  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 a promising electromagnetic
counterpart to gravitational-wave detection, the luminous signals are promising
more generally as potentially detectable phenomena, such as short gamma-ray
bursts. 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 ~10^16 statvolts with a projected luminosity of 10^42 ergs/s for an
M=10M_sun black hole, a neutron star with a B-field of 10^12 G, and an orbital
velocity ~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 jets, beamed
radiation, or even a hot spot on the neutron star crust.
