One-dimensional linear analysis and numerical simulations of Alfvén waves in a force-free magnetosphere around a Kerr black hole

The metric of the Boyer–Lindquist coordinates $x^{\mu}_{\rm BL}=(t,r,\theta,\varphi)$ is given by

where Greek indices such as $\mu$ and $\nu$ run from 0 to 3, while Roman indices such as $i$ and $j$ run from 1 to 3, $\displaystyle\alpha^{2}=-g_{tt}+\gamma_{ij}\beta^{i}\beta^{j}=\frac{\Sigma\Delta}{A}$, $\displaystyle\gamma_{rr}=\frac{\Sigma}{\Delta}$, $\displaystyle\gamma_{\theta\theta}=\Sigma$, $\displaystyle\gamma_{\phi\phi}=\frac{A}{\Sigma}\sin^{2}\theta\equiv R^{2}$, $\displaystyle\gamma_{ij}=0,(i\neq j)$, $\displaystyle\beta^{\phi}=-\Omega$, $\displaystyle\beta^{r}=\beta^{\theta}=0$, $\displaystyle\Delta=r^{2}-2Mr+a^{2}$, $\displaystyle\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $\displaystyle A=(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta$, and $\displaystyle\Omega=-\frac{g_{r\varphi}^{\rm BL}}{g_{\varphi\varphi}^{\rm BL}}=\frac{2Mar}{A}$. Here, the spin parameter of a black hole $a$ is given by $a=J/M$, where $M$ and $J$ are the mass and angular momentum of the black hole, respectively. The dimensionless spin parameter $a_{\ast}$ is given by $a_{\ast}=a/M$. The determinant of the metric tensor for the Boyer–Lindquist coordinates $(g_{\mu\nu}^{\rm BL})$ is given by $g_{\rm BL}\equiv\det(g^{\rm BL}_{\mu\nu})=-\Sigma^{2}\sin^{2}\theta$, and the 4-velocity of the normal observer (Zero-Angular-Momentum Observer: ZAMO) is given by

Using Eq. (2), we obtain the steady-state solution of the force-free magnetic field around the equatorial plane ($\theta\sim\pi/2$) of the black hole

where $\bar{\phi}_{i}$ ($i=1,2$) represents the steady-state solution, and $\Omega_{\rm F}$ and $I$ are constants along a field line. These constants mean that $\Omega_{\rm F}$ is the angular velocity of the magnetic field line, and $I$ is the total current, which is obtained from the regularity of $F^{\mu\nu}F_{\mu\nu}$ at the black hole horizon as,

The solution given by Eq. (5) corresponds to the field of the magnetic monopole around the equatorial plane. Here, the strength of the stationary magnetic field is normalized without loss of generality. Note that $\phi_{m}$ ($m=1,2$) corresponds to the stream function and the contour surface of $\bar{\phi}_{m}$ at a certain time represents the steady-state magnetic surfaces. In the present case, $\bar{\phi}_{1}=-\cos\theta=0$ represents the steady magnetic surface along the equatorial plane. The magnetic flux surface $\bar{\phi}_{2}=$ const. is vertical to the equatorial plane and contains the magnetic field line on the equatorial plane.
The corotating Killing vector of the field line, $\xi_{(\rm F)}^{\nu}\equiv\xi_{(t)}^{\nu}+\Omega_{\rm F}\xi_{(\varphi)}^{\nu}$ is parallel to the 4-velocity of an observer corotating with the magnetic field line, where $\xi^{\nu}_{(t)}$ is the Killing vector in the time direction and $\xi^{\nu}_{(\phi)}$ is the Killing vector in the azimuthal direction. The norm of $\xi_{(\rm F)}^{\nu}$ is

where the roots of $\Gamma=0$ give the location of the inner and outer light surfaces.

To simplify the form of the calculations, we utilize the coordinates $(T,X,\rho,z)$, defined as

The coordinate frame does not rotate, although it is similar to the coordinate frame used by Noda et al. (2022), which rotates with the same angular velocity as the magnetic field lines. In this coordinate system, $X$ is the coordinate along the magnetic field line, so we call this coordinate system $(T,X,\Psi,\rho)$ the “magnetic natural frame”. The contravariant component of the metric tensor of the magnetic natural frame $(T,X,\Psi,\rho)$ is as follows:

The determinant of the matrix $(g_{\mu\nu})$ is given by $g=\det(g_{\mu\nu})=-\Sigma^{2}$. In the magnetic natural frame, the steady-state force-free solution given by Eq. (5) is simply

In the background magnetosphere given by Eq. (13), we consider a perturbation of the magnetic field; that is, the propagation of Alfvén waves. We already know that an Alfvén wave induces a fast wave represented by a second-order perturbation, as suggested by the calculations in BTZ spacetime (Koide et al., 2022). Therefore, we should consider not only the Alfvén wave but also the fast wave using second-order perturbation. The infinitesimally small perturbation to the Euler potential $\phi_{i}\longrightarrow\bar{\phi}_{i}+\delta\phi_{i}$ is given by the displacement (polarization) vector $\zeta^{\lambda}$ as $\delta\phi_{i}=\zeta^{\lambda}\partial_{\lambda}\phi_{i}$. First, we consider the Alfvén wave with a first-order perturbation. We focus on a pure Alfvén wave propagating on the magnetic surface at the equatorial plane ($\Psi=-\cos\theta=0$) and oscillating perpendicularly to the magnetic surface: $\delta\zeta^{\lambda}=(0,0,\delta\zeta^{\Psi},0)$. This wave is the “pure” Alfvén wave because the polarization direction of the oscillation $\delta\zeta^{\lambda}$ is perpendicular to that of the propagation which is parallel to the equatorial plane $\Psi=\bar{\phi}_{1}=0$. Then, the perturbed Euler potentials are

because $\delta\phi_{2}=\delta\zeta^{\mu}\partial_{\mu}\bar{\phi}_{2}=\delta\zeta^{\Psi}\partial_{\Psi}(\rho-\Omega_{\rm F}T)=0$. Hereafter, we denote the stationary background (zeroth-order) term, first-order perturbation, second-order perturbation, $\cdots$ of a variable $A$ as $\bar{A}$, $\delta A$, $\delta^{2}A$, $\cdots$, i.e., $A=\bar{A}+\delta A+\delta^{2}A+\cdots$. Without loss of generality, we assume the perturbation for the Alfvén wave is axisymmetric, i.e., the perturbation is independent of $\rho$:

where the Alfvén wave propagates along a magnetic field line and never interacts with Alfvén waves propagating along other magnetic field lines.

In nonrotating natural coordinates, the linearization of Eq. (2) with $i=2$ yields

and can be calculated easily as

where $Z^{\lambda\alpha}=W^{\lambda\alpha\mu\beta}\partial_{\mu}\bar{\phi}_{2}\partial_{\beta}\bar{\phi}_{2}=W^{\lambda\alpha\rho\rho}-\Omega_{\rm F}(W^{\lambda\alpha\rho T}+W^{\lambda\alpha T\rho})+\Omega_{\rm F}^{2}W^{\lambda\alpha TT}$. ³³3$Z^{\mu\nu}$ is proportional to the projection operator onto the constant-$\bar{\phi}_{2}$ hypersurface, ${\cal P}^{\mu\nu}\equiv g^{\mu\nu}-\partial^{\mu}\bar{\phi}_{2}\partial^{\nu}\bar{\phi}_{2}/|\partial\bar{\phi}_{2}|^{2}=|\partial\bar{\phi}_{2}|^{-2}Z^{\mu\nu}$, where $|\partial\bar{\phi}_{2}|=\sqrt{\partial_{\mu}\bar{\phi}_{2}\partial^{\mu}\bar{\phi}_{2}}$. Using the variables $\lambda=-\sqrt{-g}Z^{TT}$, $S=\sqrt{-g}Z^{XX}$, $K=\sqrt{-g}Z^{\Psi\Psi}$, and $V=-\sqrt{-g}Z^{TX}$, we rewrite Eq. (17) as

This equation is identified by the equation of displacement of a string with a nonuniform line density $\lambda$ and nonuniform tension $\displaystyle S+\frac{V^{2}}{\lambda}$, which moves with a nonuniform velocity $\displaystyle\frac{V}{\lambda}$ and is bounded by springs with a spring constant per unit length $K$. ⁴⁴4Because Eq. (18) is expressed by $\displaystyle\lambda\left(\frac{\partial}{\partial T}+\frac{V}{\lambda}\frac{\partial}{\partial X}\right)^{2}\psi-\frac{\partial}{\partial X}\left[\left(S+\frac{V^{2}}{\lambda}\right)\frac{\partial\psi}{\partial X}\right]+K\psi+\frac{\partial V}{\partial X}\left(\frac{\partial\psi}{\partial T}+\frac{V}{\lambda}\frac{\partial\psi}{\partial X}\right)=0$. When we neglect the nonuniformity of $V$, we obtain the equation of a moving string, $\displaystyle\lambda\left(\frac{\partial}{\partial T}+\frac{V}{\lambda}\frac{\partial}{\partial X}\right)^{2}\psi-\frac{\partial}{\partial X}\left[\left(S+\frac{V^{2}}{\lambda}\right)\frac{\partial\psi}{\partial X}\right]+K\psi=0$. Using the metric tensor of the nonrotating natural coordinates (Eq. (12)) and the nature of the perturbation $(\partial_{\Psi}^{2}\psi=-\psi)$, we obtain the explicit expressions for the parameters of the analogous string,

where we have the identity $\displaystyle V^{2}+\lambda S=\frac{\Sigma K}{\sin^{2}\theta}$. Equation (20) shows that the string analogous to Eq. (18) is strange because the tension $\displaystyle S+\frac{V^{2}}{\lambda}=\frac{\Sigma}{\lambda}K$ becomes negative outside the region between the inner and outer light surfaces, as $S=-\Gamma$ at the equatorial plane.

In the eikonal limit, that is, the short wavelength limit, Eq. (18) yields the dispersion relation of the Alfvén wave, Eq. (A1). Equation (A1) reveals not only the propagation of the Alfvén wave but also the instability of the wave outside the outer light surface. In Appendix A, it is shown that the Alfvén wave is stable over the entire radial range ($r_{\rm H}<X<\infty$) when $\displaystyle\Omega_{\rm F}<\frac{2M}{2M+r_{\rm H}}|\Omega_{\rm H}|\equiv\Omega_{\rm c}$. Otherwise ($\Omega_{\rm F}>\Omega_{\rm c}$), the Alfvén wave is unstable at $X>r_{\rm m}$, where $X=r_{\rm m}$ is the solution for $K=0$. In the numerical calculations in this paper, we use the parameters of the case where the Alfvén wave is stable over the entire radial range ($\Omega_{\rm F}<\Omega_{\rm c}$).

The induction of a fast wave by an Alfvén wave is explained by a relativistic mechanism, where the angular momentum of the Alfvén wave changes in time (Fig. 2). The Poynting flux of the Alfvén wave is directed in the propagation direction of the wave. The Poynting flux is proportional to the electromagnetic momentum density of the Alfvén wave. Then, generally speaking, the Alfvén wave propagating along the magnetic field line has its angular momentum as a second-order perturbation. Therefore, if the magnetic field line is curved azimuthally ($I\neq 0$) or rotates ($\Omega_{\rm F}\neq 0$), a torque on the Alfvén wave is required so that the Alfvén wave traces the magnetic field line. This means the force-free condition is broken at second order and the azimuthal component of the Lorentz force becomes finite. To change the angular momentum of the Alfvén wave, the Alfvén wave should be subject to an external torque to trace the rotating magnetic field line. Thus, we take into account additional first- and second-order perturbations $\delta^{2}\phi_{1}$, $\delta^{2}\phi_{2}$ as

where $\chi\equiv\delta^{2}\phi_{2}=\delta^{2}\zeta^{\lambda}\partial_{\lambda}\bar{\phi}_{2}=\delta^{2}\zeta^{\lambda}\partial_{\lambda}(\rho-\Omega_{\rm F}T)=\delta^{2}\zeta^{\rho}-\Omega_{\rm F}\delta^{2}\zeta^{T}$. This perturbation corresponds to the fast wave because the polarization of the oscillation of the magnetic surface $\delta^{2}\zeta^{i}=(0,0,\delta^{2}\zeta^{\rho})$ is parallel to the propagation surface of the wave, i.e. the equatorial plane. Using $\chi$, we find that the force-free condition is recovered with the second-order perturbation. For simplicity, we assume $\partial_{\Psi}\chi=\partial^{2}_{\Psi}\chi=0$ without any contradiction.

From Eq. (2), the time evolution equation of $\chi$ is obtained as

where

with $Y^{\lambda\alpha\mu}\equiv W^{\lambda\alpha\mu\nu}\partial_{\nu}\bar{\phi}_{2}=W^{\lambda\alpha\mu\rho}-\Omega_{\rm F}W^{\lambda\alpha\mu T}$ and $\displaystyle\Theta\equiv\sqrt{-g}Y^{T\alpha X}\partial_{\alpha}\psi=-\frac{IX^{2}}{\alpha^{2}}\partial_{T}\psi-R^{2}(\Omega_{\rm F}-\Omega)\partial_{X}\psi$ (see the derivation in Appendix C). The right-hand side of Eq. (25), $s$, is recognized as the source term of the wave equation of $\chi$ as the second-order perturbation $\sim\psi^{2}$. Therefore, $\chi$ in Eq. (25) represents the wave induced by the Alfvén wave. Introducing $\chi$, we confirm that the force-free condition holds. The linear inhomogeneous equation (Eq. (25)) shows the wave described by $\chi$ propagates isotropically in space as by a fast wave (Bellman, 2006), while the complex formalism of the homogeneous linear equation of $\psi$ (Eq. (16)) suggests strong anisotropy of the wave described by $\psi$, which is a characteristic property of an Alfvén wave.

When $\xi^{\mu}$ is the Killing vector, we have the conservation law

When $\xi^{\nu}f^{\rm L}_{\nu}$ vanishes, $\xi_{\nu}T^{0\nu}$ represents the density of the conserved value and $\xi_{\nu}T^{i\nu}$ represents the density of the conserved quantity flux.

For the time-like Killing vector $\xi^{\mu}_{(T)}=(1,0,0,0)$ and the axial Killing vector $\xi^{\mu}_{(\rho)}=(0,0,0,1)$ in magnetic natural coordinates, we have the energy and angular momentum conservation laws

where we assume axisymmetry and $z$-direction translation symmetry, and $S^{\mu}=-\xi^{(T)}_{\nu}T^{\mu\nu}$ and $M^{\mu}=\xi^{(\rho)}_{\nu}T^{\mu\nu}$ are the 4-energy flux and 4-angular momentum flux, respectively, up to second order.

We now consider the energy density and energy flux to determine the energy conservation law on the equatorial plane around a black hole. The Killing vector for the energy conservation law is $\xi_{(T)}^{{\mu}}=(1,0,0,0)$. Thus, the 4-energy flux density is

The energy conservation law is expressed as

Hence, we have the energy density of the equilibrium, first-, and second-order perturbations with respect to the linear Alfvén wave (with $\delta^{1+1}$) and the second-order fast wave (with $\delta^{2+0}$),

where

In the numerical calculations, we use the above formulation (Eqs. (41)-(43)) with the tortoise coordinates (see Appendix E). We can obtain the energy flux density of the equilibrium, first-, and second-order perturbations of the linear Alfvén wave,

where

Eventually, we obtain the energy conservation law of the Alfvén wave and the induced fast mode,

where $\delta^{2}e^{\infty}=\delta^{1+1}e^{\infty}+\delta^{2+0}e^{\infty}$, $\delta^{2}S_{\rm P}=\delta^{1+1}S_{\rm P}+\delta^{2+0}S_{\rm P}$, $\delta^{1+1}e^{\infty}=\delta^{1+1}S^{T}$, $\delta^{2+0}e^{\infty}=\delta^{2+0}S^{T}$, $\delta^{1+1}S_{\rm P}=\delta^{1+1}S^{X}$, and $\delta^{2+0}S_{\rm P}=\delta^{2+0}S^{X}$. In the numerical calculations, we use the formulation (Eqs. (46)–(49)) with the tortoise coordinates (see Appendix E).

Next, we discuss the conservation law of the angular momentum. The axial Killing vector $\xi^{\mu}_{(\rho)}=(0,0,0,1)$ yields the 4-angular momentum flux density $M^{\mu}=\xi^{\nu}_{(\rho)}T^{\mu}_{\nu}=F^{\mu\lambda}F_{\rho\lambda}$ ($\mu\neq\rho$), and we obtain the angular momentum conservation law in the nonrotating natural coordinates,

We evaluate the angular momentum and the Lorentz force density up to the second order:

Equation (51) yields

which is identical to the equation for $\chi$ (Eq. (25)). When and only when $I$, $\Omega_{\rm F}$, and $a_{\ast}$ are all zero, the first term of the central part of Eq. (54) vanishes (see the detailed calculation in Eq. (B21)). In the case of $I=0$, $\Omega_{\rm F}=0$, or $a_{\ast}=0$, the Alfvén wave never induces the fast wave.
Eventually, we obtain the angular momentum conservation law with respect to the Alfvén wave and the induced fast mode,

where $\delta^{2}l_{z}=\delta^{1+1}l_{z}+\delta^{2+0}l_{z}$, $\delta^{2}M^{X}=\delta^{1+1}M^{X}+\delta^{2+0}M^{X}$, $\delta^{1+1}l_{z}=\delta^{1+1}M^{T}$, and $\delta^{2+0}l_{z}=\delta^{2+0}M^{T}$.