Test of quantum atmosphere in the dimensionally reduced Schwarzschild black hole

It has been suggested by Giddings that the origin of Hawking radiation in black holes is a quantum atmosphere of near-horizon quantum region by investigating both the total emission rate and the stress tensor of Hawking radiation. Revisiting this issue in the exactly soluble model of a dimensionally reduced Schwarzschild black hole, we shall confirm that the dominant Hawking radiation in the Unruh vacuum indeed occurs at the quantum atmosphere, not just at the horizon by exactly calculating the out-temperature responsible for outgoing Hawking particle excitations. Consequently we show that the out-temperature vanishes at the horizon and has a peak at a scale whose radial extent is set by the horizon radius, and then decreases to the Hawking temperature at infinity. We also discuss bounds of location of the peak for the out-temperature in our model.


I. INTRODUCTION
Hawking radiation as information carrier [1] leads to black hole complementarity in such a way that there would be no contradictory physical observations between static and freely falling observers [2]. In connection with black hole complementarity, one of the solutions to the firewall paradox [3] is that the infalling observer crossing the horizon could find the firewall of high frequency quanta after the Page time [4]. Thus, the Hawking radiation at the horizon should be highly excited beyond the Planckian scale. The existence of the firewall might also be explained by the Tolman temperature [5,6] since the Hawking radiation at infinity is ascribed to the infinitely blue-shifted radiation at the horizon.
On the other hand, Unruh showed numerically that the process of thermal particle creation is low-energy behavior so that the highest frequency mode does not matter for the thermal emission [7]. It was also claimed that the Hawking radiation can be retrieved by an alternative Boulware accretion scenario without recourse to a pair creation scenario at the horizon [8]. Recently, Giddings raised a refined question regarding the origin of the Hawking radiation in the Unruh vacuum [9]. He investigated both the total emission rate and the stress tensor of Hawking radiation and then concluded that the origin of Hawking radiation is the near-horizon quantum region of the quantum atmosphere whose radial extent is set by the horizon radius scale. Subsequently, there have been some related works to the quantum atmosphere; analyses of the stress tensor and the effective temperature [10][11][12], and calculations of emission rate of Hawking radiation in arbitrary dimensions [13].
In particular, from the analysis of a stress tensor, it was claimed that in the Unruh vacuum the Tolman temperature near the horizon does not originate from the out-going particles but the in-going particles from the fact that the negative influx would transition to the positive outward flux over the quantum region outside the horizon [9]. It means that the out-going Hawking radiation originates from the quantum atmosphere. This claim was also discussed by employing the local temperature responsible for the out-going particles [10]; thus the local temperature related to the out-going particles must be finite over the whole region, in particular, it has a peak at a macroscopic distance outside the horizon.
The crucial difference from conventional results comes from a modification of the Stefan-Boltzmann law for the out-going particles. In Ref. [11], the authors also advocated the quantum atmosphere with two different arguments. Heuristically, the first was based on the gravitational Schwinger effect for particle production by the tidal force outside a black hole horizon. Next, the second argument of our concern made use of a calculation of the stress tensor to derive the energy density for an observer at a constant Kruskal position in order to investigate the quantum atmosphere. However, the second argument relied on the observer in Kruskal coordinates, which are not free-fall coordinates at a finite distance from the horizon. Subsequently, this issue was resolved by using a free-fall coordinate system without acceleration [12]. Furthermore, the adiabaticity of the test field modes, which allows us to test where the Wentzel-Kramers-Brillouin breaks down, was taken as an indicator of a particle creation process. The authors found a peak in the violation condition for the field modes; then the condition agrees to the peak of the discrepancy between the energy density and the effective temperature. This discrepancy was regarded as a signal for the location of quantum atmosphere where a particle creation process is taking place.
In this work we would like to revisit the spatial origin of Hawking radiation by investigating the local temperature responsible for the outgoing Hawking particles in a dimensionally reduced Schwarzschild black hole which is more or less realistic and exactly soluble model. The organization of this paper is as follows. In Sec. II we study expectation values of the stress tensor for scalar fields from the one-loop effective action in the dimensionally reduced model based on Refs. [14][15][16][17]. Then in Sec. III the stress tensor in thermodynamic equilibrium states proposed by Balbinot and Fabbri [18] will be extended to the stress tensor in the Unruh vacuum of nonequilibrium state. In Sec. IV we first calculate the local temperature in the Hartle-Hawking vacuum and then find it is finite everywhere. Next we show that the local temperature in the Unruh vacuum is just the Tolman temperature, but it consists of the influx and outward flux. Since the only outgoing modes contribute to the Hawking temperature at spatial infinity, we should discard the influx from the Tolman temperature and then justify the out-temperature purely characterized by the outgoing Hawking radiation. Eventually the out-temperature turns out to be the same as the local temperature in the Hartle-Hawking vacuum; it vanishes at the horizon and has a peak at a scale whose radial extent is set by the horizon radius in the quantum atmosphere, and then approaches the Hawking temperature at infinity. Furthermore we find the lower bound and the upper bound of the peak. Finally conclusion will be given in Sec. V.

II. DIMENSIONAL REDUCTION OF THE EINSTEIN-HILBERT ACTION
Let us start with the Einstein-Hilbert action and the matter action defined by where f i is a scalar field and N is the number of the scalar fields. In the spherically symmetric space, the four-dimensional line element can be written as ds 2 (4D) = ds 2 +(1/λ 2 )e −2φ dΩ 2 2 with ds 2 = g µν (x)dx µ dx ν and φ = φ(x), where λ 2 = π/(2G), e −φ is the radius, and dΩ 2 is the two-dimensional solid angle. Then, the four-dimensional actions (1) and (2) reduce to Solving the equations of motion from the actions (3) and (4), one can obtain the twodimensional vacuum solution for a static black hole described by where f (r) = 1 − (2GM )/r, φ = − ln(λr), and f i = 0. The event horizon is located at r h = 2GM . In terms of the light-cone coordinates, the line element (5) is also written as where ρ = (1/2) ln f and σ ± = t ± r * . The tortoise coordinate is defined by r * = r + r h ln (rf /r h ), and its inverse relation is given by On the other hand, the one-loop effective action for the scalar fields in Eq. (4) is obtained and the trace anomaly of the semi-classically quantized stress tensor reads as [14][15][16][17] However, the flux in the Hartle-Hawking vacuum is unfortunately negative at spatial infinity.
In order to evade the negative flux at infinity [19,20], Balbinot and Fabbri [18] considered the Weyl invariant action not affecting the trace anomaly such as which is arbitrary but phenomenologically sensible in that Eq. (9) will provide the desired expression of the Hawking radiation at infinity. The constant b was determined as b = 2 √ 3, which can be identified with b = 4 √ 3π 1 in Ref. [18]. Combining Eq. (7) with Eq. (9), we will take the one-loop effective action as where the two auxiliary scalar fields ψ and χ satisfy From the localized action (10), the quantum-mechanical stress tensor is easily obtained In the conformal gauge (6), Eq. (11) is written as and the general solutions for ψ and χ are easily obtained as where ξ = ln (r/r h ) √ f and ω ± (σ ± ) and η ± (σ ± ) are arbitrary holomorphic/antiholomorphic functions determined by some boundary conditions of vacuum states. Plugging Eqs. (14) and (15) into Eq. (12), the components of the stress tensor (12) are expressed as where we defined Here, in order for the stress tensor (17) to be static, one of the simplest solutions in Eqs. (14) and (15) might be chosen as ω ± = ±(1/2)c 1 σ ± + d 1 and η ± = ±(1/2)c 2 σ ± + d 2 [18], where c 1 , d 1 , c 2 , d 2 are constants. The two constants of d 1 and d 2 can be written as one constant without affecting the stress tensor due to the translation symmetry in it; however, this choice of ω ± and η ± always results in t + = t − in Eq. (17), which describes only equilibrium states such as the Hartle-Hawking and the Boulware vacuum. In order to incorporate non-equilibrium vacuum, we impose a less strong condition for the auxiliary fields as which also successfully makes Eqs. (16) and (17) static, where C and D are constants.

III. THREE VACUUM STATES
We consider three vacuum states; the Boulware [21], the Hartle-Hawking [22,23] which determines C B = −12/r h and D B = 12. Consequently, we get where f = 1 − x with x = r h /r. All constants can be completely fixed in the Boulware vacuum thanks to the additional condition (21). Note that Eqs. (22) and (23) are exactly the same as the results in Ref. [18]. As expected, the flux at infinity is well-defined as T ±± HH → N (π/12)T 2 H , where T H = 1/(4πr h ). Finally in the Unruh vacuum there is no influx at spatial infinity and the outward flux is finite at horizon, specifically, T −− U vanishes at infinity and T ++ U is regular on the horizon, so that the boundary conditions are chosen as t U + = 0 and t U − = −(4r h ) −2 with C U = −3/r h . Then the resulting stress tensor can be newly obtained as Near the horizon the influx is negative finite as T ++ U → −N/(192πr 2 h ), while T −− U → 0. At spatial infinity T ++ U → 0, while T −− U → N (π/12)T 2 H .

STATE
In this section, we calculate the local temperature in order to study the origin of Hawking radiation in the evaporating black hole. For this purpose, the proper velocity of radiation flow can be found by solving the geodesic equation of u α ∇ α u µ = 0, where the proper velocity u µ is defined by u µ = dx µ /dτ with a proper time τ . Then the proper velocity can be obtained If the frame is released from rest at an arbitrary point of r = r 0 , then Eq. (26) reduces to Next, we define the local quantities related to the stress tensor as where ε, p, and F are the proper energy density, pressure, and flux, respectively. Note that n µ is a spacelike unit vector normal to u µ given by n µ = (0, √ f ). In the light-cone coordinates, the energy density and flux in Eq. (27) can also be expressed as where we used u ± = 1/ √ f and n ± = ±1/ √ f in the light-cone coordinates.
Before studying the local temperature in the Unruh vacuum, we first investigate the Stefan-Boltzmann law in the Hartle-Hawking vacuum in order to relate the proper energy density (28) to the local temperature along the line of Refs. [5,6]; however, we will take into account the trace anomaly in deriving the Stefan-Boltzmann law as compared to the conventional procedure. Let us start with the first law of thermodynamics written as dU = T dS −pdV where U , S, V , and T , and p are the internal energy, entropy, volume, temperature, and pressure of a system. At a fixed temperature, the first law of thermodynamics can be rewritten as where the left-hand side is the energy density, i.e., ε = (∂U/∂V ) T . Using the Maxwell relation of (∂S/∂V ) T = (∂p/∂T ) V , one can see that Eq. (30) becomes where D c ≈ 23.03, the temperature is real in the whole region, but unfortunately it is not decreasing monotonically. Finally if we take D HH ≥ D c , then the temperature is not only real in the whole region but also decreasing monotonically as r increases after it reaches a maximum value at r c ≈ 1.43r h (see Fig. 1). In this respect we will take D HH ≥ D c from now on.
Let us now find the locations of peaks from ∂ r T HH | r=r peak = 0 with assuming D HH ≥ D c , where r peak is a position at which the maximum temperature occurs, then we obtain the relation between D HH and r peak as If r peak occurs at r h , then D HH = −3, which is prohibited by the assumption that D HH ≥ D c ≈ 23.03. Hence, for D HH ≥ D c , r peak should be at a finite distance from the horizon and increases to (3/2)r h monotonically as D HH increases. As a result, the peaks of the local temperatures in equilibrium should lie in Now, in the Unruh vacuum, one can calculate the local temperature by substituting Eqs. (24) and (25) where σ = γ/2. The local temperature in the Unruh vacuum takes exactly the Tolman's form, which is not new; however, it is worth noting that the local temperature (39) consists of the negative influx and the positive outward flux. Defining we obtain where T in and T out are related to the influx and outward flux, respectively. Note that the divergence of the Tolman temperature at the horizon comes from T in since the influx (24) Therefore, the local temperature associated with the outgoing Hawking radiation can be finite everywhere, and its peak occurs at a macroscopic distance outside the horizon, which means that the main excitations occur not at the horizon but at the peak in the quantum atmosphere.

V. CONCLUSION
In the dimensionally reduced Schwarzschild black hole, we found that the divergence of the temperature at the horizon comes from the infinite blue-shift of the negative influx, and the out-temperature responsible for the Hawking radiation is always finite, more importantly, its peak occurs in the quantum atmosphere bounded by 1.43r h r peak < 1.5r h , which is indeed the size of the black hole. It means that the main excitations of Hawking particles dominantly happens at the peak but it spreads throughout the whole region well outside the horizon. Consequently, in the spherically reduced Schwarzschild black hole, we confirmed that the origin of the Hawking radiation is the quantum atmosphere not just at the horizon from the viewpoint of the local temperature in the semi-classical regime.
As a first comment, we discuss what it happens when the trace anomaly (8) is ignored in our calculations. In Eq. (33), if we replace T µ µ by q T µ µ where q is 0 or 1, the temperature is obtained as T 2 HH = T 2 H [f (1 + 2x + x 2 (9 + 4D HH + 36 ln x)) + 8(1 − q)x 3 ]. At the horizon, T HH = 2 √ 2T H √ 1 − q so that in the usual case of q = 0 the temperature does not vanish. In our case of q = 1 the temperature vanishes at the horizon, which means that the modified Stefan-Boltzmann law (34) induced by the trace anomaly is essential in our atmosphere argument.
Finally, one might wonder what the physical meaning of D is, in consideration of the fact that three regimes are identified in Sec. IV and thus only D ≥ D c seems to be physically relevant. Unfortunately, we are not well aware of the meaning of the constant D generated from the semiclassical treatment although D plays the important role in our calculations.
Instead, we notice some intriguing points for further study: (i) D arises from the solution of the auxiliary scalar fields to localize the non-local effective actions (7) and (9). What needs to be answered is that it can be a quantum-mechanical hair or not. Otherwise, we must find a way to fix the constant like the case of Boulware state. (ii) For D = D , the Hartle-Hawking states such as |HH; D and |HH; D are degenerated at the horizon, and thus the local temperatures are coincident at the horizon; however, they depend on D in the bulk region. We hope these issues will be resolved elsewhere.