ′ ) W ( ϕ + ϕ ′ ) . (51)As is common in the operator algebraic approach to algebraic quantum field theory (cf. [Ha96]and in the present context, see also [DMP17, KW91, GLRV01, SV96]) one can introduce afamily { W ( G ) } of C ∗ algebras indexed by open, relatively compact subsets G of R × S ∗ bydefining W ( G ) as the C ∗ -subalgebra generated by all W ( ϕ ) with supp( ϕ ) ⊂ G . Then it iseasy to see that { W ( G ) } fulfills the condition of isotony , meaning that W ( G ) ⊂ W ( G ) if G ⊂ G , and it fulfills also a condition of locality , which in the present case means that W ( G ) and W ( G ) commute elementwise if G ∩ G = ∅ . Furthermore, there are certainsymmetries that act covariantly on the manifold R × S ∗ : The dilations S τ ( U, ν ) = (e − κ ∗ τ U, ν ) ,the translations L a ( U, ν ) = ( U + a, ν ) , and rotations R( U, ν ) = ( U, R ν ) , where τ, a ∈ R and R ∈ SO (3) . The actions of these symmetry operations can be lifted to D S ∗ by setting S τ ϕ = ϕ ◦ S − τ , L a ϕ = ϕ ◦ L − a and R ϕ = ϕ ◦ R − . Each of those is a symplectomorphism withrespect to the symplectic form ς , i.e. one has ς ( S τ ϕ, S τ ϕ ′ ) = ς ( ϕ, ϕ ′ ) for all ϕ, ϕ ′ ∈ D S ∗ , etc.This implies that these symplectomorphisms can be lifted to C ∗ -algebraic morphisms α ( τ,a,R ) of W ( D S ∗ , ς ) , given by α ( τ,a,R ) W ( ϕ ) = W ( S τ L a R ϕ ) . (52)We also adopt the notation to write α τ for α ( τ, , and α a for α (0 ,a, etc whenever no ambiguitycan arise. It is plain that thereby, a represention of the group of symmetries generated bydilations, translations and rotations by automorphisms of W ( D S ∗ , ς ) is established. It is alsoeasily seen that these automorphisms act covariantly (or geometrically) on the family { W ( G ) } in the sense that α τ ( W ( G )) = W (S τ G ) , α a ( W ( G )) = W (L a G ) , α R ( W ( G )) = W (R G ) . (53)We recall that a linear functional ω : W ( D S ∗ , ς ) → C is a state if it is positive, i.e. ω ( A ∗ A ) ≥ for all A ∈ W ( D S ∗ , ς ) , and normalized, i.e. ω ( ) = 1 . Moreover, every state ω induces the asso-ciated GNS-representation ( H ω , π ω , Ω ω ) of W ( D S ∗ ) , characterized by the properties that π ω is a20nital ∗ -representation of W ( D S ∗ , ς ) by bounded linear operators on the Hilbert space H ω , and Ω ω is a unit vector in H ω so that π ω ( W ( D S ∗ , ς ))Ω ω is dense in H ω and h Ω ω , π ω ( A )Ω ω i = ω ( A ) for all A ∈ W ( D S ∗ , ς ) (on writing h ξ, ψ i for the scalar product of ξ, ψ ∈ H ω ). Thus, once givena state ω on W ( D S ∗ , ς ) , one can introduce the system { N ( G ) } of local von Neumann algebras in the GNS-representation of ω given by N ( G ) = π ω ( W ( G )) = π ω ( W ( G )) ′′ (54)where the overlining means taking the weak closure in B ( H ω ) (the set of bounded linear oper-ators on H ω ); the double prime denotes the double commutant: For X ⊂ B ( H ω ) , X ′ = { B ∈ B ( H ω ) : AB − BA = 0 for all A ∈ X } is the commutant of X , and X ′′ = ( X ′ ) ′ . Whenever X contains the unit operator, it holds that X = X ′′ . For full details on these operator algebraicfacts, see [BR87, BR97, Bo00].The 2-point function Λ induces a quasifree state ω Λ on W ( D S ∗ , ς ) defined by linear extension ofthe assignment ω Λ ( W ( ϕ )) = e − Λ( ϕ,ϕ ) / . We denote the associated local von Neumann algebrasagain by N ( G ) (unless a more detailed notation is required). Of particular interest are the vonNeumann algebras N R = N (( −∞ , × S ∗ ) and N L = N ((0 , ∞ ) × S ∗ ) .Several important properties of ω Λ have been established and are well-known, from re-lated contexts or from investigations of chiral conformal quantum field theory. We collectsome of those properties here; proofs and further exposition can be found in [DMP17, KW91,GLRV01, SV96]. For notational simplicity, the GNS representation of ω Λ will be denoted by ( H Λ , π Λ , Ω Λ ) .(1) The state ω Λ is invariant under the action α : ω Λ ◦ α ( τ,a,R ) = ω Λ . Consequently, there is aunitary action U ( τ,a,R ) π Λ ( A )Ω Λ = π Λ ( α ( τ,a,R ) A )Ω Λ ( A ∈ W ( D S ∗ , ς )) implementing theaction of α in the GNS-representation of ω Λ with U ( τ,a,R ) Ω Λ = Ω Λ .(2) ω Λ is a ground state for the translations α a , i.e. there is a non-negative selfadjoint gener-ator H in H Λ so that U a = e i H a . (We are here using the same convention as previouslyexplained for α to write U a = U (0 ,a, , etc.)(3) Ω Λ is a cyclic and separating vector for the von Neumann algebras N R and N L . Let ∆ R denote the modular operator with respect to N R and Ω Λ . Then it holds that ∆ iτR = U βτ with β = 2 π/κ ∗ ( τ ∈ R ) (55)(4) The previous relation (55) can equivalently be expressed as stating that the state ω Λ re-stricted to the C ∗ -subalgebra W R = W (( −∞ , × S ∗ ) of W ( D S ∗ , ς ) is a KMS-state forthe action of the α τ at inverse temperature β = 2 π/κ ∗ . Analogously, ω Λ restricted to W L = W ((0 , ∞ ) × S ∗ ) is a KMS state for the action of the α τ at inverse temperature β = − π/κ ∗ . 21 .2 Thermal interpretation of the 2-point function Λ We will now point out that the thermal properties expressed in (3) and (4) at the end of theprevious subsection can be directly read off from the Fourier spectrum of Λ with respect to theKodama time parameter, analogously as in [MP12].To this end we recall the results presented in Section 2.4 and the action of the Kodama flowon the scaling limit state discussed in Section 4.3. In particular, points of T ∗ outside the Horizon H can be parametrized by ( u, ν ) where here the coordinate u is related to U by the followingcoordinate transformation U = − e − κ ∗ u , U < . (56)In particular, we have seen in Proposition 4.1 that the Kodama flow in the scaling limit, de-scribed by S τ , acts as u -translation, u u + τ . Thus, if ϕ and ϕ ′ are both contained in C ∞ (( −∞ , × S ∗ , R ) , i.e. they are supported on U < , one obtains Λ( ϕ, ϕ ′ ) = lim ǫ → + − r ∗ κ ∗ π Z ϕ ( u, ν ) ϕ ′ ( u ′ , ν )sinh (cid:0) ( u − u ′ ) κ ∗ + iǫ (cid:1) du du ′ d Ω ( ν ) . (57)A similar relation holds if ϕ and ϕ ′ are both supported on U > , on using the coordinatetransformation U = e κ ∗ u . The Fourier transform along u − u ′ of that distribution can be directlycomputed, see e.g. the Appendix of [DMP11]; it yields Λ( ϕ, ϕ ′ ) = 2 r ∗ Z ˆ ϕ ( E, ν ) ˆ ϕ ′ ( E, ν )1 − e − βE E dE d Ω ( ν ) , β = 2 π/κ ∗ , (58)if ϕ and ϕ ′ are both supported either on U < or U > , where the Fourier transform withrespect to u has been denoted by a hat, ˆ ϕ ( E, ν ) = 1 √ π Z e − iEu ϕ ( u, ν ) du . (59)The appearance of the Bose thermal distribution factor (1 − e − βE ) − for the Fourier “energies”in the integral expression (58) manifestly shows the thermal Fourier spectrum of the 2-pointfunction for an observer moving along the Kodama flow, where the inverse temperature is givenby β = 2 π/κ ∗ . Again proceeding as in [MP12], we now look at the Fourier transformed expression for the2-point function Λ in the case that ϕ is supported on U < , i.e. outside of the outer trappinghorizon, while ϕ ′ is supported on its inside, on U > . The result is (cf. [MP12], Sec. 3.3 b) ) Λ( ϕ, ϕ ′ ) = r ∗ Z ˆ ϕ ( E, ν ) ˆ ϕ ′ ( E, ν )sinh( β E ) E dE d Ω ( ν ) . (60)22his formula is the basis for estimating the tunneling probability or rather, transition probabilitybetween a one-particle state inside, and another one outside of the outer trapping horizon H inthe scaling limit. To this end, we choose some E > , and define, for small a > , ˆ η a ( E ) = 1 if | E − E | < a , and ˆ η a ( E ) = 0 otherwise (i.e. ˆ η a is the characteristic function of an interval ofwidth a around E ). We furthermore choose any non-zero, real, integrable, bounded function b on S ∗ and define ˆ h a ( E, ν ) = 1 √ a ˆ η a ( E ) b ( ν ) , ˆ ϕ a ( E, ν ) = ˆ h ( E, ν ) || h a || (Λ) (61)where || h a || = Λ( h a , h a ) = 2 r ∗ Z | ˆ h a ( E, ν ) | − e − βE E dE d Ω ( ν ) , β = 2 π/κ ∗ , (62)defines the one-particle norm on D S ∗ which is evidently finite for any ˆ h a . Therefore, h a (theinverse Fourier transform of ˆ h a ) defines an element in D S ∗ (Λ) , the completion of D S ∗ withrespect to the norm || . || (Λ) , supported on U < , the outside of H . Consequently ϕ a , theinverse Fourier transform of ˆ ϕ a , is an element of D S ∗ (Λ) which is supported on U < andwhich is normalized, || ϕ a || (Λ) = 1 . An a -parametrized family ϕ ′ a of elements in D S ∗ (Λ) with || ϕ ′ a || (Λ) = 1 , but supported on U > , is defined in complete analogy. We note that for each a ,there is a sequence ϕ ( n ) a ∈ D S ∗ ( n ∈ N ) supported on U < with || ϕ a − ϕ ( n ) a || (Λ) −→ n →∞ . Thesame holds for primed counterparts of the functions involved, supported on U > .It also follows from the properties of the GNS representations of W for quasifree states that,defining “one-particle vectors” ψ [ ϕ ( n ) a ] = − i ln( π Λ ( W ( ϕ ( n ) a )))Ω Λ in H Λ , || ψ [ ϕ ( n ) a ] − ψ [ ϕ ( m ) a ] || H Λ = || ϕ ( n ) a − ϕ ( m ) a || (Λ) (63)holds. Therefore, the one-particle vectors ψ [ ϕ ( n ) a ] form a Cauchy sequence, converging for anyfixed a to a unit vector, denoted by ψ [ ϕ a ] , in H Λ .We may now insert the expressions for h a and the analogously defined h ′ a into (58) and (60).Making use of the fact that | ˆ h a | , | ˆ h ′ a | and ˆ h a ˆ h ′ a are delta-sequences with respect to E peakedat E as a → , one finds in the limit a → for the transition probability lim a → |h ψ [ ϕ a ] , ψ [ ϕ ′ a ] i| = lim a → | Λ( ϕ a , ϕ ′ a ) | = 14 (cid:18) − e − βE sinh( βE / (cid:19) . (64)If E is sufficiently large, one obtains (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − e − βE sinh( βE / (cid:19) − e − βE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − βE (65)23howing that in the limit of large E , the transition probability becomes exponentially sup-pressed as is characteristic of a thermal occupation of energy levels at inverse temperature β = 2 π/κ ∗ . This observation is in agreement with [DNVZZ07, HDNVZ09, DHNVZ09]. Given the state ω Λ , its associated coherent states are of the form ω ϕ ( A ) = ω Λ ( W ( ϕ ) ∗ AW ( ϕ )) ( A ∈ W ( D S ∗ , ς )) . (66)This definition applies, in the first place, for all ϕ ∈ D S ∗ , but it is easy to see that it maybe extended to all functions which lie in the completion D S ∗ (Λ) of D S ∗ with respect to thenorm || ϕ || (Λ) = Λ( ϕ, ϕ ) / . It is also easy to check that D S ∗ (Λ) contains, e.g., C ∞ ( R , R ) ⊗ L R ( S ∗ , d Ω ) (algebraic tensor product without completion). There is a particular feature sharedby all coherent states: They are completely uncorrelated with respect to the spatial (i.e. spher-ical) degrees of freedom . This means, if there are finitely many subsets G j = I j × Σ j ( j =1 , . . . , n ; n ≥ where the I j are real open intervals (admitting the full real line) and the Σ j are open subsets of S ∗ ≃ S which are pairwise disjoint, Σ j ∩ Σ k = ∅ if j = k , then ω ϕ ( A A · · · A n ) = ω ϕ ( A ) · ω ϕ ( A ) · · · ω ϕ ( A n ) (67)holds for all A j ∈ W ( G j ) . This relation generalizes to the case that A j ∈ N ( G j ) , on extending ω Λ in the GNS representation to B ( H Λ ) as ω Λ ( B ) = h Ω Λ , B Ω Λ i ( B ∈ B ( H Λ )) . Note that ω Λ itself is a coherent state (corresponding to ϕ = 0 ).For coherent states, the relative entropy can be easily calculated. Without going into full de-tails at his point, the relative entropy of a faithful, normal state on a von Neumann algebra withrespect to another faithful, normal state was introduced by Araki [Ar76] (see also [Ul77]). It isa concept with an information theoretic background, see e.g. [Do86, OP93] for further discus-sion. If ω ϕ is any coherent state on W ( D S ∗ , ς ) as just described, then in the GNS representation ( H Λ , π Λ , Ω Λ ) it is induced by the unit vector Ω ϕ = π Λ ( W ( ϕ ))Ω Λ . If ϕ is compactly supportedin ( −∞ , × S ∗ so that π Λ ( W ( ϕ )) is contained in N R , then it is not difficult to see that Ω ϕ is a standard vector for N R , meaning that Ω ϕ is cyclic and separating for N R . In this case, thedefinition of relative entropy in the sense of Araki applies for any pair of coherent states. Inparticular, the relative entropy of ω ϕ with respect to ω Λ on N R is given as [Lo19, Ho19] S ( ω Λ | ω ϕ ) = i ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h Ω ϕ , ∆ itR Ω ϕ i (68)where ∆ R is, as above, the modular operator with respect to N R and Ω Λ .24o calculate S ( ω Λ | ω ϕ ) in the case at hand (cf. again [Lo19, Ho19] for similar calculations),we use (55) to obtain S ( ω Λ | ω ϕ ) = i ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h Ω ϕ , ∆ itR Ω ϕ i = i ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h Ω ϕ , Ω ϕ t i (69)where ϕ t ( U, ν ) = ( S πt/κ ∗ ϕ )( U, ν ) = ϕ (e πt U, ν ) . (70)Then we observe h Ω ϕ , Ω ϕ t i = ω Λ ( W ( − ϕ ) W ( ϕ t )) = e i ς ( ϕ,ϕ t ) ω Λ ( W ( ϕ t − ϕ ))= e i ς ( ϕ,ϕ t ) e − Λ( ϕ t − ϕ,ϕ t − ϕ ) / (71)Now we note that ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Λ( ϕ t − ϕ, ϕ t − ϕ ) = (cid:0) Λ( ddt ( ϕ t − ϕ ) , ϕ t − ϕ ) + Λ( ϕ t − ϕ, ddt ( ϕ t − ϕ )) (cid:1)(cid:12)(cid:12) t =0 = 0 (72)since ϕ t | t =0 = ϕ . Hence, we find S ( ω Λ | ω ϕ ) = 12 ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ς ( ϕ t , ϕ ) = − πr ∗ Z ( −∞ , × S U ( ∂ U ϕ ) ( U, ν ) dU d Ω ( ν ) . (73)In order to relate this entropy with the energy content of the coherent state measured by anobserver moving along the Kodama flow, we rewrite the relative entropy formula with respectto the coordinate (56). We then obtain S ( ω Λ | ω ϕ ) = β E ϕ (74)where E ϕ = r ∗ Z R × S ( ∂ u ϕ ) ( u, ν ) du d Ω ( ν ) (75)is the energy content of the coherent state ω ϕ measured by the Kodama observer and β = πκ isthe inverse temperature of the KMS state ω Λ (cf. Sec. 6.4 in [KW91]).25 .5 Relative entropy is proportional to outer trapped horizon surface area The previous equality (73) establishes a proportionality between the relative entropy of coherentstates of the scaling-limit-theory and the cross-section S ∗ of the outer trapping horizon, havingthe geometrical area πr ∗ , with respect to which the scaling limit and the restriction to T ∗ of thequantized scalar field on the ambient spacetime are taken. This is justified if, for different suchcross-sections, say S ∗ and S ∗ , with respective radii r ∗ and r ∗ , the associated coherent states ω Λ and ω ϕ are identified. This is certainly very natural for the scaling limit state ω Λ , but for ω ϕ that may, at first sight, not appear compelling. Let us therefore provide further motivationwhy the proportionality between the relative entropy of coherent states and the surface area ofthe cross-section of the outer trapping horizon at which the scaling-limit-theory is consideredarises naturally. The key point lies in the fact that the coherent states in the scaling-limit-theoryare completely correlation-free across spatial (i.e. spherical) separation as expressed in (67),together with the additivity of the relative entropy with respect to correlation-free states.To discuss this in more detail, fix a horizon cross-section S ∗ with radius r ∗ and considerthe corresponding scaling limit Weyl-algebra W ( D S ∗ , ς ) with the scaling limit state ω Λ , itsGNS representation ( H Λ , π Λ , Ω Λ ) and the von Neumann algebras N ( G ) for open subsets G of R × S ∗ as introduced in Sec. 5.1. Specifically, for open subsets Σ of S ∗ ≃ S , we define thevon Neumann algebras N R (Σ) = N (( −∞ , × Σ) . (76)We recall that N R (Σ) is the von Neumann algebra contained in B ( H Λ ) generated by the π Λ ( W ( ϕ )) where supp( ϕ ) ⊂ ( −∞ , × Σ . Hence, on account of (55), the N R (Σ) are in-variant under the adjoint action of the modular group ∆ itR ( t ∈ R ) with respect to N R and Ω Λ .When we denote by ω ϕ, Σ the state on N R (Σ) given by A ω ϕ, Σ ( A ) = h Ω ϕ , A Ω ϕ i ( A ∈ N R (Σ)) , (77)i.e. the restriction of the coherent state ω ϕ defined previously to N R (Σ) , and if likewise therestriction of ω Λ to N R (Σ) is denoted by ω Λ , Σ , then we find for the relative entropy in the sameway as before, S ( ω Λ , Σ | ω ϕ, Σ ) = − πr ∗ Z ( −∞ , × S U ( ∂ U ϕ ) ( U, ν ) dU d Ω ( ν ) = S ( ω Λ | ω ϕ ) . (78)Then, if Σ and Σ are any two disjoint open subsets of S , and if supp( ϕ j ) ⊂ ( −∞ , × Σ j j = 1 , ), and setting ϕ = ϕ + ϕ , one finds S ( ω Λ , Σ ∪ Σ | ω ϕ , Σ ∪ Σ ) = S ( ω Λ | ω ϕ )= − πr ∗ Z ( −∞ , × S U ( ∂ U ϕ ) ( U, ν ) dU d Ω ( ν )= − πr ∗ Z ( −∞ , × S U (cid:2) ( ∂ U ϕ ) ( U, ν ) + ( ∂ U ϕ ) ( U, ν ) (cid:3) dU d Ω ( ν )= S ( ω Λ | ω ϕ ) + S ( ω Λ | ω ϕ )= S ( ω Λ , Σ | ω ϕ , Σ ) + S ( ω Λ , Σ | ω ϕ , Σ ) (79)where we passed from the 3rd equality to the 4th since ϕ and ϕ are assumed to have disjoint ν -supports. This shows that the relative entropy of coherent states in any scaling limit is additivewith respect to angular separation ; actually, a corresponding additivity of the relative entropyacross angular separation holds upon replacing the two open, disjoint subsets Σ and Σ of S by finitely many Σ , . . . , Σ N , and similarly ϕ and ϕ by finitely many ϕ , . . . , ϕ N with supp( ϕ j ) ⊂ ( −∞ ) × Σ j .In fact, this can be seen to be, more generally, a consequence of the fact that the coherentstates in the scaling limit are correlation-free across angular separation, and the additivity of therelative entropy of correlation-free states. One can show that there is a joint unitary equivalence ω ϕ , Σ ∪ Σ ≃ ω ϕ , Σ ⊗ ω ϕ , Σ and ω Λ , Σ ∪ Σ ≃ ω Λ , Σ ⊗ ω Λ , Σ , where the correlation-free productstate ω ϕ , Σ ⊗ ω ϕ , Σ is the state defined on N R (Σ ) ⊗ N R (Σ ) by linear extension of A ⊗ A ω ϕ , Σ ⊗ ω ϕ , Σ ( A ⊗ A ) = ω ϕ , Σ ( A ) · ω ϕ , Σ ( A ) . (80)For (faithful, normal) correlation-free product states, the equation S ( ω Λ , Σ ⊗ ω Λ , Σ | ω ϕ , Σ ⊗ ω ϕ , Σ ) = S ( ω Λ , Σ | ω ϕ , Σ ) + S ( ω Λ , Σ | ω ϕ , Σ ) (81)holds (cf. [OP93], eq. (5.22)), whereupon one may conclude that S ( ω Λ , Σ ∪ Σ | ω ϕ , Σ ∪ Σ ) = S ( ω Λ , Σ | ω ϕ , Σ ) + S ( ω Λ , Σ | ω ϕ , Σ ) . (82)Therefore, the scaling of the relative entropy of coherent states proportional to the geometricarea of the horizon cross-section arises naturally. This is seen particularly cleary when consid-ering coherent states corresponding to elements ϕ ∈ D S ∗ (Λ) which are of the form ϕ = h ⊙ χ Σ where ( h ⊙ χ Σ )( U, ν ) = h ( U ) · χ Σ ( ν ) ( U ∈ ( −∞ , , ν ∈ S ) (83)27ith h ∈ C ∞ (( −∞ , , R ) and χ Σ the characteristic function of an open, or more generally,measurable subset Σ of S . In this case, S ( ω Λ | ω h ⊙ χ Σ ) = − π Z −∞ U ( ∂ U h ) ( U ) dU · r ∗ Z Σ ⊂ S d Ω ( ν )= − π Z −∞ U ( ∂ U h ) ( U ) dU · A (Σ r ∗ ⊂ S ∗ ) (84)where A (Σ r ∗ ⊂ S ∗ ) is the geometrical area of Σ viewed as subset of the horizon cross-section S ∗ which is a copy of the 2-dimensional sphere with radius r ∗ , i.e. the area of Σ as subset of S ,scaled by r ∗ .In the light of these observations, it is entirely natural to identify, if Σ = S , the coher-ent states ω h ⊙ for different horizon cross-sections S ∗ with different radii r ∗ , which renders aproportionality of the relative entropies with the horizon cross-section area A ( S ∗ ) , S ( ω Λ | ω h ⊙ ) = − π Z −∞ U ( ∂ U h ) ( U ) dU · A ( S ∗ ) (85)for the coherent states of the said type, when considering the scaling-limit-theory taken at S ∗ ,arising from any Hadamard state of the quantum field theory on the underlying sphericallysymmetric spacetime with an outer trapping horizon. Remark
Without the factor 2 inserted twice in the scaling limits (38), (41), one would obtainthat the relative entropy S ( ω Λ | ω h ⊙ ) equals one quarter of the horizon cross-sectional area times − π R −∞ U ( ∂ U h ) ( U ) dU , where the latter is the relative entropy of the coherent state inducedby h of the free chiral conformal quantum field theory defined on the real line with the vacuumtwo-point function Λ (1) ( h, h ′ ) = lim ε → − π Z h ( U ) h ′ ( U ′ )( U − U ′ + iε ) dU dU ′ ( h, h ′ ∈ C ∞ ( R , R )) (86)This would fully match the classical derivation where black hole entropy is equated to onequarter of the cross-sectional horizon area. In this paper we have investigated the scaling limits of Hadamard 2-point functions on the light-like submanifold T ∗ of a spherically symmetric outer trapping horizon generated by lightlikegeodesics traversing the outer trapping horizon. The scaling limit 2-point function Λ was foundto have a universal form, independent of which Hadamard 2-point function of the quantum field28heory on the underlying spherically symmetric spacetime is initially chosen. The projectedKodama flow acts in the scaling limit like a dilation, and the scaling limit 2-point function Λ shows a thermal spectrum with respect to the projected Kodama flow at inverse temperature β = 2 π/κ ∗ where κ ∗ is the surface gravity of the horizon cross-section S ∗ where the lightlikegenerators of T ∗ traverse the outer trapping horizon. Consequently, one can derive a tunnelingprobability in the scaling limit for Fourier modes peaked at Fourier energy E with respect tothe Kodama time behaving like e − βE for large E , analogous to a thermal distribution of energymodes. These results are in agreement with earlier, related results for stationary black horizonsor bifurcate Killing horizons, in particular [MP12] (see also [KW91, DNVZZ07, DHNVZ09]),and also with the first law of non-stationary black hole dynamics discussed by Hayward [Ha97], M ′ = κ π A ′ + w V ′ mentioned in the Introduction.Furthermore, the scaling limit 2-point function Λ defines a quantum field theory on each T ∗ , the scaling-limit-theory, determined by the horizon cross-section S ∗ . The thermal Fourierspectrum with respect to the Kodama time in the scaling limit is equivalent to the KMS propertyof the scaling limit state ω Λ induced by Λ when restricted to observables localized on the partof T ∗ lying either inside or outside of the outer trapping horizon. Furthermore, the state ω Λ aswell as all the coherent states ω ϕ in the scaling-limit-theory are correlation-free product stateswith respect to separation in the angular coordinate ν of S ∗ , and we have seen that this leadsnaturally to a proportionality of the relative entropy S ( ω Λ | ω ϕ ) with πr ∗ , the area of the cross-section S ∗ defining the scaling-limit-theory. Again, this is in keeping with the classical theoryof black hole thermodynamics [BCH73, Ha97, Be73]. We emphasize that this is a consequenceof our scaling limit analysis and seems to be the first such result in the setting of quantum fieldtheory in curved spacetime (apart from the related arguments of [HI19]).We should remark that our scaling limit consideration is akin to an adiabatic limit in thesense that effectively, in the scaling limit all processes or dynamical changes at finite time-scales are being scaled away. In this sense, our concepts of inverse temperature and of relativeentropy in the scaling-limit-theory are not dynamical, and that is a considerable limitation of ourapproach. The entropy concept, in our the scaling limit, bears some similarity to that in Beken-stein’s early article [Be73] on the subject: When an object (e.g. a table, a chair or a tankard)traverses the horizon, then the information about the object is lost outside of the horizon. In[Be73], the example of beams of light entering a black hole horizon is used. Our scaling-limit-theory can be seen as a bunch of elementary theories for such beams of light, namely, a freechiral conformal field theory, one for each point on S ∗ . As the area of S ∗ is increased, forexample, it accomodates for more such ingoing light beams as measured by the area, and corre-spondingly a larger amount of information along “elementary light beams” passing the horizonthrough S ∗ can be absorbed, which corresponds to the scaling of entropy – a measure for thelost amount of information – proportional to the area of S ∗ .We think that similar results can also be obtained for other types of horizons, like cosmo-logical horizons [Da20] or isolated horizons. A greater challenge is to attempt to obtain a more29ynamical concept of temperature and entropy for dynamical black hole horizons in the set-ting of quantum field theory in curved spacetimes and semiclassical gravity, in the spirit of theapproach of [HI19] which takes dynamical metric perturbations around a static Schwarzschildblack hole horizon into account. It would also be of interest to see if the notions of temperatureand entropy in the context of our semiclassical approach to the temperature and entropy of blackhole horizons can be linked to more “holographic” entropy concepts [HRT07]. Acknowledgments
F.K. thanks the IMPRS at the Max-Planck-Institute for Mathematics in the Sciences, Leipzig,for financial support. N.P. thanks the ITP of the University of Leipzig for the kind hospitalityduring the preparation of this work and the DAAD for supporting this visit with the program“Research Stays for Academics 2017”.
In this Appendix, we present the proof of Thm. 4.1. The proof will be facilitated by the follow-ing auxiliary result.
Lemma 7.1.
Let ( ̺ α ) <α k ≤ ( k = 1 , . . . , n ) be a family of measurable functions ̺ α : R m → C ,indexed by α = ( α , . . . , α n ) ∈ R n , so that for any compact subset C of R m there is some < a ≤ with sup <α k ≤ a sup y ∈ C | ̺ α ( y ) | ≤ (87) Furthermore, let ( j α ) <α k ≤ be a family of continuous functions j α : R m → C , with the proper-ties:(i) supp( j α ) ⊂ J m with some fixed compact real interval J ;(ii) | j α ( y ) | ≤ b for all α and all y ∈ J m with a fixed finite constant b > .Then the following statements hold. (A) There are some some < a ≤ and a finite positive constant κ so that, if < δ ≤ , itholds that (writing y = ( y , y ) and d m y = dy d m − y ) sup <α k ≤ a Z R m − Z δ (cid:12)(cid:12) ln | ̺ α ( y ) + y | (cid:12)(cid:12) | j α ( y ) | dy d m − y ≤ κδ / . (88)30 B) There are some < a ≤ and a finite constant κ so that, if δ ≤ , it holds that (writing y = ( y , y , y ) and d m y = dy dy d m − y ) ) sup <α k ≤ a Z R m − Z | y − y | <δ (cid:12)(cid:12) ln | ̺ α ( y ) + y − y | (cid:12)(cid:12) | j α ( y ) | dy dy d m − y ≤ κδ / . (89) Proof of Lemma 7.1
Part (A)
Making use of the integration coordinate substitution z = y , thus dz = 2 y dy , Z R m − Z δ (cid:12)(cid:12) ln | ̺ α ( y ) + y | j α ( y ) (cid:12)(cid:12) d m y = Z R m − Z δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln | ̺ α ( √ z, y ) + z | j α ( √ z, y )2 √ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz d m − y . (90)H¨older’s integral inequality with p = 3 , q = 3 / (so that /p + 1 /q = 1 ) with respect to the z -integration yields Z R m − Z δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln | ̺ α ( √ z, y ) + z | j α ( √ z, y )2 √ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz d m − y (91) ≤ Z R m − "Z δ | ln | ̺ α ( √ z, y ) + z | | dz / "Z δ z − / | j α ( √ z, y ) | / dz / d m − y Choosing < a ≤ so that sup <α k ≤ a sup y ∈ J m | ̺ α ( y ) | ≤ , (92)then for < α k ≤ a , the last integral can be estimated by sup < | ̺ |≤ / "Z δ | ln | ̺ + z | | dz / "Z δ z − / b / dz / | J | m − (93)where | J | is the length of the interval J .We observe that since | ̺ | ≤ / and < z ≤ δ with δ ≤ / , we obtain | Re( ̺ ) + z | + | Im( ̺ ) | = | ̺ + z | < . Consequently, under the integral, | ln | ̺ + z | | ≤ | ln | Re( ̺ ) + z | | .This results in sup < | ̺ |≤ / "Z δ | ln | ̺ + z | | dz / ≤ sup < | ̺ |≤ / "Z δ | ln | Re( ̺ ) + z | | dz / = sup < | r |≤ / "Z δ − r − r | ln | z | | dz / ≤ K (94)31ith a finite, positive real constant K . On the other hand, we obtain "Z δ z − / dz / = 4 / δ / . (95)Putting all the previous steps together, we find sup <α k ≤ a Z R m − Z δ (cid:12)(cid:12) ln | ̺ α ( y ) + y | (cid:12)(cid:12) | j α ( y ) | dy d m − y ≤ / b | J | m − Kδ / . (96)This proves the statement of Part (A) , with κ = / b | J | m − K .Part (B) First we note that the set | y − y | < δ in the y - y -plane can be split into the four parts H ( δ ) = {| y | ≤ y < q y + δ } , H ( δ ) = {− q y + δ < y ≤ −| y |} (97) H ( δ ) = {| y | ≤ y < q y + δ } , H ( δ ) = {− q y + δ < y ≤ −| y |} (98)The sets overlap only at their boundaries, y ± y = 0 . Therefore, Z R m − Z | y − y | <δ (cid:12)(cid:12) ln | ̺ α ( y ) + y − y | (cid:12)(cid:12) | j α ( y ) | dy dy d m − y = X ℓ =1 Z R m − Z H ℓ ( δ ) (cid:12)(cid:12) ln | ̺ α ( y ) + y − y | (cid:12)(cid:12) | j α ( y ) | dy dy d m − y (99)The integrals involving the H ℓ ( δ ) all have a very similar structure and thus it suffices to showthat, e.g., sup <α k ≤ a Z R m − Z H ( δ ) (cid:12)(cid:12) ln | ̺ α ( y ) + y − y | (cid:12)(cid:12) | j α ( y ) | dy dy d m − y ≤ κ δ / (100)since similar estimates for the other H ℓ ( δ ) can be deduced by analogous arguments. Carryingout a substitution z = y followed by a H¨older-type integral inequality similarly as in the proof32f Part (A) , we find, on making a small enough so that (92) holds, for all < α k ≤ a , Z R m − Z H ( δ ) (cid:12)(cid:12) ln | ̺ α ( y ) + y − y | (cid:12)(cid:12) | j α ( y ) | dy dy d m − y = Z R m − Z J Z √ y + δ | y | (cid:12)(cid:12) ln | ̺ α ( y ) + y − y | (cid:12)(cid:12) | j α ( y ) | dy dy d m − y ≤ Z R m − Z J "Z y + δ y (cid:12)(cid:12) ln | ̺ α ( √ z, y , y ) + z − y | (cid:12)(cid:12) dz / × (101) × "Z y + δ y | j α ( √ z, y , y ) | / z / dz / dy d m − y ≤
12 sup | ̺ | < / "Z δ | ln | ̺ + z || dz / b | J | m − sup y ∈ J "Z y + δ y z − / dz / (102)where an obvious substitution of z by z − y has been carried out in the integral involving thelogarithm. It is easy to check that sup y ∈ J "Z y + δ y z − / dz / ≤ "Z δ z − / dz / (103)and therefore we see that the integral expression in (102) can be estimated by / b | J | m − Kδ / (104)just as in the proof of Part (A) . This concludes the proof of Part (B) ✷ Proof of Theorem 4.1.
It will be convenient to introduce the following abbreviations, refer-ring to adapted coordinates ( U, V, ν ) near the chosen S ∗ : x =( U, V, ν ) , x ′ = ( U ′ , V ′ , ν ′ ) , x λ = ( λU, V, ν ) , x ′ λ = ( λU ′ , V ′ , ν ′ ) , (105) dX = dX ( x ) = dU dV d Ω ( ν ) = dU dV sin( ϑ ) dϑ dϕ (106)using ν = ( ϑ, ϕ ) in spherical angular coordinates as before; dX ′ is defined analogously. An-other abbreviations that we will use are ˜ F λ ( U, V, ν ) = η − ( U, V ) F λ ( U, V, ν ) , F λ ( U, V, ν ) = ( u λ ∂ U f )( U, V, ν ) (107)and analogously for symbols endowed with primes.33ecalling (32), we have w (2) ( F λ , F ′ λ ) = ˜ w (2) ( ˜ F λ , ˜ F ′ λ )= lim ε → Z ˜ w ε ( x, x ′ ) ˜ F λ ( x ) ˜ F ′ λ ( x ′ ) d vol ˜ g ( x ) d vol ˜ g ( x ′ )= lim ε → Z ˜ w ε ( x, x ′ ) F λ ( x ) F ′ λ ( x ′ ) P ( U, V, U ′ , V ′ ) dX dX ′ (108)having made use of d vol ˜ g ( x ) = η ( U, V ) A ( U, V ) r ∗ dX in the adapted coordinates for S ∗ which,as we recall, is a copy of a 2-sphere with radius r ∗ . We introduce on a smooth partition of unityon S × S , consisting of two functions χ and χ ⊥ , as follows: Choose some < δ < π / and choose a non-negative C ∞ function χ , bounded by 1, on S × S so that χ ( ν , ν ′ ) = 1 if s ( ν , ν ′ ) /r ∗ ≤ δ / , and χ ( ν , ν ′ ) = 0 if s ( ν , ν ′ ) /r ∗ ≥ δ . We then write χ ⊥ ( ν , ν ′ ) =1 − χ ( ν , ν ′ ) . Note that χ = χ δ and χ ⊥ = χ ⊥ δ depend on the choice of δ . With this notation, wecan write ˜ w ε ( x, x ′ ) = ˜ w ε ( x, x ′ ) χ ( ν , ν ′ ) + ˜ w ε ( x, x ′ ) χ ⊥ ( ν , ν ′ ) . (109)In a further step, we observe that, on a change of the U and U ′ integration coordinates, Z ˜ w ε ( x, x ′ ) χ ⊥ ( x, x ′ ) F λ ( x ) F ′ λ ( x ′ ) P ( U, V, U ′ , V ′ ) r ∗ dX dX ′ (110) = Z " ˜ ψ ( x λ , x ′ λ ) π ˜∆ / ( x λ , x ′ λ )˜ σ ε ( x λ , x ′ λ ) + ln(˜ σ ε ( x λ , x ′ λ )) ˜ Y ( x λ , x ′ λ ) ! + ˜ Z ( x λ , x ′ λ ) ×× P ( λU, V, λU ′ , V ′ ) χ ⊥ ( ν , ν ′ ) F ( x ) F ( x ′ ) dX dX ′ . In view of the particular form of the half of the squared geodesic distance (31), we have ˜ σ ε ( x λ , x ′ λ ) = ˜ σ ( L ) ( λU, V, λU ′ , V ′ ) + s ( ν , ν ′ ) + 2 iεt ( λU, V, λU ′ , V ′ ) + ε (111)Since in the integral on the right hand side of (110), s ( ν , ν ′ ) ≥ δ > owing to the presence of χ ⊥ , the integrand functions remain uniformly bounded in the limits as ε → and λ → , andthey converge almost everywhere to an integrable function. Therefore, one obtains lim λ → lim ε → Z ˜ w ε ( x, x ′ ) χ ⊥ ( x, x ′ ) F λ ( x ) F ′ λ ( x ′ ) P ( U, V, U ′ , V ′ ) dX dX ′ = Z h ( V, V ′ , ν , ν ′ ) ∂ U f ( U, V, ν ) ∂ U ′ f ′ ( U ′ , V ′ , ν ′ ) dX dX ′ = 0 (112)for some bounded L function h ; the integral on the right hand side vanishes since, after thelimit λ → , h is independent of U and U ′ , and f and f ′ have compact support (in particular,34ompact support with respect to U , respectively U ′ ). Note that this holds no matter how small δ > has been chosen.Next we notice that s ( ν , ν ′ ) is invariant under rotations R ∈ SO (3) , s ( R ν , R ν ′ ) = s ( ν , ν ′ ) ;similarly, the surface-integration form d Ω is invariant unter the rotations, d Ω ( R ν ) = d Ω ( ν ) .Therefore, given ν ′ , we can regard it as obtained from a standard “north pole point” ◦ ν ′ by asuitable rotation R ν ′ ∈ SO (3) so that ν ′ = R ν ′ ◦ ν ′ , hence s ( ν , ν ′ ) = s ( ν , R ν ′ ◦ ν ′ ) = s ( R − ν ν , ◦ ν ′ ) .The relation between ν ′ and R ν ′ is bijective and smooth as long as ν ′ is bounded away by afinite distance from the antipode point − ◦ ν ′ (on identifying ◦ ν ′ = (0 , , ∈ R ). In the followingintegrals we will consider this is always the case owing to the presence of the function χ ( ν , ν ′ ) .Introducing the abbreviations ˜ σ ( L )[ λ ] = ˜ σ ( L ) ( λU, V, λU ′ , V ′ ) , t [ λ ] = t ( λU, V, λU ′ , V ′ ) , (113)we thus have, for any λ -parametrized family q λ ( x, x ′ ) of bounded, compactly supported C functions, writing R ν ′ x = ( U, V, R ν ′ ν ) for x = ( U, V, ν ) Z q λ ( x, x ′ )˜ σ ε ( x λ , x ′ λ ) χ ( ν , ν ′ ) dX dX ′ = Z q λ ( x, x ′ )˜ σ ( L )[ λ ] + s ( R − ν ′ ν , ◦ ν ′ ) + 2 iεt [ λ ] + ε χ ( ν , ν ′ ) dX dX ′ = Z q λ ( R ν ′ x, x ′ )˜ σ ( L )[ λ ] + s ( ν , ◦ ν ′ ) + 2 iεt [ λ ] + ε χ ( R ν ′ ν , ν ′ ) dX dX ′ . (114)Using the standard spherical angular coordinates ( ϑ, ϕ ) = ν , with ϑ = 0 corresponding to the“north pole point” = ◦ ν ′ , the half of the squared geodesic distance on the sphere with radius r ∗ takes the simple form s ( ϑ, ϕ, ◦ ν ′ ) = r ∗ ϑ , (115)and we thus obtain, on writing ξ λ ( x, x ′ ) = χ ( R ν ′ ν , ν ′ ) q λ ( R ν ′ x, x ′ ) , Z q λ ( R ν ′ x, x ′ )˜ σ ( L )[ λ ] + s ( ν , ◦ ν ′ ) + 2 iεt [ λ ] + ε χ ( R ν ′ ν , ν ′ ) dX dX ′ = Z Z δϑ =0 ξ λ ( x, x ′ )˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε sin( ϑ ) dϑ dϕ dU dV dX ′ (116)since in the polar coordinates chosen, r ∗ ϑ / is the half of the squared geodesic distance be-tween ◦ ν ′ and ν on the sphere with radius r ∗ , and ξ λ ( x, x ′ ) = 0 if ϑ > δ by the properties of χ .35ow carrying out a partial integration with respect to ϑ and observing σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε = 1 r ∗ ϑ ∂ ϑ ln(˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε ) , (117)we are led to Z Z δϑ =0 ξ λ ( x, x ′ )˜ σ ( L )[ λ ] + r ∗ ϑ / − iεt [ λ ] + ε sin( ϑ ) dϑ dϕ dU dV dX ′ = Z (cid:20) ln(˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε ) ξ λ ( x, x ′ ) sinc( ϑ ) r ∗ (cid:21) δϑ =0 dϕ dU dV dX ′ (118) − Z Z δϑ =0 ln(˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε ) ∂ ϑ (cid:18) ξ λ ( x, x ′ ) sinc( ϑ ) r ∗ (cid:19) dϑ dϕ dU dV dX ′ . (119)We note that ln(˜ σ ε ( x, x ′ )) = ln | ˜ σ ( L ) ( U, V, U ′ , V ′ ) + s ( ν , ν ′ ) + 2 iεt ( x, x ′ ) + ε | + i arg(˜ σ ( L ) ( U, V, U ′ , V ′ ) + s ( ν , ν ′ ) + 2 iεt ( x, x ′ ) + ε ) (120)so that Lemma 7.1 applies to the expression in (119), with α = ( λ, ε ) ∈ R , on noting that theargument function part stays uniformly bounded in ( λ, ε ) , resulting in a contribution in (119)which is O ( δ ) as δ → . Therefore, supposing that λ and ε have been chosen sufficiently small,and likewise that δ > is sufficiently small, we can conclude that sup λ,ε (cid:12)(cid:12)(cid:12)(cid:12)Z Z δϑ =0 ln(˜ σ ( L )[ λ ] + r ∗ ϑ / − iεt [ λ ] + ε ) ∂ ϑ (cid:18) ξ λ ( x, x ′ ) sinc( ϑ ) r ∗ (cid:19) dϑ dϕ dU dV dX ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ δ / (121) with a suitable positive constant κ .In a similar manner we find, provided that λ, ε and δ are sufficiently close to 0, on accountof Lemma 7.1 sup λ,ε (cid:12)(cid:12)(cid:12)(cid:12)Z ln(˜ σ ε ( x λ , x ′ λ )) ˜ Y ( x λ , x ′ λ ) χ ( ν , ν ′ ) F ( x ) F ( x ′ ) P ( λU, V, λU ′ , V ′ ) dX dX ′ (cid:12)(cid:12)(cid:12)(cid:12) (122) ≤ sup λ,ε (cid:12)(cid:12)(cid:12)(cid:12)Z Z δϑ =0 ln(˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε ) k λ ( x, x ′ ) dϑ dϕ dU dV dX ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ κ δ / with a family of smooth functions k λ ( x, x ′ ) of x and x ′ which is uniformly bounded and uni-formly compactly supported in λ ; κ > is a suitable constant.36urthermore, since ˜ Z ( x, x ′ ) is C ∞ , we see that sup λ,ε (cid:12)(cid:12)(cid:12)(cid:12)Z ˜ Z ( x, x ′ ) χ ( ν , ν ′ ) F λ ( x ) F λ ( x ′ ) P ( U, V, U ′ , V ′ ) r ∗ dX dX ′ (cid:12)(cid:12)(cid:12)(cid:12) = O ( δ ) (123)if λ and δ are small enough.Summarizing our findings up to this point, we see that, on choosing q λ ( x, x ′ ) = ˜ ψ ( x λ , x ′ λ ) ˜∆ / ( x λ , x ′ λ )8 π F ( x ) F ( x ′ ) P ( λU, V, λU ′ , V ′ ) (124)in (114), we obtain lim λ → w (2) ( F λ , F ′ λ ) (125) = lim λ → lim ε → Z (cid:20) ln(˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε ) ξ λ ( x, x ′ ) sinc( ϑ ) r ∗ (cid:21) δϑ =0 dϕ dU dV dX ′ + O ( δ / ) for any sufficiently small δ > . However, the integral expression is independent of δ : Recallingthat ξ λ ( x, x ′ ) = 0 if ϑ > δ , the evaluation of the integral expression at ϑ = δ vanishes, and theresulting expression, as we will see, is independent of χ δ which is contained in the definition of ξ λ ( x, x ′ ) . Therefore, since δ may be chosen arbitrarily small, we now obtain lim λ → w (2) ( F λ , F ′ λ ) (126) = lim λ → lim ε → Z − ln(˜ σ ( L )[ λ ] + r ∗ ϑ / iεt [ λ ] + ε ) ξ λ ( x, x ′ ) sinc( ϑ ) r ∗ (cid:12)(cid:12)(cid:12)(cid:12) ϑ =0 dϕ dU dV dX ′ . Evaluating the integral expression at ϑ = 0 , observing ξ λ ( x, x ′ ) = χ ( R ν ′ ν , ν ′ ) q λ ( R ν ′ x, x ′ ) ,results in lim λ → w (2) ( F λ , F ′ λ ) (127) = lim λ → lim ε → Z − ln(˜ σ ( L )[ λ ] + 2 iεt [ λ ] + ε ) q λ ( U, V, ν ′ , U ′ V ′ , ν ′ ) 2 πr ∗ dU dV dU ′ dV ′ d Ω ( ν ′ ) To see this, note first that, in the coordinates chosen, x | ϑ =0 = ( U, V, ◦ ν ′ ) , implying s ( R ν ′ ν , ν ′ ) = s ( ν , ◦ ν ′ ) = s ( ◦ ν ′ , ◦ ν ′ ) = 0 and therefore, χ ( R ν ′ ν , ν ′ ) = 1 . On the other hand, ν = ◦ ν ′ also means that there is no ϕ -dependence in the integrand, and the integral with respect to ϕ can be carried out, just contributing a factor π . Moreover, it implies that q λ ( R ν ′ x, x ′ ) = q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) in the last integral. 37e are thus left with having to evaluate the limits of the right hand side in (127). We note that ˜ σ ( L )[ λ ] (cid:12)(cid:12)(cid:12) λ =0 = ˜ σ ( L ) ( λU, V, λU ′ , V ′ ) (cid:12)(cid:12) λ =0 = 0 (128)and therefore, the Taylor expansion of ˜ σ ( L )[ λ ] in λ at λ = 0 up to second order yields ˜ σ ( L )[ λ ] = λ ( U − U ′ )( ˜ V − ˜ V ′ ) + R λ ( U, V, U ′ , V ′ ) (129)with R λ = O ( λ ) uniformly on compact sets in U, U ′ and V, V ′ while ˜ V (and ˜ V ′ ) is a geodesicparameter of the lightlike curves V (0 , V, ν ) with respect to the conformally transformedmetric ˜ g ab chosen such that dU a = ˜ g ab ( ∂/∂ ˜ V ) b . This can be seen from eqns. (3.3) and (3.4) in[PPV11]; note that V is an affine parameter for the said lightlike curves with respect to g ab butnot necessarily with respect to the conformally transformed metric ˜ g ab . Using the form of the“Lorentzian” part of the conformally transformed metric − η ( U, V ) A ( U, V ) dU dV , (130)it is not difficult to check that ˜ V = ˜ V ( V ) has the property ˜ V − ˜ V ′ = Z V ′ V η (0 , V ) dV implying ˜ V − ˜ V ′ = ( V − V ′ ) γ ( V, V ′ ) (131)with a positive, jointly continuous function γ ( V, V ′ ) where γ ( V, V ′ ) and /γ ( V, V ′ ) are boundedwhen V and V ′ range over compact sets. Consequently, we have that lim λ → w (2) ( F λ , F ′ λ )= lim λ → lim ε → Z − ln( λ ( U − U ′ )( V − V ′ ) γ ( V, V ′ ) + R λ ( U, V, U ′ , V ′ ) + 2 iεt [ λ ] + ε ) ×× q λ ( U, V, ν ′ , U ′ V ′ , ν ′ ) 2 πr ∗ dU dV dU ′ dV ′ d Ω ( ν ′ )= lim λ → lim ε → Z − [ln( λγ ) + ln(( U − U ′ )( V − V ′ ) + ̺ λ ( U, V, U ′ , V ′ ) + λ − γ − [2 iεt [ λ ] + ε ])] ×× q λ ( U, V, ν ′ , U ′ V ′ , ν ′ ) 2 πr ∗ dU dV dU ′ dV ′ d Ω ( ν ′ ) (132) with ̺ λ = λ − γ − R λ , and we have abbreviated γ = γ ( V, V ′ ) .Let us first consider the ε -independent part in (132). We split ln( λγ ) = ln( λ ) + ln( γ ) . Thenthe limits can be carried out to yield lim λ → Z − ln( γ ( V, V ′ )) q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ = 0 (133) We reinsert the abbreviation dX ′ = dU ′ dV ′ d Ω ( ν ′ ) U and U ′ dependence in q ( U, V, U ′ , V ′ ) comes from ∂ U f ( U ) and ∂ U ′ f ′ ( U ′ ) and therefore, since f and f ′ are compactly supported, we find that the resulting integral van-ishes. On the other hand, we also have q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) = q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) + q (1) λ ( U, V, U ′ , V ′ ; ν ′ ) where q (1) λ ( U, V, U ′ , V ′ ; ν ′ ) = O ( λ ) uniformly on compact sets in U, U ′ and V, V ′ . Then, lim λ → Z ln( λ )( q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) + q (1) λ ( U, V, U ′ , V ′ ; ν ′ )) 2 πr ∗ dU dV dX ′ = 0 (134)since, as in the previous argument, q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) depends on U and U ′ only through ∂ U f and ∂ U ′ f ′ , and ln( λ ) q (1) λ ( U, V, U ′ , V ′ ; ν ′ ) → as λ → uniformly on compact sets.Therefore, setting β λ,ε ( U, V, U ′ , V ′ ) = ̺ λ ( U, V, U ′ , V ′ ) + γ − [2 iεt [ λ ] + λε ] , we obtain lim λ → w (2) ( F λ , F ′ λ ) = lim λ → lim ε → Z − ln(( U − U ′ )( V − V ′ ) + β λ,ε ( U, V, U ′ , V ′ )) ×× q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ (135)where we have made use of the fact that, owing to the limit ε → being taken prior to λ → , one may redefine ε as ε/λ without changing the result of the limits.Then we choose δ > and split the integration over the U, U ′ and V, V ′ coordinates into thedomains D > ( δ ) = | ( U − U ′ )( V − V ′ ) | ≥ δ , D < ( δ ) = | ( U − U ′ )( V − V ′ ) | < δ . (136)Furthermore, we change from the ( U, V ) coordinates to ( T, Y ) coordinates where T = 12 ( U + V ) , Y = 12 ( U − V ) (137)and using a coordinate substitution ¯ T = T − T ′ , ¯ Y = Y − Y ′ , one arrives at Z D < ( δ ) ln(( U − U ′ )( V − V ′ ) + β λ,ε ( U, V, U ′ , V ′ )) q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ (138) = Z Z | ¯ T − ¯ Y | <δ ln( ¯ T − ¯ Y + ¯ β λ,ε ( ¯ T , ¯ Y , T ′ , Y ′ ))¯ q λ ( ¯ T , ¯ Y , T ′ , Y ′ ; ν ′ ) 2 πr ∗ d ¯ T d ¯ Y dT ′ dY ′ d Ω ( ν ′ ) where ¯ β λ,ε and ¯ q λ are the ¯ T , ¯ Y , T ′ , Y ′ -coordinate versions of β λ,ε and q λ . We see that theright hand side is in a form to which Part (B) of Lemma 7.1 applies (with α = ( λ, ε ) ), andarguing analogously as with (119) before, we conclude that, once ε, λ and δ have been chosensufficiently small, it holds that sup λ,ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D < ( δ ) ln(( U − U ′ )( V − V ′ ) + β λ,ε ( U, V, U ′ , V ′ )) q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( δ / ) (139) lim λ → w (2) ( F λ , F ′ λ ) = lim δ → lim λ → lim ε → Z D > ( δ ) ln(( U − U ′ )( V − V ′ ) + β λ,ε ( U, V, U ′ , V ′ )) ×× q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ (140)The limits can now be carried out performing the limit ε → followed by λ → prior tointegration, to yield lim λ → w (2) ( F λ , F ′ λ ) = Z [ln( | ( U − U ′ ) | ) − iπθ ( U − U ′ )] q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ (141)where θ denotes the Heaviside function. To see this, we note that lim λ → lim ε → ln(( U − U ′ )( V − V ′ ) + β λ,ε ( U, V, U ′ , V ′ )) q λ ( U, V, ν ′ , U ′ , V ′ , ν ′ ) (142) = [ln( | ( U − U ′ )( V − V ′ ) | ) + iπθ (( U ′ − U )( V − V ′ ))sign( γ − ( V − V ′ ))] q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) Using ln( | ( U − U ′ )( V − V ′ ) | ) = ln | U − U ′ | + ln | V − V ′ | , one can see again that the ln | V − V ′ | term gives a vanishing contribution on integration with respect to U and U ′ be-cause q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) depends on U and U ′ only through ∂ U f and ∂ U ′ f ′ . Moreover,since γ − = γ − ( V, V ′ ) > , we have θ (( U ′ − U )( V − V ′ ))sign( γ − ( V − V ′ )) = θ (( U ′ − U )sign( V − V ′ ))sign( V − V ′ ) ; and since Z [ θ ( U − U ′ ) + θ ( U ′ − U )] ∂ U f ( U, V, ν ′ ) ∂ U ′ f ′ ( U ′ , V ′ , ν ′ ) dU dU ′ = 0 (143)we can conclude that Z iπθ (( U ′ − U )( V − V ′ ))sign( γ − ( V − V ′ )) q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) dU dV dX ′ = − Z iπθ ( U − U ′ ) q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) dU dV dX ′ (144)showing that (141) holds.It is well-known – or can easily be derived by arguments analogous to those given in theproof up to now, using partial integrations with respect to U and U ′ – that Z [ln( | ( U − U ′ ) | ) − iπθ ( U − U ′ )] q ( U, V, ν ′ , U ′ , V ′ , ν ′ ) 2 πr ∗ dU dV dX ′ = lim ε → Z q ( U, V, ν ′ , U ′ , V ′ , ν ′ )( U − U ′ + iε ) πr ∗ dU dV dX ′ . (145)40inally, we observe that ˜ ψ (0 , V, ν ′ , , V ′ , ν ′ ) = 1 because the points (0 , V, ν ′ ) and (0 , V ′ , ν ′ ) are causally related and lie, by assumption, in a causal normal neighbourhood. Therefore, wehave shown that lim λ → w (2) ( F λ , F ′ λ ) = lim ε → Z q ( U, V, ν ′ , U ′ , V ′ , ν ′ )( U − U ′ + iε ) πr ∗ dU dV dX ′ (146) = − r ∗ π Z f ( U, V, ν ) f ′ ( U ′ , V ′ , ν )( U − U ′ − iε ) Q ( V, V ′ , ν ) dU dU ′ dV dV ′ d Ω ( ν ) as claimed in the statement (I) of the Theorem.The second statement is easily proved since the µ → limit can be taken directly as the distri-bution L only involves integrations against continuous, bounded functions with respect to the V and V ′ coordinates. In fact, the limits ε → in the definition of L (resp., Λ ) can be interchangedwith the limit µ → used to pass from L to Λ . 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