Zero, Normal and Super-radiant Modes for Scalar and Spinor Fields in Kerr-anti de Sitter Spacetime

The Lagrangian and field equations for complex scalar fields $\Phi(x)$ in Kerr-AdS spacetime is given by:

	$\displaystyle\mathcal{L}=g^{\mu\nu}\partial_{\mu}\Phi^{*}(x)\partial_{\nu}\Phi(x)-\bar{\mu}^{2}\Phi^{*}(x)\Phi(x),$		(2.2)
	$\displaystyle(\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}-\bar{\mu}^{2})\Phi(x)=0,$		(2.3)

where $\bar{\mu}$ denotes the mass of scalar fields. If we write $\Phi(x)$ as

$\displaystyle\Phi={\rm e}^{-i\omega t}{\rm e}^{im\phi}S(\theta)R(r),$

(2.4)

then the scalar field equations are separated into angular and radial field equations respectively:

	$\displaystyle\left(\frac{\partial_{\theta}\sin{\theta}\Delta_{\theta}\partial_{\theta}}{\sin{\theta}}-\frac{(a\omega\sin{\theta}-\Xi m/\sin{\theta})^{2}}{\Delta_{\theta}}-\bar{\mu}^{2}a^{2}\cos^{2}{\theta}+\lambda\right)S(\theta)$	$\displaystyle=$	$\displaystyle 0$		(2.5)
	$\displaystyle\left(\partial_{r}\Delta_{r}\partial_{r}+\frac{((r^{2}+a^{2})\omega-\Xi am)^{2}}{\Delta_{r}}-\bar{\mu}^{2}r^{2}-\lambda\right)R(r)$	$\displaystyle=$	$\displaystyle 0,$		(2.6)

where $\omega,m$, and $\lambda$ denote the frequency, azimuthal angular momentum of scalar fields, and the separation parameter, respectively.

Now we study the normal modes as bound states with Dirichlet or Neumann boundary conditions on the horizon:

$\displaystyle R(r)=0\ \ \ \mbox{or}\ \ \partial_{r}R(r)=0\ \ \ {\rm on}\ \ \ r=r_{\rm H},$

(2.7)

with asymptotic condition:

$$R(r)\rightarrow 0\ \ \ {\rm as}\ \ \ r\rightarrow\infty.$$

(2.8)

¹¹1Note that we introduce a small positive constant $\epsilon$ in eq. (2.7) as $r_{+}+\epsilon$ for regularization in case of necessity.

The following two identities can be obtained from eqs. (2.5) and (2.6):

	$\displaystyle\int_{0}^{\pi}d\theta\sin{\theta}\left((\omega^{*}-\omega^{\prime})\frac{a^{2}\sin^{2}{\theta}(\omega^{*}+\omega^{\prime})-2a\Xi m}{\Delta_{\theta}}-(\lambda^{*}-\lambda^{\prime})\right)$
	$\displaystyle\hskip 40.00006pt\times S_{\omega,m,\lambda}^{*}(\theta)S_{\omega^{\prime},m,\lambda^{\prime}}(\theta)=0,$		(2.9)

for angular part and

	$\displaystyle\int_{r_{\rm H}}^{\infty}dr\left((\omega^{*}-\omega^{\prime})\frac{(r^{2}+a^{2})^{2}(\omega^{*}+\omega^{\prime})-2(r^{2}+a^{2})\Xi am}{\Delta_{r}}-(\lambda^{*}-\lambda^{\prime})\right)$
	$\displaystyle\hskip 20.00003pt\times R_{\omega,m,\lambda}^{*}(r)R_{\omega^{\prime},m,\lambda^{\prime}}(r)=0,$		(2.10)

for radial part.

Let us define $X_{(\omega,\omega^{\prime},\lambda,\lambda^{\prime})}$ as follows

	$\displaystyle X_{(\omega,\omega^{\prime},\lambda,\lambda^{\prime})}:={\displaystyle\int_{r_{\rm H}}^{\infty}}dr\int_{0}^{\pi}d\theta$
	$\displaystyle\times\frac{\sqrt{-g}}{\rho^{2}}\left(\left(-\frac{a^{2}\sin^{2}{\theta}}{\Delta_{\theta}}+\frac{(r^{2}+a^{2})^{2}}{\Delta_{r}}\right)(\omega+\omega^{\prime})+\left(\frac{1}{\Delta_{\theta}}-\frac{r^{2}+a^{2}}{\Delta_{r}}\right)2\Xi am\right)$
	$\displaystyle\times S_{\omega,m,\lambda}^{*}(\theta)S_{\omega^{\prime},m,\lambda^{\prime}}(\theta)R_{\omega,m,\lambda}^{*}(r)R_{\omega^{\prime},m,\lambda^{\prime}}(r).$		(2.11)

Then from eqs. (2.9) and (2.10) we can obtain the following equations:

	$\displaystyle(\omega^{*}-{\omega}^{\prime})X_{(\omega,{\omega}^{\prime},\lambda,{\lambda}^{\prime})}$	$\displaystyle=$	$\displaystyle 0,$		(2.12)
	$\displaystyle(\lambda^{*}-{\lambda}^{\prime})X_{(\omega,{\omega}^{\prime},\lambda,{\lambda}^{\prime})}$	$\displaystyle=$	$\displaystyle 0,$		(2.13)

which reduce to the real values of $\omega,\lambda$ and the orthonormal relations:

$\displaystyle X_{(\omega,\omega^{\prime},\lambda,\lambda^{\prime})}=\delta_{\omega,\omega^{\prime}}\delta_{\lambda,\lambda^{\prime}}.$

(2.14)

Let us define the inner product notation of $A$ and $B$ as follows

$\displaystyle<A\ ,\ B>:=\int_{\Sigma}d^{3}x\sqrt{-g}(-ig^{t\nu})\left(A^{*}(t,x)\partial_{\nu}B(t,x)-\partial_{\nu}A^{*}(t,x)B(t,x)\right)\ .$

The orthonormal relations are expressed:

	$\displaystyle<f_{\alpha}\ ,\ f_{\alpha^{\prime}}>$	$\displaystyle=$	$\displaystyle-<f_{\alpha}^{*}\ ,\ f_{\alpha^{\prime}}^{*}>=\delta_{\alpha,\alpha^{\prime}}^{(3)}$
	$\displaystyle<f_{\alpha}^{*}\ ,\ f_{\alpha^{\prime}}>$	$\displaystyle=$	$\displaystyle<f_{\alpha}\ ,\ f_{\alpha^{\prime}}^{*}>=0\ ,$		(2.15)

with the eigenfunctions

$$f_{\omega,m,\lambda}:=\frac{1}{\sqrt{2\pi}}{\rm e}^{-i\omega t}{\rm e}^{im\phi}S_{\omega,m,\lambda}(\theta)R_{\omega,m,\lambda}(r)\ .$$

where $\alpha=(\omega,m,\lambda)$ and $\alpha^{\prime}=(\omega^{\prime},m^{\prime},\lambda^{\prime})$ denote set of quantum numbers.

Let us expand the scalar field $\Phi$ and its conjugate momentum $\Pi$ in eigenfunctions $f_{\alpha},f_{\alpha}^{*}$ as follows

	$\displaystyle\Phi(t,x)$	$\displaystyle=$	$\displaystyle\sum_{\alpha}\left(a_{\alpha}f_{\alpha}(t,x)+b_{\alpha}^{\dagger}f_{\alpha}^{*}(t,x)\right)\ ,$
	$\displaystyle\Pi(t,x)$	$\displaystyle=$	$\displaystyle\frac{\partial\mathcal{L}}{\partial\partial_{t}\Phi}=-i\sum_{\alpha}(g^{tt}\omega-g^{t\phi}m)(a_{\alpha}^{\dagger}f_{\alpha}^{*}(t,x)-b_{\alpha}f_{\alpha}(t,x)).$		(2.16)

The inverse relations can be found to

	$\displaystyle a_{\alpha}$	$\displaystyle=$	$\displaystyle\int_{\Sigma}d^{3}x\sqrt{-g}\left(if_{\alpha}^{*}(t,x)\Pi^{\dagger}(t,x)-(g^{tt}\omega-g^{t\phi}m)f_{\alpha}^{*}(t,x)\Phi(t,x)\right)\ ,$
	$\displaystyle b_{\alpha}^{\dagger}$	$\displaystyle=$	$\displaystyle-\int_{\Sigma}d^{3}x\sqrt{-g}\left(if_{\alpha}(t,x)\Pi^{\dagger}(t,x)+(g^{tt}\omega-g^{t\phi}m)f_{\alpha}(t,x)\Phi(t,x)\right),$		(2.17)

using the completeness relations:

$$\sum_{\alpha}(-ig^{t\nu})\left(f_{\alpha}^{*}(t,x)\partial_{\nu}f_{\alpha}(t,x^{\prime})-\partial_{\nu}f_{\alpha}^{*}(t,x)f_{\alpha}(t,x^{\prime})\right)=\frac{1}{\sqrt{-g}}\delta^{(3)}(x-x^{\prime}).$$

(2.18)

Note that any states, including quasi-normal modes and super-radiant modes, can express in terms of eigenfunctions due to the completeness relation (2.18).

Canonical quantization relations are given by

$\displaystyle\left[\Phi(t,x),\Pi(t,x^{\prime})\right]=\left[\Phi^{\dagger}(t,x),\Pi^{\dagger}(t,x^{\prime})\right]=\frac{i}{\sqrt{-g}}\delta^{(3)}(x-x^{\prime}),$

(2.19)

which lead to the quantization of creation and annihilation operators:

$$[a_{\alpha},a_{\alpha^{\prime}}^{\dagger}]=[b_{\alpha},b_{\alpha^{\prime}}^{\dagger}]=\delta^{(3)}_{\alpha,\alpha^{\prime}}\ .$$

Energy and angular momentum of scalar fields are given by

	$\displaystyle E$	$\displaystyle=$	$\displaystyle\int_{\Sigma}d^{3}x\sqrt{-g}(-g^{tt}\partial_{t}\Phi^{\dagger}\partial_{t}\Phi+g^{\phi\phi}\partial_{\phi}\Phi^{\dagger}\partial_{\phi}\Phi$
			$\displaystyle+g^{rr}\partial_{r}\Phi^{\dagger}\partial_{r}\Phi+g^{\theta\theta}\partial_{\theta}\Phi^{\dagger}\partial_{\theta}\Phi)$
	$\displaystyle L$	$\displaystyle=$	$\displaystyle\int_{\Sigma}d^{3}x\sqrt{-g}(g^{tt}(\partial_{t}\Phi^{\dagger}\partial_{\phi}\Phi+\partial_{\phi}\Phi^{\dagger}\partial_{t}\Phi)+2g^{t\phi}\partial_{\phi}\Phi^{\dagger}\partial_{\phi}\Phi),$		(2.20)

which can be also written as

$\displaystyle E=\sum_{\alpha}\omega(a_{\alpha}^{\dagger}a_{\alpha}+b_{\alpha}b_{\alpha}^{\dagger})\ \ \ ,\ \ \ L=\sum_{\alpha}m(a_{\alpha}^{\dagger}a_{\alpha}+b_{\alpha}b_{\alpha}^{\dagger})\ .$

The effective energy of scalar field, as measured by the co-rotating observer, is given by

$\displaystyle E-L\Omega_{\rm H}=\sum_{\alpha}(\omega-\Omega_{\rm H}m)(a_{\alpha}^{+}a_{\alpha}+b_{\alpha}b_{\alpha}^{\dagger}),$

(2.21)

where

$$\Omega_{\rm H}=a\Xi/(r_{\rm H}^{2}+a^{2}),$$

(2.22)

denotes the angular velocity of BH on the horizon $r_{\rm H}$.

Note that for normal modes $(\omega-\Omega_{\rm H}m\neq 0)$ near the horizon, we have ingoing and outgoing solutions with the momentum $p_{\rm H}$ with respect to a radial variable $r_{*}=\int dr(r^{2}+a^{2})/\Delta_{r}$ as $\exp{(-ip_{\rm H}r_{*})}$ and $\exp{(ip_{\rm H}r_{*})}$. The effective frequency $\omega_{\rm H}$ is the same in magnitude with the momentum $p_{\rm H}$ because scalar field is treated as massless near the horizon:

$$\omega_{\rm H}=p_{\rm H}=\omega-\Omega_{\rm H}m.$$

(2.23)

Now we consider the existence or non-existence of zero mode defined by $\omega-\Omega_{\rm H}m=0$. Radial equation for zero mode becomes:

$\displaystyle\left(\frac{d}{dr}\Delta_{r}\frac{d}{dr}+\frac{(r^{2}-r_{\rm H}^{2})^{2}\omega_{*}^{2}}{\Delta_{r}}-\bar{\mu}^{2}r^{2}-\lambda\right)R_{\rm zero}(r)=0,$

(2.24)

where $\omega_{*}=\Omega_{\rm H}m$ denotes the frequency of zero mode. The zero mode equation near horizon becomes from eq. (2.24):

$$\left(\frac{d}{dr}(r-r_{\rm H})\frac{d}{dr}-{\alpha}\right)R_{\rm zero}(r)\simeq 0,$$

(2.25)

where $\alpha=(\bar{\mu}^{2}r_{\rm H}^{2}+\lambda)/X(r_{\rm H})$ with $X(r)=\partial_{r}\Delta_{r}$. General solution of eq. (2.24) is given by, using the Frobenius method:

$$R_{\rm zero}(r)\simeq c_{1}(1+\alpha(r-r_{\rm H}))+c_{2}(\log(r-r_{\rm H})\,(1+\alpha(r-r_{\rm H}))-2\alpha(r-r_{\rm H})),$$

(2.26)

where $c_{1},c_{2}$ denote integration constants. If we impose the Dirichlet boundary condition, then from eq. (2.26), we find that $c_{1}=c_{2}=0$. Therefore non-trivial solution to eq. (2.25) which satisfies the Dirichlet boundary condition does not exist. We can also show the zero mode solution to eq. (2.25) does not satisfy the Neumann boundary condition either.

From this result we can derive the spectrum condition of normal modes for scalar fields with Dirichlet or Neumann boundary conditions as follows:

• (i)

  When the specific rotation parameter $a=J/M=0$, the zero mode line is on $\omega=0$ and the allowed physical modes are $\omega>0$ in $\omega-m$ plane as in the standard scalar field theory in flat Minkowski spacetime.

• (ii)

  When $a=J/M\neq 0$, the zero mode is defined by the line $\omega-\Omega_{\rm H}m=0$ in $\omega-m$ plane and the allowed physical modes should satisfy the spectrum condition:

  $\displaystyle\omega-\Omega_{\rm H}m>0.$

  (2.27)

  The reason is that any normal modes cannot across the zero mode line where the radial functions cannot satisfying the Dirichlet or Neumann boundary conditions. In this argument we take account of the analyticity with respect to rotation parameter $a$ under conditions that the outer horizon $r_{\rm H}$ should be well separated zero of $\Delta_{r}$ from other zeros and the rotation parameter should be less than the cosmological parameter $a<\ell$ for regularity of $\Delta_{\theta}$ and $\Xi$.

Let us comment on super-radiant modes and the brick wall model.

• (1)

  We can apply the spectrum condition for normal modes to super-radiant modes because any modes can express in terms of eigenfunctions due to the completeness relation in eq. (2.18). Then allowed types of super-radiant modes taking account of general super-radiant condition given in equation (1.1) are the following two:

  1. type 1

     $(\omega>0,\omega-\Omega_{\rm H}m<0)$ is unphysical and does not occur because this type contradicts with the spectrum condition of (2.27).

  2. type 2

     $(\omega<0,\omega-\Omega_{\rm H}m)>0$ are physical which coincides with both equations (1.1) and (2.27).

  In type 2 super-radiance, annihilation operators of particles with $\omega<0,m<0$ are understood as creation operators of antiparticles with $\omega>0,m>0$ as the interpretation in quantized field theory. Also, we mention that the results obtained here are consistent with our previous work basen on the Bargmann-Wigner formulation for scattering problem in Kerr spacetime [18].

• (2)

  The partition function in the Kerr-AdS black hole background spacetime becomes well-defined:

  $\displaystyle Z={\rm Tr}\exp{(-\beta_{\rm H}(E-L\Omega_{\rm H}))}$

  (2.28)

  considering the expression of effective energy (2.21) and the spectrum condition (2.27).

The Dirac equation in general curved spacetime is given by

$\displaystyle(\gamma^{\mu}(\partial_{\mu}+\frac{1}{4}\omega^{ij}_{\mu}\gamma_{ij})+{\bar{\mu}})\Psi(x)=0$

(3.2)

where $\omega^{ij}_{\mu}$, $\gamma_{ij}$, and $\bar{\mu}$ denote the spin connection, the anti-symmetric product of Dirac gamma matrices, and the mass of spinor fields, respectively. The spin connections are divided into two terms:

$\displaystyle\gamma^{\mu}\frac{1}{4}\omega^{ij}_{\mu}\gamma_{ij}$

$\displaystyle=$

$\displaystyle\frac{1}{2\sqrt{-g}}\gamma^{i}\partial_{\mu}(\sqrt{-g}b^{\mu}_{i})+\frac{1}{4}b^{\mu}_{i}b^{\nu}_{j}\partial_{\mu}b_{\nu k}\gamma^{ijk},$

(3.3)

whose derivation is in Appendix C. The first term of the right-hand side of eq. (3.3) is geometric in origin and the second term has spinorial origin. The totally anti-symmetric product of Dirac matrices is denoted by $\gamma^{ijk}$. Each term is calculated as

	$\displaystyle\frac{1}{2\sqrt{-g}}\gamma^{i}\partial_{\mu}(\sqrt{-g}b^{\mu}_{i})$	$\displaystyle=$	$\displaystyle\frac{\gamma^{\tilde{r}}}{2\rho}(\frac{r\sqrt{\Delta_{r}}}{\rho^{2}}+\frac{\partial_{r}\Delta_{r}}{2\sqrt{\Delta_{r}}})$		(3.4)
		$\displaystyle+$	$\displaystyle\frac{\gamma^{\tilde{\theta}}}{2\rho}(-\frac{a^{2}\cos{\theta}\sin{\theta}}{\rho^{2}}\sqrt{\Delta_{\theta}}+\cot{\theta}\sqrt{\Delta_{\theta}}+\frac{\partial_{\theta}\Delta_{\theta}}{2\Delta_{\theta}}),$
	$\displaystyle\frac{1}{4}b^{\mu}_{i}b^{\nu}_{j}\partial_{\mu}b_{\nu k}\gamma^{ijk}$	$\displaystyle=$	$\displaystyle\frac{a}{2\rho^{3}}(\gamma^{rt\phi}r\sin{\theta}\sqrt{\Delta_{\theta}}-\gamma^{\theta t\phi}\cos{\theta}\sqrt{\Delta}_{r}),$		(3.5)

Then the explicit form of Dirac equation in Kerr-AdS spacetime in the Boyer-Lindquist coordinate becomes:

			$\displaystyle\Bigl{(}\frac{\gamma^{0}}{\rho\sqrt{\Delta_{r}}}{((r^{2}+a^{2})\partial_{t}+a\Xi\partial_{\phi})}+\frac{\gamma^{2}}{\rho\sqrt{\Delta_{\theta}}}(a\sin{\theta}\partial_{t}+\frac{\Xi}{\sin{\theta}}\partial_{\phi})$		(3.6)
		$\displaystyle+$	$\displaystyle\frac{\gamma^{3}\sqrt{\Delta_{r}}}{\rho}\partial_{r}+\frac{\gamma^{1}\sqrt{\Delta_{\theta}}}{\rho}\partial_{\theta}$
		$\displaystyle+$	$\displaystyle\frac{\gamma^{3}}{2\rho}(\frac{r\sqrt{\Delta_{r}}}{\rho^{2}}+\frac{\partial_{r}\Delta_{r}}{2\sqrt{\Delta_{r}}})+\frac{\gamma^{1}}{2\rho}(-\frac{a^{2}\cos{\theta}\sin{\theta}}{\rho^{2}}\sqrt{\Delta_{\theta}}+\cot{\theta}\sqrt{\Delta_{\theta}}+\frac{\partial_{\theta}\Delta_{\theta}}{2\Delta_{\theta}})$
		$\displaystyle+$	$\displaystyle\frac{a\gamma_{5}}{2\rho^{3}}(\gamma^{1}r\sin{\theta}\sqrt{\Delta_{\theta}}+\gamma^{3}\cos{\theta}\sqrt{\Delta}_{r})+{\bar{\mu}}\Bigr{)}\Psi(x)=0.$

The representation of Dirac matrices are:

	$\displaystyle\gamma^{\tilde{t}}$	$\displaystyle=$	$\displaystyle\gamma^{0}=-i\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),\ \ \gamma^{\tilde{\theta}}=\gamma^{1}=\left(\begin{array}[]{cc}0&-i\sigma_{1}\\ i\sigma_{1}&0\end{array}\right),$		(3.11)
	$\displaystyle\gamma^{\tilde{\phi}}$	$\displaystyle=$	$\displaystyle\gamma^{2}=\left(\begin{array}[]{cc}0&-i\sigma_{2}\\ i\sigma_{2}&0\end{array}\right),\ \ \gamma^{\tilde{r}}=\gamma^{3}=\left(\begin{array}[]{cc}0&-i\sigma_{3}\\ i\sigma_{3}&0\end{array}\right),$		(3.16)
	$\displaystyle\gamma_{5}$	$\displaystyle=$	$\displaystyle-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-\left(\begin{array}[]{cc}0&I\\ I&0\end{array}\right),$		(3.19)

where $\sigma_{a}\ (a=1,2,3)$ are Pauli matrices.

With the ansatz

$\displaystyle\Psi(x)=\frac{\exp{(-i\omega t+im\phi)}}{(\Delta_{r}\Delta_{\theta}\rho^{2}\sin^{2}{\theta})^{1/4}}F(r,\theta),$

(3.20)

where $\omega,m$ denote the frequency and azimuthal quantum number, respectively, the Dirac equation becomes

			$\displaystyle\Bigl{(}\frac{-i\gamma^{0}}{\rho\sqrt{\Delta_{r}}}{((r^{2}+a^{2})\omega-a\Xi m)}+\frac{-i\gamma^{2}}{\rho\sqrt{\Delta_{\theta}}}(a\sin{\theta}\omega-\frac{\Xi m}{\sin{\theta}})$		(3.21)
		$\displaystyle+$	$\displaystyle\frac{\gamma^{3}\sqrt{\Delta_{r}}}{\rho}\partial_{r}+\frac{\gamma^{1}\sqrt{\Delta_{\theta}}}{\rho}\partial_{\theta}$
		$\displaystyle+$	$\displaystyle\frac{a\gamma_{5}}{2\rho^{3}}(\gamma^{1}r\sin{\theta}\sqrt{\Delta_{\theta}}+\gamma^{3}\cos{\theta}\sqrt{\Delta}_{r})+{\bar{\mu}}\Bigr{)}F(r,\theta)=0,$

where the first term on the right-hand side of eq. (3.3) is eliminated. Furthermore, let us expand the spinor wave function $F(r,\theta)$ in chiral eigenfunctions $f_{\pm}$ as follows:

$\displaystyle F(r,\theta)=(\frac{\chi^{*}}{\chi})^{1/4}f_{+}+(\frac{\chi}{\chi^{*}})^{1/4}f_{-},\ \ \ \gamma_{5}f_{\pm}=\pm f_{\pm},$

(3.22)

with $\chi=r+ia\cos{\theta}$. Then the Dirac equation becomes

			$\displaystyle{\chi^{*}}^{1/2}\Bigl{(}\frac{-i\gamma^{0}}{\rho\sqrt{\Delta_{r}}}{((r^{2}+a^{2})\omega-a\Xi m)}+\frac{-i\gamma^{2}}{\rho\sqrt{\Delta_{\theta}}}(a\sin{\theta}\omega-\frac{\Xi m}{\sin{\theta}})$		(3.23)
		$\displaystyle+$	$\displaystyle\frac{\gamma^{3}\sqrt{\Delta_{r}}}{\rho}\partial_{r}+\frac{\gamma^{1}\sqrt{\Delta_{\theta}}}{\rho}\partial_{\theta}+{\bar{\mu}}\Bigr{)}f_{+}(r,\theta)$
		$\displaystyle+$	$\displaystyle{\chi}^{1/2}\Bigl{(}\frac{-i\gamma^{0}}{\rho\sqrt{\Delta_{r}}}{((r^{2}+a^{2})\omega-a\Xi m)}+\frac{-i\gamma^{2}}{\rho\sqrt{\Delta_{\theta}}}(a\sin{\theta}\omega-\frac{\Xi m}{\sin{\theta}})$
		$\displaystyle+$	$\displaystyle\frac{\gamma^{3}\sqrt{\Delta_{r}}}{\rho}\partial_{r}+\frac{\gamma^{1}\sqrt{\Delta_{\theta}}}{\rho}\partial_{\theta}+{\bar{\mu}}\Bigr{)}f_{-}(r,\theta)=0,$

where the second term on the right-hand side of eq. (3.3) is eliminated. Let $f_{\pm}(r,\theta)$ be such that

	$\displaystyle f_{+}(r,\theta)$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\zeta\\ -\zeta\end{array}\right),\ \ \ f_{-}(r,\theta)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\eta\\ \eta\end{array}\right),$		(3.28)
	$\displaystyle\eta(r,\theta)$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{c}R_{1}(r)S_{1}(\theta)\\ R_{2}(r)S_{2}(\theta)\end{array}\right),\ \ \ \zeta(r,\theta)=\left(\begin{array}[]{c}R_{2}(r)S_{1}(\theta)\\ R_{1}(r)S_{2}(\theta)\end{array}\right).$		(3.33)

Then the Dirac equation reduces to the following set of the ordinary differential equations:

	$\displaystyle(\sqrt{\Delta_{\theta}}\partial_{\theta}+\frac{a\omega\sin^{2}{\theta}-{\Xi m}}{\sqrt{\Delta_{\theta}}\sin{\theta}})S_{1}(\theta)=(-a{\bar{\mu}}\cos{\theta}+\kappa)S_{2}(\theta),$		(3.34)
	$\displaystyle(\sqrt{\Delta_{\theta}}\partial_{\theta}-\frac{a\omega\sin^{2}{\theta}-{\Xi m}}{\sqrt{\Delta_{\theta}}\sin{\theta}})S_{2}(\theta)=(-a{\bar{\mu}}\cos{\theta}-\kappa)S_{1}(\theta),$		(3.35)
	$\displaystyle(\sqrt{\Delta_{r}}\partial_{r}-i\frac{(r^{2}+a^{2})\omega-a\Xi m}{\sqrt{\Delta_{r}}})R_{1}(r)=(\kappa-i{\bar{\mu}}r)R_{2}(r),$		(3.36)
	$\displaystyle(\sqrt{\Delta_{r}}\partial_{r}+i\frac{(r^{2}+a^{2})\omega-a\Xi m}{\sqrt{\Delta_{r}}})R_{2}(r)=(\kappa+i{\bar{\mu}}r)R_{1}(r),$		(3.37)

where $\kappa$ is the separation parameter. The Dirac equations of the form (3.13)-(3.16) are consistent with other previous works for Schwarzschild and Kerr spacetime [20, 21, 22, 23, 24].

Now we study zero mode for spinor field. Radial equations for zero mode $(p_{\rm H}=\omega-\Omega_{\rm H}m=0)$ become

	$\displaystyle(\sqrt{\Delta_{r}}\partial_{r}-i\frac{(r^{2}-r_{\rm H}^{2})\omega_{*}}{\sqrt{\Delta_{r}}})R_{1\rm zero}(r)=(\kappa-i{\bar{\mu}}r)R_{2\rm{zero}}(r),$
	$\displaystyle(\sqrt{\Delta_{r}}\partial_{r}+i\frac{(r^{2}-r_{\rm H}^{2})\omega_{*}}{\sqrt{\Delta_{r}}})R_{2\rm zero}(r)=(\kappa+i{\bar{\mu}}r)R_{1\rm{zero}}(r),$		(3.38)

where $\omega_{*}=\Omega_{\rm H}m=ma\Xi/(r_{\rm H}^{2}+a^{2})$. Near the horizon, zero mode equations (3.17) become

$\displaystyle\sqrt{r-r_{\rm H}}\partial_{r}R_{1\rm zero}(r)\simeq\beta R_{2\rm{zero}}(r),\ \ \sqrt{r-r_{\rm H}}\partial_{r}R_{2\rm zero}(r)\simeq\beta^{*}R_{1\rm{zero}}(r),$

(3.39)

where $\beta=(\kappa-i\bar{\mu}r_{\rm H})/\sqrt{X(r_{\rm H})}$ with $X(r)=\partial_{r}\Delta_{r}$. General solutions to eq. (3.18) are given by

	$\displaystyle R_{1\rm zero}(r)$	$\displaystyle\simeq$	$\displaystyle d_{1}\exp{(2|\beta|\sqrt{r-r_{\rm H}})}+d_{2}\exp{(-2|\beta|\sqrt{r-r_{\rm H}})},$
	$\displaystyle R_{2\rm zero}(r)$	$\displaystyle\simeq$	$\displaystyle(\beta^{*}/\beta)^{1/2}(d_{1}\exp{(2|\beta|\sqrt{r-r_{\rm H}})}-d_{2}\exp{(-2|\beta|\sqrt{r-r_{\rm H}})}),$		(3.40)

where $d_{1}$ and $d_{2}$ are integration constants.

The Dirichlet and Neumann boundary conditions for spinor fields require the spacial component (in the present case, the radial component) of conserved current on the horizon to vanish:

$\displaystyle J^{r}(x)=i\sqrt{-g}\bar{\Psi}(x)\gamma^{r}\Psi(x)=0\ \ \ {\mbox{on}}\ \ \ r=r_{\rm H}$

(3.41)

where $\bar{\Psi}=i\Psi^{\dagger}\gamma^{0}$. The explicit expression for $J^{r}$ is given by

$\displaystyle J^{r}=\frac{1}{\sqrt{\Delta_{\theta}}\Xi}(|R_{1}|^{2}-|R_{2}|^{2})(|S_{1}|^{2}+|S_{2}|^{2}),$

(3.42)

where we used eqs. (3.8), (3.10) and (3.12). There are two solutions of $J^{r}=0$,

$$({\rm i})\ \ \ R_{1}=(\beta/\beta^{*})^{1/2}R_{2},\ \ \ ({\rm ii})\ \ \ R_{1}=-(\beta/\beta^{*})^{1/2}R_{2},$$

(3.43)

which correspond to the Dirichlet and Neumann boundary conditions, respectively. ²²2 The Dirichlet and Neumann boundary conditions for spinor fields in eq. (3.22) are certified by the Bargmann-Wigner formulation too. Here a constant phase factor $(\beta/\beta^{*})^{1/2}$ is introduced, where $\beta$ is in eq. (3.18). (Note that the additional phase factor reduces to unity in the massless limit: $\beta/\beta^{*}\rightarrow 1$ as $\bar{\mu}\rightarrow 0$). The zero mode solution in equation (3.19) satisfies neither (i) nor (ii) in eq. (3.22). This means that the zero mode of spinor fields does not exist as physical states.

From the non-existence of zero mode for spinor fields we can derive the spectrum condition of normal modes for spinor fields satisfying $p_{\rm H}=\omega-\Omega_{\rm H}m>0$ as in the scalar field case.

As a consequence of the spectrum condition, type 1 super-radiance ($\omega>0,\omega-\Omega_{\rm H}m<0$) cannot occur whereas type 2 super-radiance ($\omega<0,\omega-\Omega_{\rm H}m>0$) does occur for spinor fields as well as scalar fields.