Unruh Effect for Inertial Observers Through Vacuum Correlations

Eur. Phys. J. C (2018) 78:433 https://doi.org/10.1140/epjc/s10052-018-5911-0

Regular Article - Theoretical Physics

Unruh effect for inertial observers through vacuum correlations

Kinjalk Lochan1,2,a, Sumanta Chakraborty2,3,b , T. Padmanabhan2,c 1 Department of Physical Sciences, IISER Mohali, Manauli 140306, India 2 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411007, India 3 Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata 700032, India

Received: 18 September 2017 / Accepted: 21 May 2018 / Published online: 30 May 2018 © The Author(s) 2018

Abstract We study a dynamic version of the Unruh effect This arises because the modes of the quantum field in the in a two dimensional collapse model forming a black hole. In region inaccessible to the observer are traced out. Similarly, this two-dimensional collapse model a scalar field coupled a uniformly accelerated observer in the right wedge of the flat to the dilaton gravity, moving leftwards, collapses to form spacetime will associate a thermal nature to the global, iner- a black hole. There are two sets of asymptotic (t →∞) tial, vacuum state, again obtained by tracing out the modes observers, around x →∞and x →−∞. The observers at inaccessible to her (on the left wedge).1 Both situations are the right null infinity witness a thermal flux of radiation asso- time-reversal invariant; while the relevant observers see an ciated with time dependent geometry leading to a black hole ambient thermal radiation, they do not associate a flux of formation and its subsequent Hawking evaporation, in an particles with this radiation. In the rest of the analysis, by the expected manner. We show that even the observers at left null word “radiation” we will imply scenarios where the wave infinity find themselves in thermal ambiance, without expe- modes are thermally populated but there may not be any riencing any change of spacetime geometry all along their physical energy momentum tensor (and hence flux) associ- trajectories. They remain as inertial observers in a flat region ated with it. of spacetime where curvature tensor identically vanishes in A somewhat different situation arises in the case of a a portion of full spacetime. These observers find the state of black hole formed by collapsing matter, with the quantum the quantum field in a late time thermal configuration, with state being the Unruh vacuum (or, the In-vacuum) at very exactly the same temperature as measured by the observers early times. In this case, at late times, observers far away at right null infinity, despite being inertial in flat spacetime from the collapsing body detect a flux of particles with a region throughout their history. This is very closely related spectrum which is thermally populated. The energy carried to the standard Unruh effect in the flat spacetime, except away by the particles is ultimately obtained from the mass for a key difference Ð since they are inertial throughout and of the collapsing body and this leads to the concept of black have no causally connected source in the past light cone to hole evaporation. The mathematical description of this pro- account for what they see. The result arises from quantum cess takes into account; (1) the change in geometry due to correlations which extend outside the past light cone and is the collapse process and (2) the formation of event horizon conceptually similar to the EPR correlations. leading to inaccessibility of a region from future asymptotic observers. With future applications in mind, let us briefly recall the key concepts. In standard 3 + 1 dimensional col- 1 Introduction lapse, in which classical matter collapses to form a black hole, an apparent horizon is formed, which grows and Ð at a Hawking radiation from a black hole [1Ð10] and the Unruh certain stage Ð an event horizon is formed thereby producing radiation [3,6Ð8,11,12] in the Rindler frame have very simi- a black hole region. The last null ray originating from the lar mathematical properties. In the context of an eternal black past null infinity J − at the null co-ordinate u = d,inthe hole, the HartleÐHawking vacuum state of a quantum field double null co-ordinate system [1,3,6Ð8], reaching future will appear as thermal for static observers in the right wedge. null infinity J + defines the location of the event horizon. a e-mail: [email protected] 1 By tracing out we mean that the relevant mode function will have only b e-mail: [email protected] a compact support on the Cauchy surface and the behaviour outside the c e-mail: [email protected] region of support would not contribute in integrations to be performed. 123 433 Page 2 of 14 Eur. Phys. J. C (2018) 78 :433

by only a portion of J −, rather than the direct effects of change in the geometry, which is already discarded in the ray-optics approximation. This is the mathematical reason why the dynamics of the collapse is irrelevant to obtain the temperature and one obtains the same result as in the case of an eternal black hole and HartleÐHawking vacuum state. The exact Bogoliubov coefficients, obtained by going beyond the ray optics, will, of course, be sensitive to the geometry change as well. Further, it is the loss of time reversal invariance during the collapse which leads to a non-zero flux of energy, which is absent in the case of an eternal black hole and HartleÐHawking vacuum state. A combination of these effects arise in the case of CGHS black hole in 1+1 dimension [5,13]. The collapse of quantum Fig. 1 Penrose diagram For Schwarzschild collapse matter leads to the standard black hole evaporation scenario +( →∞, →+∞) with observers at iR t x detecting a flux of thermal radiation, which is well-known in the literature. This Note that we have taken u as the advanced time coordinate, situation is mathematically identical to what happens in the which is different from the standard choices, where usually case 1 + 3 spherically symmetric collapse. But it turns out v is referred to as advanced coordinates. We will follow this that there is another effect in the same spacetime which is convention throughout. An asymptotic observer on J + has unnoticed in the literature, which is conceptually very sim- no causal connection with events inside the event horizon. + ilar to Unruh effect in its structure: Observers at future left Further, whole of J derives its complete past causal sup- + asymptotic J (t →∞, x →−∞) find themselves in a port (e.g., the thick orange line in Fig. 1) from only a part of L − 2 thermal environment at the same temperature as seen by J , which lies prior to the ray forming event horizon. + observers at J (but without any associated flux!), how- A wave mode which originates from J − and reaches R ever these observers do not have to necessarily accelerate J + also experiences a change in background geometry in unlike their right moving counterparts. This result is rather the process. In order to obtain the Bogoliubov coefficients curious because these are inertial observers in a flat region of between the asymptotic observers3 on J − and J +, we need the spacetime who see no change in geometry as they reside to re-express a set of complete mode functions vω suited to in the causal past of the collapsing shell. This shows one observers on J + in terms of complete set of mode functions − instance of Unruh effect in flat spacetime without any classi- uω, defined equivalently on J . When we trace back the + cal source, which we will argue originates through quantum out-going modes vω on J through the center, (i.e., through correlations in analogy with the standard EPR phenomenon. r = 0 with r being the Schwarzschild radial co-ordinate) There are cases, in which a geodesic observer witnesses onto J −, we see that the mode functions suited to J + have − particle excitations in the vacuum state, e.g., thermal radi- support only on the portion of mode functions uω on J . ation witnessed by an inertial observer in flat spacetime in Therefore the Bogoliubov coefficients are obtained by tak- − presence of a receding mirror [3]. The acceleration of the mir- ing inner products of uω and vω on a portion of J , i.e., below ror modifies the boundary conditions of the quantum field, the line u = d [1,3,7,8]. (This has an effect similar to that which the inertial observer will ascribe as the source of the of partial tracing out of modes in the case of eternal black radiation. Other instances of geodesic observers experienc- hole.) Furthermore, we now also have to take into account ing radiation arise in special geodesic trajectories in curved the change of geometry experienced by the mode function spacetime, like e.g., the r = 0 observer in de Sitter space- uω when we trace it back to the past null infinity. However, time (for another interesting example, see [14]). But if the if we are interested only in the late time behavior of uω, observer measures the curvature tensor at r = 0 in de Sitter we can use ray-optics approximation [1,3,7,8] for tracing spacetime she will find it to be non-zero and thus will know back the mode functions. Thus, the late time behavior of the that something nontrivial is happening to the geometry [15]. out going asymptotic future modes is essentially controlled Here gravity plays the role of acceleration, as evident from the equivalence principle. 2 Since the rays suffering gravitational lensing do not undergo any fre- The effect we discuss in this paper on the other hand, arises quency shift, they are typically discarded in the standard Hawking effect for geodesic observers in a flat region of spacetime, i.e., if the and we will also not be concerned about them in this paper hereafter. observers measure the curvature tensor in a finite neighbor- 3 We can think of observers moving with large speeds and reaching future timelike infinity. These observers, in an approximating limit, hood around their trajectory, they will find that it is zero. The may be mimicked through late time observers on J +. effect arises due to oblivion of the physics in the portion of 123 Eur. Phys. J. C (2018) 78 :433 Page 3 of 14 433 interest, to a part of the modes on past Cauchy surface (which J − we will be calling R ) and the mathematics closely parallels standard Unruh effect. But, unlike the standard Unruh effect, the necessity to trace over modes arises due to the dynamics of collapse, which takes place in a region of the spacetime which is not causally connected to the observers! Thus, what we have is a setting where:

1. A family of inertial observers in a flat portion of spacetime experience thermal ambiance in the vacuum state of the field. 2. There is no classical source, (e.g., moving mirror, a dis- tant black hole, local curvature, acceleration, time depen- Fig. 2 Rindler trajectory in Minkowski spacetime dent geometry etc.) which can classically explain the source of such a thermal effect. π ω β =− 1 − −i i , ω π ω exp exp log This effect (which, as far as we know, has been missed in 2 a 2a a a a the CGHS literature) constitutes a dynamics realization of (2) Unruh effect in flat spacetime. with a being the acceleration of the Rindler observer [6]. Therefore the Rindler observer sees the vacuum of the inertial observer, denoted by |0inertial to be thermally occupied, i.e., 2 A picture book representation | | = |β |2 = 1 . inertial 0 N 0 inertial ω 2π (3) a − In order to explain the concepts involved in this, rather pecu- ω e 1 liar result, we will first provide a picture-book description of how the result arises, which should demystify it. where N stands for densitized particle number operator, We start with the Minkowski spacetime, for which the essentially removing an overall volume factor from the com- Penrose diagram corresponds to Fig. 2. The full spacetime is putation. The thermal ambiance of the Rindler observer can bounded by four null lines depicting future and past left/right further be verified through correlations among the various J + J − J + J − modes which can be shown to be thermal as well. In partic- null infinities, viz R , R , L and L . In this spacetime ular for a two-point correlation we obtain any inertial (geodesic) observers will start from past time- ⎛ ⎞ ⎛ ⎞ like infinity i− at t =−∞and would reach future time- + like infinity i at t =∞. Two such geodesic observers | | = ⎝ |β |2⎠ ⎝ |β |2⎠ inertial 0 N1 N2 0 inertial 1ω 2ω moving leftwards and rightwards respectively, are shown by ω ω dashed curves in Fig. 2. In addition, there can also be some = 1 1 . π π (4) accelerated observers. An important set of such accelerated 2 1 2 2 observers is the eternally accelerating Rindler observer. The e a − 1 e a − 1 J − trajectory of the Rindler observer starts on L and acceler- Since we are considering a free field theory, using Wick’s J + ates along a hyperbolic path to reach L (shown in the thick theorem for free fields, one can compute all the higher order green curve in Fig. 2). Let us consider the past causal support correlators as well using the two point correlator and verify of the trajectories of the Rindler observer vis-á-vis the iner- that these are exactly thermal [5]. tial observer. The inertial observer has causal access to the Alternatively, the observations carried out by the Rindler full spacetime, whereas the spacetime region accessed by the observer can be described using a density matrix obtained Rindler observer is only a part of the full Minkowski space- by tracing out modes inaccessible to her. If the field is in time. Thus the vacuum state for the inertial observer (who the inertial vacuum state, the resulting density matrix will can access the full spacetime) would be different from that be thermal. As an illustration let us consider Fig. 3, where of the Rindler observer (for whom only a part of the space- three observers are shown. The dashed observer is an inertial time is allowed) leading to non zero Bogoliubov coefficients, observer and, as in the earlier figure, has access to the full given as spacetime; the dotted trajectory is that of the standard, eternally accelerating Rindler observer; the third trajectory (rep- 1 π i ω i resented by thick magenta line) represents an observer who αω = exp exp − log , 2πa ω 2a a a a originally started as inertial but then changed her mind and J + (1) accelerates uniformly to end up on L just like the Rindler 123 433 Page 4 of 14 Eur. Phys. J. C (2018) 78 :433

Fig. 4 Observers in a hypothetical spacetime Fig. 3 Non-geodesic observer in Minkowski spacetime

observer. This observer also has past causal connection only J + to the past domain of dependence of L , which being only a portion of the full manifold, will lead to a mixed density matrix having thermal character at late times. At finite times, its trajectory will be different from the Rindler observer and unlike the Rindler observer will have a future causal link to the right wedge. Hence the density matrix will be different. The same can be seen through the Bogoliubov coefficients, which will be similar to those of the Rindler observer for high frequencies at late times only. The exact form of the Bogoliubov coefficients will differ for small frequencies as Fig. 5 Penrose diagram For Schwarzschild null shell collapse the large wavelengths will take them on different trajectories. Let us now consider a hypothetical scenario in which some portion of Minkowski spacetime becomes inaccessible, as The situation as depicted in Fig. 4 appears unphysical as shown in Fig. 4 by the dashed triangular region. (Right now presented Ð because we have artificially removed part of the this is the Penrose diagram of some hypothetical spacetime; spacetime —, but we will later discuss a situation in which we will soon see how it actually arises in the CGHS case.) a portion of spacetime is indeed dynamically denied to a Further, we assume this truncation of spacetime is such that geodesic observer, very much like in the spirit of the observer it requires left-moving geodesic observers to terminate on in Fig. 4 (shown in thick magenta curve). + + But before we do that, we will consider another example iL instead of i as in Fig. 3. Since these observers derive J + in the next section which might make all these less surprising. their causal support only from the past of L , they are com- J − This will involve the collapse of a null shell forming a black pelled to trace over a portion of field configuration on R and hence they will end up using a density matrix which is hole (see Fig. 5). In the case, a timelike geodesic observer at = mixed. Whenever any observer, geodesic or not, who derives r 0, will remain entirely in the flat spacetime until being her past causal support from a subset of the full spacetime, eaten up by the singularity and receives causal signals only J − the global vacuum state will appear to be non-vacuous at late from a part of (see Fig. 5 below). This study will provide times. us useful insights towards the constructs to be used in the The geodesic part of the above statement might appear later sections of the paper. perplexing. One may wonder how a geodesic observer can ever experience such a nontrivial effect usually associated with accelerated observers. The role of acceleration is only 3 A null shell collapse to make part of the spacetime region inaccessible; if we can achieve this by some other means, we will still have the same For a closer look at the above mentioned features, we will result. Indeed, there can be various other scenarios where first consider a null shell collapse forming a Schwarzschild geodetic observers can witness thermal radiation. However, hole [3,7,16Ð18] (see Fig. 5). In this case, the singularity gets in this construction, the left moving observers are unable to originated from the co-ordinate u = ui , the co-ordinate point even associate any classical source to these radiations. of introduction of the null-shell. (This collapse model will be 123 Eur. Phys. J. C (2018) 78 :433 Page 5 of 14 433

J + the modes reaching R derive their past causal support from the region I in Fig. 6. No event in the region II is connected J + to R . An exactly similar picture is there for the left-moving modes. In the standard black hole analysis, the out-going mode J + functions on R are given as

1 −iωu vω = √ e ; u ∈ (−∞, ∞). (5) 2ω These mode functions provide an orthonormal basis across J + J − complete R . Similarly the right-moving modes on L are Fig. 6 Schwarzschild in isotropic co-ordinates spanned by

1 −iωv uω = √ e ; v ∈ (−∞, ∞). (6) 2ω relevant for comparison with the null shell collapse in 1 + 1 dimensional dilatonic gravity, forming a black hole, which The Bogoliubov transformation coefficients between these modes can be evaluated by taking the covariant inner products is discussed later on.) The singularity starts forming from − = on J . However, in order to do that we need to express mode the co-ordinate u ui , while the event horizon is located L − = functions vω on J in terms of uω. For that purpose we at r 2M, with M being the mass of the shell. The co- L + = = need to track the out-going modes at J all the way down to ordinate r 2M can be reflected back through r 0onto − R − J J at u = d. As discussed previously, the out-going modes L . In principle, this a formidable job, since the exact form + vω on J derive their past causal support completely from of modes in the whole spacetime is complicated at best, if the portion u < d on J −. Therefore the Bogoliubov coeffi- obtainable in closed form. However, appealing to ray-optics cients of mode transformation, should be evaluated through approximation [1,7] for modes very close to the horizon, we ω < in a sense, avoid this issue. This approximation gives us the the portion of mode functions u in the regime u d. Fur- − − expression of modes vω close to |x |= , i.e., u = d in ther, all the null rays emanating from J region u < ui also M experience a change of geometry, i.e., they start moving in terms of uω. We obtain the Bogoliubov coefficients readily flat spacetime inwards, get reflected at the origin (r = 0) as and then encounter the collapsing shell to feel the geometry 0 − ∗ changed into a Schwarzschild one. αω =−2i dx u∂−uω, This can be more clearly seen in an isotropic co-ordinate −∞ system (see Fig. 6) which uses a Cartesian x coordinate with 0 −∞ < x < ∞ instead of the usual radial coordinate r − βω = 2i dx u∂−uω. (7) with 0 < r < ∞. The relevant Penrose diagram is shown in Fig. 6. In these coordinates, (see Fig. 6) there will be two past −∞ (J −) (J −) null infinities, namely left L and right R ,aswellas However, as we discussed this integration has to be truncated J + J + < two future null infinities L and R . The location of event to within the region u d, which gives it a Rindler kind of horizon in these co-ordinates is marked to (say) M , which appearance, making it [1,3,7] corresponds to r = 2M in Schwarzschild co-ordinates. It is 1 π usual to define the modes as left moving or right moving in αω = exp − πκ ω κ these co-ordinates. A right moving mode originates from J 2 2 + L and ends up on J and vice versa for the left-moving modes. i ω i R × exp [i( − ω)d] exp log − , Due to spherical symmetry, consideration of any set of past κ C κ and future asymptotic observer pairs will be equivalent to 1 π βω =− exp − any other. A time-like observer (thick green curves in Fig. 6), 2πκ ω 2κ similarly starting from past time like infinity and escaping the + + i ω i black hole region, either end up on i or on i depending × [i( + ω)d] − . L R exp exp κ log κ (8) upon whether the observer moves leftwards or rightwards. C Let us consider a set of modes moving rightwards. Any where C = C1C2 is a product of affine parameters for in- J − J + κ = / null ray originating from L and ending up on R will expe- going (C1) and out-going (C2) null geodesics with 1 4M rience a change of geometry after it crosses the collapsing being the surface gravity associated with the black hole hori- J + null shell. For observers on R , the isotropic co-ordinate zon. Subsequent computation of the expectation value of den- − |x |=M marks the location of event horizon. Therefore, sitized number operator in the ‘in’ vacuum yields, 123 433 Page 6 of 14 Eur. Phys. J. C (2018) 78 :433 + − | | = |β |2 = 1 x ∈ (0, ∞), and x ∈ (−∞, 0). (11) in 0 N 0 in ω π (9) exp( 2 ) − 1 ω κ For the matter fields, the classical solutions are those in + − which is definitely thermal. Along identical lines one can which, fi = fi+(x )+ fi−(x ). Given some particular mat- immediately verify that the higher order correlations will also ter fields one can obtain corresponding solutions for φ and be thermal in nature and justifying the thermal ambiance of ρ respectively from the equations of motion. A simple static −2ρ −2φ 2 + − black holes. From the rather general nature of the analysis we solution corresponds to e = e = (M/λ) − λ x x , expect this result to give the Bogoliubov coefficients for the representing a black hole of mass M, with a line element timelike geodesic observer stationary at r = 0 if we replace d dx+dx− by ui . Such an observer is engulfed by a true singularity when ds2 =− . (12) = M 2 + − the shell collapse to r 0, it results in geodesic incomplete- λ − λ x x ness, something we will come back to, in a subsequent work. In the limit M → 0, the portion being denied causally, In absence of the mass, M = 0 and we obtain linear dilaton becomes small. Further the geometry change as well as the vacuum line element from Eq. (12). formation of the singularity does not occur, making J + a We consider, for the collapsing scenario, a simplistic case Cauchy surface, which is not the case when M = 0. Since where only one scalar field f is present. The matter moving the effect of causal denial is intimately tied with the effect leftwards collapses to form a black hole [13,23]. If the matter + + of geometry change, it is difficult to account for these effects distribution starts at xi and extends up to x f , then the line individually in a general case in 1 + 3 dimension. Neverthe- element corresponding to this matter configuration turns out less, one can argue that the “tracing over” of modes, which to be, are causally not accessible, alone leads to Unruh effect with + − zero flux, whereas the geometry change makes the thermal 2 dx dx ds =− + , (13) radiation more ‘real’ with a non-zero flux [19Ð22]. The non M(x ) 2 + − + + + λ − λ x x − P (x )x vanishing of flux can be associated with moving of the geometry away from being a flat one. We will see below that these where the functions M(x+) and P+(x+) correspond to the + two effects can nicely be segregated in a 1 1 dimensional integrals, collapse model in dilaton gravity. x+ + + + + M(x ) = dy y T++(y ), (14) 4 (1 + 1)-dimensional dilatonic black hole + xi x+ The CGHS black hole [5,13]isa1+ 1 dimensional gravita- +( +) = + ( +). tional collapse model of a dilatonic field φ interacting with P x dy T++ y (15) gravity in the presence of cosmological constant λ and matter + xi fields fi , described by the action, The matter field satisfies the null energy condition and hence + the quantity defining the mass M(x ) remains positive semi- √ + 1 2 −2φ 2 2 ≡ A = d x −g e R + 4(∇φ) + 4λ definite. The region outside x f is a black hole of mass M 2π + −ρ M(x ) e = f [5]. There is a curvature singularity at 0. The N 1 singularity hides behind an event horizon (which is located at 2 − + 2 − (∇ fi ) . (10) x =−P /λ ) for future null observers receiving the out- 2 i=1 moving radiation. The location of the event horizon can be obtained starting from the location of the apparent horizon, This action, when compared with the standard Hilbert Ein- see Fig. 7. This can be obtained using ∂+ A ≤ 0, where stein action, can be viewed as written in a conformally related A stands for the transverse area of the horizon in a 3 + 1 frame. In standard Einstein frame, obtained through confor- dimensional setting. Borrowing this idea to 1 + 1 dimension, mal transformation, the dilaton field makes appearance as a this equality would lead to the location of the event horizon canonical scalar field. Since all two dimensional space-times − =− +/λ2 + > + at x P , in the out region, i.e., after x x f . are conformally flat the metric ansatz will involve a single Thermodynamics as well as Hawking evaporation of such ρ unknown function, the conformal factor , which is written black hole solutions have been extensively studied [5,23Ð27]. in double null coordinates as, ± J − We introduce a co-ordinate set z suited for the in-region L ρ + − ± ±λ ± ds2 =−e2 dx dx , with, ± λx = e z , (16) 123 Eur. Phys. J. C (2018) 78 :433 Page 7 of 14 433

∞ † ∗ f = dω aωuω + aωuω ,(in) (21) 0 ∞ † ∗ ˆ ˆ† ∗ = dω bωvω + bωvω + bωvˆω + bωvˆω ,(out) (22) 0

† where aω corresponds to creation operator appropriate for the † ˆ† ‘in’ region. Similarly bω and bω stand for the creation operators for the ‘out’ region and the black hole interior region ∗ respectively. The inner product between v and uω corre- Fig. 7 Penrose diagram for a CGHS black hole sponds to,

− − zi zi J − − ∈ (−∞, ∞) which maps the entire L into z .Wealso i − ∗ 1 ω − J + σ ± ∈ αω =− dz v∂−uω = dz introduce another co-ordinate system suited for R as out π 2π (−∞, ∞) where the transformation between (z+, z−) and −∞ −∞ + − + (σ ,σ ) is given by: i − P out out × e−λz − + iωz− exp λ ln λ + (−ω)/λ + + − 1 −λσ − P ω P+ i z = σ ; z =− ln e out + . (17) = 1 out λ λ πλ λ 2 i iω i − + 2 × B − + , + , The horizon located at x =−P /λ , will get mapped λ λ 1 λ (23) − = − =−1 ( +/λ) to z zi λ log P in these co-ordinates. ‘In-’ J − state modes are defined on the asymptotically flat region L while the inner product between v and uω gives J + moving towards R and the convenient basis modes defined − − z z correspond to, i i i − 1 ω − βω = dz v∂−uω = dz π 2π = √1 −iωz− , −∞ −∞ uω e (18) + 2ω i − P × e−λz − − iωz− exp λ ln λ + where ω>0. The ‘out’ region corresponds to J which + i(+ω)/λ − R 1 ω P receives the state from J after the black hole has formed. = L + 2πλ λ The basis modes in the out region at J are R i iω i ×B − − , + , λ λ 1 λ (24) − − 1 −iωσ vω(σ ) = √ e out ; (19) out 2ω with B(x, y) being the Beta function. Therefore, we can ver- − − 1 −iωσ (z−) − − ify that the late time right moving observers (such a observer vω(z ) = √ e out (z − z ), (20) 2ω i is depicted by the thick magenta curve in Fig. 7) do obtain a thermal spectrum [5] with a temperature λ/2π, such that the where is the usual step function marking the fact that the expectation value of the densitized number operator in the J − ‘in’ vacuum becomes, out modes are supported by states on L between the region (−∞, −) zi only. These mode functions provide a complete 2 1 + − | | = |βω| = , J σ in 0 N 0 in 2π (25) orthonormal basis on R in terms of out, however. Again exp( ) − 1 J − J + ω λ the field can be specified fully on L or jointly on R and the event horizon HR. Since the mode functions at HR cor- which is certainly thermal. Similarly the higher correlations respond to part of the field falling into the singularity and are also thermal with identical temperature. In two dimen- J + λ −1 such interior modes cannot be observed by observers at R , sions, the parameter has dimension L and is a legitimate they need to be traced over. Thus, the precise form of mode quantity to set the scale of the frequencies of emission. Fur- decomposition on HR does not affect physical results for ther, λ also turns out to be the surface gravity of the black J + R . Therefore, we can expand the dilaton field in different hole. Interestingly, the effect of mass of the black hole can mode basis as, be eliminated in the redefinition of the co-ordinates in the 123 433 Page 8 of 14 Eur. Phys. J. C (2018) 78 :433

+ 1 − 1 x =− ; x =− , (26) λy+ λy−

can be written as dy+dy− ds2 =− . (27) −λ2 y+ y− The singularity curve originates from the co-ordinate of the + point of matter introduction, i.e., from xi which is marked + =−/λ + through Eq. (26)asyi 1 xi . Under another set of J − co-ordinate transformations, the metric on R can also be J − brought into flat form. On R we adopt Fig. 8 Left moving modes in a CGHS black hole + e−λχ =−y+, λχ− − e = y , (28) region exterior to the black hole [5]. As a result, M does not such that the metric becomes explicitly appear in the expression for the temperature, which + − ds2 =−dχ dχ , (29) is essentially related to the surface gravity. Further, all references to the matter content which formed with χ ± ∈ (−∞, ∞). Given the spacetime metric as in Eq. the black hole, appears only in the phase [see Eq. (23) and (29) we can construct a co-ordinate system {x ≡ (T, X)} + Eq. (24)], through P and gets wiped out when we take the + − χ = T + X modulus. If we follow a null ray originating from J or a L − timelike trajectory and moving rightwards, it suffers a change χ = T − X (30) of geometry once it crosses the collapsing null shell, as in to write Eq. (29)as the case for Schwarzschild hole formation. Therefore, the experiences of these observers are inclusive of both tracing ds2 =−dT2 + dX2 (31) over and the geometry change. Hence, as associated with a where, T plays the role of inertial time coordinate in this black hole formation, there is an associated flux of radiation spacetime. The left moving inertial observer is given by the as measured by the observers attached to such trajectories trajectory [5,23]. However, this model being asymmetric under left- right exchange, provides an opportunity of studying the effect dT = γ ; dX =−vγ, of tracing over separately, if we consider the left-moving dτ dτ modes, which we do next. where v is the inertial velocity and γ is the corresponding Lorentz factor. It is clear that in flat spacetime such an observer will have zero two-acceleration. We note that an inertial observer sitting at X =−a, with a 1 will have a J − J + trajectory closely hugging L and L . 5 Dynamically generated Unruh effect for inertial Therefore the complete set of left-moving mode functions + observers in CGHS black hole spacetime corresponding to the field f+(χ ) can be written in these co- ordinates as The CGHS model is not symmetric under left-right exchange, − + + 1 −iωχ+ and hence the experiences of null ray originating from J or uω (χ ) = √ e . (32) R 2ω a timelike trajectory moving leftwards, will be different from J + what we discussed above. Such trajectories do not suffer any Whereas on L we adopt to change in geometry in their course, hence, as we will see, the + −λχ˜ =−( + − +), only effect a late time observer (the dotted curve in Fig. 8)on e y yi + λχ˜ − − J finds, is rooted only in the tracing over a part of Cauchy e = y , (33) L − surface J . Therefore, these observers will also obtain a R χ˜ ± ∈ (−∞, ∞) thermal expectation value in Eq. (25) as we will see, but such that these new co-ordinates range in + < + there will be no associated flux for these thermal spectrum. in the region x xi . The metric in these new co-ordinates + The spacetime metric in the region x+ < x is given as flat, becomes i + − given by Eq. (13) with M, P+ → 0,whichonusingthe dχ˜ dχ˜ ds2 =− . (34) co-ordinates − + λχ˜ + 1 yi e 123 Eur. Phys. J. C (2018) 78 :433 Page 9 of 14 433

Given the metric in the (χ˜ +, χ˜ −) coordinates, one can simply solve the field equation for the minimally coupled massless scalar field in these co-ordinates using conformal invariance of the matter field in two dimensions. The resulting complete + set of left-moving modes corresponding to the field f+(χ˜ ) are again given as

+ + 1 −iωχ˜ + vω (χ˜ ) = √ e . (35) 2ω Notably, the timelike observers corresponding to these null co-ordinates χ˜ ± = T˜ ± X˜ will find the metric to be dynamic, but will be well suited observers to define positive frequency Fig. 9 Unavailability of Cauchy surface for left-moving modes modes of the field. Thus, the relevant questions to ask to the inertial observers are how many particle modes (i.e. e−iωT + excitations) were there at remote past and how many positive time response, i.e., the observers reaching iL , we need to take − ω ˜ (ω/λ ) frequency particle modes (i.e. e i T ) excitations are present the high frequency behaviour 1 of the Bogoliubov at late times? coefficients J −,v+ Clearly, on R ω has support only in the region + + + + + 1 ω + i(−ω) π x < x . The point x is mapped to χ ≡ χ = α −→ | | λ i i i ω yi exp 1 + 2πλ 2λ − λ log (−y ). Therefore, the Bogoliubov transformation i ω coefficients between these two set of observers can be × −i i ; exp λ log λ λ obtained exactly as in Eqs. (23) and (24) but with the replace- | +|↔ +/λ ment yi P as 1 ω + i(+ω) π βω −→ − |y | λ exp − 2πλ i 2λ χ+ χ+ i i i ω i i + ∗ 1 ω + × exp − log , (38) αω =− dχ v∂−uω = dχ λ λ λ π 2π −∞ −∞ which are (upto overall phases) exactly the same as for the i + + × e−λχ −|y | + iωχ+ exp λ ln i Rindler observer Eq. (2). The overall phase factors are not really important for identifying the field content, as seen in 1 ω + i(−ω) = |y | λ Eq. (4). Therefore, just like the Rindler observer, the left- πλ i 2 moving observer at high frequencies, at late times sees the i iω i × B − + , + , ‘in’ vacuum to be thermally occupied λ λ 1 λ (36) | | = |β |2 = 1 . in 0 N 0 in ω 2π (39) λ − while ω e 1

χ+ χ+ Again, as before, the thermal ambiance of such an observer i i i + 1 ω + can be verified through correlations among the various modes βω = dχ v∂+uω = dχ π 2π which also turn out to be thermal in these settings, −∞ −∞ + 2 2 i −λχ + + in0|N N |0in = |β ω| |β ω | × e −|y | − iωχ 1 2 1 2 exp λ ln i ω ω 1 1 ω (+ω) = . 1 + i π π (40) = |y | λ 2 1 2 2 πλ i e λ − 1 e λ − 1 2 i iω i Similarly, any observable can be evaluated in the ‘in’ vacuum × B − − , 1 + . (37) λ λ λ using these Bogoliubov coefficients and the result will turn out to be as for a thermal state. Thus, the experiences of the This form of the Bogoliubov coefficients results from the geodesic observer start resembling that of a Rindler observer J + integration of the positive frequency modes over L ,having and the observer finds the field content to be thermal (albeit J − λ/ π support in the past only on a portion of R . The horizon at a temperature 2 ). J + + = + for L is given as y yi , which also marks the point of This effect can be more clearly understood using non- singularity. More importantly, if one is interested in the late availability of Cauchy surfaces for future asymptotic obser- 123 433 Page 10 of 14 Eur. Phys. J. C (2018) 78 :433

The spacetime, they move in, throughout, is flat and such observers do not receive any flux of radiation, as the vacuum expectation value of the stress energy tensor which was van- J − + < + ishing on R , stays put on zero, in the region y yi .As we discussed previously Eq. (25), the right moving observers tuned to right moving modes, witness a thermal radiation flux at a temperature 1/λ which is independent of the matter content of the forming black hole and depends only on the other dimensionful parameter in the theory. J + Following an identical path, the observers at L coupled to left-moving modes, observe a similar kind of Bogoliubov Fig. 10 Freely falling observers in CGHS black hole spacetime coefficients as their right-moving counterparts Eqs. (23), | +|↔ +/λ (24) but with the parameter exchange yi P which mark the corresponding event horizons for such observers vers. For that purpose let us consider a left moving timelike and appear as overall phases in the transformation coeffi- or null trajectory. Due to conformal flatness, a complete set cients. Hence, we see that such form of Bogoliubov coef- of orthonormal mode functions on any surface orthogonal to ficients Eqs. (36), (37) are entirely due to tracing over of them, can always be obtained as plane waves under a proper modes which lie in the causally denied region of spacetime, (−∞, ∞) I+ co-ordinatization (which extends as ) of that sur- to L and not due to any geometry change. One can check face. If we consider an orthogonal null surface for the left that if the fraction of tracing over vanishes, which is marked | +|→ moving modes, before the formation of singularity (the red by yi 0, the Bogoliubov coefficients assume a trivial surface labeled ‘2’ in Fig. 9), the Bogoliubov transformation form, i.e., αω → δ( − ω) while βω → 0. Similarly, the J − coefficients between that surface and R will be trivial, i.e., Bogoliubov coefficients for the right-moving observers Eqs. + (αω = δ(− ω), βω = 0). However, once the singularity (23), (24), assume a trivial form in the limit P → 0, mark- J − forms, any null surface relating a point on L to the sin- ing the vanishing of event horizon, as well as, the amount gularity (e.g., green surface labeled ‘1’ in Fig. 9)), fails to of change of geometry suffered by such observers. Hence, causally connect with the entire spacetime. Also, any time- for such observers effects of tracing over is indistinguish- like observer has a domain of dependence, which is not the able from geometry change. Both these effects vanish simul- J − full spacetime. A portion of R (right to the dashed green taneously in that limit. Therefore, we see that the vacuum line) is causally denied to the past of such surfaces. Therefore, response for both left-moving or right-moving observers is the Bogoliubov coefficients assume a non-trivial form, such indistinguishable from each other. Late time environment for J + J + as in Eqs. (23), (24)orEqs.(36), (37), with a phase factor such observers on L or on R for vacuum state (of a test capturing the information about the region of the traced over field) is thermal. modes. This is dynamically similar to a Rindler trajectory However, unlike the left-moving observers, the right mov- in a Minkowski spacetime, see Fig. 2, where an accelerated ing observers also associate a flux with the radiation and observer (thick green curve) reaches the future null infinity hence the black hole region shrinks as a result of the evap- rather than future timelike infinity. For such observers too, oration. While left-moving observers do not associate any the domain of dependence is only a part of the full space- mass to the region beyond their horizon, the location of their time. There is a horizon masking a portion of spacetime and horizon does not shrink and the “black hole region” does not hence the Bogoliubov transformation coefficients between evaporate for them. Classically speaking it creates a casual J − J + J + R and L (rather than the full L ) is non-trivial [3,6,8] dilemma for the left moving inertial observers since they and the accelerated observers obtain a thermal spectrum for a can not associate any source to the radiation they observe. J − vacuum state defined on R . However, there is no flux asso- Such observers are oblivious to an event horizon formed due ciated with this spectrum as the vacuum expectation value of to collapsing matter, from which they are unable to get any the stress tensor vanishes identically in the flat spacetime. causal communication; nor does the geometry reveal such a Similarly in the CGHS model, the region of spacetime development. However, quantum correlations of the global + > + y yi is dynamically made inaccessible to any left- vacuum state seep through the horizon, which these observers moving time-like or null trajectories, which also is the are able to tap (reminiscent of what happens in the famous case with such trajectories in the null collapse forming a EPR paradox). It is these correlations which the left-moving Schwarzschild black hole Fig. 6. However, in contrast to observers attribute, from their point-of-view, as due to an the Schwarzschild case, such observers in the CGHS model, excited state defined on the Hilbert space on their accessible + > + do not see any change of geometry hence they do not asso- region. Had there been no dynamics in the region y yi , ciate any mass to the “black hole region” as seen by them. the full Cauchy surface would have become accessible to 123 Eur. Phys. J. C (2018) 78 :433 Page 11 of 14 433 the left-moving observers (as well), making them standard the vacuum state for the semiclassical equations due to the inertial observers. Thus, the dynamics of the spacetime has anomaly, which we consider somewhat unphysical. There is generated an Unruh (like) effect for a set of inertial observers. no unique way of handling this issue because, as we stressed All the geodesic observers moving rightwards or remain- before, Gab = 0inD = 2 and the left hand side which ing stationary at any finite value (with the exception of sta- we work with to mimic Gab =Tab depends crucially on tionary observer at left infinity), on the other hand, end up the model we use, with the hope that it can mimic aspects in singularity and in this process have to undergo a geom- of D = 4 gravity. We have chosen simple, physically well- etry change (thick cyan curves in Fig. 10), so they witness motivated choices to do this as we describe next. a combined effect and hence a flux of radiation, as in the To analyze the right hand side, we need a scheme for defin- Schwarzschild scenario [17,18,28]. Whereas the left-moving ing a c−number stress tensor from the quantum operator and + timelike observers end up at iL , do not see any geometry the vacuum state has to be motivated from some specific (geo- change but encounter flux-free thermal atmosphere due to a metric) considerations. We have defined the vacuum state by dynamical emergence of a horizon, which washes out infor- a natural assumption: viz. that the geometry must remain flat mation of a section of initial field configuration, exactly in when sourced by such a (vacuum) state. This is important the spirit of the Rindler observer, as discussed previously. because as we discussed, in two-dimensional spacetime, the Finally, we will comment on the back-reaction due to expectation value of the stress energy tensor has a conformal black hole evaporation and its implications for our result. anomaly term which can violate this criterion in general, if To compute any kind of back-reaction in the spacetime in this term is not accounted for in the source terms in the right Einstein gravity, one needs to use an equation of the form hand side. We will choose the state of the matter field such = + Gab Tab . Neither side of this equation can be handled that one recovers the classical flat geometry prior to xi ,in without additional assumptions and, obviously, the result will order to remain true to the classical consideration. Therefore, 2 + − depend on the additional assumptions! The left-hand-side we require exp(ρ) = exp(ρflat) = 1/λ x x in the region = + < + identically vanishes in D 2 because Einstein tensor iden- x xi . Thus, the classical values of the T±± are being real- tically vanishes in two dimensions. The right hand side is: ized by T±±. We can further choose the boundary condi- (1) divergent and needs to be regularized and (2) depends tions for the set of initial states judiciously such that the con- on the quantum state which is chosen. The best we can do, tribution due to conformal anomaly exactly cancels out in the therefore, is to give prescriptions to define both sides and the region of interest leading to a flat spacetime semi-classically, results will obviously depend on these prescriptions.Wedis- see for instance [29Ð31]. For this judicious choice of fam- + < + cuss our set of choices while describing the pertinent issues ily of quantum states [32], the spacetime region x xi in two-dimensional gravity below. remains unaffected by the conformal anomaly term. There- To get a non-zero left hand side, one postulates some non- fore the results discussed in this work, use stability of the Einsteinian form of the gravity action (like the CGHS action solution under these class of states. A more detailed analysis in Eq. (10)) and vary the metric to get the equations of motion. of conformal anomaly modified evaporating black holes can Obviously, this is not Gab (which, of course, is zero) but can be found in [5]. Our scheme to deal with such terms will be possibly act as a proxy for the same. We stress the fact that we reported in details elsewhere [33]. are trying to model some gravitational features by some suitable dilatonic action and one cannot ignore the implicit ad- hocness in the procedure. For example, in such an approach, 6 FAQ: demystifying some aspects of the analysis one also often takes the the normal ordered, non-covariant, expectation value of the stress energy tensor as the classi- In this section we provide answers to some FAQ, which will cal source. However, in a two-dimensional spacetime, the help the reader to understand the content of this work from expectation value of the stress energy tensor comes with a a better perspective. conformal anomaly term [5] as well. The anomaly term can then act as an extra source of stress energy and may lead to an 1. What does the non-trivial Bogoliubov coefficients for the evolution different from the classical case, if not accounted left patch signify ? for properly. A prescription used in the literature is to add The Bogoliubov coefficients essentially signify the “par- some extra terms (e.g., Polyakov action, RST action terms) ticle content” of a field configuration as measured by to the standard CGHS action corresponding to the confor- suitable annihilation and creation operators between two mal anomaly terms and obtain these modified equations of sets of observers who are entitled to declare positive fre- motion, as a result of the variational principle. But this new quency excitations of the field. (In general, this “particle action will correspond to a different physical system, though content will be quite different from the particle content one which can also be thought of as a proxy for gravity. But determined by detectors; for a discussion see e.g. [34]). in this prescription, flat spacetime will not be a solution to If two set of observers are connected by trivial Bogoli- 123 433 Page 12 of 14 Eur. Phys. J. C (2018) 78 :433

ubov coefficients, they must agree on the field content, trivial Bogoliubov coefficients of a Rindler observer, one and, in particular, the notion of the vacuum state. In the can not object that the Unruh radiation lets the observer case we are interested in, the Bogoliubov coefficients are know what it will be doing in the future (i.e., observer non-trivial for the left moving observer and hence (s)he will be eternally accelerating, making her incapable of will ascribe the ‘in’ vacuum state of the quantum field for changing her mind of whether to accelerate or not) in the spacetime to be filled by particles as determined by violation of causality, one can not object to causality in the expectation value of a number operator. Notably, such this set-up as well. observers may well be left moving inertial observers. 5. Then how should one interpret these Bogoliubov coeffi- 2. Will such observers, therefore find a swarm of particles cients? hitting them ? In the same spirit as the Bogoliubov coefficients in the The answer, as in the case of Unruh effect, is no. Since the case of Rindler observer. For example if we consider an left moving observer lives in a static patch of the space- accelerated observer in the right wedge of the Minkowski time, having time reversal symmetry, there is no outgoing spacetime, that observer will be oblivious to the field con- flux for the relevant observers. These observers will find figuration in the left wedge. Still if the field configuration the state of the field to be just thermally populated. This is distorted only in the left wedge, it will carry imprints of is very much analogous to the experience of the Rindler that in the right wedge too, without breaking any causal- observer, witnessing Unruh effect, but for the fact that ity. The crucial thing is that no information can propagate the left moving observers in this CGHS spacetime are from left to the right wedge. inertial observers. 6. How does the collapsing shell affect the left moving 3. There are other inertial observers too, such as asymptotic observers then? observers in Schwarzschild spacetime, r = 0 observer in The collapsing shell ensures that the left moving observers de Sitter. What is so special in this particular observa- do not have infinite proper time in the future. The global tion? geometry is modified due the collapse of the null shell, Indeed there are other observers who do witness ther- making the left moving observers trajectory only finitely mal radiations, in one way or another, despite being iner- extendible in the future. Therefore, the mode-functions tial. However, they are mostly in a spacetime with non- which provided a complete basis in past do not remain vanishing curvature. Any exploration of the spacetime relevant in the future and one needs to find mode func- region in the vicinity of their location will reveal the non- tions which provide a complete orthonormal basis for left zero curvature to them. On the other hand, the observers moving observers in the future. Therefore the vacuum discussed in this paper, live completely in a flat space- viewed in the light of new orthonormal basis appears as time region throughout their lifetime. They do not have if it is excited. Again to stress the central point, the quan- any spacetime curvature or acceleration to explain their tum correlations between events in the future and past of experiences. the shell remain non-zero, but their commutator vanishes, thus there is no causal violation. 4. The non-trivial Bogoliubov coefficients tell us that these observers declare the initial vacuum as excited state. This 7. What are the requirements which go into deriving the happens because of a null shell collapsing in their causal form of mode functions suited for future observers? future. Is this not in violation of causality? For future observers, one needs to get a complete basis in The single most important thing to notice is that, the the region accessible to them (just like Rindler observers, notion of Bogoliubov coefficients is global rather than or for late time observers in black hole case for that mat- local. In order to evaluate the Bogoliubov coefficients, ter). Further, we require a complete range of co-ordinates one must have a knowledge regarding the full global over which the variable characterizing the plane wave geometry beforehand. So it is premature to contest runs, in order to get a complete basis (i.e., eikx provides causality from the non-triviality of the Bogoliubov coef- a complete basis only if x ∈ (−∞, ∞). Given this fact, ficients. Causality in quantum field theory is protected by one can decompose the positive frequency modes in the the commutation relations, which in this context, across future in terms of positive frequency and negative energy the horizon, still vanishes. So based on that one can con- modes in the past to obtain the Bogoliubov coefficients clude that there is no breakdown of causality. The left between them. In the co-ordinate system which yield a moving observers just have access to (or get a signal complete basis, we also have a positive frequency mode- from) the field configuration in a portion of the space- function in terms of co-ordinate time (just like e.g., early time, which remains insufficient for them to conclude the field to be in the vacuum state. Just like based on non-

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time mode function during inflation). Though this co- of the vacuum state defined on their Hilbert space would have ordinate time is not a Killing one. had different signatures. This thermality she gets is not through any geometry 8. What does the reduced density matrix for the left moving change, or through any artifact of her non-inertial motion observer look like? but through quantum correlations of the ambient field. How- Again, since the left moving observers are totally igno- ever by no means is this a case of causality violation any more rant about the field configuration in the region which than there is causality violation in the EPR situation, with the is future to the shell, they will have to trace over those counterpart of the thermal excitations lying beyond the hori- configurations to obtain a reduced density matrix. How- zon. Only a pair of observers across the horizon will have the ever, the Bogoliubov coefficients are of the form that complete information about the vacuum structure, not any they pronounce thermality only for high frequencies (like one of them individually. This is, of course, in stark contrast in Hawking radiation). For low frequencies, i.e at long to the right moving observers, who, knowing the field equa- wavelengths, one is able to see that the region explored tions of dilaton gravity, will be able to associate a change of by left moving observers is different from an eternally geometry and hence interpret it as the source, essentially as accelerated observer, leading to departure from thermal- the phenomenon of black hole evaporation with no apparent ity. The reduced density matrix is also expected to bear surprise. these features. Thus, we have shown quantum correlations as a probe of 9. Can such effects be present in other dimensions ? maximal extendibility of the spacetime. Any probe of such Our result depends essentially on chirality of the dynam- quantum correlations will reveal the differences between the ics in 2-dimensional dilaton gravity. In D = 2wehavea vacuum of a maximally extended and maximally extensi- solution in which observers moving in one direction do ble spacetime, which remain hidden classically. In addition not feel the presence of black hole which has formed as to these correlations, if some classical channels [35,36]are a result of a collapse. Currently, we do not know of any also present, we can in principle utilize them to probe the other solutions in any theory of gravity which has such geometry of causally inaccessible regions more effectively a feature. Evidently, due to the existence of standard no- in higher dimensions as well. We will pursue these issues in hair theorems, Einstein gravity will not lead to such an a subsequent work. asymmetric black hole configuration. However, whether any other theory of gravity can lead to such a solution Acknowledgements The authors thank Alessandro Fabbri for useful in higher dimensions, is a question which we hope to comments on the preprint version of the paper. Research of KL is supported by INSPIRE Faculty Fellowship grant by Department of Science address in a future study. and Technology, Government of India. Research of SC is funded by a SERB-NPDF Grant (PDF/2016/001589) from SERB, Government of India. The research of TP is partially supported by the J.C. Bose research 7 Concluding remarks grant of the Department of Science and Technology, Government of India.

In this work, we have shown that pure quantum correlations Open Access This article is distributed under the terms of the Creative can play the role of a source of a thermal bath in a flat region of Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, spacetime. There is no classical source otherwise, to describe and reproduction in any medium, provided you give appropriate credit the thermal ambiance of the quantum field, which a section to the original author(s) and the source, provide a link to the Creative of inertial observers in the spacetime and into. The inertial Commons license, and indicate if changes were made. 3 observers in the flat region are unaware of an event horizon Funded by SCOAP . formation due to the collapsing matter, which lies to their causal future. Therefore, unlike their right moving counterparts, they are unable to witness any black hole formation, and do not have a explanation to their thermal environment References other than describing the state to be a non-vacuum one. It is well known that, quantum correlations (e.g., of the global 1. S. Hawking, Particle creation by black holes. Commun. Math. Phys. vacuum state) do not vanish beyond the light cone and it is 43, 199Ð220 (1975). https://doi.org/10.1007/BF02345020 2. S. Hawking, Black hole explosions. Nature 248, 30Ð31 (1974). these fluctuations, which these observers are actually detect- https://doi.org/10.1038/248030a0 ing. It is the partially traced over quantum correlations (due to 3. N. Birrell, P. Davies, Quantum Fields in Curved Space.Cam- presence of an horizon the existence of which these observers bridge Monographs on Mathematical Physics (Cambridge Univer- do not know), which the left moving observers falsely asso- sity Press, Cambridge, 1982) 4. A.D. Helfer, Do black holes radiate? Rept. Prog. Phys. 66, ciate with an excited state, defined on the Hilbert space on 943Ð1008 (2003). https://doi.org/10.1088/0034-4885/66/6/202. their accessible region, since clearly the quantum fluctuations arXiv:gr-qc/0304042 [gr-qc] 123 433 Page 14 of 14 Eur. Phys. J. C (2018) 78 :433

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