Can a QED photon propagate out of the black-hole horizon?

TL;DR

This paper demonstrates that in the vicinity of an evaporating black hole, quantum electrodynamics predicts photons can effectively propagate outwards due to a negative mass-squared, challenging traditional views on black hole horizons.

Contribution

It introduces a one-loop quantum correction showing photons can have a negative gauge-invariant mass-squared near black hole horizons, suggesting possible information escape mechanisms.

Findings

Photon acquires negative gauge-invariant mass-squared near horizon

Potential for information retrieval from black hole interior

Challenges traditional horizon concept in quantum field theory

Abstract

We show at one-loop approximation that the QED photon acquires a negative gauge-invariant mass-squared in the near-horizon region of an evaporating Schwarzschild black hole. This result may imply that information about the inner structure of evaporating black holes is accessible, even without invoking the modification of the basic QFT and GR principles.

Equations

$$S(x,x^{\prime })=S_M(x,x^{\prime })+S_R(x,x^{\prime })$$

$$S_{R}(x,x^{\prime})\approx-\frac{27}{32\pi^{2}}{\int\limits_{0}^{+\infty}}\frac{dp\,p}{e^{\beta p}+1}\,i\not{\nabla}\cos\big{(}p\Delta{v}\big{)}.$$

$$S_{R}(\bar{x},\bar{x}^{\prime})\approx-\frac{27}{8\pi^{2}}{\int\limits_{0}^{+\infty}}\frac{dp\,p}{e^{2\beta p}+1}\,i\not{\partial}\cos\big{(}p(\Delta\tau+\Delta{x})\big{)}\,,$$

$$f_e(\mathbf{x},\mathbf{p})$$

$$\langle T_{e\bar{e}}^{\mu\nu}(\bar{x})\rangle={\int}\frac{d^{3}\mathbf{p}}{|\mathbf{p}|}\,p^{\mu}p^{\nu}\big{(}f_{e}(\mathbf{x},\mathbf{p})+f_{\bar{e}}(\mathbf{x},\mathbf{p})\big{)}\,.$$

$$m_{\gamma}^{2}=2e^{2}{\int}\frac{d^{3}\mathbf{p}}{|\mathbf{p}|}\big{(}f_{e}(\mathbf{x},\mathbf{p})+f_{\bar{e}}(\mathbf{x},\mathbf{p})\big{)}\approx-\frac{27}{16}\frac{e^{2}T_{H}^{2}}{6}\,,$$

$$S(\bar{x},\bar{x}^{\prime})={\int}\frac{d^{4}p}{(2\pi)^{4}}\,\Big{(}S_{M}(p)+S_{R}(p)\Big{)}e^{-ip(\bar{x}-\bar{x}^{\prime})}\,,$$

$$S_M(p)$$

$$S_R(p)$$

$$\langle\textrm{tr}\big{(}\bar{\psi}\gamma_{\mu}\partial_{\nu}\psi\big{)}\rangle=-\lim_{\bar{x}\rightarrow\bar{x}^{\prime}}\textrm{tr}\big{(}\gamma_{\mu}\partial_{\nu}S(\bar{x},\bar{x}^{\prime})\big{)}\,,$$

$$m_{\gamma }^2\approx \underset{|\mathbf{k}|\to {\omega }_{\mathbf{k}}}{\lim }\underset{0}{\overset{+\infty }{\int }}\frac{dp{p}^3}{e^{2\beta p}+1}\Pi (k,p)=-\frac{27}{16}\frac{e^2{T}_H^2}{6},$$

$$\Pi (k,p)\equiv -\frac{27e^2}{{\pi }^2}\frac{4{\omega }_{\mathbf{k}}|\mathbf{k}|\cos \theta +({\omega }_{\mathbf{k}}^2+{\mathbf{k}}^2)(1+{\cos }^2\theta )}{4p^2({\omega }_{\mathbf{k}}+|\mathbf{k}|\cos \theta {)}^2-({\omega }_{\mathbf{k}}^2-{\mathbf{k}}^2{)}^2},$$

$$|{\psi }_{\gamma }(\bar{x})⟩=\int \frac{d^3\mathbf{p}}{(2\pi {)}^3}\frac{1}{2{\omega }_{\mathbf{p}}}{e}^{ip\bar{x}}{\gamma }_{\mathbf{k}}(\mathbf{p})|\mathbf{p}⟩,$$

$${\gamma }_{\mathbf{k}}(\mathbf{p})=N\exp \left[-\frac{(\mathbf{p}-\mathbf{k}{)}^2}{4\sigma }\right]$$

$$A_{\gamma }(\bar{x},{\bar{x}}^{\prime })=⟨{\psi }_{\gamma }(\bar{x})|{\psi }_{\gamma }({\bar{x}}^{\prime })⟩,$$

$${\eta }_{\mu \nu }(\bar{x}-{\bar{x}}^{\prime }{)}^{\mu }(\bar{x}-{\bar{x}}^{\prime }{)}^{\nu }\approx -\frac{|m_{\gamma }^2|}{{\mathbf{k}}^2}(\tau -{\tau }^{\prime }{)}^2,$$

$$\langle\psi_{\gamma}|\hat{N}|\psi_{\gamma}\rangle=\left\{\begin{array}[]{ll}1\,,&\textrm{supp}(N)\cap\textrm{supp}(\psi_{\gamma})\neq\varnothing\,,\\[8.53581pt] 0\,,&\textrm{supp}(N)\cap\textrm{supp}(\psi_{\gamma})=\varnothing\,.\end{array}\right.$$

$$⟨vac|[\hat{N}(\bar{x}),\hat{N}({\bar{x}}^{\prime })]|vac⟩=0.$$

$$|hor\rangle=\left\{\begin{array}[]{ll}|vac\rangle\,,&\bar{x}\,\notin\,\textrm{supp}(\psi_{\gamma})\,,\\[5.69054pt] |\psi_{\gamma}\rangle\,,&\bar{x}\,\in\,\textrm{supp}(\psi_{\gamma})\,.\end{array}\right.$$

$$⟨hor|[\hat{N}(\bar{x}),\hat{N}({\bar{x}}^{\prime })]|hor⟩=A_{\gamma }(\bar{x},{\bar{x}}^{\prime }),$$

$${\eta }_{\mu \nu }\sim R_{\mu \lambda \nu \rho }({\bar{x}}^{\prime })(\bar{x}-{\bar{x}}^{\prime }{)}^{\lambda }(\bar{x}-{\bar{x}}^{\prime }{)}^{\rho }$$

$$\frac{|\mathbf{k}|}{\omega_{\mathbf{k}}}-1\approx\left\{\begin{array}[]{ll}10^{-6}\,,&r_{H}|\mathbf{k}|\rightarrow 10\,,\\[5.69054pt] 0\,,&r_{H}|\mathbf{k}|\rightarrow\infty\end{array}\right.$$

Full text

black holes, quantum electrodynamics, quantum field theory in curved spacetime

I Introduction

Physical black holes are expected to evaporate in quantum theory Hawking . The evaporation process is, however, featureless. This circumstance leads to the information loss in black-hole physics. The outstanding question is if a semi-classical mechanism exists which allows to access the information fell in black holes (for a recent critical review of various proposals towards the resolution of the information-loss problem, see Unruh&Wald ).

Inspired by the proposal of non-violent nonlocality near the event horizon of black holes Giddings-1 ; Giddings-2 , we intend to study in this article the photon dispersion relation at one-loop order in quantum electrodynamics (QED) near the horizon, with the black-hole-evaporation effect taken into account.

Through this article, we take $c=G=k_{B}=\hbar=1$, unless otherwise stated.

II Photon mass-squared

In the Minkowski vacuum in the absence of external fields, the photon dispersion relation reads $\omega_{\mathbf{k}}=|\mathbf{k}|$ at any order of perturbation theory due to the gauge and Lorentz symmetries, where $\omega_{\mathbf{k}}$ and $\mathbf{k}$ are, respectively, the photon’s energy and momentum. This relation typically changes in mediums/non-trivial quantum states and/or in the presence of external fields. In particular, the photon gains a gauge-invariant mass-squared $m_{\gamma}^{2}$ equaling to $\frac{1}{6}e^{2}T^{2}$ in the neutral electron-positron plasma of temperature $T\gg m_{e}$, where $e>0$ is the elementary charge and $m_{e}$ is the electron mass. Specifically, it turns out that $\omega_{\mathbf{k}}\approx(|\mathbf{k}|^{2}+m_{\gamma}^{2})^{\frac{1}{2}}$ if the condition $eT\ll|\mathbf{k}|\ll T$ is fulfilled Weldon . In case of the cold plasma, $T\ll m_{e}$, the thermal photon mass is still non-vanishing, but suppressed by the Boltzmann factor $\exp(-m_{e}/2T)$.

The method used in Weldon to compute $m_{\gamma}$ is diagrammatic, i.e. based on the computation of the photon self-energy at one-loop order in perturbation theory. The same result can be derived within the kinetic-theory approach Blaizot&Iancu , as these turn out to be equivalent Mrowczynski&Thoma . We shall employ below both methods.

The local quantum physics in the near-horizon region of an evaporating Schwarzschild black hole is described by the following Feynman propagator of the massless Dirac field:

[TABLE]

with $S_{M}(x,x^{\prime})$ locally corresponding to the Minkowski propagator and $S_{R}(x,x^{\prime})$ being a non-singular solution of the Dirac equation, which gives the evaporation effect:

[TABLE]

where $\beta\equiv 4\pi r_{H}\equiv 1/T_{H}$ is the inverse Hawking temperature Hawking with the black-hole Schwarzschild radius $r_{H}$, and $v\equiv t+r_{*}$ is the in-going Eddington-Finkelstein coordinate, where the Regge-Wheeler radial coordinate $r_{*}$ is equal to $r+r_{H}\ln(r/r_{H}-1)$. It should be pointed out that we have employed in (2) the DeWitt approximation for the transmission amplitude, which explains the factor $27$ DeWitt . Moreover, the basic structure of (2) can be envisaged from the fact that $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\propto(\partial/\partial{u})(\partial/\partial{v})$ holds close to the horizon, where $u\equiv t-r_{*}$ is the out-going Eddington-Finkelstein coordinate, as well as the non-singularity condition for $S_{R}(x,x^{\prime})$ in the coincidence limit $x^{\prime}\rightarrow x$, and that (2) has to give non-trivial stress-energy tensor, which does not diverge on the horizon.

In the Fermi normal coordinates $\bar{x}\equiv(\tau,\mathbf{x})\equiv(\tau,x,y,z)$, we have $\Delta{v}\approx\frac{1}{2}(\Delta\tau+\Delta{x})$, where the local region $x>\tau\,(x<\tau)$ is outside (inside) the black-hole horizon. Replacing $p$ by $2p$ in (2) and, then, rewriting (2) in terms of the Fermi normal coordinates, we obtain the following result:

[TABLE]

which holds in the local Minkowski frame freely falling near the black-hole horizon. The form of $S_{R}(\bar{x},\bar{x}^{\prime})$ is valid for any value of $p$, assuming that $|\bar{x}-\bar{x}^{\prime}|\ll r_{H}$ holds.

II.1 Quantum-statistical approach

In the quantum-kinetic-theory approach Emelyanov-17-a ; Emelyanov-17-b , $S_{R}(\bar{x},\bar{x}^{\prime})$ in (3) gives the following “one-particle” distribution:

[TABLE]

with the on-mass-shell condition $p_{0}=|\mathbf{p}|$. This distribution function $f_{e/\bar{e}}(\mathbf{x},\mathbf{p})$ refers to the massless electron/positron field, where the momentum space associated with the point $(\tau,\mathbf{x})$ is defined according to Bunch&Parker .

Physically, the distribution function $f_{e/\bar{e}}(\mathbf{x},\mathbf{p})$ describes an inward negative-energy-density flux (due to the electron/positron field) into the event horizon, which leads to $\dot{r}_{H}<0$, where the dot stands for the differentiation with respect to the Schwarzschild time. This can be directly shown by calculating the stress-energy tensor in the local Minkowski/Fermi frame:

[TABLE]

The negative sign of $f_{e/\bar{e}}(\mathbf{x},\mathbf{p})$ is non-typical for classical many-particle systems, but consistent with the computation of the stress-energy tensor near the horizon in the Schwarzschild frame Candelas . This is also in agreement with the fact that the inward quantum flux of the negative energy density originates well outside of the event horizon Unruh . This means that this flux cannot be understood in classical terms Emelyanov-17-a .

Employing the kinetic-theory approach for the computation of one-loop effects in quantum electrodynamics Blaizot&Iancu , we obtain in the near-horizon region that

[TABLE]

where we have used $f_{e/\bar{e}}(\mathbf{x},\mathbf{p})$ given in (4). The negative value of $m_{\gamma}^{2}$ is the main result of this article.

II.2 Diagrammatic approach

The photon mass-squared $m_{\gamma}^{2}$ can also be obtained by computing the one-loop photon self-energy, which provides the lowest-order radiative correction to the photon propagator. To this purpose, we need the electron-positron propagator in the momentum-space representation near the black-hole horizon:

[TABLE]

with the definitions

[TABLE]

where $\mathbf{n}\equiv(1,0,0)$ is an outward unit three-vector being locally identical with the radial direction from the centre of the black hole Emelyanov-17-a ; Emelyanov-17-b .

With the identity

[TABLE]

one can derive a part of the renormalised stress-energy tensor, which originates from the second term in round brackets on the right-hand side of (7). This gives rise to the inward flux of the negative energy density, in complete agreement with (5). The rest part of the total stress tensor comes from the trace anomaly, which arises from the renormalization, that could be based, e.g., on the Hadamard-parametrix subtraction.

Furthermore, the correction to the Minkowski propagator in (7) leads to the change of the pole structure of the photon propagator. It should be noted that $S_{M}(p)$ alone does not change this structure at any order of perturbation theory due to its Lorentz-invariant form and the gauge symmetry. Following Weldon and Emelyanov-17-b , we obtain at one-loop approximation that

[TABLE]

where

[TABLE]

where $\theta$ is the angle between $\mathbf{n}$ and $\mathbf{k}$. The equation (11) is, thus, identical to (6), in accordance with Mrowczynski&Thoma . This implies that the result $m_{\gamma}^{2}\sim-e^{2}T_{H}^{2}$ is accurate within the approximation considered.

For the one-loop result (6) to be reliable, the de Broglie wavelength of the photon must be much less than the black-hole size $r_{H}$, otherwise no notion of the photon exists in the Wigner sense, which appears to be the only one that is a posteriori physically relevant in elementary particle physics. In other words, the photon, we consider herein, is high-energy enough (with energy being still very much less than the Planck energy $(\hbar c^{5}/G)^{\frac{1}{2}}\approx 1.2{\times}10^{19}\,\textrm{GeV}$). The energy scale of order $1/r_{H}$ provides then a natural IR cutoff below which the one-loop computation is unreliable.111The IR cutoff is necessarily present, if the Minkowski-space approximation is used, but locally immaterial in particle physics, because particles in QFT are localized field excitations.

In fact, the electron-positron field is not massless in nature. Nevertheless, the result (6) holds for black holes of mass $M\ll 10^{16}\,\textrm{g}$, because then $m_{e}\ll T_{H}$ is fulfilled, i.e. the electron-positron field can be considered as effectively massless (the hard-thermal-loop approximation). We obtain that $m_{\gamma}^{2}$ is suppressed by the Boltzmann factor for larger black holes, but $m_{\gamma}^{2}$ is still non-vanishing and negative.

III Physical consequence of $m_{\gamma}^{2}<0$

In order to find out the physical consequence of the modified dispersion relation, $\omega_{\mathbf{k}}=(\mathbf{k}^{2}-|m_{\gamma}^{2}|)^{\frac{1}{2}}>0$, one needs to consider a wave package $\psi_{\gamma}$, which describes the real photon with the momentum $|\mathbf{k}|\gg 1/r_{H}$ in the space-time region with $\textrm{supp}(\psi_{\gamma})\neq\varnothing$ close to the black-hole horizon.

In quantum theory, this wave package gives rise to the following one-photon state:

[TABLE]

where the photon-polarisation index has been suppressed. The state $|\mathbf{p}\rangle\equiv\sqrt{2\omega_{\mathbf{p}}}\hat{a}_{\mathbf{p}}^{\dagger}|vac\rangle$ is the momentum-operator (non-normalisable) eigenstate and

[TABLE]

with the normalisation factor $N$ and the momentum variance $\sigma$. In what follows, we choose that $\sigma\ll|\mathbf{k}|^{2}$ in order to have the photon of the momentum $\mathbf{k}$. The momentum variance $\sigma$ implies that $|\psi_{\gamma}(\bar{x})\rangle$ is described by the position variance of order $1/\sigma$ in the vicinity of the point $\bar{x}$, in accordance with the Heisenberg uncertainty relation. Assuming that $r_{H}|\mathbf{k}|\gg 1$, we obtain $|\psi_{\gamma}(\bar{x})\rangle$ is localised at $\bar{x}$ with the support of size much less than $r_{H}$. This all means that $|\psi_{\gamma}\rangle$ (with the unspecified position in spacetime) is the one-photon state which is sufficiently localised near the black-hole horizon of radius $r_{H}$, such that we are allowed to apply the powerful machinery of QFT used in elementary particle physics to the photon $|\psi_{\gamma}\rangle$.

The position-space probability amplitude for the photon to propagate from $\bar{x}^{\prime}$ to $\bar{x}$ reads

[TABLE]

which reaches its maximal value if the photon propagates along the classical trajectory: $\mathbf{x}=\mathbf{x}^{\prime}+(\mathbf{k}/\omega_{\mathbf{k}})(\tau-\tau^{\prime})$. The space-time interval $s(\bar{x},\bar{x}^{\prime})$ is then given by

[TABLE]

where we have taken into account that $|\mathbf{k}|\gg|m_{\gamma}|$ for the photon described by $|\psi_{\gamma}\rangle$. Since $(\bar{x}-\bar{x}^{\prime})^{2}<0$ at one-loop level near the event horizon, it is tempting to conclude that this high-energy photon may be able to propagate out of the black-hole horizon if it was initially inside.

The result (16) is in apparent tension with the causality in QED. The causality principle belongs to one of the postulates in the Wightman axiomatic approach to QFT Haag . Specifically, the notion of the Einstein causality in local quantum field theory refers to the circumstance that two local observables $\hat{O}(\bar{x})$ and $\hat{O}(\bar{x}^{\prime})$ commute with each other if these are space-like separated, i.e. $(\bar{x}-\bar{x}^{\prime})^{2}<0$. But the commutator $[\hat{O}(\bar{x}),\hat{O}(\bar{x}^{\prime})]$ is generically not a c-number in interacting QFTs Haag and, thus, depends on a quantum state under consideration.

As an example, one can choose $\hat{O}(\bar{x})$ be the particle-number operator $\hat{N}(\bar{x})$, which gives the number of photons in the vicinity of the point $\bar{x}$. This means that

[TABLE]

In practice, $\hat{N}$ corresponds to a certain single-photon detector (SPD), which clicks if the photon and the detector have overlapping supports.222In case of charged particles, $\hat{N}$ can be realised, for instance, with the help of the Wilson cloud chamber, where $\textrm{supp}(N)$ corresponds to the space-time domain where that is located. A charged particle can then be observed if it penetrates inside the chamber. On the contrary, if this charged particle does not pass through the space-time region where the cloud chamber is, then the particle remains unobserved. In the vacuum state $|vac\rangle$, one has that

[TABLE]

The result is trivial for any $\bar{x}$ and $\bar{x}^{\prime}$, because there are no real photons in $|vac\rangle$ and SPD cannot, therefore, click. On the other hand, one can consider the following state:

[TABLE]

This describes a quantum state in the near-horizon region, corresponding to one photon of the momentum $\mathbf{k}$, which hovers somewhere near the black-hole horizon. In the non-vacuum state $|hor\rangle$, one has that

[TABLE]

where we have assumed that both SPDs located at $\bar{x}$ and $\bar{x}^{\prime}$ click. Normally, $A_{\gamma}(\bar{x},\bar{x}^{\prime})$ is essentially zero if $\bar{x}$ and $\bar{x}^{\prime}$ are space-like separated, which holds for any particle from the Standard Model. That is what we experience on earth. In the near-horizon region, $A_{\gamma}(\bar{x},\bar{x}^{\prime})$ gets, however, its maximum for $\bar{x}$ and $\bar{x}^{\prime}$, which are slightly space-like separated as follows from (15) and (16).333In the electron-positron plasma, $A_{\gamma}(\bar{x},\bar{x}^{\prime})$ is, in contrast, essentially non-vanishing for time-like separated $\bar{x}$ and $\bar{x}^{\prime}$, because the effective QED-photon mass is real there Weldon . This means that the Einstein causality may be violated near the horizon of evaporating Schwarzschild black holes, as a result of the non-classical quantum properties of $|vac\rangle$ at $r\sim r_{H}$.

On the other hand, it is usually expected that the negative (Lagrangian) mass-squared $-\mu^{2}<0$ inevitably leads to the vacuum instability. The prime example is the spontaneous symmetry breaking in the standard electroweak theory. This happens, because the Higgs-field modes with $|\mathbf{k}|^{2}$ less than the negative-mass-squared absolute value, $\mu^{2}$, correspond to imaginary-frequency modes, which grow exponentially with time. But if these imaginary-frequency modes are not taken in the quantum-field expansion, then the quantum vacuum is stable, but the theory is non-causal Aharonov&Komar&Susskind .

In the case under consideration, we have that $\mu=|m_{\gamma}|$, where $|m_{\gamma}|\sim eT_{H}$ holds, i.e. the origin of $\mu\neq 0$ is quantum in nature as, first, $T_{H}\propto\hbar$ and, second, stems from one-loop order in perturbation theory, which is non-existent in the classical limit, i.e. $\hbar\rightarrow 0$. Moreover, $1/\mu$ is much larger than the curvature length near the horizon, which is of the order of $r_{H}$. For two space-time points $\bar{x}$ and $\bar{x}^{\prime}$ with $|\bar{x}-\bar{x}^{\prime}|\sim r_{H}$, the local curvature is non-negligible and, thus, the local Minkowski geometry is no longer a good approximation, e.g.

[TABLE]

for $|\bar{x}-\bar{x}^{\prime}|\sim r_{H}$ near the horizon. This means that the notion of the photon described by the wave package of size of order $r_{H}$ and larger is meaningless within the framework of elementary particle physics in the near-horizon region. Since the one-loop result $m_{\gamma}^{2}\neq 0$ has been obtained within that framework, it cannot be applied to those low-energy photons. These photons cannot even be defined as belonging to one of the unitary and irreducible representations of the Poincaré group, which is a isometry group of the local Minkowski geometry. In general, the Lehmann-Symanzik-Zimmermann reduction formula relating physical particle states with poles in the Feynman propagators is then no more working and, thus, elementary particle physics becomes physically obscure.

Consequently, the perturbative result $m_{\gamma}^{2}<0$ does not imply that the quantum-vacuum instability, that normally happens due to the frequency modes with $|\mathbf{k}|<\mu$ in theories with the negative (Lagrangian) mass-squared term, has to occur in the case considered, because the one-loop computation providing $m_{\gamma}^{2}\neq 0$ is not applicable to wave packages with $r_{H}|\mathbf{k}|<1$. But $m_{\gamma}^{2}\neq 0$ is applicable to the high-energy photons, which are described by wave packages of the type $|\psi_{\gamma}\rangle$.444It is worth noting that this observation also agrees with the expectation that radiative corrections should be small in perturbation theory. In particular, this means that the dispersion relation in perturbation theory can be only marginally modified: $\omega_{\mathbf{k}}=(|\mathbf{k}|^{2}-|m_{\gamma}^{2}|)^{\frac{1}{2}}=|\mathbf{k}|+\textrm{O}(e^{2})$. This circumstance also appears to exclude an extremely rapid black-hole-instability scenario, which occurs in field theories with the faster-than-light phase velocity Barbado&Barcelo&Garay&Jannes (incidentally, the result (6) leads to the slower-than-light phase velocity).

If the causality-violation effect happens in nature near evaporating black holes, then the real photons we could be interested in must not be ultra-high-energy ones in order to be able to propagate out of the black-hole horizon. This extra condition appears from the expression for the photon group velocity, $\mathbf{k}/\omega_{\mathbf{k}}$, for which we have that

[TABLE]

in the case of primordial evaporating black holes of today’s mass $M\ll 10^{16}\,\textrm{g}$. For astrophysical black holes, the upper bound for the photon group velocity is extremely close to unity. Therefore, the Einstein acausality, in favour of which our result turns out to decide, should not imply any tension with the up-to-date observations on earth.

IV Conclusions

As a consequence of the black-hole evaporation, the high-energy-enough QED photon acquires a gauge-invariant mass-squared at one-loop order in perturbation theory. The mass-squared $m_{\gamma}^{2}$ turns out to be of order $e^{2}T_{H}^{2}$ and negative. The latter is directly related to the negativeness of the energy density of the inward quantum flux in the near-horizon region. This implies that the group velocity (with which real elementary particles propagate) of the photon is slightly larger than the speed of light in the Minkowski vacuum in the absence of external fields, i.e. $|\partial\omega_{\mathbf{k}}/\partial\mathbf{k}|>c$. The high-energy photons may, thus, potentially escape from the interior region of black holes. This kind of strongly red-shifted photons reaching the observer at spatial infinity can probably bring some information about the inner structure of evaporating black holes.