# Mining the quantum vacuum: quantum tunnelling and particle creation

Ian G. Moss and Piotr Z. Stasiak  School of Mathematics, Statistics and
Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Particle production from the vacuum is a remarkable aspect of particle
physics. Prime examples are the Schwinger process of particle production in
strong electric fields and the Hawking process of particle production from
black holes. These processes can be viewed as quantum tunnelling of particles
from the vacuum. The tunnelling approach, and the closely related instanton or
complex path approaches, are reviewed here with emphasis on paths in the
complex coordinate plane. The method is applied to particle production from a
black hole in a magnetic field, where ultra-high energy charged particles are
produced.

##  1 Introduction

The quantum vacuum is alive with virtual particles that only emerge into
reality in extreme conditions near black holes or in powerful external fields.
This particle creation can be described using various techniques, but the one
we focus on here is quantum tunnelling from the vacuum. Each methodology has
its various strengths, but there are situations where the tunnelling approach
is especially useful. One particular application where this is the case is the
production of particles from a magnetic black hole.

The tunnelling approach is influenced by an early description of particle
production from black holes that appeared in the work of Hartle and Hawking
[1]. They suggested that the amplitude for particle production could be
related to a particle path from the future singularity to the black hole
exterior, as in Fig. 1. There is no such classical path, but in the analysis,
they used analytic continuation of the time coordinate to show that the
probability $P$ of particle production and absorption for a Schwarzschild
black hole where related by

$P(\hbox{particle emmission})=e^{-\beta E}P(\hbox{particle absorption})$ (1)

where $\beta$ is the inverse Hawking temperature. This relation is enough to
guarantee that the black hole can be in equilibrium with a heat bath at the
Hawking temperature.

Figure 1: Particle production on a black hole spacetime [1]. The amplitude for
particle production at the point $C$ can be related to paths $BAC$ and $AD$ by
analytic continuation.

Hartle and Hawking also extended their relation to charged and rotating black
holes. In the period since their pioneering work, analytic continuation has
been used to deliver more detailed information about the particle production
rate beyond the simple relation Eq. (1), for example with charged black holes
[2, 3, 4]. The approach is often employed when a quantum field theory approach
is problematic, for example for back reaction problems [5] and for problems
with extremal horizons [6]. The combination of quantum tunnelling and particle
pair creation actually preceded the theory of black hole pair production,
first introduced in the context of alternating electric fields [7], and later
developed into a fully consistent theory of the Schwinger process [8, 9, 10,
11]. We aim to show that these situations have features in common that make it
reasonable to refer to them all as quantum tunnelling phenomena.

Astrophysical applications of vacuum breakdown of are somewhat restricted. A
rotating black hole with a magnetic field of around $10^{13}{\rm G}$ could in
principle induce electric field strengths $1.3\times 10^{18}{\rm V}{\rm
m}^{-1}$ needed for electron pair creation. Holes like this may arise from the
collapse of a magnetar to form a black hole, for example [12]. However, such
systems would be scenes of complex astrophysical phenomena, and secondary pair
production processes from high energy synchrotron photons $\gamma\to
e^{+}e^{-}$ would likely be prevalent. Nevertheless, the vacuum production
process would generate currents near the horizon and it may be important to
include these in fluid simulations.

Simple estimates of the vacuum breakdown near a black hole can easily be found
by taking the pair creation rates in flat space using the local electric field
value in some suitable chosen reference frame [2]. Here we shall improve on
this simple approach and include the effects of curvature on the particle
production. The wave equations for a charged particles around a magnetic
rotating hole are not separable, but the quantum tunnelling approach proves
invaluable. It turns out that the flat space effect overestimates the particle
particle production. We shall also be able to determine the dynamical
parameters of the electrons that are produced and examine their trajectories
in some detail.

The first sections of this paper aims to clarify some of the aspects of
particle production using instantons. In particular, we explore the difference
between an instanton that describes vacuum breakdown and an instanton that
describes Hawking radiation from an event horizon. We shall also make
extensive use of Hamiltonian methods and contours in the complex coordinate
plane, whose importance for particle production where extensively studied by
Srinivasan and Padmanabhan [13]

This paper uses a small modification of SI units in which the distance unit is
chosen so that the velocity of light $c=1$.

##  2 The instanton approach to quantum tunnelling

We start with a review of the instanton approach to quantum tunnelling through
a potential barrier, in order to bring out some of the features that will be
important later on. We will introduce Hamilton’s principle function and see
how this replaces the usual action, and we will empahsise the roles of branch
cuts in the complex coordinate plane.

In the simplest situation, a particle tunnels from a localised initial state.
The particle is prepared at time $t=0$ ‘inside’ the barrier, i.e. to the left
of the maximum of the the potential $V$ shown in figure 2. The probability of
finding the particle inside the barrier decays exponentially with a rate
$\Gamma$, which we identify as the vacuum decay rate. A simple analysis of the
decay rate using the WKB approximation to the Schrödinger equation gives

$\Gamma\approx\frac{\omega}{2\pi}\exp\left\\{-\frac{2}{\hbar}\int_{a}^{b}\left\\{2m(V-E)\right\\}^{1/2}dx\right\\}$
(2)

where $\omega^{2}=V^{\prime\prime}(0)/m$ and $E=(n+\frac{1}{2})\hbar\omega$
for some integer $n$.

Figure 2: A simple scenario for quantum tunnelling. The decay rate is
dominated by the WKB approximation with energy $E$ given by an harmonic
oscillator state to the left of the barrier. The exponential factor can be
expressed as a contour integral around the contour $C$.

Banks and Bender [14] demonstrated (in a more general context) that the
exponent in the decay rate could be obtained from a classical trajectory
$x_{b}(t_{I})$ with imaginary time $t_{I}=it$. The trajectory, or instanton,
runs from $x=a$ to $x=b$ in figure 2 and back to $x=a$. Consider the classical
action

$S[x]=\int\left\\{\frac{m}{2}\left(\frac{dx}{dt}\right)^{2}-V\right\\}dt.$ (3)

Switching to imaginary time,

$S[x]=i\int\left\\{\frac{m}{2}\left(\frac{dx}{dt_{I}}\right)^{2}+V\right\\}dt_{I}.$
(4)

Note that, along the instanton trajectory,

$\frac{m}{2}\left({dx_{b}\over dt_{I}}\right)^{2}-V=-E.$ (5)

It is now possible to relate the exponent in the tunnelling rate to the
instanton solution. First, we introduce Hamilton’s principle function $W$,

$W[x_{b}]=S[x_{b}]+E\int_{\cal C}dt,$ (6)

where the contour ${\cal C}$ goes around the path in imaginary time. From Eq.
(5), this can be simplified to

$W[x_{b}]=i\int_{C}\left(V-E\right)dt_{I}=2i\int_{a}^{b}\left\\{2m(V-E)\right\\}^{1/2}dx,$
(7)

Comparing with the WKB result (2) gives an important relation between the
tunnelling rate and the principle function,

$\Gamma\approx\frac{\omega}{2\pi}\exp\left\\{-\frac{W_{I}[x_{b}]}{\hbar}\right\\},$
(8)

where $W_{I}={\rm Im}\,W$. We could stop at this point, but suppose that
$V(x)$ is an analytic function, then the expression for $W[x_{b}]$can also be
expressed as a contour integral in the complex $x$ plane,

$W[x_{b}]=\int_{C}\left\\{2m(E-V)\right\\}^{1/2}dx.$ (9)

In this form, we can distort the contour of integration as long as it goes
exactly once around the branch cut in the integrand. We shall show later that
branch cuts and singularities in the complex coordinate plane play an
important role in distinguishing different types of quantum process.

It is useful at this point to compare the result to the theory of vacuum decay
[15]. Suppose we take Eq. (8) and expand in powers of $E/V$. We find,

$\Gamma\approx
A\left(\frac{S_{I}[x_{b}]}{2\pi}\right)^{1/2}\exp\left\\{-\frac{S_{I}[x_{b}]}{\hbar}\right\\},$
(10)

where $S_{I}={\rm Im}\,S$ and the factor $A$ depends on the detailed shape of
the potential. If we approach the same problem as a vacuum decay problem, we
obtain the same result with the factor $A$ determined by an operator
determinant. Although the two approaches are similar, we note there are
important differences. The result using the function $W$ does not assume $E/V$
is small and gives a simpler expression for the factor in front of the
exponential when we have a finite number of degrees of freedom.

In most of the applications considered below we have some ignorable
coordinates. As an example, suppose in the quantum tunnelling problem there
are two extra spatial dimensions $y$ and $z$, but the potential only depends
only on $x$. The wave function factorises, and the WKB analysis of the
tunnelling rate at fixed values of the momenta $p_{y}$ and $p_{z}$ is the same
as the one dimensional case. The formula (8) is still valid provided we modify
the definition of the principle function to remove the ignorable coordinates,

$W=S+Et-yp_{y}-zp_{z}.$ (11)

A similar correction should be applied and the modified principle function
used whenever there are conserved momenta.

##  3 The Schwinger process

The Schwinger process is the pair creation of charged particles, usually
electron positron pairs, in an electric field. Schwinger’s original
discussion, used heat-kernel methods, and gave an early example of a non-
perturbative result in quantum field theory. We shall review the tunnelling
approach to the Schwinger process with the aim of obtaining some general rules
for the tunnelling instanton.

Figure 3: The instanton for the Schwinger process and Vilenkin’s ‘ex nihilo’
version.

Consider a particle with mass $m$ and charge $e$. The particle world-line
$x^{\mu}(\tau)$ is parameterised by proper time $\tau$. The action can be
expressed in Hamiltonian form with momenta $p_{\mu}$,

$S=\int\left(\dot{x}^{\mu}p_{\mu}-H\right)d\tau.$ (12)

Given the metric $g_{\mu\nu}$, the vector potential $A_{\mu}$ and charge $e$,

$H=\frac{1}{2m}g^{\mu\nu}(p_{\mu}+eA_{\mu})+\frac{m}{2}.$ (13)

We take flat spacetime with a constant electric field ${\cal E}$ in the $x$
direction, associated with a potential $A_{t}={\cal E}x$. The resullting
Hamiltonian is

$H=-\frac{1}{2m}\left(p_{t}+e{\cal
E}x\right)^{2}+\frac{1}{2m}p_{x}^{2}+\frac{1}{2m}p_{y}^{2}+\frac{1}{2m}p_{z}^{2}+\frac{m}{2}.$
(14)

Normalisation of four-velocity $\dot{x}^{\mu}$ imposes a constraint $H=0$ on
the Hamiltonian. Furthermore, ignorable coordinates $t$, $y$ and $z$ imply
that the energy $E=-p_{t}$ and momenta $p_{y}$, $p_{z}$ are conserved. With
these restrictions, the modified principle function (11) reduces to

$W=S+Et-yp_{y}-zp_{z}=\int p_{x}dx$ (15)

For convenience, we introduce a new parameter $x_{0}$ related to the energy by
$E=e{\cal E}x_{0}$, then the constraint $H=0$ implies

$p_{x}^{2}=(e{\cal E})^{2}\left(x-x_{0}\right)^{2}-m^{2}-p_{\perp}^{2},$ (16)

where $p_{\perp}$ is the momentum perpendicular to the $x$ direction. Note
that, for real values of position $x$, we take the positive square root for
$p_{x}$.

In the tunnelling approach, we evaluate the tunnelling exponent Eq. (15) for a
solution of the equations of motion that runs along a closed complex contour
${\cal C}$ in the complex $x$ plane. The tunnelling exponent using Eq. (16)
is,

$W=\int_{\cal C}\left((e{\cal
E})^{2}\left(x-x_{0}\right)^{2}-m^{2}-p_{\perp}^{2}\right)^{1/2}dx,$ (17)

The integrand has a branch cut between $x_{0}\pm\kappa$, where
$\kappa=(m^{2}+p_{\perp}^{2})^{1/2}/|e{\cal E}|$. In order to find a suitable
integration contour we start from the general solution to the equations of
motion in real time,

$\displaystyle x-x_{0}$ $\displaystyle=\kappa\cosh\frac{\tau}{\kappa}$ (18)
$\displaystyle t-t_{0}$ $\displaystyle=\kappa\sinh\frac{\tau}{\kappa},$ (19)

Consider the complex contour

$\tau=-i\kappa\phi+\epsilon,$ (20)

where the real parameter $\phi$ lies on a circle and an $i\epsilon$
prescription has been used to avoid the branch cut in Eq. (17). In the plane
with axes ${\rm Re}(x)$ and ${\rm Im}(t)$, the contour is a circle, as shown
in Fig. 3. The direction has been chosen so that the principle value of the
square root will result in a positive imaginary part for the integral. If we
use the negative root in Eq. (17) then we take a counter-clockwise contour. An
interesting interpretation of the instanton has been suggested by Vilenkin
[16]. Combining the bottom half of the instanton to the real time evolution of
the particle worldlines for $t>0$ produces the picture on the right. From the
point of view of an observer in real time, the electron positron pair suddenly
appears as if we have ‘creation from nothing’. Strange behaviour should be
expected when we try to interpret a quantum phenomenon in purely classical
terms.

Figure 4: Alternative views of the instanton for the Schwinger process. The
complex space plane (left) and the complex time plane (right).

The first diagram in Fig. 4 shows how the instanton contour goes around the
branch cut in the complex $x$ plane. Note that any contour which circles the
branch cut clockwise exactly once gives the same value of the tunnelling rate,
so that the only ambiguity in the result lies in the winding number of the
contour. The second diagram in Fig. 4 shows the contour in the complex $t$
plane. In this picture the tunnelling contour can be split into a particle
line and an antiparticle line. Each line contributes half of the closed
instanton path, and an instanton with winding number $-n$ would represent the
production of $n$ particle-antiparticle pairs.

Integrating (17) along the contour around the branch cut gives the tunnelling
exponent,

$\frac{W_{I}}{\hbar}=\frac{\pi m^{2}}{|e{\cal E}|\hbar}+\frac{\pi
p_{\perp}^{2}}{|e{\cal E}|\hbar}.$ (21)

The prefactor for the tunnelling rate in the barrier penetration case was
$(V^{\prime\prime}/2\pi m)^{1/2}$, and we will divide this by the Compton
wavelength $\hbar/mc$ to get the correct dimensions. Putting in a phase space
factor in addition gives an estimate for the particle production $d\Gamma$
with transverse momentum $p_{\perp}$,

$d\Gamma=\frac{|e{\cal
E}|}{2\pi\hbar}\left(\frac{dp_{\perp}}{2\pi\hbar}\right)^{2}e^{-W_{I}/\hbar}$
(22)

After integrating the particle production rate $d\Gamma$ over the transverse
momenta, we obtain the correct formula for the particle production rate
$\Gamma$ per unit volume [9] ,

$\Gamma={1\over\pi}\left({e{\cal E}\over 2\pi\hbar}\right)^{2}e^{-\pi
m^{2}/\hbar|e{\cal E}|}$ (23)

An exponent $\pi m^{2}/\hbar|e{\cal E}|\approx 1$ for electrons corresponds to
an electric field strength $4.157\times 10^{18}\,{\rm V}{\rm m}^{-1}$. Pair
production is heavily suppressed for smaller field strengths. On the other
hand, ordinary perturbation theory can be used to describe pair production for
larger field strengths. The result is only useful over a limited range of
field strengths.

In conclusion, the Schwinger process is represented by a closed contour around
a branch cut in the complex coordinate plane. Two halves of the contour with
single winding number in the complex time plane represent production of a
particle and an antiparticle.

###  3.1 The thermal Schwinger process

The production of particles in an electric field at finite temperature gives
another application of the tunnelling approach [17, 18, 19]. Thermal
tunnelling rates in quantum mechanics are related to the imaginary part of the
free energy [20]. In the path integral approach, we find the free energy by
imposing a periodicity $\beta=\hbar/k_{B}T$ on the action in imaginary time.
We do the same for calculating the particle creation rate. As before, the the
main focus here will be on the choice of contour for the instanton
approximation.

Figure 5: The instanton for the thermal Schwinger process. Pasting the
circular instanton on to the periodic manifold gives a non-differentiable path
(left). Moving the pieces together gives the differentiable path (right).

In the Schwinger process, the periodicity in imaginary time cuts off the top
and bottom of the circular instanton as shown in figure 5. The contour would
be continuous on the periodic manifold, but not differentiable. In order to
obtain a differentiable contour we move the left and right segment together as
on the right side of figure 5. This adjustment is essential for obtaining the
correct value of the instanton action.

The right segment is centred at $x_{0}$ and the left segment at $x_{1}$. The
corresponding integrals are denoted by $W_{R}$ and $W_{L}$, and evaluated
using the angle $\phi=i\tau/\kappa$ as independent variable in Eqs. (17) and
(18). The contributions are

$\displaystyle W_{R}$
$\displaystyle=im\kappa\int_{-\phi_{0}}^{\phi_{0}}\sin^{2}\phi\,d\phi-
imx_{0}\int_{-\phi_{0}}^{\phi_{0}}\cos\phi\,d\phi$ (24) $\displaystyle W_{L}$
$\displaystyle=im\kappa\int_{\pi-\phi_{0}}^{\pi+\phi_{0}}\sin^{2}\phi\,d\phi-
imx_{1}\int_{\pi-\phi_{0}}^{\pi+\phi_{0}}\cos\phi\,d\phi,$ (25)

where $\sin\phi_{0}=\beta/2\kappa$. The final result is independent of $x_{0}$
and $x_{1}$ because of the identity $x_{0}-x_{1}=2\kappa\cos\phi_{0}$. The
total functon $W=W_{R}+W_{L}$,

$W=2i\kappa m\left\\{\phi_{0}+\frac{1}{2}\sin 2\phi_{0}\right\\}$ (26)

The tunnelling exponent [17],

$\frac{W_{I}}{\hbar}=\frac{2\kappa
m}{\hbar}\left\\{\arcsin\left(\frac{\beta}{2\kappa}\right)+\frac{\beta}{2\kappa}\left[1-\frac{\beta^{2}}{4\kappa^{2}}\right]^{1/2}\right\\},$
(27)

where $\kappa=(m^{2}+p_{\perp}^{2})^{1/2}/|e{\cal E}|$. This reproduces the
Schwinger result (21) in the zero temperature limit. In the high temperature
limit, $W_{I}/\hbar\to 2\beta m$, which represents the probability of finding
a particle-antiparticle pair at high temperature. In future, whenever we see
$W_{I}/\hbar\to\beta E$ we will interpret this as a signal of thermal particle
production at temperature $k_{B}T=\hbar/\beta$.

##  4 The Fulling-Davies-Unruh effect

The next example is the detection of thermal particles by an accelerating
detector. We take the detector to be at rest in a two-dimensional accelerating
frame with acceleration $g$. We shall review the tunnelling description to see
what features of tunnelling instantons are typical of thermal particle
production.

The accelerating frame is associated with a set of Rindler coordinates
$(x,t)$, and metric

$ds^{2}=-g^{2}x^{2}dt^{2}+dx^{2}.$ (28)

The Hamiltonian (13) for particle motion $x(\tau)$ and $t(\tau)$, is

$H=-\frac{1}{2m}\frac{p_{t}^{2}}{g^{2}x^{2}}+\frac{1}{2m}p_{x}^{2}+\frac{m}{2}.$
(29)

The energy $E=-p_{t}$ is conserved and the Hamiltonian is constrained to
$H=0$. As before, the tunnelling is related to the principle function
integrated around a closed contour,

$W=\int_{\cal C}p_{x}dx,$ (30)

Using the Hamiltonian constraint,

$W=\int_{\cal C}\left(\frac{E^{2}}{g^{2}}-m^{2}x^{2}\right)^{1/2}\frac{dx}{x}$
(31)

To investigate the integration contour, we take the general solution to the
equations of motion,

$\displaystyle x$
$\displaystyle=\left(\frac{E^{2}}{m^{2}g^{2}}-\tau^{2}\right)^{1/2},$ (32)
$\displaystyle t-t_{0}$
$\displaystyle=\frac{1}{g}\log\left(\frac{E+mg\tau}{E-mg\tau}\right)^{1/2}.$
(33)

Consider proper time contour

$\tau=-\frac{E}{mg}+\epsilon e^{2i\phi},$ (34)

where $\phi$ lies on the circle. This gives a circular contour in the complex
$x$ plane around the horizon $x=0$. Any closed contour which goes around the
horizon singularity once will give the same value for the tunnelling exponent.
In the complex $t$ plane, the contour goes between $t_{0}\pm 2\pi i/g$. The
metric is regular and the contour is closed if we impose periodicity of the
metric in imaginary time.

Figure 6: Two views of the instanton for the Unruh process. The complex space
plane (left) and the complex time plane (right).

Integrating around the singularity using the residue theorem gives exponent

$\frac{W_{I}}{\hbar}=\frac{2\pi E}{\hbar g}.$ (35)

This has the thermal interpretation $W_{I}/\hbar=E/k_{B}T$ as in relation (1),
where the Unruh temperature

$T=\frac{\hbar g}{2\pi k_{B}}.$ (36)

We should also examine what happens if we use a different coordinate system,
specifically putting the metric in Boyer-Lindquist form with $r=gx^{2}/2$,

$ds^{2}=-fdt^{2}+f^{-1}dr^{2},\hbox{ where }f=g^{2}r^{2}.$ (37)

The same contour (32) which wound once around the horizon in the complex $x$
plane now winds twice around the horizon in the complex $r$ plane, though the
periodicity in the complex $t$ plane and the particle production rate remain
the same.

In conclusion, horizon radiation is represented by a closed contour around a
singularity in the complex coordinate plane. The contour is closed in the
complex time plane only when we impose impose periodicity in imaginary time.
The contour has winding number two in the complex $r$ plane when we use the
Boyer-Lindquist coordinates.

##  5 Charged black holes

Radiation from charged black holes can include contributions from thermal
radiation with the Hawking temperature $T_{h}$ and breakdown of the vacuum due
to the electric field outside the black hole. All forms of radiation are
included in the simple expression for the particle flux obtained from a mode
decomposition of the Dirac or the wave equation [21],

$F=\sum_{l,m}\int_{0}^{\infty}\frac{d\omega}{2\pi}\left(1-|A_{lm}|^{2}\right)\frac{1}{e^{\beta\omega_{h}}-1},$
(38)

where the inverse temperature $\beta=\hbar/T_{H}$. The frequency
$\omega_{h}=\omega-e\Phi_{h}$, where $\Phi_{h}$ is the electrostatic potential
at the horizon. The amplitude $A_{lm}$ represents reflection of the particle
modes with angular wave numbers $l$ and $m$ back into the black hole. This
amplitude can only be obtained numerically, or using approximate methods for
various regimes.

The quantum tunnelling approach to particle creation can be used to obtain
closed expressions in the regime $|\beta\omega_{h}|\gg 1$, when it is related
to using WKB approximations to the reflection amplitude. In this limit, the
flux integral can be decomposed into two parts:

The super-radiant regime $\omega<e\Phi_{h}$ where the flux becomes

$F_{\rm
super}\approx\sum_{l,m}\int_{0}^{e\Phi_{h}}\frac{d\omega}{2\pi}\left(|A_{lm}|^{2}-1\right)$
(39)

It is in this regime that electromagnetic breakdown of the vacuum can occur.

The non-super-radiant regime $\omega>e\Phi_{h}$, where

$F_{\rm
thermal}\approx\sum_{l,m}\int_{e\Phi_{h}}^{\infty}\frac{d\omega}{2\pi}\left(1-|A_{lm}|^{2}\right)e^{-\beta\omega_{h}},$
(40)

which we can regard as the Maxwell-Boltzmann approximation to the thermal
Hawking flux filtered by a grey-body factor. We shall now show how the quantum
tunnelling approach reproduces these results.

###  5.1 The tunnelling approach

The spacetime is described by the Reissner-Nordstrom metric

$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$
(41)

where

$f=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$ (42)

Geometric mass $M=GM_{*}$ and Geometric charge
$Q=G^{1/2}Q_{*}/(4\pi\epsilon_{0})^{1/2}$ are related to the physical mass and
charge $M_{*}$ and $Q_{*}$. The electrostatic potential $\Phi$ at radius $r$
is

$\Phi=\frac{Q_{*}}{4\pi\epsilon_{0}r}.$ (43)

Figure 7: The complex $r$ plane showing different contours.

Due to the rotational symmetry, it will be sufficient to start from the
Hamiltonian (13) for a particle of charge $e$ in the equatorial plane, with
conserved momenta $E=-p_{t}$ and $L=p_{\phi}$,

$H=-\frac{(E-e\Phi)^{2}}{2mf}+\frac{fp_{r}^{2}}{2m}+\frac{1}{2m}\left(m^{2}+\frac{L^{2}}{r^{2}}\right).$
(44)

The modified principle function $W=S+Et-L\phi$ is,

$W=\int_{\cal C}p_{r}dr=\int_{\cal
C}\frac{1}{f}\left\\{\left(E-e\Phi\right)^{2}-\left(m^{2}+\frac{L^{2}}{r^{2}}\right)f\right\\}^{1/2}dr.$
(45)

There are poles at the outer and inner horizons $r_{h}$ and $r_{c}$, as well
as possible branch cuts. A typical representation of the complex $r$ plane is
shown in figure 7. The contribution from each of the contours will be denoted
by a subscript, e.g. $W_{\infty}$ for the large outer contour.

From the large radius limit, we find

$W_{\infty}=\frac{4\pi
iM}{(E^{2}-m^{2})^{1/2}}\left\\{\left(E^{2}-\frac{e\Phi_{h}r_{h}}{2M}\right)-\frac{m^{2}}{2}\right\\}$
(46)

The horizon integrals are obtained from the residue theorem,

$\displaystyle W_{h}$ $\displaystyle=\frac{\pi
i}{\kappa_{h}}\left(E-e\Phi_{h}\right)$ (47) $\displaystyle W_{c}$
$\displaystyle=\frac{\pi i}{\kappa_{c}}\left(E-e\Phi_{c}\right)$ (48)

where the surface gravities $\kappa_{h}=f^{\prime}(r_{h})/2$ and
$\kappa_{c}=f^{\prime}(r_{c})/2$. Integrals around the branch cuts can be
deduced from the other integrals using Cauchy’s theorem.

###  5.2 Black hole Schwinger process

The Schwinger process for electron-positron production is represented by a
contour which goes around the branch cut. This contribution is independent of
the Hawking temperature and we identify it with the super-radiant flux (39).
From Cauchy’s theorem, the principle function $W_{S}=W_{h}+W_{c}-W_{\infty}$.
The imaginary part,

$W_{I}=\frac{4\pi
M}{(E^{2}-m^{2})^{1/2}}\left\\{\left(E-(E^{2}-m^{2})^{1/2}\right)\left(E-\frac{e\Phi_{h}r_{h}}{2M}\right)-\frac{m^{2}}{2}\right\\}.$
(49)

In the large energy limit, the tunnelling exponent at leading order of $m/E$
is

$\frac{W_{I}}{\hbar}=\frac{\pi m^{2}e\Phi_{h}r_{h}}{\hbar E^{2}}.$ (50)

The angular momentum $L$ only appears in the location of the branch cut. If
$L\ll Er_{h}$, then the branch cut is narrow with centre at the radius where
$E-e\Phi=0$. Physically, this represents the radius $r(E)$ at which the
particles of energy $E$ are created. The electric field at the centre of the
branch cut is

$e{\cal E}=\frac{e\Phi}{r}=\frac{E^{2}}{e\Phi_{h}r_{h}}$ (51)

Hence

$\frac{W_{I}}{\hbar}=\frac{\pi m^{2}}{\hbar e\cal E}.$ (52)

This recovers the Schwinger result, but with the local electric field ${\cal
E}$ at the radius where the particles are created. We conclude that the
particle production is sufficiently localised for the equivalence principle to
hold. Furthermore, we can use the Schwinger result to infer the pre-factor for
the particle production rate per unit volume,

$\Gamma={1\over\pi}\left({e{\cal E}\over 2\pi\hbar}\right)^{2}e^{-\pi
m^{2}/e\hbar{\cal E}}.$ (53)

As with the flat spacetime result, this is only valid for large electric
fields.

Figure 8: Evolution of the geometric mass and charge parameters due to
particle production is downwards along the red lines in this plot. Thermal
emission is tiny and not included. Units are solar Schwartzchild radii (2.9
km).

The total luminosity of the black hole can be obtained by integrating the
particle production for the region outside of the horizon. Because of the
relation between the location of particle creation $r$ and the energy, this is
equivalent to integrating over the energy. First, we rewrite the particle
production rate in terms of radius $r$ using (51) and (52),

$\Gamma={1\over 4\pi\alpha^{2}}\left({mr_{h}\over\hbar r}\right)^{4}e^{-\alpha
r^{2}/r_{h}^{2}},$ (54)

where

$\alpha=\frac{\pi m^{2}r_{h}}{\hbar e\Phi_{h}}.$ (55)

The evaporation rate is then

$\frac{dM_{*}}{dt}=\int_{r_{h}}^{\infty}dr\,4\pi r^{2}\,\Gamma
E=\frac{\pi}{\alpha^{2}r_{h}}\left(\frac{mr_{h}}{\hbar}\right)^{5}\Gamma(-2,\alpha),$
(56)

where $\Gamma(a,x)$ is the incomplete Gamma function. The charge evaporates at
a rate

$\frac{dQ_{*}}{dt}=\int_{r_{h}}^{\infty}dr\,4\pi r^{2}\,\Gamma e=\frac{\pi
e}{\alpha^{3/2}r_{h}}\left(\frac{mr_{h}}{\hbar}\right)^{4}\Gamma(-3/2,\alpha)$
(57)

The relative rates of (geometric) charge and mass evaporation has a simple
expression,

$\frac{dQ}{dM}=\frac{Q}{r_{h}}\frac{\Gamma(-2,\alpha)}{\alpha^{1/2}\Gamma(-3/2,\alpha)}$
(58)

It is a known result that the black hole looses charge due to super-radiance
at a far higher rate than it looses mass [22]. However, having an expression
in closed form is a success of the tunnelling approach.

###  5.3 Black hole Hawking process

The Hawking flux has two contributions. For $E>e\phi_{h}$, there is a
contribution from the contour which circles the horizon and represents
particle production at the horizon. We may also have contributions from branch
cut outside the horizon which now represents the transmission term
$|A_{lm}|^{2}$ through the potential barrier. The horizon contribution has
winding number two in the coordinate system in use, as we saw earlier in the
context of the Fulling-Davies-Unruh effect. The integral $W_{H}=2W_{h}$ gives
a particle creation rate

$\Gamma_{H}\propto e^{-\beta(E-e\Phi_{h})/\hbar}$ (59)

which agrees with the first term in (40), at the Hawking temperature
$T_{H}=\hbar\kappa_{h}/2\pi$.

##  6 Particle production on a magnetic rotating black hole background

In this section we apply the tunnelling method to the production of electron-
positron pairs from the vacuum around a rotating black hole in an external
magnetic field. The Hawking radiation is insignificant for large black holes,
and so with astrophysical applications in mind we consider only the Schwinger
process. However, we take an idealised vacuum situation with no other
particles present.

###  6.1 Geometry

For a solar-mass black hole, the back-reaction of the magnetic field on the
geometry is small when $B\leq 10^{15}\,G$ and the Kerr metric can be used,

$ds^{2}=-\frac{\Delta}{\rho^{2}}\omega^{t\
2}+\frac{s^{2}}{\rho^{2}}\omega^{\phi\
2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2},$ (60)

where

$\omega^{t}=dt-as^{2}d\phi,\qquad\omega^{\phi}=(a^{2}+r^{2})d\phi-adt,$ (61)

The metric functions are $\Delta=r^{2}+a^{2}-2Mr$, $s=\sin\theta$ and
$\rho^{2}=r^{2}+a^{2}-a^{2}s^{2}$. The geometric mass $M$ is $G/c^{2}$ times
the physical mass.

We will take a magnetic field with rotational symmetry about the black hole
axis and assume the simplest dipole field that approaches a constant field
with strength $B$ in the $z$ direction at large distances. Furthermore, we
will assume that the movement of charged particles leaves the black hole with
a net charge that neutralises the electromotive force (EMF). The
electromagnetic potential for this zero EMF field has components

$A_{t}=-\frac{Mars^{2}}{\rho^{2}}B,\qquad A_{\phi}=\frac{As^{2}}{2\rho^{2}}B,$
(62)

where $A=(r^{2}+a^{2})^{2}-\Delta a^{2}s^{2}$. Although the EMF vanishes,
there is an electric field in the non-rotating (zero angular momentum) frame
defined in Ref. [23]. We shall see that this electric field is associated with
the particle production.

### Dynamics

Some basic dynamical notions will be needed for the particle production
calculation. The four-momentum $p_{\mu}$ for a particle with mass $m$, charge
$e$ and four velocity $u^{\mu}$ is

$p_{\mu}=mg_{\mu\nu}\left(u^{\nu}+eA^{\nu}\right)$ (63)

Along the Killing directions, we set

$E=-p_{t},\qquad L=p_{\phi},$ (64)

The momenta are related by the constraint

$g^{\mu\nu}p_{\mu}p_{\nu}=-m^{2},$ (65)

After inserting the metric components,

$\frac{\Delta}{\rho^{2}}p_{r}^{2}+\frac{1}{\rho^{2}}p_{\theta}^{2}+V=0,$ (66)

where the effective potential $V$ is given by

$V=-\frac{A}{\Delta\rho^{2}}\left(E-\Omega
L\right)^{2}+\frac{\rho^{2}}{As^{2}}\left(L-\frac{eBAs^{2}}{2\rho^{2}}\right)^{2}+m^{2}.$
(67)

The local rotation rate $\Omega=2Mra/A$.

###  6.2 Tunnelling exponents

The tunnelling exponent is given by ${\rm Im}W/\hbar$, where the modified
principle function $W=S+Et-L\phi$. Inserting the action leaves

$W=\int_{\cal C}\left(p_{r}dr+p_{\theta}d\theta\right).$ (68)

Unlike in the previous examples, there are two remaining coordinates $r$ and
$\theta$, but both are implicitly functions of the proper time $\tau$. The
complex contour $C$ for the Schwinger process surrounds a branch cut and gives
an imaginary value to $W$. This happens in a region where classical
trajectories are forbidden because $V$ is negative, and the momenta are
therefore complex. Tunnelling occurs inside a potential barrier that ends at
points PQ as shown in Fig 9.

Figure 9: A region of the $(r,\theta)$ plane showing regions of positive
potential (grey) and negative potential (white). The line PQ shows the
trajectory of an instanton, with the particle-antiparticle pair produced at P
and Q.

A crucial observation is that the tunnelling only occurs with any significant
rate for very small values of $W_{I}$ compared to the astrophysical scales set
by the mass of the black hole. This requires both brackets in the potential
(67) to be very small, and restricts the values of the energy and angular
momentum. The centre of the barrier $r=r_{c}$, $\theta=\theta_{c}$ is located
where both brackets vanish,

$E=\Omega_{c}L,\qquad L=\frac{eBs_{c}^{2}A_{c}}{2\rho^{2}_{c}}$ (69)

These relate both the energy and angular momentum to $r_{c}$ and
$s_{c}=\sin\theta_{c}$. Because the barrier is extremely narrow, we can think
of pair creation for particles with energy $E$ happening along the circle at
$r_{c}(E)$ and $\theta_{c}(E)$.

In the $x^{i}=(r,\theta)$ sector, the Hamiltonian that generates the field
equations is

$H=\frac{1}{2m}g^{ij}p_{i}p_{j}+\frac{1}{2m}V$ (70)

In the region of the barrier, we introduce small quantities $\delta
q^{1}=r-r_{c}$ and $\delta q^{2}=\theta-\theta_{c}$, and we use a quadratic
approximation to the Hamiltonian,

$H=\frac{1}{2m}g^{ij}p_{i}p_{j}+\frac{1}{4m}V_{,ij}\delta q^{i}\delta
q^{j}+\frac{m}{2},$ (71)

where the Hessian of the potential is evaluated at the centre of the barrier
$(r_{c},\theta_{c})$. We diagonalise the Hamiltonian by solving the eigenvalue
problem for basis vectors $e_{n}^{i}$,

$V_{,ij}e_{n}^{j}=2m^{2}\lambda_{n}g_{ij}e_{n}^{j}.$ (72)

Introduce normal mode coordinates $x^{n}$, where

$\delta q^{i}=x^{n}\,e_{n}^{i}$ (73)

In terms of the normal modes,

$H=\frac{1}{2m}\delta^{mn}p_{m}p_{n}+\frac{1}{2}m\Lambda_{mn}x^{m}x^{n}+\frac{m}{2},$
(74)

where $\Lambda_{mn}={\rm diag}(\lambda_{1},\lambda_{2})$. For a compact
instanton, we must use the mode $x$ which has a negative eigenvalue
$\lambda=-\omega^{2}$. This is the mode that corresponds to the line PQ in
figure 9. For this mode,

$H=\frac{1}{2m}p_{x}^{2}-\frac{1}{2}m\omega^{2}x^{2}+\frac{m}{2}=0$ (75)

The principle function is

$W=\int_{\cal C}p_{x}dx=im\int_{\cal C}(1-\omega^{2}x^{2})^{1/2}dx,$ (76)

where the contour winds once around the branch cut for a single pair creation
event. Hence

$W=\frac{im\pi}{\omega}.$ (77)

A better feel for the result can be obtained by scaling out the dimensionfull
quantities from $\omega$,

$\omega=\frac{eB}{m}\hat{\omega},$ (78)

where $\hat{\omega}\equiv\hat{\omega}(r_{c}/M,a/M,\theta_{c})$ is
dimensionless, and obtained by solving the eigenvalue problem (72) with
$\lambda=-\omega^{2}$. The particle production rate at $(r_{c},\theta_{c})$ is
$\propto\exp(-W_{I}/\hbar)$, where

$\frac{W_{I}}{\hbar}=\frac{\pi m}{\hbar\omega}=\frac{\pi
m^{2}}{\hbar\hat{\omega}eB}={\pi B_{0}\over\hat{\omega}B},$ (79)

and $B_{0}=m^{2}/e\hbar=4.4\times 10^{13}\,G$ for electrons. In general, the
factor $\hat{\omega}$ has to be obtained numerically, but in the special case
of equatorial particle production the value has a closed form,

$\hat{\omega}=\frac{1}{r^{2}}\left\\{a^{2}(r+M)^{2}-\Delta
r^{2}\right\\}^{1/2}.$ (80)

###  6.3 Particle fluxes

The factor $\pi/\hat{\omega}$ that determines the particle production rate has
been plotted in figures 10, where we see the the relative amount of particle
production for different values of $r$ and $\theta$. Particle production is
concentrated close to the horizon. Initially, $p_{r}=p_{\theta}=0$, and the
the particles move in circular orbits. As more particles are produced, a
current loop will build up which produces a field counteracting the original
field. Gradually, the instability in the normal mode ‘$x$’ direction drives
particles away from their circular orbits, and into a chaotic ones [24].

Figure 10: A region of the $r$ and $\theta$ plane showing contours of constant
particle production rate. The colours represent the factor $\pi/\hat{\omega}$.
The extremal case $a=M$ on the left and $a=0.7M$ on the right.

Comparing the exponents in the particle production rates Eq. (23) and Eq. (79)
suggest that there is an ‘effective’ Schwinger process electric field, which
we denote by $E_{s}$,

$E_{s}=\hat{\omega}B$ (81)

Figure 11 shows a comparison between the actual electric field strength,
$E_{r}$, in the locally non-rotating frame and the field strength $E_{s}$
inferred by actual rates. The two agree at the horizon, but as we move away
from the horizon the flat space Schwinger result overestimates the production
rate.

Figure 11: The electric field strength $E_{r}$ in the non-rotating (zero
angular momentum) frame and the effective field strength $E_{s}$, inferred
from the Schwinger particle production formula. The field strengths are
evaluated in the equatorial plane for an extremal black hole.

Although the Schwinger process in flat space does not give the exact exponent,
it should still be accurate enough for calculating the pre-factor in the
particle production rate, especially if we use the effective field strength
$E_{s}$ in the Schwinger result (23). The particle production rate per unit
proper volume and time should therefore be

$\Gamma={1\over 4\pi^{3}}\left({eB\hat{\omega}\over\hbar}\right)^{2}e^{-\pi
B_{0}/\hat{\omega}B}$ (82)

The particle production depends on radius $r$ and angle $\theta$. The
electrons and positrons move in circular orbits at near-light speed and
generate current density $J^{\mu}$, with

$dJ^{\mu}=2e\Gamma u^{\mu}d\tau,$ (83)

where $d\tau$ is the proper time interval in the non-rotating frame. The rate
of change of azimuthal current $I$ in the Boyer-Lindquist frame is obtained by
a volume integral of $dJ^{\phi}$,

$\frac{dI}{dt}=\int_{r_{h}}^{\infty}dr\int_{0}^{\pi}d\theta\,4\pi\rho^{2}\sin\theta\,e\Gamma
u^{\phi}$ (84)

For a detailed calculation, we can find the value of the velocity $u^{\phi}$
by expanding about the centre of the barrier as before. Consider the velocity
components $u^{\alpha}$, where $x^{\alpha}=\\{t,\phi\\}$. When expressed in
terms of the velocity, the potential $V$ in Eq. (67) becomes

$V=m^{2}g_{\alpha\beta}u^{\alpha}u^{\beta}+m^{2},$ (85)

where the $u^{\alpha}$ velocity components are regarded as functions of $r$
and $\theta$, given in terms of the constant momenta by
$u^{\alpha}=g^{\alpha\beta}(p_{\beta}-eA_{\beta})/m$. We defined the centre of
the barrier $r_{c}$, $\theta_{c}$ as the point where these functions vanish.
At the ends of the barrier, we use the normal mode $e^{i}$ from Eq. (73) and
define $u^{\alpha}{}_{,x}=u^{\alpha}{}_{,i}e^{i}$,

$u^{\alpha}=u^{\alpha}{}_{,i}\delta q^{i}=xu^{\alpha}{}_{,x}$ (86)

The other components $u^{i}$ vanish at the ends of the barrier by Eq. (66).
Substituting back into the potential (85) gives

$u^{\alpha}=\frac{u^{\alpha}{}_{,x}}{|u_{,x}|}$ (87)

where
$|u_{,x}|^{2}=-g_{\alpha\beta}u^{\alpha}{}_{,x}u^{\beta}{}_{,x}=\omega^{2}$.

Finally, we can obtain a rough estimate by expanding $\hat{\omega}$ about the
horizon, where $\hat{\omega}=2$. This gives

$I\approx\left(\frac{2eBr_{h}}{\pi\hbar}\right)^{2}e^{-\pi B_{0}/2B}\Delta
t\approx I_{0}B_{13}^{2}e^{-\pi B_{0}/2B}\Delta t$ (88)

where $B_{13}$ is the magnetic field strength in units of $10^{13}{\rm G}$,
$I_{0}=3.4\times 10^{51}{\rm A}$ and $\Delta t$ is the time over which the
particles remain in circular orbits. The system can only maintain equilibrium
if the magnetic field generated by this current is smaller than the external
field $B$. The induced field near the horizon $B_{I}\approx\mu_{0}I/2r_{h}$.
Requiring $B_{I}<B$ gives

$B\lesssim 1.0\times 10^{12}{\rm G}$ (89)

Note that this is rather less than the field $B_{0}=4.4\times 10^{13}{\rm G}$.
Nevertheless, the energy of the particles from Eq. (69) is still very high,
$E\approx 3.00\times 10^{20}\,B_{13}r_{\rm km}\,{\rm eV}$ for particles
produced close to the horizon $r_{\rm km}$ measured in kilometers.

We can obtain information about the trajectories by looking at the potential
diagrams in 12. Initially, due to the closeness to the point $r_{c}$,
$\theta_{c}$ where the potential gradients vanish, the forces moving the
particles away from the circular orbits are very small. From the potentials,
we see that the particles produced on the inner edge of the instanton always
end up inside the hole. Particles produced on the outside edge can eventually
move off to infinity. Depending on the initial radius, the particle may cross
the equatorial plane, but the ones that avoid the equatorial plane drift away
in the direction along the axis of rotation.

Figure 12: A region of the $r$ and $\theta$ plane showing positive potential
(grey) and negative potential (white). Particles are confined to a white
region and produced at the edge. On the left $r_{c}=1.5M$ and the outgoing
particle cannot cross the equatorial plane. On the right $r_{c}=1.4M$ and the
particle can cross the equatorial plane.

The electrons are produced in high energy circular orbits and should be
significant sources of synchrotron radiation. In flat space, the synchrotron
emission rate $dE/dt\propto\gamma^{2}$ where $\gamma$ is the Lorentz factor.
In curved space, the Lorentz factor $\gamma\sim u^{t}$ and we saw earlier that
the tunnelling process requires special values of the energy and angular
momenta, and these imply $u^{t}\sim 1$. Consequently, synchrotron emission in
the circular orbits is very highly suppressed, and has a negligible affect on
the motion. However, this only covers the initial circular orbits, and
gradually as the particles drift away from the potential barrier, they will
accelerate to higher speeds and the emission will increase.

##  7 Conclusion

We have seen some of the tricks employed when using the instanton approach to
particle creation in moderately strong fields and curved spacetimes. It can be
viewed as a method for obtaining quick results from any situation where WKB
analysis would be appropriate. The results are non-perturbative, but the range
of usefulness is restricted and prefactors to the exponential rates are often
difficult to obtain.

An example is the rate of vacuum breakdown due to the Schwinger effect near a
black hole in a magnetic field. The system is very limited because it ignores
collisions between particles in the surrounding medium, in particular the
production from high energy photons via $\gamma\to e^{+}e^{-}$. Nevertheless,
it is clear that fields in excess of $4.4\times 10^{11}\,G$ will copiously
produce electrons of energy above $10^{18}{\rm eV}$. The particles produced by
the Schwinger mechanism can form a current loop around the black hole before
drifting off along the rotation axis, whilst emitting significant synchrotron
radiation as they speed up. One application of the results may be to add
particle production terms to relativistic MHD simulations, using the Schwinger
particle production rate in a zero angular momentum frame.

The examples illustrate some interesting features of the instanton approach to
particle creation. In particular, the importance of Hamiltonian methods and
the distinction between branch cuts which signal vacuum breakdown and
singularities that signal horizon radiation.

This work was supported by the UK Science and Technology Facilities Council
(STFC) [grant ST/T000708/1].

## References

## References

  * [1] Hartle J B and Hawking S W 1976 Phys. Rev. D 13(8) 2188–2203 URL https://link.aps.org/doi/10.1103/PhysRevD.13.2188
  * [2] Preparata G, Ruffini R and Xue S S 1998 Astron. Astrophys. 338 L87–L90 (Preprint astro-ph/9810182)
  * [3] Kim S P and Page D N 2005 Nuovo Cim. B 120 1193–1208 (Preprint gr-qc/0401057)
  * [4] Ruffini R, Salmonson J D, Wilson J R and Xue S S 1999 Astronomy and Astrophysics 350 334–343 (Preprint %****␣BHParticles.bbl␣Line␣25␣****astro-ph/9907030)
  * [5] Parikh M K and Wilczek F 2000 Phys. Rev. Lett. 85 5042–5045 (Preprint hep-th/9907001)
  * [6] Angheben M, Nadalini M, Vanzo L and Zerbini S 2005 JHEP 05 014 (Preprint hep-th/0503081)
  * [7] Brezin E and Itzykson C 1970 Phys. Rev. D 2(7) 1191–1199 URL https://link.aps.org/doi/10.1103/PhysRevD.2.1191
  * [8] Affleck I K, Alvarez O and Manton N S 1982 Nucl. Phys. B 197 509–519
  * [9] Kim S P and Page D N 2002 Phys. Rev. D 65 105002 (Preprint hep-th/0005078)
  * [10] Kim S P and Page D N 2006 Phys. Rev. D 73 065020 (Preprint hep-th/0301132)
  * [11] Dunne G V and Schubert C 2005 Phys. Rev. D 72 105004 (Preprint hep-th/0507174)
  * [12] Nathanail A, Most E R and Rezzolla L 2017 Mon. Not. Roy. Astron. Soc. 469 L31–L35 (Preprint 1703.03223)
  * [13] Srinivasan K and Padmanabhan T 1999 Phys. Rev. D 60 024007 (Preprint gr-qc/9812028)
  * [14] Banks T and Bender C M 1973 Phys. Rev. D 8(10) 3366–3378 URL https://link.aps.org/doi/10.1103/PhysRevD.8.3366
  * [15] Coleman S R 1977 Phys. Rev. D 15 2929–2936 [Erratum: Phys. Rev. D 16, 1248 (1977)]
  * [16] Vilenkin A 1982 Phys. Lett. B 117 25–28
  * [17] Medina L and Ogilvie M C 2017 Phys. Rev. D 95 056006 (Preprint 1511.09459)
  * [18] Gould O, Rajantie A and Xie C 2018 Phys. Rev. D 98 056022 (Preprint 1806.02665)
  * [19] Gould O, Mangles S, Rajantie A, Rose S and Xie C 2019 Phys. Rev. A 99 052120 (Preprint 1812.04089)
  * [20] Affleck I 1981 Phys. Rev. Lett. 46 388
  * [21] Gibbons G W 1975 Commun. Math. Phys. 44 245–264
  * [22] Hiscock W A and Weems L D 1990 Phys. Rev. D 41 1142
  * [23] Bardeen J M, Press W H and Teukolsky S A 1972 Astrophys. J. 178 347
  * [24] Takahashi M and Koyama H 2009 Astrophys. J. 693 472–485 (Preprint 0807.0277)