Extracting Hawking Radiation Near the Horizon of AdS Black Holes

We study how the evaporation rate of spherically symmetric black holes is affected through the extraction of radiation close to the horizon. We adopt a model of extraction that involves a perfectly absorptive screen placed close to the horizon and show that the evaporation rate can be changed depending on how close to the horizon the screen is placed. We apply our results to show that the scrambling time defined by the Hayden-Preskill decoding criterion, which is derived in Pennington's work (arXiv:1905.08255) through entanglement wedge reconstruction is modified. The modifications appear as logarithmic corrections to Pennington's time scale which depend on where the absorptive screen is placed. By fixing the proper distance between the horizon and screen we show that for small AdS black holes the leading order term in the scrambling time is consistent with Pennington's scrambling time. However, for large AdS black holes, we find the leading order term differs from Pennington's results. The leading order Log contains the Bekenstein-Hawking entropy of a cell of characteristic length equal to the AdS radius rather than the entropy of the full horizon.


Introduction
The AdS/CFT correspondence is a conjecture that relates gravitational systems in asymptotically AdS spacetimes to conformal field theories in one fewer spatial dimension [1][2][3]. This provides an ideal setting to resolve the black hole information paradox [4][5][6][7]. In particular, it suggests that information thrown into a black hole is not lost. The reason for this is that the AdS black hole undergoing evaporation is dual to the unitary time evolution of a thermal state on the CFT side of the duality, which does not allow for information loss. The information thrown into a black hole is thus argued to be scrambled by some kind of unitary dynamics and then remitted via Hawking radiation [8][9][10]. The question of how long one needs to wait for information thrown into a black hole to emerge in the subsequent Hawking radiation was first addressed in [8]. It stated that information thrown into a black hole after the Page time would re-emerge within a scrambling time scale which is given by: t scr ∼ β ln(S), (1.1) where β is the inverse Hawking temperature and S is the number of degrees of freedom in the black hole which take part in scrambling. Usually in the context of AdS/CFT one considers black holes well beyond the Hawking Page transition. These black holes, often referred to as large AdS black holes, are dual to large N gauge theories [11,12]. They have a horizon radius, r s , that satisfies r s L where L is the AdS radius. A peculiar property of large AdS black holes is that they are thermally stable. This is due to the confining potential which comes from the asymptotics of AdS spacetimes. In such a case any Hawking radiation that the black hole emits reaches the conformal boundary and bounces back being reabsorbed into the black hole. Eventually the black holes reaches stable equilibrium with the surrounding Hawking radiation and will not evaporate [13,14]. This makes large AdS black holes ill-suited to discuss the information paradox. To remedy this issue, it has been suggested to start with a large AdS black hole and then couple the bulk fields to an auxiliary field (called the evaporon) which carries energy away from the AdS black hole into an auxiliary system thereby allowing the black hole to evaporate [15,16].
In such constructions, it is the joint system of the reservoir and black hole which satisfy unitarity. Such constructions have been of recent interest in explorations of the information paradox. For example, [17,18] rely on such setups to show how information from the black hole gets released in the Hawking radiation. They use entanglement wedge reconstruction to show how information inside a black hole after the Page time scale is encoded in the subsequent Hawking radiation. In particular, Pennington showed that a small amount of information thrown into a black hole (after the Page time) will re-emerge in Hawking radiation after a time scale given by: where C can be thought of as the radial distance away from the horizon that one expects the Rindler description to hold, dr s /dt is the average rate of change of the horizon radius during evaporation, and β is the inverse Hawking temperature. Moreover, as we shall review in Section 3, he concludes that t emerge is the scrambling time scale discussed in [8]. A key assumption that was made in the calculation was that radiation was being extracted close to the horizon by some unspecified means. In the calculation, he assumes that if radiation is extracted sufficiently close to the horizon such that greybody factors are ignored then he can use the 2D Stefan-Boltzmann law for the evaporation rate: where c evap represents the number of modes being extracted near the horizon. Using this evaporation in conjunction with the first law of black hole thermodynamics gave an information emergence time of the form 1 : This resembles the result derived for 2D black holes in Jackiw-Teitelboim (JT) gravity studied in [18]. Which is given by: where c is the central charge (a measure of the degrees of freedom of a CFT) of a CFT that describes bulk matter in the 2D gravity theory. In light of the two results in Eqs. (1.4 -1.5) for the emergence time, it is tempting to make a rough identification of c ∼ c evap . The central charge, c, in Eq. (1.5) seems to be a fixed parameter which does not appear to have any kind of dependence on quantities that characterize the black hole such as temperature. However, it is clear that in Pennington's setup c evap depends on details of where and how radiation is extracted near the horizon. For example, c evap should depend on how close one is extracting radiation near the horizon. The closer we are, the larger c evap can get. Furthermore, c evap will depend on the means by which one extracts radiation from the horizon; if we choose to place a surface at a radial distance δr from the horizon with perfectly absorbing boundary conditions then c evap would be larger than if we chose some kind of semi-reflective boundary conditions. All these details will have some effect on the value of c evap and therefore on the evaporation rate.
In light of these observations, we explore how the evaporation rate of a black hole depends on how close we extract radiation from the horizon. In this paper, we extracting radiation at an absorptive screen. We assume that the screen can be understood from the prospective of the holographic renormalization group in AdS/CFT [19]. At infinity we have a full UV complete theory, and "moving" the boundary deeper into the bulk amounts to coarse-graining the degrees of freedom of the UV theory and considering an effective theory at a lower energy scale. We should be able to roughly model the situation by introducing an absorptive screen where the effective coarse grained version of the full boundary theory lives.
To simplify considerations we assume a toy model for the screen. We assume that it will absorb any radiation that reaches it 2 . In Section 2.1, we review how to calculate the average evaporation rate of a black hole and discuss how greybody factors affect this rate. By doing this we are able to clearly identify Pennington's c evap in terms of an infinite sum over angular momentum modes. We discuss how in two dimensions c evap in Eq. (1.4) can be reasonably identified with c in Eq. (1.5) with no further dependence on parameters that characterize the black hole but in higher dimensions such a naive identification is not valid. We introduce the notion of a generalized greybody factor which quantifies the fraction of radiation that gets to a point at a radial distance δr away from the horizon. At this distance away we introduce a perfectly absorbing screen which will absorb any radiation that hits it. We then write down an expression for the evaporation rate in terms 2 By doing this we are not actually defining the effective theory living on the screen that is consistent with some UV completed theory on the boundary. If we did make the effective theory on the screen consistent with a UV completed theory, we should not expect a perfectly absorptive screen. However, we still believe that a perfectly absorptive screen near the horizon is a reasonable approximation. In Section 4, we discuss a more rigorous way of defining how the screen should absorb radiation. of the generalized greybody factor. After doing this we restrict ourselves to massless scalar perturbations and write down a model for the generalized greybody factor which treats the effective potential as a "hard wall." In Section 2.2, we apply the hard wall model to AdS Schwarzschild black holes and find the evaporation rate. In Section 2.3, we discuss why the hard wall model for the generalized greybody factor is not sufficient for near extremal AdS ReissnerâĂŞNordstrÃűm (RN) black holes. We motivate a correction that "softens" the wall and accounts for radiation being able to tunnel into the classically forbidden region. We then provide an estimate using this modified model for the evaporation rate of near extremal AdS RN black holes. In Section 3, we review Pennington's calculation of t emerge and then use the modified evaporation rates that we calculated in Sec. 2 and find that t emerge takes on a slightly different form from what Pennington claims. In particular, for AdS black holes with r s /L 1 we find results that agree with Pennington's calculation up to some logarithmic correction which depends on how far we choose to extract radiation. However, in the case of r s /L 1 we find results that differ from Pennington's in the sense that the argument that goes into the Log is not the entropy of the entire horizon, but rather the entropy of a cell of size L controlled by the AdS radius (in addition to the usual logarithmic correction which depends on extraction radius). In Section 4, we discuss the subtleties in choosing how the subleading Log correction to t emerge depends on how we choose to fix the radial position of the screen. In particular, we find that by fixing the proper distance between the screen and horizon, t emerge is consistent with the scrambling time for near extremal black holes (up to a Log correction that has no further dependence on the temperature of the black hole). We speculate that by fixing the proper radial distance to the screen we are fixing the energy scale of the effective theory on the screen. We then conclude this work by discussing limitations of our calculations and suggestions for future exploration.

Modelling Hawking Radiation Extraction Through Generalized Greybody Factors
It is well known that close to the horizon, a black hole will emit radiation as a black body. However, by the time this radiation reaches an observer very far away from the black hole the spectrum of the radiation is modified. This is because the black hole generates a non-trivial potential that perturbations travelling through the background will experience resulting in partial reflection and transmission of perturbations. These effects are contained in greybody factors and they have a non-trivial effect on the evaporation rate of a black hole. Here we will review the basics of how greybody factors will affect the the evaporation rate. We will then introduce the notion of a generalized greybody factor which will depend on how far one is extracting radiation from the horizon. We begin with the well known result which describes the occupation number distribution of Hawking quanta emitted by a black hole, not accounting for greybody factors: The plus is for fermionic Hawking quanta and the minus is for bosonic Hawking quanta. For the sake of simplicity we will restrict ourselves to bosonic quanta. To find the total evaporation rate of the black hole we calculate: where N b is the number of different bosonic species and N is the degeneracy of theth hyper-spherical harmonic 3 . Note that we recover the 2D Stefan-Boltzmann law used by Pennington with c evap = N b N . This is only finite in 2D where the sum over disappears and we are left with c evap = N b which does not depend on the parameters that characterize the black hole (or even the exact position of the screen) this is similar to the behaviour of c in Eq. (1.5) which we discussed in Section 1.
In higher dimensions the sum persists and will be divergent resulting in an infinite evaporation rate. The effective potential near the horizon is essential for understanding how the divergence is regulated in higher dimensions. Generally speaking, if we extract Hawking radiation a finite radial distance δr from the horizon we should expect some fraction of the total radiation emitted by the black hole to to reach r = r s +δr. This is due to the fact that the effective potential is only zero at the horizon and strictly increases (at least in some neighborhood of the horizon). The larger is the more quickly it increases, this causes the higher angular momentum modes to reflect back into the black hole, effectively placing a cutoff over the sum of angular momentum modes. As we will see this cutoff contains information of where we place our screen and also on various parameters that characterize the black hole.
To begin we define the generalized greybody factor, γ (ω, δr), for each . It quantifies the fraction of radiation that gets to some surface a finite distance δr from the horizon 4 . If the surface sitting at r = r s + δr is a purely absorptive surface or screen then, the generalized greybody factor would represent the rate at which energy from the -th mode is being absorbed by the screen. Then the total rate at which the black hole losses mass is given by: Here the generalized greybody factor will be essential in regulating the infinite sum over . In general, we can compute γ (ω, δr) is done by considering the wave equation on the black hole background. However, doing this analytically is very difficult. To circumvent this issue we will introduce models for the generalized greybody factor which will capture the essential physics of the situation near the horizon. 3 To understand why N is present recall that the solution to the massless scalar wave equation in a spherically symmetric background can be decomposed as a product Ψ(t, where Φ are hyper-spherical harmonics for a given angular momentum mode there are N degenerate eigenfunctions. In particular, we identify Pennington's cevap = N b N . 4 In particular lim δr→∞ γ (ω, δr) will reproduce the greybody factors that are usually discussed in the context of an observer sitting at asymptotic infinity.
For the sake of concreteness we will consider the massless scalar wave equation for a spherically symmetric black hole background in d + 1 dimensions 5 , although we only consider ≥ 1, i.e. exclude monopole Hawking radiation 6 . We are interested in the radial part of the solution which can be shown to obey.
where V is the effective potential given by: If we choose to extract radiation close to the horizon (i.e. r − r s r s ) we can approximate V to linear order as: where β is the inverse Hawking temperature. We place a perfectly absorbing surface at r − r s = δr where δr/r s 1. Now consider the quantity: As long as ω 2 V we should be able to ignore the effects of V and expect that the radiation should experience little to no hindrance to get to the absorbing boundary we place near the horizon (i.e. γ (ω, δr) ∼ 1). However, once ω 2 ≤ V we should expect most of the radiation to be reflected back into the black hole and reabsorbed (i.e. γ (ω, δr) ∼ 0). We depict the scenario in Figure. 1.
We model this sort of "hard wall" potential by introducing the following generalized greybody factor: where Θ is the Heaviside step function. Using this model the evaporation rate using Eq. (2.3) is given by: where ω min, satisfies: (2.10) 5 These black hole spacetimes will generally have a metric of the following form It is widely believed that there is no global symmetry in Quantum Gravity (e.g., [20]). Since such a symmetry (e.g., shift or conformal symmetry) would be necessary to keep a massless scalar radiatively stable, we exclude the possibility of monopole radiation.

Figure 1.
Above is a depiction of how perturbations behave near the horizon with a generalized greybody factor given in Eq. (2.8). Near the horizon the Potential V (r) is linear and is depicted by the solid blue line. The slope of the blue line increases with . The absorptive boundary is depicted by the vertical red line at r = r s + δr. The thick black line is a lower bound for the frequency of radiation that gets absorbed. Everything below the thick line has frequency ω < ω min, = V (r s + δr) and cannot get to the absorbing surface, it bounces off the potential and gets reabsorbed. Everything above the thick line has frequency ω > ω min, and is able to reach the absorptive surface and gets completely absorbed.
In the next section, we use this model to find how the evaporation rate of AdS Schwarzschild Black holes is affected. We will also use a similar model with some adjustments to calculate the evaporation rate of a near extremal AdS RN black hole.

AdS Schwarzschild Black Holes
We have provided an argument for the convergence of the expression in Eq. (2.9) in Appendix A. Now we will estimate the evaporation rate as follows. To begin note that the convergence of the sum involving this has to do with the exponential suppression of the integrand for very large frequencies. In particular, the integral over x of the integrand x/(e x − 1) is well approximated by integrating up to some finite value x max which is of order unity. This places a rough bound of the form ω max ∼ 1/β. Having an upper bound on the frequency also implies an upper bound on the angular momentum modes you are extracting. This upper bound can be estimated by solving T 2 H − V max (r s + δr) = 0 using the linear approximation in Eq. (2.7) gives the following estimate: So now we are cutting off the sum up to max . Furthermore we know that x min, is suppressed by δr/r s , which is much less than one when max and it becomes of order unity around ∼ max . We have the following approximation: For very large the degeneracy of the -th hyper-spherical harmonic roughly goes as N ∼ d−2 . By approximating the sum as an integral it is straightforward to see that: Notice that the result above essentially modifies the 2D blackbody emission rate through a dimensionless pre-factor which measures the the number of angular momentum modes that are being extracted. The closer we are to the horizon the larger the mode numbers we can absorb. This results in a faster rate of evaporation than one would expect if radiation was absorbed very far from the horizon. If we go back to the evaporation rate that Pennington used one can see that c evap has non-trivial dependence on β. We will see later that this β dependence changes the the length scale that goes inside the Log for the expression in t emerge when r s /L 1.

Near Extremal AdS RN Black Holes
Now lets consider near extremal RN black holes. We want to analyze how the evaporation rate depends on where we extract radiation near the horizon. In this case we should expand V to second order. This is because the first order expansion of V is proportional the temperature which will go to zero in the extremal limit. Sufficiently close to the extremal regime the second order term will dictate the leading order behaviour of the potential close to the horizon. (2.14) The expansion above will be valid if r − r s r s . Sufficiently close to the extremal regime we will have the leading order contribution equal to: As before we can consider placing a perfectly absorbing surface a radial distance δr from the horizon. If we decide to use the Heaviside step model in Eq. (2.8) then we will need to do the integral in Eq. (2.9) with the lower bound: Unlike, the non-extremal case we discussed previously the lower bound is much larger than unity sufficiently close to the extremal regime. This means that we are well into the exponentially decaying tail of the integrand. Recall that the Heaviside step function model was used to simulate the effective potential as a "hard" wall. In reality we know that the waves can actually enter the classically forbidden region. The amplitude of the solution will decay through some power law in the classically forbidden region. By the time a wave with ω < ω min, reaches the absorptive surface its amplitude would be power law suppressed as depicted in  For ω min, the model is unchanged and everything is absorbed. However, for ω < ω min, we account for the wave-like behaviour of the solution which allows for the solution to tunnel into the classically forbidden region. The amplitude the the solution would decay as some power law after the classical turning point. We estimate the amount of energy that tunnels to the absorptive surface by taking the ratio between the amplitude of the solution at the turning point and the amplitude at the absorptive surface. Doing this gives a power law suppression of the generalized greybody factor for ω < ω min, in Eq. (2.18).
The Heaviside model completely disregards these effects. This would be okay if the contribution of modes with ω ≥ ω min, was not exponentially suppressed, but since it is suppressed in the near extremal regime we need to consider the effects of ω < ω min, . Therefore, for a near extremal black hole we need a generalized Greybody factor of the form: where q( ) is some function of which will be determined by analyzing the dynamics of the perturbations near the horizon and gives us the power law decay we need. The details of how to obtain a reasonable model for q( ) for scalar wave perturbations is detailed in Appendix B. The result is: where f ext (r s ) is the second derivative of f (r) evaluated at the horizon radius r s , in the limit where the Hawking temperature goes to zero. Using this we the expression, the evaporation rate is given by: We want to approximate the values of these integrals under the assumption that βω min, 1 (i.e. we are sufficiently close to the extremal regime). Lets begin with the first term(s) in Eq. (2.19) which describes modes with ω ≤ ω min, . The term(s) read: (2.20) To begin, we note that the integrated will generally have a local maximum. As long as βδrf ext (r s ) is sufficiently large (this is true when we are sufficiently close to extremality) we are guaranteed to have a sharply peaked local maximum within the interval of integration. This means that we can easily extend the range of integration from ∈ (0, 1) to ∈ (0, ∞) without changing the value of the integral. Such an integral can be done in full generality shown below: This function will diverge as α → 0 and the divergent behaviour takes the form: The function will continue to monotonically decrease to zero. In particular, at sufficiently low temperatures the the function will decay very quickly and be very close to zero when α 1. Therefore, it is reasonable to cut off the sum over when α ∼ 1. This means we should sum modes ≤ max .
In Appendix C we show that the the second term(s), which involve ω > ω min, , are exponentially suppressed and will have a much smaller contribution to the evaporation rate than the first term(s) sufficiently close to extremality. Therefore, by ignoring the exponentially suppressed terms we have the following estimate for the evaporation rate: (2.23) Before proceeding we need to note that in order to get to this point it was important to make the assumption that βω min, 1 this implies that δr r s / β f ext (r s ) . We introduce a dimensionless parameter Λ and write δr = . Then we have the following estimate in terms of Λ: where, 1 Λ β f ext (r s ). Here, Λ will control where the absorptive surface near the horizon is located and the upper bound on Λ comes from the condition that δr/r s 1. Unsurprisingly, we see that the evaporation rate increases as we extract closer to the horizon. We will now proceed by considering the specific case of a near extremal AdS RN black hole.
We begin by estimating max as follows. First note that: (2.25) From our previous analysis we know α 2 max ∼ 1 this gives: (2.26) Now we can approximate the evaporation rate for these two cases.
Case 1: r s /L 1 We know f ext (r s ) ∼ L −2 so we have: where 1 Λ β/L. Case 2: r s /L 1 We know f ext (r s ) ∼ r −2 s so we have: where 1 Λ β/r s .

Review of Pennington's Calculation
As we discussed in the introduction, it was shown in [8] that after the Page time a small amount of information thrown into a black hole could be reconstructed from subsequent Hawking radiation after the scrambling time scale. The works [17,18] are able to reproduce this result in a holographic setting. The setup is to have the usual black hole in AdS which is dual to some CFT on the boundary. This is then supplemented by some type of absorbing boundary condition at the boundary which allows the radiation emitted by the black hole to be absorbed and stored. The radiation in the reservoir purifies the the black hole CFT state. There are two entanglement wedges in this scenario, one corresponds to the entanglement wedge of the black hole and the other is the entanglement wedge of the reservoir where radiation is absorbed. As the black hole evaporates these entanglement wedges have time dependence and it can be shown that information that is initially sitting in the entanglement wedge of the black hole a scrambling time in the past (assuming we are considering a time after the Page time) will end up in the entanglement wedge of the reservoir. This is equivalent to saying that information thrown into a black hole, after a Page time has elapsed, will re-emerge in the subsequent radiation after a scrambling time as is claimed in the Hayden-Preskill decoding protocol [8].
In this section, we review the details of how this scrambling time scale appears in Pennington's calculations [17]. It comes from trying to find the location of a classical "maximin" surface in the spacetime of a spherically symmetric evaporating black hole (which happens to be a good approximation for where the quantum extremal surface is after a Page time has elapsed.). The determination of the location of the surface eventually comes down to the following calculation. The first step is to start with a static spherically symmetric black hole metric of the form: Then one defines ingoing Eddington-Finkelstein coordinates v = t + r * where dr = f (r)dr * .
With some simple manipulations one arrives at the following metric: Upon doing this one approximates the metric of an evaporating black hole by introducing time dependence into f by allowing the Schwarzschild radius r s to become time dependent (i.e. r s = r s (v)). One then wants to consider radial null geodesics on this evaporating black hole spacetime. The radial coordinate r lc describing the trajectory of these null geodesics satisfy: The right most expression comes from expanding f (r lc ) to first order and β = T −1 H = 4π/f (r s ). Define a coordinate r lc = r lc − r s then we will find: Under the assumption that dr s /dv < 0 and approximately constant the equation can be integrated to find: where C is an integration constant. It is clear that when C = 0, then r lc is constant (up to corrections caused by dr s /dv not being constant.) this defines the horizon of the evaporating black hole which is given by: To determine |dr s /dv|, Pennington makes the assumption that Hawking quanta emitted by the black hole is assumed to be extracted sufficiently close to the horizon so that one can use the 2D Stefan-Boltzman law: and N f are the number of bosonic and fermionic modes respectively. Using the first law of black hole thermodynamics the rate of energy loss can be related to dr s /dv the final result is: This results in: , for near extremal BH. (3.12) So after the Page time, information thrown into the black hole reemerges after waiting for the time scale |v 0 | = t emerge in Eq. (1.2). Note that in the near extremal case the expression written down above is valid for small near extremal AdS black holes. For large near extremal AdS black holes C ∼ L 2 /β so there will be some awkward L dependence inside the Log. As we will see in the following sections, by properly understanding c evap for large AdS black holes, the correct length scale in the Log should be L not r s .

Information Emergence Time for AdS Schwarzschild Black Hole
Using our newly derived evaporation rate in Eq. (2.13) along with the first law of black hole thermodynamics and the area law for entropy of a black hole we will get: To avoid clutter in our expressions we drop Ω d−1 and other irrelevant dimensionless factors. Plugging this into Eq. (1.2) we find for non-extremal black holes: 7 The length scale of C was chosen by analyzing how far the expansion f (r) near the horizon is valid to first order. In particular, it is not hard to see that C ∼ 1 βf (rs) . For small AdS black holes f (rs) ∼ r −2 s (as noted by Pennington) and for large AdS black holes f (rs) ∼ L −2 (not discussed by Pennington). Where rs and L are the horizon and AdS radius respectively. (3.14) For very large AdS Schwarzschild black holes (r s L) and the inverse temperature goes as β ∼ L 2 /r s . Plugging this into Eq. (3.14) we find that Pennington's scrambling time scale results in: where we assume that L/ p r s /δr. The interesting thing to note here is that the leading order term is not the the Log of the entropy of the horizon of the black hole. It is actually the entropy of a small cell on the horizon which has the size of the AdS radius L. We can do a similar calculation for very small AdS black holes (r s L) in this case β ∼ r s and we will obtain a more familiar result that Pennington got up to a Log correction that depends on where we place our absorptive screen: As we can see from Eq. (3.15), by understanding the explicit dependence of c evap on β we find that t emerge contains the Bekenstein-Hawking entropy of a cell on the horizon of characteristic length L inside the Log. This reasonable and consistent with the scrambling time discussed in [9] for large AdS black holes dual to large N gauge theories 8 .

Information Emergence Time for Near Extremal AdS RN Black Hole
Now lets consider what happens for near extremal AdS RN black holes. We can compute |dr s /dt| using the first law: We now can compute |dr s /dt| using the evaporation rates in Eqs. (2.27 -2.28). We can then plug this into Eq. (1.2). Case 1: r s L In this case we have: where 1 Λ β/r s .

Case 2: r s L
In this case we have: where 1 Λ β/L. If we make the assumption that Λ has no additional β dependence then we see the for small AdS black hole the first term matches what Pennington had but for large AdS black holes we again we L instead of r s . In the next section we will discuss an ambiguity that Λ presents us with which is related to where we place our absorptive screen.

Discussion and Conclusion
To summarize our results, we found that by placing a screen that absorbs Hawking radiation a radial distance δr from the horizon the time scale after which information re-emerges for AdS Schwarzschild black holes is given as: (4.1) For near extremal AdS RN black holes (d ≥ 4) we have: In the case of AdS Schwarzschild black holes we should assume the following hierarchy of scales that p δr r s . By doing this it is clear the the dependence on δr for the re-emergence time is sub-leading to the first term in the limit where p → 0. We can reasonably identify t emerge with the scrambling time scales discussed in [8,9].
The case of a near extremal AdS RN black holes is much more subtle. For near extremal AdS RN black holes we have an additional length scale that we did not have for the AdS Schwarzschild case. This length scale is β and causes problems when we try to decide on where the screen should be placed. To understand the issue, recall that we introduced Λ through the following definition which relates it to δr: where we require that 1 Λ β f ext (r s ). In particular, there are a number of choices we can make for the β-dependence of Λ. In Pennington's paper it is suggested that we extract radiation at a fixed distance from the horizon. There are at least two natural ways to do this.
The first way is to set the radial coordinate distance from the horizon, δr, to some constant that does not depend explicitly on β. Then it is clear that Λ ∼ β/r s . In this case we would have results that look like: , r s L.

(4.4)
These results are at odds with what Pennington has for near extremal AdS black holes and also with the literature [22,23] which discusses the scrambling time for near extremal black holes. In particular, the main difference is how β appears in the Log. One should expect β to appear in the denominator rather than the numerator. This suggests that fixing an absorptive screen at a constant coordinate distance will yield a re-emergence time that is much longer than the scrambling time, β 2π ln(S − S ext ). Now consider the second way, which is to fix the proper radial distance from the screen to the horizon. Then we can show δr ∼ (Proper Length) 2 /β 9 . By doing this, we see that Λ will have no additional dependence on β and we are be able to reasonably identify the information re-emergence time with the scrambling time for near extremal black holes. So the question is what we should be fixing, the coordinate distance or proper distance, or perhaps something else? We believe the answer lies in the idea of fixing the energy scale of our effective theory. We know in the AdS/CFT correspondence the radial direction in the bulk corresponds to the energy scale of the CFT on the boundary. So by fixing the energy scale we should unambiguously fix how δr scales with β. However, it is not clear exactly how the energy scale of the boundary theory depends on the radial distance. If it depends on the proper radial distance then we should fix the proper length between the horizon and screen. In discussions of the holographic renormalization group one usually considers metrics written in the form [19]: where z is the radial direction in the bulk and x i are coordinates on the boundary and γ ij is the induced metric on a constant z slice. The fixing of energy scales can be interpreted as the fixing of z. The way the metric is written suggests that z is the proper radial length in the bulk. Therefore, it seems that fixing the proper length between the screen and horizon seems like a reasonable way to fix the energy scale, although this may not be valid for metrics that significantly differ from (4.5).
To summarize our discussion, we found that there are many ways to fix the β dependence of where the screen is placed. Depending on how δr depends on β we can get t emerge that may or may not resemble the scrambling time for near extremal AdS black holes. In particular, we find that by fixing the proper radial distance between the horizon and absorptive screen we get an information emergence time that is consistent with the scrambling time for near extremal AdS black holes. We suggested that fixing the proper radial distance between the screen and horizon can be interpreted as fixing the energy scale of the theory on the screen. We also find an additional sub-leading Log term which contains information on 9 To see this consider the proper radial length from the horizon to a point δr from the horizon this is given by the an integral lprop = rs+δr rs dr √ f (r) , for δr min{rs, L} we can expand to first order and do the integral to find δr ∼ l 2 prop β exactly where the screen absorbs radiation (which should not explicitly depend on β). It is interesting to note that for large AdS black holes it is not the entropy of the entire horizon that goes into the Log but instead the entropy of a cell on the horizon of characteristic length L. This is reasonable if we recall that large AdS black holes are dual to large N gauge theories with N 2 ∼ L d−1 / d−1 p [21]. As we have demonstrated, the effect of extracting Hawking radiation near the horizon of a black hole generally has non-trivial consequences for the evaporation rate. In this work we extracted radiation close to the horizon using a perfectly absorbing screen that would absorb any Hawking radiation that gets to it. The rate at which energy was being absorbed by the screen for each angular momentum mode is captured through the generalized greybody factor. We did not rigorously compute this factor but instead proposed models that would capture the essential behaviour of the generalized greybody factor near the horizon. If we wanted to get more realistic models for the generalized greybody factors we could define the following effective potential which should reasonablly model the introduction of an absorptive screen placed at some radial coordinate r = r s + δr: V screen, = V (r), r s ≤ r ≤ r s + δr 0, r > r s + δr. (4.6) Then we should solve: The basic idea behind this is that we want to keep the spacetime unchanged until we arrive at the absorptive screen. The process of the screen absorbing radiation at r + δr can be thought of a gluing an asymptotically flat region just behind the screen and letting the wave escape to infinity as depicted in Figure 3.
To implement this we know that in the first region where r ∈ [r s , r s + δr] the general solution will be some linear combination of two independent solutions: ψ I, (r * ) = c 1, f (r * ) + c 2, g (r * ). (4.8) We should analyze the solution near the horizon which should take on the form of plane waves and normalize the outgoing wave to unity (i.e. we start with outgoing Hawking radiation) this will fix some type of relation between c 1, and c 2, . In region 2 where the potential is zero the solution should be purely outgoing (i.e. absorptive screen boundary condition). ψ II, (r * ) = T e iωr * . (4.9) We have 2 unknowns left now, namely T and one of the coefficients of the solution in region 1 which will represent how much of the wave is reflected back. We can fix these by requiring the solution and its first derivative at r = r s + δr be continuous. This will fix T uniquely. The generalized greybody factor is then defined by the the amplitude square of the transmission coefficient: Above is a depiction of the potential that we are considering to emulate an absorptive screen placed at r = r s + δr depicted by the dotted red line. We keep the effective potential the same as the black hole up until we get to the screen interface. We then transition to a trivial potential for a flat space which will act as a reservoir for the extracted Hawking radiation. Close to the horizon the solution takes on the form of in-going and out-going plane waves. We normalize the outgoing wave near the horizon to unity and the amplitude of the in-going plane wave is R. The absorptive screen boundary condition is enforced by only allowing outgoing plane waves in the flat region with amplitude T . We patch the solutions and uniquely determine T and R by requiring continuity of the solution and its derivative at the screen interface. Then the generalized greybody factor is defined by |T | 2 .
γ (ω, δr) = |T (ω, δr)| 2 . (4.10) This would be a more rigorous way to find the greybody factor. As one can imagine doing this analytically for any choice of δr would be difficult, however the procedure we just outlined can be implemented numerically to find the exact behaviour of the generalized greybody factors. We expect that the generalized greybody factors still mimic the behaviour of the idealized models we analyzed in this paper at least in the limit where δr min{r s , L}. It would be interesting to see how this method of extracting radiation at a finite distance from the horizon compares to other models that have been proposed to extract radiation from AdS black holes. For example, one could move the screen further from the horizon and ask how the generalized greybody factor at infinity (which is really just a greybody factor now) compares to greybody factors of models that use the evaporon [15,16] to absorb energy from the black hole.

A Argument for Convergence
In this appendix we argue for the convergence of the evaporation rate defined in Eq. (2.9). The starting point is the following integral: To begin we can do the integral exactly and find the following sum: N P olyLog 2, e −x min, − x min, ln 1 − e −x min, .
For very large we expect N ∼ d−2 we will approximate the discrete sum using an integral. So now we just need to argue for the convergence of the following integral: First we need to approximate the x min, which will satisfy: x min, We solve this to leading order in δr by using the expression in Eq. (2.7). We find: x min,l = βω min, 8π 2 δr(d − 1) r s 1 + β r s It is not difficult to see that any potential divergence in the integral over will occur in the tail behaviour of the integral going to infinity. For sufficiently large it is clear x min, ∼ α , where α is some dimensionless constant larger than zero. Now we can use that for very large : P olyLog 2, e −α e −α . (A.6) From this we can see that the integral in Eq. (A.3) will converge. Therefore we should expect the evaporation rate to be finite.

B Power Law Behaviour of Generalized Greybody Factor for Near Extremal BH
Here we present a way to get the power law behaviour for ω < ω min, in Eq. (2.18). We do this by analyzing the near horizon solution of the wave equation for an extremal black hole.
consider ω ≤ ω min, . We parameterize this in terms of 0 ≤ ≤ 1 and write ω = ω min, . Then we can express r tp * as: We also set r * at the position of interest (i.e. where the absorbing surface is): . (B.9) Now we can express T 2 in terms of : (B.10) We can do a series expansion of T 2 in α to understand the power law behaviour we find: We will use this behaviour to model the generalized greybody factor for ω < ω min, . So now we have the following for near extremal black holes: γ (ω, δr) = Θ(ω − ω min, ) + ω ω min, 2ν +1 Θ (ω min, − ω) . (B.12) C Exponential Suppression of ω > ω min, in Near Extremal Regime In this appendix we argue that the second set of terms in Eq.(2.19) are exponentially suppressed in the limit where β → ∞. We begin with the integral which accounts for modes with a frequencies higher than ω min, : This takes care of the integral and sum of the second term(s) in Eq. (2.19) we find that it is exponentially suppressed as expected. Therefore we can ignore this compared to the term(s) involving ω < ω min, .