Particle Production in the Interiors of Acoustic Black Holes

Phonon creation inside the horizons of acoustic black holes is investigated using two simple toy models. It is shown that, unlike what occurs in the exterior regions, the spectrum is not thermal. This non-thermality is due to the anomalous scattering that occurs in the interior regions.


I. INTRODUCTION
Among the many spectacular celestial objects that populate our universe, black holes (BHs) are perhaps the most intriguing. The presence of a causal horizon prevents any direct observation of the interior. According to theoretical studies based on General Relativity, the interior is a place where peculiar physical effects occur, which cannot be confirmed by astronomical observations. The situation has positively changed in recent times with the advent of so called "analog BHs" [1, 2]. These are condensed matter systems that are realizable in the laboratory, which mimic some of the essential features of gravitational BHs.
A typical example is a Bose-Einstein Condensate (BEC) fluid (see for example [3]) whose flow becomes supersonic [4][5][6]. The supersonic region, trapping sound waves inside it, is the analog of the BH interior. The sonic surface where the speed of the flow equals the local speed of sound, plays the role of the horizon. This sonic horizon however has no causal significance at all: there is nothing to prevent one from directly observing the interior region.
Indeed the first experimental observations of the analog of Hawking radiation [7] in BECs by Steinhauer et. al. [8,9] were made by performing simultaneous measurements of the density outside and inside the sonic horizon. A peak was observed in the resulting in-out density-density correlation function that was predicted in [10,11] and which is the 'smoking gun' signaling the presence of Hawking radiation. In the same spirit, one can imagine that other processes that are predicted to take place in the interior of a BH can be experimentally verified by looking at appropriate analog models.
With this as motivation, in this paper we discuss the unusual features of scattering by a potential inside the horizon of a stationary BEC analog BH and its consequences. The calculations are done in the analog spacetime using quantum field theory in curved space techniques. These are the same types of calculations that one would do to explore similar effects in the interior of a real black hole.
In Quantum Mechanics in the presence of a potential, an incident flux is split into a transmitted and a reflected part (see Fig.1). Reflection (R) and Transmission (T ) coefficients satisfy the unitary relation |R| 2 + |T | 2 = 1, which is the conservation of probability. Note that the previous relation implies that |R| 2 and |T | 2 ≤ 1.
Inside the horizon of a BH both the transmitted and the "would be reflected" part of the field are forced to propagate in the same direction, namely towards the center of the black Incident T R FIG. 1: Illustration of a plane wave incident onto a potential from the right, which then is partially transmitted to the left and also partial reflected back to the right of the barrier.

Incident T R Direction of Flow
FIG. 2: Illustration of a plane wave incident onto a potential from the right in the interior of a BH, which then is partially transmitted to the left and also partial reflected, however the reflected portion is also moving to the left of the barrier since in the interior the wave is forced to travel further into the BH.
hole (see Fig. 2). The scattering is 'anomalous' and R and T no longer satisfy the previous unitary relation. Instead they satisfy |T | 2 − |R| 2 = 1 which implies particle creation since |T | 2 ≥ 1. Another way to think about this is that, while the outside region of a nonrotating black hole is static, the interior can be thought of as a dynamical cosmology in which particle creation occurs.
We shall deal with both massless and massive quantum fields. For the latter case there is usually a mass gap, namely E ≥ m where E is the conserved (Killing) energy and m is the mass of the particle. Inside a BH the former inequality no longer holds, E can take any value, even negative ones.
In Sec. II a brief review is given of the set-up for BEC analog black holes. In Sec.
III particle production is investigated in the case of massless phonons with a double delta function potential. In Sec. IV particle production is investigated for massive phonons when the effective potential is zero and the mass term in the mode equation is approximated by two step functions. Sec. V contains a discussion of our results and comparisons with some previous work.

II. THE SETTING
Under the hydrodynamic approximation the phase fluctuation operatorφ in a BEC satisfies a covariant version of D'Alembert's wave equation (see for instance [2]) whereˆ =∇ µ∇ ν is evaluated on a fictitious curved spacetime metric, called the acoustic metric, which in our case we write as follows where n is the density of the condensate (here assumed to be constant), m a is the mass of a single atom of the BEC, and c(x) is the sound speed. The flow is assumed to be stationary and one dimensional along the x axis with the velocity v = −v ox constant and directed from right to left.
For a typical profile used in BEC analog models c(x) becomes constant in both asymptotic regions (x → ±∞) so that lim x→+∞ c(x) = c r and lim x→−∞ c(x) = c l with c r > v 0 and c l < v 0 . Thus the asymptotic regions are homogeneous and the profile monotonically decreases from right to left. The profile c(x) is chosen so that the horizon c(x) = v 0 is at x = 0. In the region x < 0, where c(x) < v 0 the metric describes the interior region of the acoustic BH while for x > 0, where c(x) > v 0 , the metric describes the exterior region of the acoustic BH. We call the exterior the r region and the interior the l region. Performing a dimensional reduction along the transverse direction and passing from the Gullstrand-Painlevé coordinates (T, x) to the Schwarzschild like ones (t, x * ) via the transformation (2.3) the wave equation (2.1) can be reduced to where the effective potential is given by (2.5) The coefficient k 2 ⊥ is related to the transverse momentum andφ (2) is the dimensionally reduced field operator (see the appendix of Ref. [12] for details). The last two terms in Eq.
(2.4), the mass-like term and V eff , cause scattering of the modes. Note that both of these terms vanish at the horizon. There the modes are effectively massless and propagate freely.
The creation and annihilation operators,â ω andâ † ω , satisfy the usual commutation relations. The modes f ω are normalized using the conserved scalar product with dΣ µ = n µ dΣ, where Σ is a Cauchy surface, n µ a future directed unit vector perpendicular to Σ, and g Σ the determinant of the induced metric. Writing and substituting into (2.4) gives (2.9) In this paper we consider two toy models for the terms in Eq. (2.9) responsible for the scattering which have the advantage of being exactly solvable while, despite their crudeness, encode all of the basic features of the process we wish to discuss.

III. DIRAC DELTA FUNCTION POTENTIALS
In the first toy model, the transverse excitations are neglected (i.e., k ⊥ = 0) and V eff is approximated by two Dirac delta functions, one in region r and one in region l. For simplicity we choose them at x * r = 0 in r and at x * l = 0 in l leading to 1 The Penrose diagram for the BH metric given in Eq. (2.2) is shown in Fig 3, where the modes representing our 'in' basis are schematically indicated.
The asymptotic behaviors of these modes are on the portion of the past horizon in region r, H r − ; on the portion of the past horizon in region l, H l − . These are positive norm modes on I r − or H − which together form a Cauchy surface for the spacetime. These modes are associated with annihilation operators in the expansion of the fieldφ (2) in Eq. (2.6).
In Eqs (3.2-3.4), u = t − x * and v = t + x * are the Eddington-Finkelstein retarded and advanced null coordinates respectively. Note the + sign in the exponent of Eq. (3.4). The conserved (Killing) energy associated with it is negative and corresponds to excitations called "partners". We need to find the explicit forms of the modes throughout the spacetime. Let us begin with the in f r I mode whose evolution is represented schematically in Fig 4. The incoming v mode of the form Eq. (3.2) coming from I r − is partially transmitted (T r I ) towards the horizon as a v mode and partially reflected (R r I ) back to infinity I r + as a u mode by the delta potential located at x * r = 0 (see Fig. 5). The transmitted part crosses the horizon, enters the black hole and is split by the second delta function potential located the r region and in the region The transmission and reflection coefficients are found by matching these solutions across the delta function potentials. In general for a potential of the form V = λδ(x * ) we require that χ(x * ) satisfies where χ| ± = lim x * →0 ± χ, and χ represents the derivative with respect to x * . The results for Vr 2iω These satisfy the relations The negative sign in front of the R l I term in (3.10b) comes from the fact that the "reflected" modes R l I e −iωu inside the BH have a negative norm (see Eq. (2.7)). The asymptotic form of the in f r I mode as x → +∞ is Following the same procedure for the in f r H modes coming out from the part of the past horizon in the r region (see Fig 7), we have  FIG. 7: Illustration of a plane wave incident onto a potential from the left, which then is partially transmitted to the right and also partial reflected back to the left of the barrier. The reflected portion then travels into the interior of the BH where it encounters the potential in the interior. There the "reflected" and "transmitted" portions travel away from the potential to the left, see inside the horizon (as shown in Fig. 8) Finally, for the modes in f l H coming from the part of the past horizon in region l, see 9, the effective transmission and reflection coefficients arẽ Having defined the "in" basis, the field operatorφ (2) can be expanded aŝ where theâ's are the annihilation operators for the respective modes.
Alternatively one can construct another basis called the "out" basis formed by modes having the asymptotic form and then obtain the following expressions of the field operator where the outâ 's are the associated annihilation operators. The "in" and "out" basis are Note that there is no contribution to in f l H from the out f r u modes. Using the scattering S-matrix formalism we can write the relation between the two basis as is the transpose of the scattering matrix S.
The notation used is borrowed from Ref [13] and is quite intuitive. For example S ur,vr indicates an incoming v mode from r leading to an outgoing u mode in r. The corresponding Bogoliubov transformation for the annihilation operators of the two bases is For the two delta functions potential the S-matrix elements can be found by inspection of We are interested in the numbers of outgoing particles in the various channels, namely for all values of ω. This is the most "natural" quantum state one can define on the extended manifold described by the Penrose diagram of Fig 3. Physically |B describes a state in which there are no incoming particles either from past right infinity I r − or from the past horizon H − . Although "natural", this does not correctly describe the quantum state of the fieldφ (2) if the BH is formed by a dynamic gravitational collapse. The collapse in fact induces the conversion of quantum vacuum fluctuations to real on shell particles, the so called Hawking radiation [7]. The state which correctly describes this process, at least at late times, is called the Unruh vacuum |U [15]. The difference between the two states can be schematically summarized as follows. For the Unruh vacuum the modes coming out from the past horizon are chosen to be positive and negative frequency, not with respect to the Schwarzschild time t, but with respect to Kruskal time. Thus instead of the mode in f r u and in f l u , the modes are chosen as where the − and + refer to the r and l regions respectively, and κ is the surface gravity of the BH horizon, which for our metric is The modes coming from past null infinity for the Unruh vacuum are chosen as in f r I . The field can then be expanded in terms of a complete set of these modeŝ The Unruh state is therefore defined aŝ for every ω and ω K . The relation between the two sets of operators is given by the following

Bogoliubov transformations
inâr where the Bogoliubov coefficients are given by (see for example [16]) Using the Bogoliubov transformations, Eq. (3.31) and Eq. (3.44), we obtain One can see the combined effect of the near horizon mixing (the α and β) encoded in the Bogoliubov transformation (3.44), which engenders Hawking thermal radiation, and the scattering caused by the potential (the S matrix element). After some calculation we obtain (3.51) The latter expression represents the numbers of the negative energy excitations created inside the BH. Here we see the usual problem of the normalization of plane waves leading to the δ(0). Using wave packets we can set δ(0) → 1 and verify that n r u + n l v = n l u . Thus the number of positive energy excitations created equals the number of negative ones as energy conservation requires. One notices immediately the striking difference between the emission in the exterior region compared to that of the interior region. In the exterior region the scattering is the standard one, n r u describes, as expected, a thermal emission at the temperature T H = κ 2πk B modulated by the gray body factor 4ω 2 4ω 2 +V 2 r which regulates the infrared divergence associated with the Planckian distribution. The gray body factor goes to one for ω V r . In the interior region the scattering is anomalous resulting in particle production; as a result we see that both n l u and n l v do not decay exponentially for large ω but as a power law as seen in Fig. 11. The emission in the interior is not thermal. It is infrared divergent, i.e, the spectrum is dominated by soft phonons. The low frequency behaviors of n r u and n l u for various values of V r > 0 and V l < 0 are shown in Fig .12. The qualitative behaviors of n v for the same cases are identical to that of n l u and thus are not depicted. In addition to the non thermal behaviors of the high frequency modes for n l u seen in Fig. 11, the plots in Fig.12 show another nontrivial feature, a peak, that arises in the quantity ωn l u ( and also occurs for ωn l v ). It appears the peak is most pronounced when |V l | >> V r ∼ κ/(2π). In this regime, the position, in ω, of the peak is proportional to V r so it moves to the right on a plot of ωn l u versus ω as V r increases. For V r >> κ/(2π) it disappears because it becomes lost in the power law decay that occurs at high frequencies. In contrast, as V r gets smaller and moves to the left on the plot, the height of the peak decreases relative to its base, which for small values of ω is the limit lim ω→0 ωn l u . If for fixed V r , |V l | decreases, but is still larger than V r , then the height of the peak also decreases. However, its location stays about the same. When V r = |V l | the peak no longer The quantity C ω n r u shows how the thermal nature of the exterior modes is modified by the gray-body factor. Here C is a scaling factor so that maximum value of C ω n r u = 1. Right: The quantity C ω n l u , where C is a different scaling constant so that C ω n l u = 1 for ω = 10 −6 . An unexpected peak in C ω n l u is visible, and exists in all 3 cases shown. The qualitative behavior of n l v is identical to n l u and so is not depicted here.
exists. This can be shown analytically by looking at the derivative of ωn l v , As the denominator is positive for all ω we can just focus at the numerator. Making the substitution ω = πω κ it can be shown that 2V r 2 κe ω 2 (ω − sinh (ω )) (3.53) which is less than zero for all ω > 0. Thus there is no peak like the one seen in Fig. 12 in the V r < |V l | case. The same can also be shown for n u but the expressions are more complicated.
This peak is also present in other, more realistic, configurations for the effective potential.
This will be shown elsewhere.
A final remark concerning the Boulware vacuum |B . This state is characterized by being a vacuum state at infinity I r ± (no incoming and no outgoing particles for x → +∞), that is singular however at H ± . Indeed if we calculate the number of particles created in the r region one finds This is not true in the BH interior region because of the particle production that occurs there resulting in

IV. THE MASSIVE MODEL
The second toy model we want to investigate is the one introduced in Ref [12], where in the field equation (2.4) V eff is neglected and the mass term k 2 ⊥ (c 2 − v 2 0 ) is approximated as two step functions (see Fig 13 ) FIG. 13: Plot from Ref [12] of the coefficient of m 2 (dashed) and the approximation to that coefficient(solid). This result is based on the sound speed profile used in [11] and [16].
where m 2 r = m 2 (c 2 r − v 2 0 ) and m 2 l = m 2 (v 2 0 − c 2 l ). Again c r and c l are the asymptotic values of c(x) as x → +∞ and x → −∞ respectively. The − sign in front of m 2 l comes from the fact that inside the BH c 2 < v 2 0 . We also choose x * 0l = 0 = x * 0r for simplicity. The field equation (2.4) simplifies to Since the construction of the "in" basis for this model has been performed in Ref [12], here we briefly sketch the basic features. The asymptotic form of the incoming v mode coming with k r ≡ ω 2 − m 2 r . This is a massive mode and it exists only if ω > m r i.e., there is, as usual, a mass gap. On the other hand on H − where these modes are massless The form of these modes throughout the spacetime can be found by enforcing continuity of the spatial part χ of the modes and their derivatives at the boundaries of the step functions with the result where k l ≡ ω 2 + m 2 l . Note that unlike k r , k l is real for any value of omega and k l ≥ m l . These modes can be illustrated schematically in the same way as the previous toy model of  Note that for ω < m r , n r u = 0 because no modes reach future null infinity in the r region. Thus there is a discontinuity in n r u at ω = m r . Right: Plot of n l u vs ω which shows the non-thermal nature of the interior particle number as the low ω behavior is shown to approach a constant and for large ω it decays as a power law. The qualitative behavior of n l v is very similar to n l u , thus it is not shown.
For completeness we can also work out the numbers of created particles in the Boulware state |B N r u = 0 (4.21) One can see that, just like in the previous case, |B is no longer a vacuum state in the interior of the BH. Moreover, unlike in the Unruh state the number of created particles diverges as ω → 0.

V. CONCLUSIONS
We have investigated scattering in the exterior of the acoustic horizon of a BEC analog BH and anomalous scattering or particle production in its interior in a simple model with massless phonons and a different one for massive phonons. We have considered both the Unruh and Boulware states. The latter is the natural vacuum state for a static star while the former gives a good approximation in the exterior region to the late time radiation produced by the black hole. As expected we find for the region outside the horizon that the spectrum at infinity is thermal modulo the graybody factor for the Unruh state and there are no particles for the Boulware state. In the massive case we find that, as expected, the emitted thermal radiation in the exterior is gapped.
In the interior anomalous scattering produces additional particle production for both massless and massive phonons and this destroys the thermal nature of the spectrum for the Unruh state. At small frequency the emission is dominated by soft phonons but only in the massless case. At high frequency one finds that, for the considered models, the particle number falls off like an inverse power of the frequency rather than exponentially.
Not surprisingly particle production also occurs for the Boulware state in the interior. So the Boulware state remains a vacuum state in the exterior and can only be considered to be an initial vacuum state in the interior.
For massless phonons an unexpected peak was found in the quantities ωn u and ωn v when they are plotted as functions of ω, with n u and n v the number of right moving and left moving particles found at future null infinity in the interior. This peak represents a clear deviation from a thermal spectrum. It occurs for a limited range of the factors V r and V in the delta function potentials (3.1).
The presence of particle creation even for the Boulware state inside a BH can be understood by the fact that the Killing vector ∂ ∂t , of which the Boulware modes "in" are eigenfunctions, is spacelike inside the horizon. The symmetry associated with it is homogeneity rather than stationarity. This is clearly seen by the switch of roles of the coordinates t and x * inside the BH; x * is timelike and t is spacelike so a potential depending on x * is a time dependent potential which as such causes particle creation.
Particle production associated with anomalous scattering induced by curvature and consequent deviation from thermality of Hawking radiation was first noticed by Corley and Jacobson [17] in a different context in the region exterior to the event horizon. Specifically, they introduced an ad hoc modification of the two dimensional wave equation for the modes propagating in a BH metric which results in a nonlinear dispersion relation, subluminal in their case, i.e. ω − vk = ± k 2 − k 4 k 2 0 . Then they analyzed the influence of the induced anomalous scattering on the spectrum of the particles radiated by the BH in the region exterior to the horizon. The fact that the anomalous scattering occurs outside the horizon is a peculiar effect of the dispersion they chose. In a genuine General Relativity framework, like the one we use, anomalous scattering and related particle production can occur only inside the horizon, outside the scattering is always the standard one giving just a gray body factor and no extra particle production.
Deviation of thermality of Hawking radiation in the context of BEC analog BHs, where the modification of the relativistic dispersion relation is superluminal, i.e. ω − vk = ± k 2 + k 4 k 2 0 , was first analyzed numerically by Macher and Parentani [5].
Our results are in the context of quantum field theory in curved space, as such they involve a strictly linear dispersion relation for which there are no superluminal/subluminal modes. The connection to actual analog BHs is that our results should be valid for long wavelength phonons for which the mode equation is approximately the same as that for a massless minimally coupled scalar field in the analog spacetime [20]. The connection of our results to real black holes is that, in the interior (where the Killing vector is spacelike) the spacetime is dynamic and there is also an effective potential, this time due to the spacetime curvature, and so nonthermal particle production should also occur.
The advantage of analog gravity is that, unlike what happens in the gravitational context, one has direct experimental access to the region inside the horizon and so the spectrum of the phonons emitted there will be observable. Our results predict that this spectrum will be completely different from the one emitted outside the horizon. In particular, it will not be thermal.