Electromagnetic waves from neutron stars and black holes driven by polar gravitational perturbations

Abstract

Neutron stars and black holes are the most compact astrophysical objects we can think of and as a consequence they are the main sources of gravitational waves. There are many astrophysically relevant scenarios in which these objects are immersed in or endowed with strong magnetic fields, in such a way that gravitational perturbations can couple to electromagnetic ones and can potentially trigger synergistic electromagnetic signatures. In a recent paper we derived the main equations for gravito-electromagnetic perturbations and studied in detail the case of polar electromagnetic perturbations driven by axial gravitational perturbations. In this paper we deal with the case of axial electromagnetic perturbations driven by polar black-hole or neutron stars oscillations, in which the energy emitted in case is considerably larger than in the previous case. In the case of neutron stars the phenomenon lasts considerably longer since the fluid acts as an energy reservoir that shakes the magnetic field for a timescale of the order of secs.

Appendix

In this appendix, we present the perturbation equations for gravitational and electromagnetic waves adopted in this paper. More details can be found in Paper I [46]. Using the well-known Regge-Wheeler gauge, the metric perturbations, $h_{\mu \nu }$, with polar parity can be decomposed into tensor spherical harmonics as

$$\begin{aligned} h_{\mu \nu } = \sum _{l=2}^{\infty } \sum _{m=-l}^{l}\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} e^{\nu } H_{0,lm} &{} H_{1,lm} &{} 0 &{} 0 \\ *&{} e^{\lambda } H_{2,lm} &{} 0 &{} 0 \\ 0 &{} 0 &{} r^2 K_{lm} &{} 0 \\ 0 &{} 0 &{} 0 &{} r^2 \sin ^2\theta K_{lm} \\ \end{array} \right) Y_{lm}, \end{aligned}$$

(8)

where $H_{0,lm}, H_{1,lm}, H_{2,lm}$, and $K_{lm}$ are functions of $t$ and $r$ only, and $Y_{lm}=Y_{lm}(\theta ,\phi )$ denotes the scalar spherical harmonics on the $2-$sphere. On the other hand, the tensor harmonic expansion for the “magnetic” multipoles (or axial parity) of the electromagnetic perturbations, $f_{\mu \nu }$, is given by

$$\begin{aligned} f_{\mu \nu }^\mathrm{(M)} = \sum _{l=2}^{\infty } \sum _{m=-l}^{l}\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} &{} 0 &{} &{} f^\mathrm{(M)}_{02,lm} {\sin ^{-1}\theta } \partial _{\phi } &{} &{}-f^\mathrm{(M)}_{02,lm} \sin \theta \, \partial _{\theta } \\ 0&{} &{} 0 &{} &{}f^\mathrm{(M)}_{12,lm} {\sin ^{-1}\theta } \partial _{\phi } &{} &{}-f^\mathrm{(M)}_{12,lm} \sin \theta \, \partial _{\theta } \\ *&{} &{} *&{} &{} 0 &{}&{} f^\mathrm{(M)}_{23,lm}\sin \theta \\ *&{} &{} *&{}&{} *&{}&{} 0 \\ \end{array} \right) Y_{lm}, \end{aligned}$$

(9)

where $f_{02}^\mathrm{(M)}, f_{12}^\mathrm{(M)}$, and $f_{23}^\mathrm{(M)}$ are also functions of $t$ and $r$ only.

1.1 Neutron-star background

For the perturbations around the neutron star background, we adopt the formalism derived by Allen et al. [58], where the perturbations are described by three coupled wave-type equations, i.e., two equations describe the perturbations of the spacetime and the other one describes the fluid perturbations. In addition to the three wave equations, there is also a constraint equation. The two wave equations for the spacetime variables are

$$\begin{aligned}&-\frac{\partial ^2 S_{lm}}{\partial t^2} + \frac{\partial ^2 S_{lm}}{\partial r_*^2} + \frac{2 e^\nu }{r^3}\left[ 2 \pi r^3 (\rho +3p) + m - (n+1)r \right] S_{lm} \nonumber \\&\quad = -\frac{4 e^{2\nu }}{r^5} \left[ \frac{(m+4\pi p r^3)^2}{r-2m} + 4\pi \rho r^3 - 3m \right] F_{lm}\end{aligned}$$

(10)

$$\begin{aligned}&-\frac{\partial ^2 F_{lm}}{\partial t^2} + \frac{\partial ^2 F_{lm}}{\partial r_*^2} + \frac{2 e^\nu }{r^3} \left[ 2 \pi r^3 (3 \rho + p) + m - (n+1)r \right] F_{lm} \nonumber \\&\quad = - 2\left[ 4 \pi r^2 (p+\rho ) - e^{-\lambda } \right] S_{lm} + 8 \pi (\rho +p) {r e^{\nu }} \left( 1- \frac{1}{C_s^2} \right) H_{lm}, \end{aligned}$$

(11)

where $r_*$ is the tortoise coordinate defined as $\partial _{r_*} =e^{(\nu -\lambda )/2}\partial _r$, $n \equiv (l-1)(l+2)/2$, $C_s$ is the fluid sound speed, and $F_{lm}$, $S_{lm}$, and $H_{lm}$ are given by

$$\begin{aligned} F_{lm}(t,r)&= rK_{lm}, \end{aligned}$$

(12)

$$\begin{aligned} S_{lm}(t,r)&= \frac{e^{\nu }}{r}\left( H_{0,lm} -K_{lm}\right) , \end{aligned}$$

(13)

$$\begin{aligned} H_{lm}(t,r)&= \frac{\delta p_{lm}}{\rho +p}. \end{aligned}$$

(14)

On the other hand, the wave equation for the perturbations of the relativistic enthalpy, $H_{lm}$, describing the fluid perturbations, is

$$\begin{aligned}&- \frac{1}{C_s^2}\frac{\partial ^2 H_{lm}}{\partial t^2} + \frac{\partial ^2 H_{lm}}{\partial r_*^2} + \frac{e^{(\nu +\lambda )/2}}{r^2}\left[ (m + 4\pi p r^3)\left( 1-\frac{1}{C_s^2}\right) + 2 (r-2m) \right] \frac{\partial H_{lm}}{\partial r_*} \nonumber \\&\qquad + \frac{2 e^\nu }{r^2} \left[ 2 \pi r^2 (\rho +p)\left( 3 + \frac{1}{C_s^2} \right) - (n+1) \right] H_{lm} \nonumber \\&\qquad \qquad = (m+4 \pi p r^3)\left( 1-\frac{1}{C_s^2}\right) \frac{e^{(\lambda -\nu )/2}}{2 r} \left( \frac{e^\nu }{r^2}\frac{\partial F_{lm}}{\partial r_*} - \frac{\partial S_{lm}}{\partial r_*} \right) \nonumber \\&\qquad \qquad \qquad + \left[ \frac{(m+4\pi p r^3)^2}{r^2(r-2m)}\left( 1+\frac{1}{C_s^2}\right) - \frac{ m+4\pi p r^3}{2 r^2} \left( 1-\frac{1}{C_s^2}\right) -4\pi r (3p+\rho ) \right] S_{lm} \nonumber \\&\qquad \qquad \qquad + \frac{e^\nu }{r^2}\left[ \frac{2(m+4\pi p r^3)^2}{r^2(r-2m)}\frac{1}{C_s^2} - \frac{ m+4\pi p r^3}{2 r^2}\left( 1-\frac{1}{C_s^2} \right) -4\pi r (3p+\rho )\right] F_{lm}.\nonumber \\ \end{aligned}$$

(15)

This third wave equation is valid only inside the star. On the other hand, the first two wave equations are simplified considerably outside the star, and they can be reduced to a single wave equation, i.e., the Zerilli equation (see [58] and §4.2). Finally, the Hamiltonian constraint,

$$\begin{aligned}&\frac{\partial ^2 F_{lm}}{\partial r_*^2} - \frac{e^{(\nu +\lambda )/2}}{r^2} \left( m + 4 \pi r^3 p \right) \frac{\partial F_{lm}}{\partial r_*} + \frac{e^\nu }{r^3} \left[ 12\pi r^3 \rho - m - 2(n+1)r \right] F_{lm} \nonumber \\&\qquad - r e^{-(\nu +\lambda )/2}\frac{\partial S_{lm}}{\partial r_*} + \left[ 8\pi r^2(\rho +p) -(n+3) + \frac{4m}{r} \right] S_{lm} \nonumber \\&\qquad + \frac{8 \pi r}{C_s^2} e^\nu (\rho +p) H_{lm} = 0, \end{aligned}$$

(16)

can be used for setting up initial data and monitoring the evolution of the coupled system.

Finally, the perturbation equation for the “magnetic-type” (axial) electromagnetic perturbations outside the neutron star is

$$\begin{aligned} \frac{\partial ^2 \Phi _{10}}{\partial t^2} - \frac{\partial ^2 \Phi _{10}}{\partial r_*^2} + \frac{2}{r^2}e^{\nu }\Phi _{10} = \mathcal{S}_{20}^\mathrm{(M)}, \end{aligned}$$

(17)

where $\Phi _{lm} \equiv f_{23,lm}^\mathrm{(M)}$ and

$$\begin{aligned} \mathcal{S}_{20}^\mathrm{(M)}&= -\frac{2}{\sqrt{15}}e^{\nu }\bigg [\left( 2B_2 + r {B_2}'\right) r^2S_{20} \nonumber \\&\quad + \left( e^{\nu }B_2 + re^{\nu }{B_2}' - \frac{2}{r}B_1\right) F_{20} + rB_2 \frac{\partial F_{20}}{\partial r_*}\bigg ]. \end{aligned}$$

(18)

1.2 Black-hole background

The “magnetic-type” (axial) electromagnetic perturbations driven by the gravitational perturbations on a black hole background are also governed by the same equation as the one derived for the exterior region of neutron stars [see Eq. (17)], that is

$$\begin{aligned} \frac{\partial ^2 \Phi _{10}}{\partial t^2} - \frac{\partial ^2 \Phi _{10}}{\partial r_*^2} + \frac{2}{r^2}e^{\nu }\Phi _{10} = \mathcal{S}_{20}^\mathrm{(M)}. \end{aligned}$$

(19)

The source term of this wave equation, $\mathcal{S}_{20}^{(\mathrm M)}$, has the same form as in Eq. (18). On the other hand, and is well-known, the spacetime metric perturbations are described by a single equation, the Zerilli equation

$$\begin{aligned}&\frac{\partial ^2 Z_{lm}}{\partial t^2}-\frac{\partial ^2 Z_{lm}}{\partial r_*^2} + V_Z(r)Z_{lm} = 0\end{aligned}$$

(20)

$$\begin{aligned}&V_{Z}(r) =\frac{2e^{\nu }\left[ n^2(n +1)r^3 + 3Mn^2 r^2 + 9M^2n r + 9M^3\right] }{r^3(nr + 3M)^2}. \end{aligned}$$

(21)

In the same way as in the case of the neutron star background, one can also adopt $F_{lm}$ and $S_{lm}$ as the perturbation variables for the spacetime oscillations. Actually, one can obtain the variables $F_{lm}$ and $S_{lm}$ from the Zerilli function $Z_{lm}$ using the following expressions:

$$\begin{aligned} F_{lm}&= r\frac{dZ_{lm}}{dr_*}+\frac{n(n+1)r^2+3 n Mr+6M^2}{r(n r+3M)}Z_{lm},\end{aligned}$$

(22)

$$\begin{aligned} S_{lm}&= \frac{1}{r}\frac{dF_{lm}}{dr_*}-\frac{(n+2)r-M}{r^3}F_{lm}+\frac{(n+1)(n r+3M)}{r^3}Z_{lm}. \end{aligned}$$

(23)