\section{Introduction}

Black holes are shown to carry soft hairs \cite{Hawking:2016msc,Compere:2016hzt,Afshar:2016wfy,Mao:2016pwq,Grumiller:2016kcp,Hawking:2016sgy,Chu:2018tzu,Choi:2018oel,Choi:2019fuq,Takeuchi:2021rrq}. The soft hairs in three dimensions are purely microscopic \cite{Afshar:2016uax,Sheikh-Jabbari:2016npa}. In four dimensions, the same feature has been verified by checking the Hawking radiation of soft-hairy black holes \cite{Chu:2018tzu,Javadinezhad:2018urv,Compere:2019rof,Lin:2020gva,Takeuchi:2021ibg}. More precisely, the Hawking fluxes from Schwarzschild black hole is unaffected by their linearized supertranslation hair \cite{Chu:2018tzu}. The spectrum of Hawking radiation of the Schwarzschild black hole is unchanged after including the dressing of asymptotic states with clouds of soft photons and soft gravitons \cite{Javadinezhad:2018urv}. Unruh effect is invariant under supertranslations \cite{Compere:2019rof}. The Hawking flux from Schwarzschild with non-linear supertranslation hair is still the same as the Schwarzschild's one \cite{Lin:2020gva} (see also \cite{Takeuchi:2021ibg}). However counter examples are presented in \cite{Chu:2018tzu,Chiang:2020lem,Wen:2021ahw}. In particular, the Hawking temperature of dynamical black holes is found to be modified by soft hair \cite{Chu:2018tzu,Chiang:2020lem}. Complementary to these investigations where only scalar particles are involved, we study in this paper the Hawking radiation of Dirac particles from soft-hairy black holes using the tunneling method.

The tunneling method proposed by Parikh and Wilczek \cite{Parikh:1999mf} is a very effective way of studying Hawking radiation. In general, the tunneling method involves calculating the imaginary part of the action for the classically forbidden path to evaluate the radiation spectrum. Applying the tunneling method, we examine the Hawking radiation of Dirac particles from Schwarzschild and Vaidya black holes with linear supertranslation hair. We find that the radiation spectrums and the Hawking temperatures in both cases are unchanged after soft hair implant on black holes. It may not be surprising that the linear supertranslation hair of Schwarzschild black hole does not affect the Hawking radiation of Dirac particles as the scalar case \cite{Chu:2018tzu}. For the Vaidya case, our result is different from the one of scalar particles in \cite{Chu:2018tzu}. When applying the  tunneling method for Dirac particles, one needs to properly take care of the angular-dependent part of the action \cite{Kerner:2007rr,Li:2008zra,DiCriscienzo:2008dm,Chen:2008vi,Kerner:2008qv,Chen:2008ge,Jiang:2008gq,Chen:2014xgj}. The simplest assumption that solving only the radial equation at a fixed angle may not be a solution of the Dirac equation. We find that the linear supertranslation hair dependence in the Hawking spectrum of the Vaidya black hole is precisely canceled by taking account the contribution from the angular-dependent part. Our results furnish an important confirmation of the microscopic nature of soft hairs in four dimensions.




\section{Schwarzschild black hole with supertranslation hair}

The line element of the supertranslated Schwarzschild black hole in the advanced Bondi coordinates reads \cite{Hawking:2016sgy}
\begin{multline}\label{sch-super}
\td s^2=-\left(V-\frac{M D^2f}{r^2}\right) \td v^2 + 2 \td v \td r - D_A \left(2 V f + D^2 f \right) \td v \td \Theta^A \\
+\left(r^2 {\bar\gamma}_{AB} + 2 r D_A D_B f - r {\bar\gamma}_{AB} D^2 f \right) \td \Theta^A \td \Theta^B \ , \quad V=1-\frac{2M}{r} \ ,
\end{multline}
where ${\bar\gamma}_{AB}$ is the metric of the unit two sphere on the boundary and $D_A$ is the covariant derivative with respect to ${\bar\gamma}_{AB}$, $\Theta^A=(z, \bar z)$ are the angular coordinates, $f$ is an arbitrary function of the angular variables that characterizes the supertanslation. Up to order $f$ term, $g_{vv}=-\frac{1}{r^2}(r-r_h)(r+r_f)$ where $r_f=\frac12 D^2 f$ and  $r_h=2M + \frac12 D^2 f$ is the location of event horizon. Since the supertranslation is infinitesimal, we drop the terms with order higher than $\cO(f)$. Physically, the solution can be considered as throwing in an asymmetric null shock into the Schwarzschild black hole.

The crucial step of computing Hawking radiation by tunneling method is the application of WKB approximation to the covariant Dirac equation \cite{Kerner:2007rr}. We consider massive spinor field $\Psi$ with mass $\mu$. The covariant Dirac equation reads
\begin{equation}\label{Dirac}
i \hbar \gamma^a e_a^\mu \nabla_\mu \Psi - \mu \Psi = 0 \ ,
\end{equation}
where $\nabla_\mu$ is the spinor-covariant derivative defined by $\nabla_\mu=\partial_\mu + \frac14 {\omega^{ab}}_\mu \gamma_{[a}\gamma_{b]}$ and ${\omega^{ab}}_\mu$ is the spin connection. The cotetrad fields $e^a_\mu$ according to the metric \eqref{sch-super} are chosen as
\begin{align}
&e^0_\mu=\left[ -\frac12 g_{vv}, -1 , - g_{vA}\right] \ ,\\
&e^1_\mu=\left[1,0,0,0\right] \ ,\\
&e^2_\mu=\left[0,0,f_1(x^\mu),f_2(x^\mu)\right] \ ,\\
&e^3_\mu=\left[0,0,0,f_3(x^\mu)\right] \ ,
\end{align}
where $f_1=\sqrt{g_{zz}},\,f_2=\frac{g_{z\bar z}}{\sqrt{g_{zz}}},\,f_3=\sqrt{g_{z\bar z} - \frac{g_{z\bar z}^2}{g_{zz}}}$ and the fundamental matrix is of the form
\begin{equation}
\eta_{ab}=\eta^{ab}=\begin{pmatrix}
0 & -1 & 0 & 0  \\ -1 & 0 & 0 & 0  \\ 0 & 0 & 1 & 0  \\ 0 & 0 & 0 & 1
\end{pmatrix}\ .
\end{equation}
We will pick for the $\gamma$ matrices
\begin{align}
&\gamma^0=\frac{1}{\sqrt{2}}\begin{pmatrix}
i & 0 \\ 0 & - i
\end{pmatrix}  + \frac{1}{\sqrt{2}} \begin{pmatrix}
0 & \sigma^3 \\ \sigma^3 & 0
\end{pmatrix}\ ,\quad \gamma^2=\begin{pmatrix}
0 & \sigma^1 \\ \sigma^1 & 0
\end{pmatrix}\ ,\\
&\gamma^1=\frac{1}{\sqrt{2}}\begin{pmatrix}
i & 0 \\ 0 & - i
\end{pmatrix}  - \frac{1}{\sqrt{2}} \begin{pmatrix}
0 & \sigma^3 \\ \sigma^3 & 0
\end{pmatrix} \ ,
\quad
\gamma^3=\begin{pmatrix}
0 & \sigma^2 \\ \sigma^2 & 0
\end{pmatrix}\ ,
\end{align}
where the matrices $\sigma^k(k = 1,2,3)$ are the Pauli matrices. The $\gamma^a$ matrices satisfy the anti-commutation relations $\{\gamma^a,\gamma^b\}=2\eta^{ab}\times \mathbb{1}$.
The spin-up and spin-down spinor fields are assumed to have the form \cite{Kerner:2007rr}
\begin{equation}
\Psi_{\uparrow}=\begin{pmatrix}
A(x^\mu)  \\  0  \\ B(x^\mu)   \\ 0
\end{pmatrix}\text{exp}\left[\frac{i}{\hbar}I_{\uparrow}(x^\mu)\right] \ ,\quad
\Psi_{\downarrow}=\begin{pmatrix}
0 \\ C(x^\mu)  \\  0  \\ D(x^\mu)
\end{pmatrix}\text{exp}\left[\frac{i}{\hbar}I_{\downarrow}(x^\mu)\right] \ .
\end{equation}
We will only demonstrate the spin-up case explicitly while the spin-down case is fully in analogue.

Inserting the spin-up ansatz into the covariant Dirac equation \eqref{Dirac} and applying the WKB approximation, one can arrive at the following four equations at the leading order in the expansion of $\hbar$:
\begin{align}
&\sqrt{2} B \left[\partial_r I (g_{vv} - 2) - 2 \partial_v I \right] + A \left[4 \mu + \sqrt2 i (2 + g_{vv}) \partial_r I - 2\sqrt2 i \partial_v I \right] = 0 \ ,\\
&B \left[ (f_2 + i f_3 ) (\partial_z I - g_{vz} \partial_r I) - f_1 (\partial_{\bar z} I - g_{v\bar z} \partial_r I ) \right]= 0 \ ,\label{2}\\
&\sqrt{2} A \left[\partial_r I (g_{vv} - 2) - 2 \partial_v I \right] + B \left[4 \mu - \sqrt2 i (2 + g_{vv}) \partial_r I + 2\sqrt2 i \partial_v I \right] = 0 \ ,\\
&A \left[ (f_2 + i f_3 ) (\partial_z I - g_{vz} \partial_r I) - f_1 (\partial_{\bar z} I - g_{v\bar z} \partial_r I ) \right]= 0 \ .\label{4}
\end{align}
For the first and third equations, the non-trivial solution for A and B exists if and only if the determinant of the coefficient matrix vanishes. Then we obtain
\begin{equation}\label{r}
g_{vv} (\partial_r I)^2 - 2 \partial_v I \partial_r I + \mu^2= 0 \ .
\end{equation}
Equations \eqref{2} and \eqref{4} yield the same equation regardless of $A$ or $B$, i.e.,
\begin{equation}\label{angle}
(f_2 + i f_3 ) (\partial_z I - g_{vz} \partial_r I) - f_1 (\partial_{\bar z} I - g_{v\bar z} \partial_r I )=0 \ .
\end{equation}
Because $\frac{\partial}{\partial_v}$ is a Killing vector in the supertranslated spacetime, we can separate the variables for $I(v,r,z,\bar z)$ as follows:
\begin{equation}
I =- \omega v + R(r,z,\bar z) + K \ ,
\end{equation}
where $\omega$ is the Dirac particle's energy and $K$ is a complex constant. Then \eqref{r} can be solved out as
\begin{align}
&R_+=\int \td r \, \frac{\omega + \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} + \Phi_+(z,\bar z) \ ,\label{+} \\
&R_-=\int \td r \, \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} + \Phi_-(z,\bar z) \ ,\label{-}
\end{align}
where $R_+$ denotes the outgoing solution and $R_-$ denotes the incoming solution. The angular-dependent part of the action for outgoing solution is usually the same as the incoming solution for rotating black holes, see, e.g., \cite{Li:2008zra,DiCriscienzo:2008dm,Chen:2008vi,Kerner:2008qv,Chen:2008ge,Jiang:2008gq,Chen:2014xgj}. Though it is not necessary to be a real function, it will be canceled out when computing the probability of tunneling. However in the present case, equation \eqref{angle} involves the $r$-derivative of the action. A priori, it is necessary to consider them as different functions. Nevertheless we will argue that $\Phi_+(z,\bar z)$ and $\Phi_-(z,\bar z)$ can have real solutions. We will consider the case that the two expressions $\partial_z I - g_{vz} \partial_r I$ and $\partial_{\bar z} I - g_{v\bar z} \partial_r I$ in \eqref{angle} are complex conjugate to each other. The complex conjugate of the first expression for the incoming solution is
\begin{equation}
\partial_{\bar z} \left[\int \td r \, \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} + \xbar\Phi_-(z,\bar z)\right] - g_{v\bar z} \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} \ ,
\end{equation}
where we have used the fact that $\int \td r \, \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}}$ is real and $\xbar\Phi_-(z,\bar z)$ is the complex conjugate function of $\Phi_-(z,\bar z)$. Comparing this equation to the second expression
\begin{equation}
\partial_{\bar z} \left[\int \td r \, \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} + \Phi_-(z,\bar z)\right] - g_{v\bar z} \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} \ ,
\end{equation}
one obtains that $\xbar\Phi_-(z,\bar z)=\Phi_-(z,\bar z)$. Hence $\Phi_-(z,\bar z)$ is real.



For the outgoing solution, the expression $\int \td r \, \frac{\omega + \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}}$ is not real. But the imaginary part of this expression is $4\pi \omega M$ \cite{Chu:2018tzu}, i.e., a constant that will be killed by a $z$ or $\bar z$ derivative. Hence, one can show that $\Phi_+(z,\bar z)$ is real following the same argument as the incoming case.

The probabilities of crossing the outer horizon each way are respectively given by
\begin{align}
&P_{\text{out}}=\text{exp}\left[-2 \text{Im}I_+\right]= \text{exp}\left[- 8\pi \omega M  - 2 \text{Im}K \right]\ ,\\
&P_{\text{in}}=\text{exp}\left[-2 \text{Im}I_-\right]=\text{exp}\left[-2 \text{Im}K\right] \ ,
\end{align}
where we set $\hbar$ to unity. Any incoming classical particles should have a $100\%$ chance of entering the black hole horizon. So we set $K=0$. This implies that the probability of a particle tunneling from inside to outside of the horizon is given by
\begin{equation}
\Gamma= \text{exp}\left[- 8\pi \omega M\right] \ ,
\end{equation}
which recovers the result of the scalar particles tunneling from the supertranslated Schwarzschild black hole \cite{Chu:2018tzu}. The Hawking radiation is independent of the supertranslation hair of the black hole. Identifying $\Gamma$ with the Boltzmann factor, exp$(-\frac{\omega}{T} )$, we can read off the Hawking
temperature as $\frac{1}{8\pi M}$.




\section{Vaidya black hole with supertranslation hair}

The line element of the supertranslated Vaidya black hole in the advanced Bondi coordinates is \cite{Chu:2018tzu}
\begin{multline}
\td s^2=-\left(V - \frac{2fM'(v)}{r} - \frac{M D^2f}{r^2}\right) \td v^2 + 2 \td v \td r -  D_A \left(2 V f + D^2 f \right) \td v \td \Theta^A \\
+\left(r^2 \gamma_{AB} + 2 r D_A D_B f - r \gamma_{AB} D^2 f \right) \td \Theta^A \td \Theta^B \ ,
\end{multline}
where $V=1-\frac{2M(v)}{r}$ and $M'(v)=\frac{\td M(v)}{\td v}$. Up to order $f$ term, $g_{vv}=-\frac{1}{r^2}(r-r_h)(r+r_f)$ where $r_f=\frac12 D^2 f$ and $r_h=2M + 2f M' + \frac12 D^2 f$ is the location of trapping horizon. Physically, the solution can be considered as collapse of a certain energy momentum tensor with soft charges.

The line element of the supertranslated Vaidya black hole has the same form as the supertranslated Schwarzschild one. Hence the covariant Dirac equation with the same choice of $\gamma$ matrices and the same ansatz for spin-up spinor fields, after applying the WKB approximation, will lead to the same formulas as the supertranslated Schwarzschild case, i.e., \eqref{r} and \eqref{angle}. Following closely \cite{Chu:2018tzu}, we define the invariant energy of particle $\omega$ in favor of the Kodama-like vector which yields $\partial_v I=-\omega$. Substituting this back to \eqref{r}, one obtains
\begin{align}
&I_+=-\omega v + \int \td r \, \frac{\omega + \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} + \Phi_+(z,\bar z) + K \ ,\label{plus} \\
&I_-=-\omega v + \int \td r \, \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}} + \Phi_-(z,\bar z) + K \ .\label{minus}
\end{align}
Since function $\int \td r \, \frac{\omega - \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}}$ is real \cite{Chu:2018tzu}, $\Phi_-(z,\bar z)$ is real applying the same argument as previous section. We set $K=0$ due to the fact that the probability of any incoming classical particles crossing the horizon is unity. A direct computation shows that \cite{Chu:2018tzu}
\begin{equation}\label{Im}
\text{Im}\int \td r \, \frac{\omega + \sqrt{\omega^2 - \mu^2 g_{vv}}}{-g_{vv}}=4\pi \omega M + 4\pi \omega f M' \ ,
\end{equation}
where higher orders in $f$ has been neglected. Applying the same argument as previous section, \eqref{Im} will lead to
\begin{equation}
\text{Im}\Phi_+(z,\bar z)=-4\pi \omega  f M' \ .
\end{equation}
Hence
\begin{equation}
\text{Im}I_+=4\pi \omega M \ .
\end{equation}
In the end, the probability of a particle tunneling from inside to outside of the horizon is given by
\begin{equation}
\Gamma= \text{exp}\left[- 8\pi \omega M\right] \ ,
\end{equation}
which is different from the result of scalar particles tunneling from the supertranslated Vaidya black hole \cite{Chu:2018tzu}. Note that the Hawking radiation is again independent of the supertranslation hair when taking into account the angular contribution. By identifying $\Gamma$ with the Boltzmann factor, exp$(-\frac{\omega}{T} )$, the Hawking
temperature is $\frac{1}{8\pi M}$.








\section{Conclusion}


In this work, we examine the semi-classical property of soft hair implants on black holes. Applying the tunneling method, we compute the Hawking radiation of Dirac particles from Schwarzschild and Vaidya black holes with linear supertranslation hair. Our calculation provides the first example that Hawking radiation of dynamical black hole is not modified by supertranslation hair. This is another evidence to support the microscopic nature of soft hairs.





\section*{Acknowledgments}

This work is supported in part by the National Natural Science Foundation of China under Grant No. 11905156, No. 11975164 and No. 11935009.



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