The Hawking effect in the particles-partners correlations

We analyze the correlations functions across the horizon in Hawking black hole radiation to reveal the correlations between Hawking particles and their partners. The effects of the underlying space-time on this are shown in various examples ranging from acoustic black holes to regular black holes.


I. INTRODUCTION
The 1974 prediction by Hawking [1] of a quantum thermal emission by black holes (BHs) is a milestone of modern theoretical physics.Being the associated temperature extremely tiny (of order 10 −8 K for a solar mass BH) no experimental evidence of this remarkable result has been given so far.Nowadays the only indication that this phenomenon can indeed exist in Nature comes surprisingly from condensed matter physics.An analog of a BH created by a Bose-Einstein condensate (BEC) undergoing a transition from a subsonic flow to a supersonic one has shown [2][3][4], in the density-density correlation function across the horizon, a characteristic peak [5,6] which is consistent with a pair-creation mechanism of a Hawking particle outside the horizon and its entangled partner inside.This has stimulated a large interest in studying quantum correlations across the horizon in relation to the Hawking effect.
In this paper we shall consider various BH metrics and analyze if and where this kind of signal does indeed appear.We shall simplify the mathematical treatment by considering two-dimensional (2D) spacetimes (some of them can be considered as the time-radial section of spherically symmetric 4D spacetimes) and a massless scalar quantum field propagating on them.We shall focus on a particular component of the quantum energy momentum tensor of the scalar field which is relevant to reveal the presence of the Hawking effect.

II. THE SETTING
The 2D BH metrics we shall consider are stationary and can be cast in the Eddington-Finkelstein (EF) form where v is a null advanced coordinate.The horizon r H corresponds to f (r H ) = 0.A retarded coordinate u can be introduced by where the Regge-Wheeler coordinate r * is One can write the metric in the double-null form where r is a function of u and v defined implicitly by We consider a massless scalar field φ(x) propagating in the spacetime of eq.(2.1) (or where □ ≡ ∇ µ ∇ µ is the covariant d'Alembertian and x is a generic space-time point.The energy momentum operator associated to φ reads The (across the horizon) correlator of this operator we will study is [7] G where r > r H and r ′ < r H .
In the double null coordinate system of eq.(2.4) The quantum state |U ⟩ in which the expectation value in eq.(2.8) is taken is the Unruh state [8].This state is defined by expanding the field φ in a base where U is Kruskal coordinate defined which, unlike u, is regular on the future horizon.In eq.(2.11) κ is the surface gravity of the horizon and − holds outside the horizon, while + inside.The state |U ⟩ describes the state of the quantum field at late retarded time u after the BH is formed and is the relevant one to discuss the Hawking BH evaporation in this limit.
The correlator eq.(2.8) is quite general.For a gravitational black hole it is related to the energy density correlator measured by geodesic observers.Its square root appears in acoustic BH giving the density-density correlator [5,6] in a BEC under the hydrodynamical approximation. 1 The two-point function for the u sector of the state |U ⟩ is from which one easily gets where This function is expected to display the correlations of the particle-partner pairs.The two, when on-shell, propagate along u = const trajectories.As can be seen from eq. (2.14) the cosh −2 term has indeed a maximum along the null trajectories u = u ′ confirming the expectations.There are however also the "geometrical" prefactors f −2 that can mask the above behaviour.

III. ACOUSTIC BH MODEL
Acoustic BH metrics are mostly characterized by a single sonic point separating the subsonic from the supersonic region of the flow.Both regions are asymptotically homogeneous.
A profile mathematically simple enough to manipulate and sufficiently representative is given by a metric for which the conformal factor reads where −∞ < r < +∞.It has an horizon at r = 0 and κ is its surface gravity.
The subsonic region is r > 0 while the supersonic one is at r < 0. In the acoustic language the profile would correspond to a flow with velocity V (r) such that and from eq. (2.2) The condition for the maximum of the cosh −2 term in G(x, x ′ ) at equal v is (see eq. (2.15)) where r > 0 and r ′ < 0, and is plotted in Fig. (2).The peak appears at u = u ′ | v=v ′ only for points r, |r ′ | ≳ 1 κ .For points located closer to the horizon no peak appears [9], the maximum disappears and it merges in the light-cone singularity at coincidence points (i.e.r = |r ′ | = 0).This behaviour corroborates the idea that the Hawking particle and its corresponding partner emerge on shell out of a region of nonvanishing extension across the horizon called "quantum atmosphere" [10][11][12].In this case it has an extension of order 1 κ .Inside this quantum atmosphere vacuum polarization and Hawking radiation are comparable and one cannot disentangle the two.

IV. SCHWARZSCHILD BH
The Schwarzschild BH is characterized by where m is the mass of the BH and r > 0. The horizon is at r h = 2m, its surface gravity is κ = 1 4m , while r = 0 is the physical singularity.In this case and The condition u = u ′ | v=v ′ is shown in Fig. (5) and it reads The 3D plot of the correlator G(r, r ′ ) is given in Fig. (6).
One does not see any structure [13].The expected peak does not show up.This can also be seen from Fig. (7) where G(r, r ′ ) is plotted as a function of r for various values of r ′ .The reason of this negative result is not that correlations between the Hawking particles and their partners do not exist in this case, simply they do not show up in the equal time correlators.

V. THE CGHS BH
We have seen that in a BH in order to see the correlations one has to catch the partner before it reaches the singularity.Here we consider a BH metric where the singularity is pushed at r = −∞.The Callan-Giddings-Harvey-Strominger BH we shall discuss appears as a solution of a 2D dilaton gravity theory (see [14] for details) and its metric reads where m is the mass of the BH and λ is a parameter interpreted as a cosmological constant.
Here −∞ < r < +∞.The horizon is located at and the corresponding surface gravity is κ = λ.The metric has a physical singularity at r = −∞ where the curvature diverges.For this metric where now Here r > r h and r ′ < r h .In Fig. (10) we plot both sides of eq.(5.5) .Note that the value of r corresponding to r ′ → −∞ is (5.6) So r ′ = −∞ is correlated to a point still inside the quantum atmosphere.As shown in Fig. (11), when the correlator G(r, r ′ ) for this metric is plotted, no correlation shows up.The explanation of this negative result relies on the fact that, although the singularity is at r = −∞, it is not "infinitely" far away.The proper distance The CGHS metric shown has therefore the same behaviour we found in the Schwarzschild metric concerning the correlation across the horizon.The partner is swallowed by the singularity before the Hawking particle emerges out of the quantum atmosphere.

VI. SIMPSON-VISSER METRIC
Our last example concerns a BH metric where the singularity has been removed, one has a "regular BH".Among many proposals in the literature for this kind of BHs, we confine our attention to a very simple metric, the so called Simpson-Visser metric [15], for which where a is a parameter, regularizing the singularity, which we choose such that a < 2m.For a = 0 we have the Schwarzschild metric with singularity at r = 0.For a ̸ = 0 the spacetime surface r = 0 is regular and represents a bounce which separates one asymptotically flat Universe (where r > 0) from an identical copy with r < 0. For this metric where − refers to r > r + and + to r < r + ; the other regular on r − where + refers to r > r − and − to r < r − .These κ is the absolute value of the surface gravity, which is the same for both horizons Note that the coordinate U (+) is regular on r + , where U (+) = 0, but is singular on r − , where We can define a Unruh state |U (+) ⟩ expanding the quantum fields in modes like (2.10) where U = U (+) .Similarly, |U (−)⟩ is defined by the expansion (2.10) where now U = U (−) .
The singularities of the coordinates (6.5), (6.6) induce singularities in the corresponding modes, for example e −iωU (+) is singular on r − .However, being the surface gravity the same in absolute value one can show that the quantum stress tensor is the same in both states |U (±) ⟩ and is regular on both horizons (see appendix A).Also the correlator, being an even function of the surface gravity (see eq. (2.14)), is the same in |U (±) ⟩.
The extremal of the cosh 2 term is now given by u = u ′ | v=v ′ which reads 2r + 4M ln r a + r a The corresponding correlator 3 is graphically represented in Fig. (15).One can further appreciate this by examining the correlator at fixed inner point r ′ , as shown in Fig. (16). 2 From Figs. (13a) and ( 14) one sees that the condition (6.8) is always satisfied, but the points with r − r + ≳ 1 κ are correlated with corresponding partners that are piling up close to r − . 3For a preliminary study see [16].17).This is understandable since both the Hawking particle and the partner now pile up along r − remaining well inside the quantum atmosphere where the correlator is dominated by the coincidence limit singularity.

VII. CONCLUSION
Hawking radiation is a genuine pair (particle-partner) production process that is expected to be a general feature of gravitational BHs and of analogue ones realized in condensed matter systems.It is indeed in these latter (and only in them) that the existence of this radiation has been (indirectly) observed.The observation consisted in detecting the correlations across the sonic horizon between the Hawking particles and their entangled partners.
In this paper we have analyzed various examples of BH spacetime metrics to see if and where these correlations show up.Indeed they can be hidden by the geometrical structure of the underlying spacetime, like the presence of singularities or inner horizons as shown explicitly here.We have seen that singularities (without inner horizons) swallow the partners before the corresponding Hawking particles emerge on shell out of the quantum atmosphere obscuring the existing correlations.On the other hand inner horizons produce a piling up of the partners along them enhancing and strongly localizing the correlations.

. 2 )
One see from Fig. (1) that very rapidly the profile becomes flat indicating an homogeneous flow.From eqs. (3.1) and (2.3) we have that in this case

rFIG. 12 :
FIG. 12: Penrose diagram of the part of the Simpson-Visser metric covered by the v, r coordinates.

. 4 )
Fig. (13a) represents the trajectories of a Hawking particle (solid line) -partner (dashed line) pair created near r + , while Fig. (13b) a pair created near r − .All particles propagate along u = const.One can see that in the first case ((13a)), while the Hawking particle