Big black hole, little neutron star: Magnetic dipole fields in the Rindler spacetime

As a black hole and neutron star approach during inspiral, the field lines of a magnetized neutron star eventually thread the black hole event horizon and a short-lived electromagnetic circuit is established. The black hole acts as a battery that provides power to the circuit, thereby lighting up the pair just before merger. Although originally suggested as an electromagnetic counterpart to gravitational-wave detection, a black hole battery is of more general interest as a novel luminous astrophysical source. To aid in the theoretical understanding, we present analytic solutions for the electromagnetic fields of a magnetic dipole in the presence of an event horizon. In the limit that the neutron star is very close to a Schwarzschild horizon, the Rindler limit, we can solve Maxwell’s equations exactly for a magnetic dipole on an arbitrary worldline. We present these solutions here and investigate a proxy for a small segment of the neutron star orbit around a big black hole. We find that the voltage the black hole battery can provide is in the range $\sim {10}^{16}$ statvolts with a projected luminosity of ${10}^{42}\mathrm{ergs}/\mathrm{s}$ for an $M=10M_{\odot }$ black hole, a neutron star with a B-field of ${10}^{12}\mathrm{G}$, and an orbital velocity $\sim 0.5c$ at a distance of $3M$ from the horizon. Larger black holes provide less power for binary separations at a fixed number of gravitational radii. The black hole/neutron star system therefore has a significant power supply to light up various elements in the circuit possibly powering bursts, jets, beamed radiation, or even a hot spot on the neutron star crust.

Figure 1

Neutron star—Rindler horizon effective circuit diagram. Magnetic field lines act as wires connecting the neutron star to the horizon. Current flows in (out) of the horizon via positively (negatively) charged particles spiraling in tight Larmor radii around magnetic field lines into the horizon.

Figure 2

Rindler space is the shaded wedge given by $T>\pm Z$, $Z>0$ on the Minkowski spacetime diagram. The dotted vertical line is the trajectory of a Minkowski observer and the dashed hyperbolic line is that of a Rindler observer as viewed by a Minkowski observer. In the frame of the Rindler observer an event horizon exists at $T=\pm Z$.

Figure 3

Diagram demonstrating the relationship between source trajectory coordinates $X_S(\tau )$, observer coordinates $X$, and the retarded proper time ${\tau }_{*}(X)$ at the intersection of the past light cone of an observer at $X$ and the source trajectory.

Figure 4

Spacetime diagrams for the infalling Rindler dipole of Sec. 4. Also shown is the worldline of a Rindler observer. The top panel is drawn by Minkowski observers; the bottom panel is drawn by Rindler observers. Note that $Z_S=z_S$ at $T=t=0$, hence the labeling of the initial source position.

Figure 5

The $x=x_S=0$ slice (plane containing the dipole) of the Poynting flux for the infalling dipole as viewed by Rindler observers. The axes are in units of $Z_S$.

Figure 6

3D visualization of the magnetic dipole field lines of a dipole falling from initial height $z_S(t=0)=Z_S$ above the Rindler horizon, denoted by the gray plane at $z=0$. The visualization region is a cube with side length $2Z_S$. On the left, magnetic field lines and the corresponding horizon current densities ${\mathbf{J}}_{\mathcal{H}}$ are plotted. On the right, electric field lines and corresponding charge densities ${\sigma }_{\mathcal{H}}$ (0 here) are plotted on the stretched horizon located at $z_H=0.01Z_S$.

Figure 7

Spacetime diagram depicting the $x$ component of the infalling, boosted dipole worldline of Sec. 5 from the Rindler observer’s perspective, for three different values of ${\beta }_S$. The $z$ component of the worldline is identical to that portrayed for the infalling dipole in the bottom panel of Fig. 4 except the light cone structure is altered. Because the infalling boosted dipole approaches the speed of light in the $z$ direction, the motion in the $x$ direction must go to zero ($dx/dt\to 0$). This is evident from the worldlines in this figure which asymptote to vertical lines.

Figure 8

3D visualization of the field lines of a dipole spiraling into the Rindler horizon from initial height $z_S(t=0)=Z_S$ with an initial boost of ${\beta }_S=0.9$ in the $x$ direction. The visualization region spans from $-2Z_S$ to $2Z_S$ in the $x$ and $y$ directions and extends $2Z_S$ above the Rindler horizon. Surface currents ${\mathbf{J}}_{\mathcal{H}}$ and surface charge densities ${\sigma }_{\mathcal{H}}$ are plotted on the stretched horizon located at $z_H=0.01Z_S$.

Figure 9

An $x=x_S=0$ slice of the Poynting flux for the infalling boosted dipole of Sec. 5 as viewed by Rindler observers for three different boost magnitudes in the $x$ direction, ${\beta }_S=0.1$, 0.5, 0.9. The Poynting flux is 0 at infinity despite outward components of the field in the region plotted here. The axes are in units of $Z_S$.

Figure 10

Current density vectors (white) overlaid on contours of charge density on the stretched horizon (${\alpha }_{\mathcal{H}}={10}^{-4}$) of the infalling boosted dipole with rest-frame magnetic moment in the $y$ direction. From top to bottom, the magnitude of the boost in the $x$ direction increases from ${\beta }_S=0.1$, 0.5, 0.9. As inferred from the last of Eq. (59), the magnitude of the charge density increases with ${\beta }_S$. Also the shape of the charge separation is squeezed in the direction of source boost as indicated by Eq. (60). All of the snapshots are taken at $g_{H}t=1$ and the contour labels are arbitrarily scaled. The gray regions are regions of steeply increasing ${\sigma }_{\mathcal{H}}$ which have been removed to more clearly view the contour structure. The axes are in units of $Z_S$.

Figure 11

3D visualization of the magnetic dipole field lines hovering at constant height $z_S$ above the Rindler horizon, denoted by the gray plane at $z=0$. The visualization region is a cube with side length $2z_S$.

Figure 12

Spacetime diagram in the Rindler frame depicting the worldline of a source boosted parallel to the Rindler horizon (Sec. 7). The light ray (dotted line) has slope $dt/dz=(g_H{z}_S{)}^{-1}$ and the worldline has slope $(v_{S,x}{g}_H{z}_S{)}^{-1}$, where $z_S$ is the constant position of the source above the Rindler horizon.

Figure 13

Streamlines of the magnetic (left) and electric (right) fields in the plane $x=0$ for a magnetic dipole with dipole moment $\mathbf{m}\propto {\widehat{\mathbf{e}}}_y$ and with three different boost velocities increasing from top to bottom $v_{S,x}=0.1$, 0.5, 0.9 in the $x$ direction. Plotted over the streamlines are contours of the $x$ components of the fields relative to the $y\mathrm{\text{−}}z$ magnitude. Darker regions represent negative values and lighter regions represent positive values. The white regions are clipped to better view the contour structure. The snapshots here are taken at $g_{H}t=1/4$; however the fields retain the same structure for all time except for their motion in the $x$ direction (out of the page). The axes are in units of $z_S$. Since the source is boosted, observers near the horizon see the fields as they would have been when the dipole was further away in the negative $x$ direction. The result is an observed dragging of the fields along the horizon in the negative $x$ direction. The observed larger radius of curvature of the dipole lobes manifests itself as the flattening of the field lines. As can be gathered from Fig. 12, this effect is intensified for larger boost factor $v_{S,x}$. In the $v_{S,x}=0.9$ case, plotted at the bottom of the figure, the 2D slice of the magnetic field loses its dipolar structure in most of the region below the source. As $v_{S,x}$ approaches 1, the slope of the source worldline approaches the light cone slope and an observer at a given $z$ will see further and further into the relative past of the dipole. Note also that the contours in the left panels show that the circulation direction of the dipole lobes changes sign at a value of $z$ which gets larger for larger $v_{S,x}$. This change in sign results since observers near the horizon see fields from further in the past when the fields were pointing in a different $x$ direction. The increase in $z$ location of this turning point for larger $v_{S,x}$ can again be understood from Fig. 12.

Figure 14

A 3D visualization of the magnetic fields lines and corresponding horizon currents ${\mathbf{J}}_{\mathcal{H}}$ (left) and electric field lines with the corresponding horizon charges ${\sigma }_{\mathcal{H}}$ (right) for the boosted Rindler dipole. The case shown is for $v_{S,x}=0.2$.

Figure 15

An identical plot to Fig. 10 but for the boosted case. Current density vectors (white) are overlaid on contours of charge density on the stretched horizon (${\alpha }_{\mathcal{H}}={10}^{-4}$) of the boosted dipole with magnetic moment in the $y$ direction. The bottom panel also plots streamlines of the currents. From top to bottom, the magnitude of the boost in the $x$ direction increases from $v_{S,x}=0.1$, 0.5, 0.9. Each snapshot is taken at $g_{H}t=10$. The configuration drags along the stretched horizon keeping a constant lag distance behind the moving source. The induced currents can be thought of as redistributing charge in order to slide the charge distribution along behind the boosted dipole. The gray regions are regions of steeply increasing ${\sigma }_{\mathcal{H}}$ which have been removed to more clearly view the contour structure. The axes are in units of $z_S$.

Figure 16

An $x=x_S=0$ slice through the Poynting flux. The plane contains the source as viewed by Rindler observers for three different boost magnitudes in the $x$ direction, $v_{S,x}=0.1$, 0.5, 0.9. The Poynting flux is 0 at infinity despite outward components of the field in the region plotted here. The axes are in units of $z_S$.

Figure 17

Log luminosity computed from Eq. (14) (blue, solid line and leftmost $y$-axis labels) and representative log-voltage drop on the horizon (red, dashed line and rightmost $y$-axis labels). Luminosities and voltages are computed for $M=10M_{\odot }$, ${10}^2{M}_{\odot }$, and ${10}^3{M}_{\odot }$ with the dipole at Rindler height $z_S=3M$ as a function of $v_{S,x}$ varying from 0.01 to 0.95. The last stable circular orbit in the Schwarzschild spacetime would have $v_{S,x}=0.5$.

Figure 18

Maximum curvature radiation energies computed from Eq. (72) (blue, solid line and leftmost $y$-axis labels) and corresponding maximum ${\gamma }_p$ [Eq. (75)] to which electrons/positrons can be accelerated (red, dashed line and rightmost $y$-axis labels). Both are computed for $M=10M_{\odot }$, ${10}^2{M}_{\odot }$, and ${10}^3{M}_{\odot }$ with the dipole at Rindler height $z_S=3M$ as a function of $v_{S,x}$ varying from 0.01 to 0.95.

Figure 19

Luminosity, voltage (top) and energy, Lorentz-factor (bottom) plots identical to those portrayed in Figs. 17, 18 respectively. Here we have plotted both panels for a ${10}^6{M}_{\odot }$ BH and at a much smaller horizon distance than in the previous figures: $z_S=R_{\mathrm{lc}}$, corresponding to the maximum separation where the BH-NS circuit remains connected.