# Stability and Observability of Magnetic Primordial Black Hole-Neutron Star
Collisions

John Estes  Department of Chemistry and Physics, SUNY Old Westbury, Old
Westbury, NY, United States  Michael Kavic  Department of Chemistry and
Physics, SUNY Old Westbury, Old Westbury, NY, United States  Steven L.
Liebling  Department of Physics, Long Island University, Brookville, NY,
United States  Matthew Lippert  Department of Chemistry and Physics, SUNY Old
Westbury, Old Westbury, NY, United States  John H. Simonetti  Department of
Physics, Virginia Tech, Blacksburg, VA, United States

###### Abstract

The collision of a primordial black hole with a neutron star results in the
black hole eventually consuming the entire neutron star. However, if the black
hole is magnetically charged, and therefore stable against decay by Hawking
radiation, the consequences can be quite different. Upon colliding with a
neutron star, a magnetic black hole very rapidly comes to a stop. For large
enough magnetic charge, we show that this collision can be detected as a
sudden change in the rotation period of the neutron star, a glitch or anti-
glitch.We argue that the magnetic primordial black hole, which then settles to
the core of the neutron star, does not necessarily devour the entire neutron
star; the system can instead reach a long-lived, quasi-stable equilibrium.
Because the black hole is microscopic compared to the neutron star, most
stellar properties remain unchanged compared to before the collision. However,
the neutron star will heat up and its surface magnetic field could potentially
change, both effects potentially observable.

###### Contents

  1. I Introduction
  2. II Collision of a magnetized PBH with a Pulsar
  3. III MPBH-NS bound states
    1. III.1 Black holes in neutron stars
    2. III.2 Neutron stars containing black hole
    3. III.3 mPBH-NS scenarios
      1. III.3.1 Small Charge: $Q<2.2\times 10^{17}$
      2. III.3.2 Medium charge: $2.2\times 10^{17}<Q<1.7\times 10^{20}$
      3. III.3.3 Large charge: $Q>1.7\times 10^{20}$
  4. IV Observables
    1. IV.1 Thermal radiation
    2. IV.2 Magnetic field
    3. IV.3 Collapse
    4. IV.4 Magnetic black hole
  5. V Discussion

##  I Introduction

The first direct detection of gravitational waves by LIGO and Virgo was
followed just a couple years later by the first concurrent detection of a
source in both gravitational and electromagnetic bands, heralding a golden age
of multimessenger astronomy. Just recently LIGO and Virgo detected the first
black hole (BH) merging with a neutron star (NS) LIGOScientific:2021qlt . In
fact, LIGO and Virgo detected two such mergers, both in a regime in which
electromagnetic counterparts are not expected (and none were observed). For
such mergers, when the black hole is much more massive than a neutron star,
the hole tends to devour the neutron star whole without much disrupted
material to source an observable electromagnetic signal.

The outcome of a BH-NS merger in the opposite regime, in which the neutron
star is much more massive than the black hole, is less straightforward.
Although black holes resulting from stellar collapse are invariably at least
as large as a NS, primordial black holes (PBHs) can be created by large
density fluctuations in the early universe at any mass, in principle down to
the Planck scale (see Sasaki:2018dmp ; Green:2020jor ; Villanueva-
Domingo:2021spv  for recent reviews). While the idea of PBHs is not new, the
suggestion that large black holes recently observed by LIGO/Virgo may have
primordial origin Bird:2016dcv  has spurred renewed interest.

PBHs are a possible dark matter component, subject to a number of
phenomenological bounds. As a result of Hawking radiation, very light PBHs,
with $M_{\mathrm{PBH}}<5\times 10^{11}\,\rm kg$, would have completely
evaporated by now,111The end stage of the BH evaporation process is poorly
understood, and certain proposals for resolving the BH information problem
suggest that, rather than completely evaporating, the BH could evolve into
some sort of very light remnant. For a review, see Chen:2014jwq . and somewhat
more massive PBHs with $M_{\mathrm{PBH}}<10^{14}\,\rm kg$ would still emit
sufficient radiation that their abundance is quite limited. The abundance of
PBHs in the planetary to stellar mass range, $10^{19}\,{\rm
kg}<M_{\mathrm{PBH}}<10^{33}\,{\rm kg}$, is modestly constrained by the
frequency of stellar microlensing events, though for stellar mass PBHs,
$M_{\mathrm{PBH}}>5\times 10^{30}\,\rm kg$, dynamical effects, limits on
radiation from accreted gas, and gravitational wave signals impose severe
limits.

The sub-planetary mass range, $10^{14}\,{\rm
kg}<M_{\mathrm{PBH}}<10^{19}\,{\rm kg}$, however, is still quite
unconstrained. Recent work has argued that NANOgrav may have observed
stochastic gravitational wave signals from PBH formation in the early
universe, suggesting that PBHs comprise a significant fraction of dark matter
DeLuca:2020agl ; Vaskonen:2020lbd .

If a small PBH collides with a NS, it falls to the center of the NS due to
dynamical friction and begins accreting dense nuclear matter, eventually
devouring the entire NS. Numerical simulations East:2019dxt ; Richards:2021upu
; Schnauck:2021hlm  of this process demonstrated that, at least while the BH
is still small compared to the NS, the accretion rate is proportional to the
BH horizon area, indicating a Bondi-like accretion process. These simulations
assumed the black holes were large enough, however, so the effects of Hawking
radiation could be neglected.

For small black holes, however, Hawking radiation can be significant. The
Hawking temperature of a Schwarzschild black hole is inversely proportional to
its mass, implying a negative specific heat and an instability to evaporation.
However, for a BH inside a NS, as long as the initial accretion rate is larger
than the initial Hawking luminosity, the BH will grow, cool, and eventually
consume the NS. This will be the case for any PBH-NS merger unless the PBH was
coincidentally just about to completely evaporate when it collided with the
NS.

The situation, however, is quite different if the BH is charged. Near-
extremal, Reisner-Nordstrom black holes have positive specific heat and cool
due via Hawking radiation into zero-temperature, extremal BHs. While
electrically charged BHs can readily discharge via nucleation of electron-
positron pairs (or other light, charged particles), a magnetically charged BH
could be very long-lived, because the large mass of magnetic monopoles
exponentially suppresses the rate of monopole-anti-monopole pair production
Maldacena:2020skw .

A magnetized PBH (mPBH) could result from the accretion, in the early
universe, of monopoles by a PBH. For example, a PBH in a strong magnetic field
could produce a pair of magnetic monopoles either through pair production
Kobayashi:2021des  by the magnetic field itself or through the Kibble-Zurek
mechanism as the black hole evaporates Das:2021wei . The PBH could swallow one
of the magnetic monopoles becoming a mPBH, while the other is accelerated away
by the strong magnetic field. The mPBH then continues to emit Hawking
radiation, shedding its excess mass, and by the present epoch, a small mPBH
will have evolved very near to extremality.

Once created, a mPBH of charge $Q$ has many interesting and unusual properties
compared to its electric counterpart, in addition to its stability.222We work
in units where the charge $Q$ is dimensionless. For a mPBH with $Q\lesssim
10^{32}$, the large magnetic field near the horizon causes the SU(2) gauge
fields to condense, while the Higgs vacuum expectation value vanishes
Maldacena:2020skw . This near-horizon region is bounded by an electroweak
corona with varying SU(2) gauge and Higgs fields. Outside the corona, the
SU(2) and Higgs fields take their standard vacuum expectation values. The
scale of the corona is set by the charge $Q$ and the geometric mean of the
Higgs and W-boson scales. For example, a value of $Q\sim 10^{20}$ corresponds
to a corona radius of order $10^{-8}\ \rm m$. Inside the corona, the
condensates break translation symmetry in the space transverse to the magnetic
field, corresponding to the formation of vortices. Within this region, the
standard model fermions are confined to the vortices and behave as massless
two-dimensional particles, significantly enhancing the Hawking radiation rate.
This results in the whole region inside the corona heating up to the
temperature of the black hole Maldacena:2020skw .

Some recent works have discussed observable astrophysical signatures of
magnetically charged BHs Bai:2020spd ; Ghosh:2020tdu . In Bai:2019zcd , it was
suggested that mPBHs with $M_{\mathrm{PBH}}<10^{6}\,\rm kg$ could be a
dominant component of dark matter.

In this paper, we investigate the collision of a small, extremal (or near-
extremal) mPBH with a NS. We first discuss the possibility that such
collisions could be observed as pulsar glitches and anti-glitches. We find
that, with the current sensitivity of radio observations, only collisions with
large charge $Q>10^{23}$ would be observable. However, since we subsequently
determine that, for such large mPBHs, the NS would rapidly collapse, such
collisions could not explain glitches in pulsars that are observed to continue
pulsing. However, third generation gravitational wave detectors might have
sufficient sensitivity to allow observations of impacts of smaller charge BHs
which could lead to stable bound states.

After the collision, the mPBH will quickly settle down to the core and begin
accreting the dense NS material. Unlike the case of a Schwarzschild BH in a
NS, which eventually devours the entire star, we argue that a captured mPBH
can evolve to a long-lived, quasi-stationary state. The mPBH heats up as it
accretes infalling matter, then re-emits the energy via Hawking radiation.
When the outgoing radiation flux balances the infalling mass flux, the mPBH
stops growing and reaches stable equilibrium. Hawking radiation provides the
central pressure to support the NS. If the magnetic charge of the BH is not
too large, $Q<10^{17}$, the accretion rate can increase until the Hawking
pressure is sufficient to support the NS, and this mPBH-NS state will then
persist until an appreciable fraction of the NS mass has been accreted and re-
radiated by the black hole, which implies a lifetime for small magnetic charge
$Q<10^{10}$ that is longer than the age of the universe.

In Section § II below, we analyze the initial collision between a mPBH and a
NS, in particular how the rotation of the NS is affected. We then discuss the
fate of the mPBH when it reaches the NS core and the possibility of forming a
long-lived, quasi-stationary state in Section § III. We analyze the process by
which the mPBH acretes NS material and then re-radiates the energy as Hawking
radiation in Section § III.1. Then in Section § III.2, we investigate the
evolution and stability of the NS, and in Section § III.3 enumerate three
qualitative scenarios based on the mPBH charge. In Section § IV we discuss
possible ways to observationally distinguish neutron stars with and without an
embedded mPBH, followed by a discussion of open questions in Section § V.

##  II Collision of a magnetized PBH with a Pulsar

An uncharged PBH colliding with, or even passing sufficiently near, a NS will
lose kinetic energy through dynamical friction, accretion of NS matter, and
emission of gravitational waves Capela:2013yf ; Defillon:2014wla ;
Genolini:2020ejw . The PBH will get captured eventually, but only after
repeated passes through the NS, with a light BH of mass
$m_{\mathrm{PBH}}<10^{19}\,\rm kg$ taking more than $10^{4}$ years
Genolini:2020ejw .

For a magnetically charged PBH, on the other hand, electromagnetic
interactions with the NS are dominant, and even a relativistic mPBH will be
rapidly and efficiently captured Bai:2020spd .

If a magnetized PBH collides with a NS which is observed as a pulsar, one
might be able to detect the resulting change in the pulse period due to the
rapid change in the rotation rate of the NS. Such sudden changes in the
pulsar’s period are observed as glitches (for a discussion of observed
glitches see, e.g., Espinoza2011 ). Glitches refer to sudden decreases in the
pulse period, but anti-glitches, or sudden increases in pulse period, have
also been observed, e.g., Archibald2013 . Huang and Geng Huang2014  consider a
collision of a solid body with the neutron star as the possible cause of an
anti-glitch. We consider a similar scenario here, involving impact by a mPBH.

Consider an extremal mPBH of charge $Q$ moving at virial speed in the Galaxy,
i.e., $v\sim 10^{2}$ km/s $\sim 10^{5}$ m/s. The mass of the mPBH is

$M_{\mathrm{BH}}=\frac{\hbar
c\sqrt{\pi\epsilon_{0}}}{e\sqrt{G_{N}}}Q=(1.27\times 10^{-7}\,\rm{kg})\,Q.$
(1)

While it is initially very far from the NS, it eventually collides with the NS
because its impact parameter $b$ is smaller than the gravitational cross
section for collision with the NS. In other words, while it would have missed
the NS by distance $b$ if traveling along a straight path, the gravitational
attraction of the NS causes the point of closest passage of the mPBH to the NS
to be less than the radius of the NS, $R_{\mathrm{NS}}$. See Fig. 1. Take the
mass of the NS to be $M_{\mathrm{NS}}$.

Figure 1: The scenario for a mPBH colliding with a NS. The impact parameter
for the mPBH is $b$, but the mPBH collides at a closest approach of distance
$\lesssim R_{\mathrm{NS}}$, the radius of the NS. The impact parameter $b\gg
R_{\mathrm{NS}}$. Thus the mPBH speed at the collision is much greater than
its initial speed $v$ and is comparable to the escape velocity for the NS.

During the motion of the mPBH toward the NS, the angular momentum and
mechanical energy of the mPBH-NS system are conserved. Thus the angular
momentum of the NS will be changed by an amount of magnitude
$M_{\mathrm{BH}}vb$ when the NS absorbs the mPBH. To determine
$M_{\mathrm{BH}}vb$ note that

$M_{\mathrm{BH}}vb\sim mv_{c}R_{\mathrm{NS}}$ (2)

where $v_{c}$ is the speed of the mPBH when it collides with the NS. In
obtaining this result we have assumed that the $M_{\mathrm{BH}}vb$ is much
less than the angular momentum of the rotating NS. In other words, the change
in the angular momentum of the NS will be much less than its initial angular
momentum. This assumption will be proven correct below.

The escape velocity for a NS is

$v_{\mathrm{esc}}=\sqrt{\frac{2GM_{\mathrm{NS}}}{R_{\mathrm{NS}}}}\sim 10^{8}\
{\rm m/s},$ (3)

for $M_{\mathrm{NS}}\sim M_{\odot}\sim 10^{30}$ kg, and $R_{\mathrm{NS}}\sim
10^{4}$ m. Since the mPBH starts very far from the NS with speed $v\ll
v_{\mathrm{esc}}$, it collides with the NS with speed very close to
$v_{\mathrm{esc}}$. Thus the magnitude of the angular momentum imparted to the
NS is

$M_{\mathrm{BH}}vb\sim M_{\mathrm{BH}}v_{\mathrm{esc}}R_{\mathrm{NS}}.$ (4)

The fractional change in the angular momentum of the NS is therefore

$\frac{M_{\mathrm{BH}}v_{\mathrm{esc}}R_{\mathrm{NS}}}{I_{\mathrm{NS}}\omega_{\mathrm{NS}}}\sim\frac{M_{\mathrm{BH}}v_{\mathrm{esc}}P_{\mathrm{rot}}}{2\pi
M_{\mathrm{NS}}R_{\mathrm{NS}}}\sim\frac{Q}{10^{24}}10^{-10}$ (5)

where $I_{\mathrm{NS}}$ and $\omega_{\mathrm{NS}}$ are the moment of inertia
and angular rotation rate of the NS, respectively, and we have assumed the
rotation period of the NS is about 1 second. This result for the fractional
change justifies our assumption that the change in the angular momentum of the
NS will be very small. Since the fractional change in the moment of inertia of
the NS, due to the embedding of the mPBH, is roughly

$\frac{M_{\mathrm{BH}}R_{\mathrm{NS}}^{2}}{M_{\mathrm{NS}}R_{\mathrm{NS}}^{2}}\sim\frac{Q}{10^{24}}10^{-13},$
(6)

and is much smaller than the fractional change in angular momentum, then the
fractional change in the rotation period of the NS will also be

$\frac{\Delta
P_{\mathrm{rot}}}{P_{\mathrm{rot}}}\sim\frac{Q}{10^{24}}10^{-10}.$ (7)

Current observations show typical glitches have $\Delta
P_{\mathrm{rot}}/P_{\mathrm{rot}}\sim 10^{-11}-10^{-5}$ 2017A&A…608A.131F . If
at least some of these glitches are the result of mPBH-NS collisions, then
they imply $Q\sim 10^{23}-10^{29}$. As shown in Sec. § III below, if such an
mPBH is absorbed by a NS, the result would be unstable. The pulsar would soon
be unobservable. Since pulsars with these observed glitches continue to pulse,
we can say such events are not due to collisions with mPBHs of such large
values of $Q$.

However, glitches and anti-glitches also generate transient gravitational wave
signals which may be observable by future third generation detectors
Yim:2022qcn ; Lopez:2022yph . With sufficient sensitivity in band, future
gravitational wave detectors could make it possible to observe small glitches
currently unobservable via pulsar emission, in which case this collision
scenario may be viable.

Furthermore, one might hope for such observations to differentiate among the
various explanations for glitches, because, for the case considered here, a
glitch would be immediately preceded by a (very unequal mass) binary merger.
However, such a merger would be at very high frequency (a few kilohertz), a
band in which detectors tend to be less sensitive Genolini:2020ejw ;
Zou:2022wtp . The interaction of the BH’s magnetic field with that of the NS
also offers the potential of an electromagnetic precursor signal, which may
differentiate the event from an asteroid collision. Precursor electromagnetic
signals from BH-NS mergers have been predicted although with a couple caveats:
(1) such studies considered stellar sized BHs, not PBHs and (2) they assume an
uncharged BH DOrazio:2021puy ; Carrasco:2021jja ; Chen:2021sya ; East:2021spd
; Pan:2019ulx ; Paschalidis:2013jsa ; Lai:2012qe .

##  III MPBH-NS bound states

###  III.1 Black holes in neutron stars

Once a mPBH is stopped and settles down to the center of the NS, it will begin
to accrete material from the NS core. For an uncharged PBH, this accretion
process has been estimated analytically Giddings:2008gr ; Fuller:2017uyd  and
simulated numerically East:2019dxt ; Richards:2021upu  and, for a small PBH
(i.e. $M_{\mathrm{BH}}\ll M_{\mathrm{NS}}$), the Bondi-like accretion rate is
proportional to the horizon area $A_{\mathrm{BH}}$ of the PBH,333East and
Lehner East:2019dxt  simulate several NS equations of state, and the average
accretion rate is approximate $\dot{M}_{\mathrm{BH}}\approx(1.5\times 10^{15}\
{\rm\frac{g}{cm^{3}}})\ A_{\mathrm{PBH}}$ (see Fig 2 in East:2019dxt ).

$\dot{M}_{\mathrm{BH}}=\left(4.5\times 10^{26}\ {\rm\frac{kg}{m^{2}s}}\right)\
A_{\mathrm{BH}}\ .$ (8)

As the PBH grows, the accretion accelerates until the entire NS has been
consumed, leaving a stellar mass BH in its place.

The result of capturing a mPBH is quite different. An extremal mPBH with
magnetic charge $Q$ is metastable444An extremal mPBH can decay by nucleation
of monopole-anti-monopole pairs, a process which is suppressed by the high
mass of a monopole. and has zero temperature. Once it begins accreting, the
temperature will rise and the mPBH will begin emitting Hawking radiation. A
non-extremal mPBH with mass $M_{\mathrm{BH}}$ has a temperature is given by

$\displaystyle T=\frac{\hbar c^{3}}{2\pi
k_{B}G}\frac{\sqrt{M_{\mathrm{BH}}^{2}-M_{e}^{2}}}{\left(M_{\mathrm{BH}}+\sqrt{M_{\mathrm{BH}}^{2}-M_{e}^{2}}\right)^{2}}\
,$ (9)

where $M_{e}=\frac{\hbar c\sqrt{\pi\epsilon_{0}}}{e\sqrt{G_{N}}}Q=(1.27\times
10^{-7}\,\rm{kg})\,Q$ is the mass of the original extremal mPBH. We define a
dimensionless non-extremality parameter $x\equiv\Delta
M/M_{e}=(M_{\mathrm{BH}}-M_{e})/M_{e}$, which allows the temperature to be
written as

$\displaystyle T=\left(3.9\times 10^{30}\ \rm K\right)\frac{F(x)}{Q}\ ,$ (10)

where the dimensionless function $F(x)$ is

$F(x)=\frac{\sqrt{x^{2}+2x}}{(1+x+\sqrt{x^{2}+2x})^{2}}.$ (11)

For a near-extremal mPBH, the temperature grows with mass as
$\sqrt{M_{\mathrm{BH}}-M_{e}}$, or as $x^{1/2}$, yielding a positive specific
heat. However, for $M_{\mathrm{BH}}\gg M_{e}$, the black hole resembles a
Schwarzschild BH, with a temperature decreasing as $M_{\mathrm{BH}}^{-1}$ and,
consequently, a negative specific heat.

For a near-extremal mPBH with $Q<3\times 10^{32}=Q_{\mathrm{ew}}$, the
magnetic field just outside the horizon is sufficiently large that the
electroweak gauge bosons condense and the electroweak symmetry is restored. As
a result, in this region all standard model fermions are massless, and the
low-energy modes are constrained by the strong magnetic field to radial motion
along the field lines. The number of such two-dimensional massless modes
scales as $Q$ and the Hawking radiation is essentially 1+1 dimensional.

The local magnetic field decreases with distance from the mPBH, until the
field strength is equal to the square of the Higgs mass, at which point
electroweak symmetry begins breaking across a transition region called the
“electroweak corona”. Moving radially through the corona, the Higgs vacuum
expectation value increases from zero to its usual value, and outside is the
ordinary electroweak vacuum. Although this transition is not especially sharp
and the corona is not spherically symmetric, the approximate radial location
of the corona is $r_{\mathrm{C}}=(1.7\times 10^{-18}\ {\rm m})\sqrt{Q}\ $.

This mass barrier acts as a potential, trapping the standard model fermions
within the corona. When the Hawking radiation reaches the electroweak corona,
the standard model fermions are reflected back. The whole region inside the
corona heats up to the Hawking temperature set by the blackhole horizon, and
the corona radiates like a thermal blackbody.

The Stephan-Boltzmann law relates the luminosity of the EW corona to the
temperature

$L=A_{\mathrm{C}}\sigma T^{4}$ (12)

where $A_{\mathrm{C}}$ is the area of the corona and the Stephan-Boltzmann
constant
$\sigma=\frac{2\pi^{5}k_{B}^{4}}{15c^{2}h}=\frac{\pi^{2}k_{B}^{4}}{60c^{2}\hbar^{3}}$.
Inserting the expression for the Hawking temperature (10) in terms of the PBH
mass gives:

$\displaystyle L_{\mathrm{BH}}$ $\displaystyle=$
$\displaystyle\left(\frac{c^{10}\hbar}{960\pi^{2}G^{4}}\right)A_{\mathrm{C}}T^{4}$
(13) $\displaystyle=$ $\displaystyle\left(1.2\times 10^{115}\
{\rm\frac{J}{m^{2}s}}\right)\
A_{\mathrm{C}}\,\frac{F(x)^{4}}{Q^{4}}=\left(4.4\times 10^{80}\
{\rm\frac{J}{m^{2}s}}\right)\ \frac{F(x)^{4}}{Q^{3}}$ (14)

Fig. 2 shows the Hawking flux (13) plotted as a function of $x$. For a near-
extremal mPBH, where $x\ll 1$, the Hawking temperature and radiation flux
grows with $x$ and the mPBH has a positive heat capacity. However, the power
output reaches a maximum at a critical value of $x_{c}=0.155$ where the
function $F(x)$ reaches a maximum: $F(x_{c})=0.192$. For $x>x_{c}$, the
temperature and power fall as the mass increases, implying a negative heat
capacity akin to the behavior of a Schwarzschild black hole.

Figure 2: The Hawking flux $L$ for a mPBH, scaled by $Q^{3}$, as a function of
the non-extremalty parameter $x$, given by Eq. 13. The flux reaches a maximum
at $x_{c}=0.15$. For a near-extremal mPBH, $x<x_{c}$, the positive slope
indicates thermodynamic stability. Farther from extremality, $x>x_{c}$, the
black hole becomes Schwarzschild-like, with a negative specific heat.

An accreting mPBH at the center of a NS can rapidly reach an equilibrium in
which energy is emitted as Hawking radiation at the same rate that energy
falls in, in the form of NS matter. For a given accretion rate
$\dot{E}_{\mathrm{acc}}$, the condition for zero net energy flux is
$\dot{E}_{\mathrm{acc}}=L_{\mathrm{BH}}$, or

$\dot{E}_{\mathrm{acc}}=\left(4.4\times 10^{80}\ {\rm\frac{J}{m^{2}s}}\right)\
\frac{F(x)^{4}}{Q^{3}}$ (15)

which should be understood as an implicit equation that dynamically determines
the equilibrium value of the non-extremality parameter, $x_{\mathrm{eq}}$.

There are, in fact, two solutions to (15), $x_{\mathrm{eq},1}<x_{c}$ and
$x_{\mathrm{eq},2}>x_{c}$. At $x_{\mathrm{eq},1}$,
$T^{\prime}(x_{\mathrm{eq},1})>0$ implying a positive heat capacity. If some
extra mass falls in to the black hole, the temperature increases and the
Hawking emission increases to compensate, returning the system to equilibrium.
The larger solution $x_{2}$ has a negative heat capacity and is unstable to
cooling by additional accretion. When an extremal mPBH begins accreting at the
center of the NS, the mass increases from $x=0$ and the system is expected to
stabilize at $x_{\mathrm{eq},1}$.

There is, however, an upper bound on the Hawking radiation flux (13) for a
given mPBH charge, as can be seen in Fig. 2, and if the accretion rate is
above this maximum, there is no value of $x$ which satisfies (15). The
function $F(x)$ has a maximum value of $0.192$, which implies an upper bound
on the accretion rate which can yield an equilibrium

$\dot{E}_{\mathrm{acc}}<\left(6.2\times 10^{77}\ {\rm\frac{J}{m^{2}s}}\right)\
Q^{-3}.$ (16)

A mPBH which does not satisfy this condition will not be able to re-emit all
the energy that falls in and will grow without bound until the entire NS is
accreted.

###  III.2 Neutron stars containing black hole

The NS responds to the addition of an accreting, radiating mPBH to its core.
For a small enough BH, the NS heats up but is not drastically affected, and
the system can settle into a new equilibrium solution.

The NS can be effectively modeled as a spherically symmetric, hydrodynamical
system in terms of the stellar pressure $P(r)$, the energy density $\rho(r)$,
and the mass aspect function $m(r)$, which gives the mass inside a radius $r$.
The conditions for hydrostatic equilibrium given by the TOV equations:555For a
review of the TOV equations and NS solutions, see Ref. Noble:2003xx .

$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi r^{2}\rho$
(17) $\displaystyle\frac{dP}{dr}$ $\displaystyle=$
$\displaystyle-\frac{\left(\rho+P\right)\left(m+4\pi
r^{3}P\right)}{r\left(r-2m\right)}.$ (18)

For the nuclear matter of the NS, we adopt a polytropic equation of state,
$P=K\rho_{0}^{\Gamma}$, with polytropic index $\Gamma=2$ and constant
$K=10^{-2}\rm\frac{m^{5}}{kgs^{2}}$, which sets the maximum NS mass to be
$1.4M_{\odot}$. Here $\rho_{0}$ is the rest mass density in the rest frame of
the fluid so that the energy density becomes

$\rho=\left(\frac{P}{K}\right)^{1/\Gamma}+\frac{P}{\Gamma-1}.$ (19)

To solve for ordinary NS solutions, these first-order ODEs are integrated out
from the center with boundary conditions specified at $r=0$: a central
pressure, $P_{0}$, $P^{\prime}(0)=0$, $m(0)=0$, and $m^{\prime}(0)=0$. The
stellar surface is identified by the radius $R_{NS}$ at which the pressure
drops below some small threshold. The mass of the NS is given by
$M_{\mathrm{NS}}=m(R_{\mathrm{NS}})$. For a given $M_{\mathrm{NS}}$, there may
be multiple solutions. We will focus on the perturbatively stable solution,
which is the solution with the larger radius.

We use the LSODA ODE solver etde_21352532  which automatically switches
between methods for stiff and non-stiff equations. We compute numerical
residuals from a particular PBH+NS solution with increasing resolution and
display them in Fig. 3. These residuals are independent of the differencing
used by the solver, and they generally converge. There is some noisy behavior
at small radius and so we also plot the residuals for a similar NS solution.
We are confident in the NS solutions by comparison of the family of solutions
with those of Ref. Noble:2003xx , and the residuals for the PBH+NS are
comparable to those of the NS solution.

Figure 3: Convergence of numerical residuals for a particular solution. Top:
The residual of Eq. 17 for a particular PBH+NS solution and similar NS
solution. Bottom: The residual of Eq. 18 for these solutions. The residuals
for the PBH+NS generally decrease with increasing resolution, except some
noisy behavior at small radius. However, this behavior is similar to that
exhibited by the NS solution in which we have confidence by comparing the
family of solutions with those of Ref. Noble:2003xx .

For a fiducial NS of mass $M_{\mathrm{NS}}=1.35M_{\odot}$, the solutions for
$m(r)$ and $P(r)$ are illustrated in Fig. 4. The central pressure $P_{0}$ is
determined by the NS mass, and along the stable branch of solutions, $P_{0}$
increases with $M_{\mathrm{NS}}$. The fiducial $1.35M_{\odot}$ NS has a
central pressure $P_{0}=2.37\times 10^{34}\ \rm Pa$.

We now introduce a small mPBH into the center of an existing, stable NS of
mass $M_{\mathrm{NS}}$. In order to find NS solutions with a mPBH at the core,
we modify the boundary conditions. We begin the integration at the nonzero
radius of the electroweak corona, $r_{c}$. At that radius, the enclosed mass
is that of the extremal BH, $m(r_{c})=M_{\mathrm{BH}}$. The pressure at
$r_{c}$ is now generated by the outgoing Hawking radiation, which is

$P_{c}\equiv
P(r_{c})=\frac{L_{\mathrm{BH}}(x_{\mathrm{eq}})}{cA_{c}}=\left(4.0\times
10^{106}\ \rm Pa\right)\frac{F(x)^{4}}{Q^{4}}$ (20)

where $L_{\mathrm{BH}}$ is the equilibrium Hawking luminosity given by (13).
For a black hole in equilibrium, the non-extremality parameter $x$ is
determined by (15), and the luminosity is equal to the energy accretion rate
$L_{\mathrm{BH}}(x_{\mathrm{eq}})=\dot{E}_{\mathrm{acc}}$.

We look for solutions to (17) and (18) with a total mass
$M_{\mathrm{NS}}+M_{\mathrm{BH}}$. However, because we consider very small
black holes, $M_{\mathrm{BH}}\ll M_{\mathrm{NS}}$, the additional mass due to
the mPHB can safely be neglected. Some examples of these mPBH-NS solutions,
for a range of BH masses, are shown in Fig. 4, along with the corresponding
stable NS solution of the same total mass.

The mPBH-NS solutions are very close to the pure NS solutions with a small BH
placed at the center. The outer regions of the mPBH-NS solutions, beyond the
corona radius, are nearly identical to the NS solution. Near the center of the
NS, the mass aspect function is dominated by the mass of the black hole at the
center. However, as seen in the insert in Fig. 4 (Top), the NS nuclear matter
extends at roughly constant density down to the corona. Similarly, the
pressures for the mPBH-NS solutions are, to good accuracy, identical to that
of the pure NS, as can be seen in Fig. 4 (Bottom). Because, the mPBH-NS
solutions have the same mass and radial mass distribution as the pure NS, it
is not surprising that the central pressure is the same as well.

Figure 4: The fiducial $1.35M_{\odot}$ NS solution and a range of mPBH-NS
solutions with the same total mass but with varying mBH masses. Top: The mass
aspect function $m(r)$. At large radius, the solutions are identical. For the
mPBH-NS solutions, $m(r)$ levels off at small radius at $M_{\mathrm{BH}}$.
Top-middle: The pressure, $P(r)$, for the various solutions. Bottom-middle:
The energy density, $r^{2}\rho(r)$, for the various solutions. Bottom: The
(negative) gradient of the pressure shows that the pressure is not flat at
small radius. The curves all lie on top of each other, illustrating that,
outside the corona radius, the mPBH-NS solutions have the same mass
distribution as the pure NS solution.

Starting with an equilibrium NS and adding a small mPBH, to achieve an
equilibrium solution with the same total mass, we find that the Hawking
pressure at the corona must be equal to the original NS central pressure:
$P(r_{C})=P_{0}$. For the BH to also be in equilibrium, this condition
dictates a required accretion rate:

$\dot{E}_{\mathrm{acc}}^{\mathrm{eq}}=cA_{c}P_{0}$ (21)

If black hole accretion rate does not match
$\dot{E}_{\mathrm{acc}}^{\mathrm{eq}}$, the mPBH-NS will not be in hydrostatic
equilibrium, with the central pressure too high or too low for the given
amount of mass.

If the accretion rate does match (21) and the mPBH-NS is in equilibrium, then
$P(R_{c})=P_{0}$ which determines the non-extremality parameter $x$ via (15).
The mPBH will have an equilibrium temperature

$T_{\mathrm{BH}}=\left(8.7\times 10^{3}\
\rm\frac{K}{Pa^{1/4}}\right)P_{0}^{1/4}$ (22)

For the fiducial $1.35M_{\odot}$ NS, this yields $T_{\mathrm{BH}}=5.8\times
10^{12}\ \rm K$.

In solving the TOV equations (17) and (18) we have neglected temperature, but
the NS heats up due to the Hawking radiation. Because the NS interior is a
superfluid, heat is rapidly transmitted radially outward from the core. Once
the NS reaches thermal equilibrium, the energy emitted at the corona into the
NS must be emitted at the NS surface at an equal rate. From the Stefan-
Boltzmann law the power radiated by thermal radiation is $P\propto AT^{4}$.
Equating the incoming Hawking power with the outgoing thermal radiation at the
NS surface gives the NS temperature:

$T_{\mathrm{NS}}=\sqrt{\frac{r_{c}}{r_{\mathrm{NS}}}}T_{\mathrm{BH}}=\left(1.1\times
10^{-7}\ \rm\frac{K}{Pa^{1/4}}\right)Q^{1/4}P_{0}^{1/4}$ (23)

where, the equilibrium temperature of the BH is given by (22). The fiducial
$1.35M_{\odot}$ NS reaches an equilibrium temperature of $T_{\mathrm{NS}}=(73\
\rm K)Q^{1/4}$ .

The approximation that the NS temperature has negligible contribution to its
structure clearly breaks down when the NS temperature approaches the Fermi
temperature $T_{F}$, the greatest temperature at which neutron star matter can
remain degenerate. The Fermi temperature for the NS is given by

$k_{B}T_{F}=\epsilon_{F}=\frac{\hbar^{2}}{2m_{n}}\left(3\pi^{2}n_{n}\right)^{2/3},$
(24)

where $\epsilon_{F}$ is the Fermi energy Carroll2007 , and $m_{n}$ and $n_{n}$
are the mass of a neutron and number density of neutrons, respectively. For
the neutrons in a $1M_{\odot}$ NS of radius 10 km, assuming a uniform density
star, the Fermi temperature is $T_{F}\approx 5\times 10^{12}$ K.

The requirement that the equilibrium NS does not get too hot implies a bound
on the Hawking radiation rate, and therefore on the charge $Q$:

$Q\lesssim 10^{40}$ (25)

This equilibrium configuration in which the mPBH accretes matter from the NS
which it then re-emits as Hawking radiation is in fact only quasi-stationary,
with a finite lifetime. Eventually the mPBH will accrete an order-one fraction
of the NS mass. From the accretion rate (21), the time to accrete a solar mass
is approximately

$\tau=10^{31}{\rm\ yr}\ Q^{-2}.$ (26)

###  III.3 mPBH-NS scenarios

The final state resulting from the mPBH collision with a NS depends on the
charge (or equivalently the extremal mass) of the black hole. Three
qualitatively distinct outcomes are possible, depending on the size of the
magnetic charge. As we will see, only a NS that absorbs a black hole with a
small magnetic charge can form a stable system. Above a certain threshold, the
mPBH can not produce enough Hawking radiation to support the NS, and the
entire NS will eventually collapse into the black hole.

For definiteness, we will consider a mPBH colliding with a our fiducial
$1.35M_{\odot}$ NS.

####  III.3.1 Small Charge: $Q<2.2\times 10^{17}$

We first consider the case of a mPBH with small charge, $Q<2.2\times 10^{17}$.
Just after the collision, once the mPBH has settled down to the center of the
NS, it begins accreting NS matter. In the absence of numerical studies of the
accretion by a mPBH inside a NS, we will assume the accretion rate is
approximately the same as that of an uncharged PBH (8). In terms of an energy
accretion rate, this is

$\displaystyle\dot{E}=4.1\times 10^{43}\ {\rm\frac{J}{m^{2}s}}\
A_{\mathrm{PBH}}=5.1\times 10^{-25}\ {\rm\frac{J}{s}}\ Q^{2}$ (27)

As the mPBH them becomes non-extremal, it will start to emit Hawking
radiation, and the resulting outward radiation pressure would be expected to
somewhat reduce the accretion rate. The uncharged accretion rate (8) or (27),
which does not include Hawking radiation, then gives an upper bound on the
initial accretion rate.

As described above in § III.1, the mPBH reaches equilibrium when the Hawking
luminosity equals the accretion rate. Using the uncharged accretion rate, the
equilibrium condition (16) gives:

$F(x)^{4}=\left(4.4\times 10^{80}\ {\rm\frac{J}{m^{2}s}}\right)\ Q^{-3}$ (28)

As long as $Q<10^{20}$, a solution to (28) exists, with $x\leq x_{c}$.

The pressure generated by the resulting Hawking radiation is then

$P_{c}=\left(450\ {\rm Pa}\right)Q$ (29)

as shown in Fig. 5 as the upward-sloping orange line. However, for a fiducial
$1.35M_{\odot}$ NS, the needed central pressure is $P_{c}=2.37\times 10^{34}\
\rm Pa$, which is shown in Fig. 5 as a horizontal green line. Consequently,
for $Q\leq 10^{33}$, the pressure being generated by this Hawking radiation is
too low to yield an equilibrium NS solution. Numerically simulating the non-
equilibrium dynamics is beyond the scope of this paper. However, it is
reasonable to suppose that, without sufficient outward pressure, the NS core
will start to collapse, and as this happens, the accretion rate will increase.

Figure 5: Configuration space of PBH+NS solutions. The blue line gives the
maximum pressure $P_{c}^{\mathrm{max}}$ (30). The orange line gives pressure
$P_{c}$ resulting from the initial accretion rate (29). Particular PBH+NS
solutions with masses above $0.9M_{\odot}$ are shown with points whose color
is determined by their mass. Solutions that support a fiducial mass of
$1.35M_{\odot}$ have roughly constant pressure at the corona radius and are
shown with a horizontal green line.

As the accretion rate increases, the resulting Hawking luminosity and
resulting radiation pressure likewise increase. In Fig. 5, the system rises
vertically from the red line. However, as shown in Fig. 2, there is a maximum
luminosity and corresponding maximum pressure, which occurs at $x_{c}=0.15$.
From (16), the maximum radiation pressure is given by

$P_{c}^{\mathrm{max}}=\left(5.6\times 10^{106}\ {\rm Pa}\right)Q^{-4}$ (30)

This upper bound on the Hawking pressure is shown in Fig. 5 as the downward-
sloping blue line. For the small Q regime, $Q<2.2\times 10^{17}$, the
accretion rate and Hawking luminosity can rise until the mPBH-NS equilibrium
is reached.

In lieu of a dynamical simulation of this process, we can estimate the time
for the accretion rate to increase and raise the Hawking pressure enough to
stabilize the NS. An internal dynamical time scale for the NS can be set by
speed of sound, which is determined by the equation of state (19). At high
density the conformal symmetry of QCD is restored, and the speed of sound in
the NS core can be approximated by the conformal value $v_{s}\sim c/\sqrt{3}$
Altiparmak:2022bke . The sound crossing time
$\tau_{s}=R_{\mathrm{NS}}/v_{s}\sim 10^{-4}\,\rm s$ gives a lower bound on the
time for the accretion rate to increase. This process is slow compared to the
microscopic timescale on which the mPBH reaches equilibrium, but short
compared to any astrophysical timescales.

Once the mPBH-NS system has reached equilibrium, the lifetime of this quasi-
stationary state is approximated by (26), the time for an order one fraction
of the NS to be accreted and re-radiated by the mPBH, which scales as
$Q^{-2}$. For $Q<10^{10}$, the lifetime is longer than the age of the
universe: $\tau>10^{11}{\rm\ yr}$. However, for $Q\lesssim 10^{17}$, near the
upper end of this range, the lifetime is quite short $\tau\gtrsim 10^{4}{\rm\
s}$.

####  III.3.2 Medium charge: $2.2\times 10^{17}<Q<1.7\times 10^{20}$

For a somewhat larger magnetic charge, the evolution of the mPBH-NS system
begins similarly to the small $Q$ regime discussed above. The mPBH begins
accreting at the initial rate (8) and quickly comes to equilibrium, generating
an outgoing Hawking pressure $\eqref{eq:EL_pressure}$, which is insufficient
to yield an equilibrium NS.

Again, we expect the out-of-equilibrium NS to begin collapsing, driving up the
accretion rate, which in turn causes the black hole mass and the Hawking
pressure to increase. However, the necessary central pressure to support the
NS is not possible. As can be seen in Fig. 5, in this medium-$Q$ regime, the
maximum Hawking pressure $P(r_{c})^{max}$ (the blue line) is below the
necessary pressure to support a $1.35M_{\odot}$ NS (the green line). Starting
with the initial Hawking pressure $\eqref{eq:EL_pressure}$ (the orange line),
the pressure increases until the maximum $P(r_{c})^{max}$ is reached, which is
still insufficient for the NS reach equilibrium. As in the small-$Q$ case, the
timescale for this dynamical process is driven by the NS speed of sound; we
estimate the time scale for the accretion rate to increase to be $\tau\gtrsim
10^{-4}\,\rm s$.

The accretion rate continues to grow, but now, as the mPBH mass continues to
grow, the temperature and the luminosity begin to fall, with the mPBH crossing
over to Schwarzschild-like behavior. At this point, the matter begins falling
in to the mPBH faster than it can be remitted, black hole is no longer in
equilibrium, and the system becomes unstable. The subsequent accretion and
black hole growth closely resemble that of an uncharged black hole, as
described in East:2019dxt ; Richards:2021upu , and the entire NS is eventually
consumed by the black hole in a time $\tau\sim\left(5\times 10^{25}\,\rm
yr\right)Q^{-1}$ Baumgarte:2021thx , which for this range of $Q$ gives a
lifetime between $\tau\sim 10^{5}\,\rm yr-10^{8}\,\rm yr$. The final stages of
the NS collapse can be a catastrophically violent event, with the possibility
of emitting large amounts of matter or radiation, depending on the details of
the collapse process.

In the end, there remains a stellar-mass, magnetically charged black hole.
Unlike the original mPBH, this black hole is far from extremal, with a very
small Hawking temperature, but the original magnetic charge remains.

####  III.3.3 Large charge: $Q>1.7\times 10^{20}$

For large charge $Q>1.7\times 10^{20}$, Fig. 5 would seem to indicate that the
initial accretion rate (8) generates a Hawking pressure larger than
$P_{\mathrm{max}}$ which, of course, is not possible. Instead, at an accretion
rate (8), the NS matter is falling across the black hole horizon faster than
the Hawking radiation can emit it the energy back out again. As a result, the
black hole never reaches equilibrium. Starting with a highly charged extremal
mPBH, the mass grows, increasing the Hawking radiation, until the maximum
luminosity at $x_{c}=0.15$ is passed and the black hole crosses over into
Schwarzschild-like behavior.

Beyond this point, the black hole mass continues to grow, now with the
temperature and luminosity decreasing. As a result, the accretion proceeds in
much the same way as for an uncharged PBH, as in East:2019dxt ;
Richards:2021upu , with the black hole growing while accreting the entire NS
in a timescale $\tau\sim\left(5\times 10^{25}\,\rm yr\right)Q^{-1}\lesssim
10^{5}\,\rm yr$ Baumgarte:2021thx . As with the medium charge mPBH discussed
above, the NS ultimately collapses into the BH, finally ending up as a
magnetically charged solar-mass BH.

##  IV Observables

###  IV.1 Thermal radiation

We now investigate the possibility of astronomically observing the results of
a mPBH-NS collision. In the previous section, we found that the collision of a
NS and a mPBH with small charge will result in a long-lived, quasi-stable
mPBH-NS bound state. The observational challenge is to distinguish it from an
ordinary NS without a mPBH.

As discussed above, Hawking radiation from the mPBH heats up the NS which, as
a result, emits thermal radiation out into space. At thermal equilibrium, the
NS temperature is given by (23)

$T_{\mathrm{NS}}=1.5\times 10^{6}\,\rm K\left(\frac{Q}{2.2\times
10^{17}}\right)^{1/4}.$ (31)

An ordinary NS, however, has a non-zero temperature and also emits thermal
radiation. In particular, a recently-formed NS typically has a high
temperature. For example, the many examples in Page:2004fy  have $T\gtrsim
10^{6}\,\rm K$. In this case, the NS is already so hot that the additional
heat, even from a near maximal $Q\sim 10^{17}$ mBPH, would likely not be
observable.

However, as a NS ages, it cools, and can cool below 1000K in a billion-year
time scale Yakovlev:2004iq ; Page:2004fy . The additional thermal radiation
from a mPBH with $Q\gtrsim 10^{4}$ could then be observable. Assuming the age
of the NS can be reliably estimated from, for example, the spin down rate, a
higher than expected temperature could indicate the presence of a radiating
mPBH inside.

Sufficiently old and nearby neutron stars can be detected by pulsar emission
via radio telescope, such as FAST or the VLA. Follow up observation by
infrared telescope, such as the James Webb Space Telescope, the Thirty-meter
Telescope, or the Extremely Large Telescope, could then be able to accurately
measure the temperature Raj:2017wrv .

There are alternative mechanisms, however, by which a NS could be heated. For
example, dark matter accumulating and self-annihilating in the core can also
generate significant heating, up to as much as $2500\,\rm K$ Raj:2017wrv ;
Garani:2020wge . Although the thermal signal from a $Q\lesssim 10^{6}$ mPBH
might closely resemble that due to dark matter accretion, a mPBH with
$Q>10^{8}$ would be sufficiently high temperature as to exclude dark matter as
the heat source.

###  IV.2 Magnetic field

A particularly distinguishing feature of the mPBH is its characteristic
magnetic field. Near the horizon, the magnetic field is very strong. At the
corona, the magnetic field is at the value which the electroweak symmetry is
restored, $B_{c}=B_{\mathrm{EW}}=10^{20}\,\rm T$ Bai:2020spd . Assuming the
magnetic field of the mPBH outside the corona is a spherically-symmetric
monopole field, then $B\propto Q/r^{2}$. Taking the radius of the NS to be
$r_{\mathrm{NS}}=10^{4}\,\rm m$, the magnetic field at the surface is only

$B_{\rm surface}\approx(10^{-24}\ {\rm T})\,Q\lesssim 10^{-7}\ {\rm T}\ .$

Compared with the large magnetic field of a NS, which is typically at least
$10^{4}\,\rm T$, the contribution due to the mPBH would appear completely
negligible.

In the core of the NS, however, the magnetic field of the mPBH is quite
strong. Although, we have so far mostly neglected the effect of the mPBH’s
magnetic field on the NS, it could significantly affect the NS matter in the
core and change the structure of the NS. Strong magnetic fields could affect
the equation of state, in particular the hadron-quark phase transition or
transitions to hyperon or other exotic phases.

For example, the interior of the NS is expected to feature proton
superconductivity, implying the magnetic field of the mPBH outside the corona
would be confined to one or more narrow flux tubes Lander_2013 ; Graber_2015 .
As a result, the magnetic field could be highly concentrated in a few spots,
causing significant deformation of the NS magnetosphere from a dipole.
Interestingly, the NICER telescope recently observed deviations from a simple
dipole in the surface field of a NS Bilous_2019 ; Miller:2019cac .

However, the ability to make statements regarding observational consequences
of the mPBH magnetic field relies on a solid understanding of an NS without a
mPBH. Currently, even for an ordinary NS, neither the geometry nor the
strength of the internal magnetic field are well understood. The affects of
the NS’s own magnetic field on the equation of state and internal structure
are currently under investigation Soldateschi:2021cxq .

###  IV.3 Collapse

For a mPBH with a medium or large charge, $Q>10^{17}$, the NS eventually
collapses into the mPBH, a highly-energetic event which would likely have
observable consequences. For example, the collapse of the NS magnetosphere
into a BH could lead to a large transient flux of electromagnetic energy which
has been identified as a potential FRB source Fuller_2015 ; Abramowicz:2017zbp
. In particular as noted in Ref. East:2019dxt , simulations of the collapse of
magnetized neutron stars demonstrate significant electromagnetic radiation
Lehner:2011aa ; Falcke:2013xpa . Generally a dipole field is assumed, whereas
here there would be a significant monopole component which likely would change
the geometry of this emission. That the monopole field would not be shed
during collapse would also be significant, as discussed in the next section.
Other possible signatures include kilonavae Bramante:2017ulk , gamma-ray
bursts Chirenti:2019sxw , and gravitational waves Baiotti:2007np ;
Kurita:2015vga ; Abramowicz:2017zbp .

However, the collision of an uncharged PBH with NS also leads to a the NS
collapsing into the BH. Potentially, a NS collapse into a mPBH can be
observationally distinguished, but it would require a more detailed
understanding of the late collapse process and the mechanism by which the FRB
is generated. When a mPBH collides with a NS the interaction of their
magnetospheres could lead to an FRB-like prompt radio pulse. This sort of
emission mechanism has been investigated in the case of a NS-NS collision
Pshirkov_2010 . But the mPBH-NS collision would be different, in detail. Of
particular importance would be the strength of the mPBH magnetic field,
dependent on $Q$. Further investigation of this scenario is warranted.

###  IV.4 Magnetic black hole

The final end product of this collapse is a magnetically charged, solar-mass
black hole. Compared with black holes formed by stellar collapse, this
magnetic black hole would be rather light, with a mass $M\lesssim 2M_{\odot}$
smaller than any observed astrophysical black hole. But, compared to the
original, microscopic mPBH, such an object could be observed in several ways.
A rapidly spinning black hole in a magnetic field can convert rotational
energy into jets of electromagnetic radiation via the Blandford-Znajek process
Blandford:1977 . Typically, the magnetic field is generated by accreting
matter, but, in this case, the magnetic field is sourced by the black hole
itself.

This has a couple potential implications. First, the black hole could produce
Blandford-Znajek jets in the absence of any accretion disk. Alternatively, if
the magnetically charged black hole were in a binary with another black hole,
the orbital motion could also result in electromagnetic jets Liebling:2016orx
.666The distinct gravitational wave signal from a such a binary was
investigated in Takhistov:2017bpt . This contrasts with studies of uncharged
black hole mergers within an externally sourced magnetic field
Palenzuela:2010nf . In particular, black holes dynamically shed any magnetic
field higher than the monopole in accordance with the no hair conjecture.

##  V Discussion

Compared with the collision between a NS and an uncharged BH, the collision of
a NS with a magnetically charged BH can have very different dynamics and have
an entirely different outcome. Due to electromagnetic interactions, a charged
BH stops almost immediately and transfers its angular momentum to the NS,
resulting in a glitch or anti-glitch. If the NS is observed as a pulsar,
current instruments are only sensitive enough to detect the collision of a
mPBH with very large charge. Future gravitational wave detectors, however,
might be able to detect a wider range of collisions.

After the collision, different thermodynamics of charged BHs and the region of
restored electroweak symmetry just outside the horizon means that the NS does
not inevitably collapse into the BH. For small enough magnetic charge, we
argued the mPBH and NS can reach a long-lived quasi-equilibrium state in which
accretion of NS matter into the BH generates Hawking radiation which then
supports the NS. Black holes with large magnetic charge, perhaps surprisingly,
are not able to generate enough pressure to support the NS, in which case the
entire NS is eventually accreted by the BH, much like an uncharged BH.
Observationally, it may be challenging, but not impossible to distinguish a
pure NS from one with a small-charge mPBH at its core. Similarly, a NS
collapse due to a large-charge mPBH may look quite similar to that due to an
uncharged PBH, and further investigation is required to determine the
differences. Furthermore, the resulting magnetically charged solar-mass BH
could be more readily observed.

We should note that our treatment of the mPBH-NS system is highly simplified.
First, we largely ignore the detailed interplay of the Hawking radiation with
the neutron star matter, which is very complicated. In particular, we assume
that the Hawking radiation is readily absorbed by the NS matter, but we
neglect the significant radiation pressure on the infalling matter. We expect
this would decrease, perhaps significantly, the accretion rate.

Second, the dynamics of the stellar material near the black hole is extremely
complicated involving a huge magnetic field, the intense spectrum of particles
radiated via the Hawking process, and high temperatures (and concomitant
nuclear reactions). Instead, as a first step, we generally ignore these
effects, expecting them to be subdominant to fluid pressure and gravity, at
least for the qualitative aspects of the equilibrium solutions. For example,
it is known that a very large magnetic field alters the equation of state of
dense matter in such a way as to support larger masses than with no magnetic
field Watanabe:2022cuv ; Broderick:2000pe ; Suvorov:2021ymy . As mentioned in
Sec. § IV.2, the large magnetic field may also form flux tubes which could
significantly deform the neutron star away from spherical symmetry.

Third, we need to better understand the out-of-equilibrium dynamics of the NS,
specifically in the case where the Hawking radiation is insufficient to
support the NS. We assumed, in this case, that the accretion rate would
increase. However, a more sophisticated treatment, for example, a dynamical
numerical simulation, would probably be needed to study the NS evolution.

In modeling mPBH, we assumed most of the Hawking radiation emitted from the
horizon was stopped at the corona, where the electroweak symmetry breaks and
the fermions become massive. The corona was modeled as a hot radiating sphere
in thermal equilibrium with the black hole. However, an improved picture of
the Hawking radiation is needed. For example, some light particles may pass
directly through the corona.

A further limitation is that we treated the NS matter in a fluid
approximation, which breaks down around the neutron length scale of
$10^{-15}\rm m$, while we treated the mPBH as a macroscopic object in that
fluid. However, the mPBH can be extremely small, particularly for small
charge; the corona radius, for example, is $r_{c}=1.7\times 10^{-18}{\rm\ m}\
\sqrt{Q}$. For $Q<10^{6}$, this is smaller than the radius of a neutron and
our results would likely be untrustworthy. Instead, the mPBH would need to be
considered as a microscopic object interacting with individual nucleons, as
discussed in Giddings:2008gr .

###### Acknowledgements.

This work is supported by the National Science Foundation via grants
PHY-1912769 and PHY-2011383 (SLL), PHY-2000398 (ML), AST-2011757 (JHS), and
AST-2011731 (MK). Computations were performed at XSEDE.

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