# On the Non-Minimal Coupling of Magnetic Fields with Gravity in Schwarzschild
Spacetime

Kumar Ravi1111Corresponding author, Petar Pavlović2, Andrey Saveliev3,4
<EMAIL_ADDRESS>
###### Abstract

In this work we study the effects of non-minimal coupling between
electromagnetism and gravity, which are motivated by quantum effects such as
vacuum polarization. We investigate the modification of both asymptotically
dipole and uniform magnetic fields around a Schwarzschild black hole that come
as the result of non-minimal coupling. While in both cases the magnetic field
gets enhanced or suppressed with respect to the case of minimal coupling,
depending on the sign of non-minimal coupling parameter, in the case of a
background uniform magnetic field the direction of the magnetic field also
alters in the vicinity of the black hole horizon. We have discussed the
possible astrophysical and cosmological sources for which the vacuum
polarization may be at play, while also discussing the observational effects,
in particular the possibility of synchrotron radiation from the vicinity of a
black hole. We conclude that such observations could be used to constrain the
value of the non-minimal coupling parameter.

1Ramakrishna Mission Vivekananda Educational and Research Institute, Belur
Math, Howrah, West Bengal, India
2Institute for Cosmology and Philosophy of Nature, Trg svetog Florijana 16,
Križevci, Croatia
3Immanuel Kant Baltic Federal University, Ul. A. Nevskogo 14, Kaliningrad,
Russia
4Lomonosov Moscow State University, GSP-1, Leninskiye Gory 1-52, Moscow,
Russia

Keywords: non-minimal coupling, Schwarzschild spacetime, magnetic fields,
innermost stable cicular orbit, black hole, effective potential

##  1 Introduction

The description of electromagnetism on macroscopic scales in terms of Maxwell
equations represents one of the most successful, well established and also
oldest field theory. The standard description of electromagnetic fields in the
presence of gravitational fields, which is of interest in cosmology and
astrophysics, therefore assumes a direct mathematical generalization of
Maxwell’s equations on curved spacetime [1]. On the other hand, the effect of
electromagnetic fields on spacetime is described by the Einstein equation in
the standard approach, where the stress-energy tensor of electromagnetic
fields bends the spacetime in the same fashion as any other source of energy
density. This type of dependence between gravity and electromagnetism is
called minimal, since the field equations for this case are derived by a
simple addition of gravitational and electromagnetic part of the Lagrangian,
without any cross terms. There are, however, strong reasons to expect that
this picture should be changed for very strong gravitational fields, where the
non-minimal coupling – the presence of Lagrangian terms containing direct
contractions between gravitational and electromagnetic tensors – should arise.
First of all, we could expect that at sufficiently high energies gravity and
electromagnetism will get united and described by a single field, similar to
what already has been found for electricity and magnetism and electromagnetism
and weak nuclear interaction. Since in this regime the gravitational and
electromagnetic sector could change into each other and would manifest a deep
interconnection, the investigation of non-minimal coupling between
electromagnetism and gravity can serve as an effective model for such effects.

There is also a more concrete reason for consideration of non-minimal coupling
in regimes of strong gravitational fields. In [2] the effect of quantum
electrodynamics (QED) vacuum polarization onto curved spacetime was studied
for the case of a photon propagating in vacuum and it was found that it leads
to a non-minimal coupling between gravity and electromagnetism. This can be
understood as coming from the transition of a photon into an electron/positron
pair and the consequent tidal influences of the spacetime geometry along the
characteristic Compton wavelength [2, 3]. In the one-loop approach it was
demonstrated [2] that the vacuum polarization effect thus leads to the
effective Lagrangian density of the form

$\mathcal{L}=\frac{R}{\kappa}+\frac{1}{2}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\mathcal{R}^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}+\mathcal{L}_{\rm
matter}\,,$ (1)

where $\kappa=8\pi G/c^{4}$ (from now on we will set $c=G=1$), $R$ is the
Ricci scalar, $F^{\mu\nu}$ is the Maxwell tensor obeying
$F^{\mu\nu}=\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu}$, where $\nabla_{\mu}$ is
the covariant derivative, and $\mathcal{L}_{\rm matter}$ is the Lagrangian of
neutral matter. The effects of the vacuum polarization are contained in the
tensor $\mathcal{R}^{\mu\nu\rho\sigma}$ (not to be confused with the Riemann
tensor $R^{\mu\nu\rho\sigma}$) which is defined as

$\displaystyle\mathcal{R}^{\mu\nu\rho\sigma}$
$\displaystyle\equiv\frac{q_{1}}{2}(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})R$
(2)
$\displaystyle+\frac{q_{2}}{2}(R^{\mu\rho}g^{\nu\sigma}-R^{\mu\sigma}g^{\nu\rho}+R^{\nu\sigma}g^{\mu\rho}-R^{\nu\rho}g^{\mu\sigma})$
$\displaystyle+q_{3}R^{\mu\nu\rho\sigma}\,,$

where $q_{1}$, $q_{2}$ and $q_{3}$ are the coupling constants, and
$R^{\mu\nu}$ is, as usual, the Ricci tensor and $R^{\mu\nu\rho\sigma}$ is the
Riemann tensor. Physical consequences of such types of non-minimal coupling
were investigated in various settings for a long time [4, 5, 6, 7, 8, 9].

In [3] it was proposed that the signatures of non-minimal coupling between
electromagnetism and gravity could, in principle, be observed or at least
constrained by studying the magnetic fields around the event horizons of black
holes and that the same effect could be used for constraining the sizes of
primordial black holes. With this aim, the problem of modification of magnetic
fields due to the vacuum polarization effect around a Schwarzschild black hole
was for the first time studied in [3] and the consequences for the orbits of
charged particles around such black holes were for the first time studied in
[10]. In this work we try to further develop the investigation of this topic.
We study the uniform magnetic field configuration, which has not been analysed
yet in this context, and also improve and critically discuss the previously
studied dipole magnetic field case. We then investigate the motion in the
equatorial plane by using the effective potential and demonstrate that most of
the conclusions that can be reached regarding the effects of the non-minimal
coupling on the trajectories and scattering of the charged particles can be
simply understood with the help of the effective potential study.

This paper is organized as follows: In Sec. 2 we briefly review the theory of
non-minimal coupling, while in Sec. 3 we present the analytical (minimal
coupling) and numerical solutions (non-minimal coupling) of the Maxwell
equations in Schwarzschild spacetime for the cases (i) of a static and
asymptotically dipole magnetic field is being placed at the origin of the
Schwarzschild black hole, and (ii) of a Schwarzschild black hole being placed
in a static and asymptotically uniform magnetic field. Sec. 4 is dedicated to
the study of the motion of a charged test particle in the equatorial plane of
the Schwarzschild spacetime in the effective potential formalism for both of
these magnetic field configurations. In Sec. 5 we briefly explore the
energetics, collision and the possibility of acceleration of charged test
particles near the event horizon when a Schwarzschild black hole is placed in
a static and asymptotically uniform magnetic field. Subsequently, in Sec. 6,
we present possible astrophysical and cosmological scenarios for which the
considerations of vacuum polarization and hence the applications of non-
minimal coupling might be important. Also, in this section we consider the
possible observational signatures, before concluding this work with a
discussion of its results and future scopes in Sec. 7.

##  2 Non-Minimal Coupling

There are, of course, numerous ways in which the non-minimal coupling between
gravitational and electromagnetic sectors can be achieved, such as coupling
between curvature tensors of different rank and order with vector potentials,
Maxwell tensors and their contractions, etc. [4, 5, 6, 11, 12, 13, 14, 15].
Many of those options are, however, completely arbitrary and lacking an
additional motivation, and some of them lead to the violation of important
physical principles, such as gauge invariance. We will focus on the type of
non-minimal coupling discussed in Sec. 2 and considered in [2] since (i) it is
motivated by the quantum-electrodynamical one-loop corrections in curved
spacetime, (ii) it involves the contraction of all fundamental electromagnetic
and curvature tensors to the leading order, and (iii) it preserves gauge
invariance. When the variational procedure is performed on the Lagrangian (1),
we obtain the following equations of motion:

$\nabla_{\mu}\left(F^{\mu\nu}+\mathcal{R}^{\mu\nu\rho\sigma}F_{\rho\sigma}\right)=0\,.$
(3)

In this work we assume the coupling constants $q_{1},q_{2},q_{3}$ to be
phenomenological constants which should be constrained based on observations.
In [2] the authors calculated the values of those constants for the simplest
case of a photon in vacuum on curved spacetime. Since it is not simple to see
how this result can be generalized for macroscopic magnetic fields of
arbitrary configurations, our approach appears to be completely justified. For
a more detailed discussion of this issue see Sec. 6 of this work. One should
note that the freedom in the choice of these constants does not violate the
gauge invariance, i.e. the equations will stay gauge-invariant regardless of
the value of the coupling parameters. To see this clearly one should observe
that the requirement of gauge invariance was used in the computations
presented in [2] initially in order to determine that the part of the
effective action describing the effects of the virtual electron loops depends
on $F_{\mu\nu}$ and not on $A_{\mu}$ directly. The only three possible terms
leading to non-minimal coupling which respect this condition, are gauge-
invariant and which are linear in curvature invariants and $F_{\mu\nu}$ are
the ones appearing in equation (2) multiplying the coefficients
$q_{1},q_{2},q_{3}$ [16]. As this is the general result, it will not be
affected by any change of value in the coupling coefficients. In accord with
this, further calculation of their specific values for the case of photon in
vacuum presented in [2] was not based on the requirement of gauge invariance,
but was performed by comparing the action coming from the virtual electron
loops with the weak-gravitational-field limit.

The discussed non-minimal coupling will in general lead to the violation of
equivalence principle of general relativity. The equivalence principle is
essentially based on a possibility of making a local identification between an
arbitrary spacetime manifold and the flat spacetime of special relativity, in
a sufficiently small neighborhood around a point of the manifold. That is to
say, the equivalence principle assumes that there is a tangent Minkowskian
space around every point of the spacetime. However, when the effect of vacuum
polarization is introduced, it becomes impossible to make the very transition
to a flat spacetime around some point for the system of interest. Due to its
quantum nature, manifested in its transition into a virtual electron/positron
pair, a photon will acquire a characteristic length (corresponding to the
Compton length) and due to the coupling with the gravitation field, it will be
affected by tidal effects along this length. Since it, therefore, does not
simply behave like a “point-like” object, it is not possible to make a choice
of a reference frame where the effects coming from its length would vanish,
and where the propagation of a photon would be simply represented by a photon
on a flat spacetime. As a general consequence, when non-minimal coupling
coming from the vacuum polarization is introduced, it may be impossible to
reduce the description of physics of such a spacetime to the principles of
special relativity locally. A concrete consequence of such considerations was
already demonstrated in [2]. Namely, in specific reference frames photons in a
gravitational field can travel with speeds greater than unity.

If one identifies $q_{1}=1/3,\,q_{2}=-1$ and $q_{3}=1$ then the tensor
$\mathcal{R}^{\mu\nu\rho\sigma}$ turns out to be the Weyl tensor [17]. There
is an active field of research which studies the non-minimal coupling of
photon to Weyl tensor [18, 19, 20, 21, 22, 23] and which found that photons
are then subject to birefringence and apparent superluminal motion. These
consequences demand a modification/correction of the background metric
(actually two effective metrics for two polarizations of photons). In the
present work we are only focused on the situation where the background metric
is still valid and we explore the motion/energetics of charged particles. The
exploration of situations where non-minimal coupling of both the magnetic
field (and therefore the motion/energetics of charged particles therein) and
photons could be investigated together is an interesting project for the
future.

##  3 Non-Minimal Coupling in a Schwarzschild Spacetime

At this point we need to stress the most important physical approximations
which we use in our study, which were also assumed in earlier works [3, 10].

We first assume that magnetic fields are weak enough for their effect on the
spacetime being negligible, such that the latter can therefore still be
described by the Schwarzschild metric. While not necessarily asymptotically
flat, there are some exact metrics where, unlike here, the magnetic fields are
not treated as just test fields (see [24] and references therein). Finding the
complete solutions to both the non-minimally coupled Maxwell and Einstein
equations is, however, of significant interest, and should be considered in
future research. Such full solutions of the non-minimal problem, describing
both electromagnetic and gravitational sector, could be of physical importance
in the very early Universe, when the non-minimal coupling effects and the
resulting magnetic fields might have very strong (see discussion in Sec. 7).
The approximation of a weak field and a negligible feedback on the spacetime
we use here should, however, be justified for all known astrophysical systems
(see detailed discussion in [3, 25]).

The next crucial assumption we made is to ignore the electric fields and
consider only the magnetic part of the electromagnetic tensor. This is
justified because of the very high value of the conductivity of the Universe,
causing the electric fields to be completely insignificant – this assumption
therefore represents one of the standard elements in studies of astrophysical
electromagnetism. On the other hand, magnetic fields probably exist on all
scales of the observable Universe (see [26, 27, 28, 17, 29, 30, 31, 32, 33,
34, 35, 36] and references therein). This fact makes magnetic fields an
excellent candidate for investigation of the effect of non-minimal coupling
between electromagnetism and gravity. We therefore do not need to consider a
special mechanism for the creation of magnetic fields around black holes,
since every black hole will naturally be immersed at least within the galactic
magnetic field.

The physical picture we consider here is therefore the following: we assume
the existence of a magnetic field, going to its asymptotical dependence far
away from the black hole – where the spacetime is approximately flat – while
being influenced by the black hole spacetime near the black hole horizon. The
magnetic field we treat here is thus not the magnetic field of the black hole
itself, but an external magnetic field which can be galactic or belong to some
other source, which we immerse in the Schwarzschild spacetime.

The Schwarzschild metric in a coordinate system adapted to spherical symmetry
$(t,\,r,\,\theta,\,\phi)$ is

${\rm d}s^{2}=-\left(1-\frac{2M}{r}\right){\rm
d}t^{2}+\left(1-\frac{2M}{r}\right)^{-1}{\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}{\theta}{\rm d}\phi^{2}\right)\,,$ (4)

where the mass $M$ is the source of gravity. The Schwarzschild solution of the
Einstein field equation is a unique solution and is obtained using the
assumptions of spherical symmetry, asymptotic flatness, and a vacuum outside
the spherical object of mass $M$. On top of that, this solution turns out to
be static. In accordance with the already discussed weak field approximation
we will assume that the effects of magnetic fields on the spacetime are
negligible, and, as a direct consequence of this assumption, that the
deviations from the Schwarzschild vacuum solution are negligible when the
magnetic fields are introduced on that spacetime.

We assume a magnetic field, expressed in the local Lorentz frame (LLF) [37,
38],

$\displaystyle F_{\theta\phi}^{\rm LLF}$ $\displaystyle=$ $\displaystyle
B_{r}=\frac{2\mu\cos\theta}{r^{3}}\xi(r)\,,$ $\displaystyle F_{\phi{r}}^{\rm
LLF}$ $\displaystyle=$ $\displaystyle
B_{\theta}=\frac{\mu\sin\theta}{r^{3}}\psi(r)\,,$ $\displaystyle
F_{r\theta}^{\rm LLF}$ $\displaystyle=$ $\displaystyle B_{\phi}=0\,,$ (5)

where $\mu$ is the magnetic moment and the functions $\xi(r)$ and $\psi(r)$
take into account the effects of curved spacetime on the magnetic field. We
will later see that $\xi(r)$ and $\psi(r)$ may be obtained from a system of
second order linear differential equations, and so we get two linearly
independent solutions: one corresponding to a dipole magnetic field, while the
other corresponds to a uniform magnetic field. Both of these cases are the
most basic models for studying magnetic fields. For instance, the galactic
magnetic field is modeled by the dipole magnetic field, so the discussed model
should be suitable for description of black holes immersed in the galactic
magnetic field.

After doing the transformation from local Lorentz frame, the components of
Maxwell tensor in Schwarzschild spacetime are

$\displaystyle F_{\theta\phi}$ $\displaystyle=$
$\displaystyle\frac{2\mu\sin\theta\cos\theta}{r}\xi(r)\,,$ $\displaystyle
F_{\phi{r}}$ $\displaystyle=$
$\displaystyle\frac{\mu\sin^{2}\theta}{r^{2}\sqrt{1-\frac{2M}{r}}}\psi(r)\,,$
$\displaystyle F_{r\theta}$ $\displaystyle=$ $\displaystyle 0\,.$ (6)

As for any vacuum solution of the Einstein field equation, the Ricci scalar
($R$) and Ricci tensor ($R_{\mu\nu}$) are zero, such that Eq. (3) reduces to

$\nabla_{\mu}\left(F^{\mu\nu}+q_{3}R^{\mu\nu\rho\sigma}F_{\rho\sigma}\right)=0\,.$
(7)

The Maxwell tensor components are also subject to another Maxwell equation,
namely

$\nabla_{[\lambda}F_{\mu\nu]}=0\,.$ (8)

After substituting the Maxwell tensor components from Eq. (6) into Eq. (7) and
into Eq. (8), we get

$\frac{{\rm d}}{{\rm
d}r}\left[\left(1-\frac{2Mq_{3}}{r^{3}}\right)\frac{\left(1-\frac{2M}{r}\right)^{1/2}}{r^{2}}\,\psi(r)\right]+\left(1+\frac{4Mq_{3}}{r^{3}}\right)\frac{2\xi{\left(r\right)}}{r^{3}}=0\,,$
(9)

and

$\frac{{\rm d}}{{\rm
d}r}\left[\frac{\xi(r)}{r}\right]+\frac{\psi{\left(r\right)}}{r^{2}\sqrt{1-\frac{2M}{r}}}=0\,,$
(10)

respectively. These equations can easily be decoupled, resulting in

$\displaystyle\frac{{\rm d}}{{\rm
d}r}\left[\left(1-\frac{2Mq_{3}}{r^{3}}\right)\left(1-\frac{2M}{r}\right)\frac{{\rm
d}}{{\rm d}r}\left(\frac{\xi(r)}{r}\right)\right]$
$\displaystyle-\left(1+\frac{4Mq_{3}}{r^{3}}\right)\frac{2\xi(r)}{r^{3}}=0\,.$
(11)

Here, we do not write the decoupled equation for $\psi(r)$, not just due to
its complicated form, but also due to the fact that it will not be used in our
computations. In fact, it is much easier to use the solutions of Eq. (11) and
insert them into Eq. (10), thus obtaining the solutions for $\psi(r)$, rather
than solving another differential equation from the beginning.

Before getting into the solutions of Eq. (9) and Eq. (10), i.e. for the case
of non-minimal coupling, we briefly review the available solutions for the
case of minimal coupling.

###  3.1 Minimal Coupling: A Brief Review of the Analytical Solutions

As expected, with $q_{3}=0$ the Eqs. (9) and (10) reduce to the corresponding
equations of [37] and [38]. While for the case of minimal coupling (i.e.
$q_{3}=0$), getting analytical solutions is straightforward with the
application of the Frobenius series solution technique for a second order
ordinary linear homogeneous equation, we could not carry out the same for
$q_{3}\neq 0$. The solutions for the case of minimal coupling are [37, 38, 39]

$\displaystyle\xi^{\rm
d}(r)=-\frac{3r^{3}}{8M^{3}}\left[\ln\left(1-\frac{2M}{r}\right)+\frac{2M}{r}\left(1+\frac{M}{r}\right)\right],$
$\displaystyle\psi^{\rm
d}(r)=\frac{3r^{2}}{4M^{2}}\Bigg{[}1+\left(1-\frac{2M}{r}\right)^{-1}+\frac{r}{M}\ln\left(1-\frac{2M}{r}\right)\Bigg{]}\sqrt{1-\frac{2M}{r}}\,,$
(12)

and

$\displaystyle\xi^{\rm u}(r)=\frac{B_{0}}{2\mu}r^{3}\,,$
$\displaystyle\psi^{\rm u}(r)=-\frac{B_{0}}{\mu}r^{3}\sqrt{1-\frac{2M}{r}}\,,$
(13)

where $B_{0}$ is the constant of integration and can be interpreted as the
magnitude of the asymptotically uniform background magnetic field. We have
used the superscripts ‘d’ and ‘u’ for the dipole solution and for the uniform
solution, respectively. It can easily be checked that for the dipole solution
(Eq. (12)), asymptotically for $r\rightarrow\infty$, the two functions behave
as $\xi(r)\rightarrow 1$ and $\psi(r)\rightarrow 1$, such that Eq. (5) reduces
to the familiar flat-spacetime solution. Therefore, in LLF the expressions for
the asymptotically dipole magnetic field solution are

$\displaystyle B_{r}^{\rm d}$ $\displaystyle=$
$\displaystyle-\frac{3\mu\cos\theta}{4M^{3}}\left[\ln\left(1-\frac{2M}{r}\right)+\frac{2M}{r}\left(1+\frac{M}{r}\right)\right],$
$\displaystyle B_{\theta}^{\rm d}$ $\displaystyle=$
$\displaystyle\frac{3\mu\sin\theta}{4M^{2}r}\Bigg{[}1+\left(1-\frac{2M}{r}\right)^{-1}+\frac{r}{M}\ln\left(1-\frac{2M}{r}\right)\Bigg{]}\,,$

while for the asymptotically uniform magnetic field we obtain

$\displaystyle B_{r}^{\rm u}$ $\displaystyle=$ $\displaystyle
B_{0}\cos\theta\,,$ $\displaystyle B_{\theta}^{\rm u}$ $\displaystyle=$
$\displaystyle-B_{0}\sin\theta\sqrt{1-\frac{2M}{r}}\,.$ (15)

It can be seen that solutions (LABEL:MagnF1) contain a singularity at $r=2M$,
which is a coordinate singularity associated with crossing the horizon in the
given coordinates. This naturally raises the question whether this singularity
implies the violation of the assumption of weak magnetic fields and suggests
that magnetic fields significantly affect the metrics, thus making the whole
treatment problematic. Indeed, the very black hole horizon and the points in
the very vicinity of it cannot be covered by the analysis provided here, since
they depart from the regime of weak magnetic fields. However, for distances
that are still near the horizon (for instance around the value of $r=1.5\times
r_{0}$) magnetic fields will still be weak and will not violate the initial
assumption of not affecting the metrics. This happens because the singularity
contained in (LABEL:MagnF1) is actually a “mild” logarithmic singularity,
leading to significant increase of the field only in the very small
neighborhood around $r_{0}$. This can, for instance, clearly be seen in Figs.
1 and 2 which demonstrate that the solutions remain small for the
astrophysically interesting region around the horizon. As we are studying the
propagation of charged particles around the black hole, this behavior at the
very horizon is not of concern, but one should note that our treatment cannot
be applied there.

We should note that $\xi(r)$ and $\psi(r)$ basically represent the ratios of
magnetic field components in a curved (here Schwarzschild) spacetime to those
in the flat spacetime (e.g. the quantities on vertical axis in the Fig. 1 of
[3]) and these ratios are precisely what we are interested in. Now we will
proceed with the investigations for the non-minimal coupling scenario.

###  3.2 Non-Minimal Coupling: Numerical Solutions

In order to numerically solve the coupled system of differential equations (9)
and (10) we need two initial/boundary conditions, namely $\xi(r_{0})$ and
$\psi(r_{0})$ at some $r_{0}$. For a chosen value of $q_{3}$, we can choose
large enough $r_{0}$ such that $4Mq_{3}/r_{0}^{3}\ll 1$ and then safely assume
the solutions there, at $r_{0}$, to be equal to those of minimal coupling
scenario with excellent precision. Therefore, in order to obtain the
asymptotically dipole solution numerically we can choose the initial
conditions to be compatible with Eq. (12), and for the asymptotically uniform
solution with Eq. (13). For instance, $r_{0}\simeq 100\times 2M$,
$\xi(r_{0})\simeq 1$ and $\psi(r_{0})\simeq 1$ are valid initial conditions
for seeking the dipole solution, whereas for the uniform solution
$\xi(r_{0})\simeq{\rm const.}\times r_{0}^{3}/2$ and $\psi(r_{0})\simeq-{\rm
const.}\times r_{0}^{3}\sqrt{1-2M/r_{0}}$ are a reasonable approach. We define
a dimensionless form of the coupling parameter as $\tilde{q}=q_{3}/(2M)^{2}$.

Figure 1: The radial component of a dipole magnetic field in Schwarzschild
spacetime ($B_{r}^{\rm curved}$) scaled by the same quantity in flat spacetime
($B_{r}^{\rm flat}$) as a function of $r$ for different values of $\tilde{q}$.
The enhancement/suppression of the magnetic field component near the horizon
for positive/negative values of $\tilde{q}$ compared to the case of minimal
coupling ($\tilde{q}=0$) can be seen. Figure 2: The azimuthal component of a
dipole magnetic field in Schwarzschild spacetime ($B_{\theta}^{\rm curved}$)
scaled by the same quantity in flat spacetime ($B_{\theta}^{\rm flat}$) as a
function of $r$ for different values of $\tilde{q}$. The
enhancement/suppression of the magnetic field component near the horizon for
positive/negative values of $\tilde{q}$ compared to the case of minimal
coupling ($\tilde{q}=0$) can be seen.

Following the numerical recipe described above, we have obtained the magnetic
field solutions for the non-minimal coupling case. In Figs. 1 and 2 we have
plotted the ratio of the radial and azimuthal component of the asymptotically
dipole magnetic field, modified by gravity to that of a dipole magnetic field
in flat spacetime, both for the case of minimal and non-minimal coupling
scenarios, i.e. $\xi(r)$ and $\psi(r)$, respectively. We can see from these
figures that for a positive (negative) coupling constant $\tilde{q}$ there is
an enhancement (suppression) of magnetic field present near the horizon.

We should compare the results of [3, 10] with the present work. Although the
main conclusions and the qualitative behavior is the same in these previous
works and the one in this study, there are still some quantitative differences
present. Though far from the event horizon, in the asymptotically flat region,
both this presentation and that of [3, 10] do not differ significantly, and
both have the same typical dipole form ($B_{r}(r,\theta)=2{\rm
const.}\times\cos\theta/r^{3}$ and $B_{\theta}(r,\theta)={\rm
const.}\times\sin\theta/r^{3}$), however they differ near the event horizon.
The radial dependence of both the components $B_{r}(r,\theta)$ and
$B_{\theta}(r,\theta)$ in [3, 10], namely $B_{\rm rad}(r)$, is identical, but
here we have different radial dependences for these two components, namely
$\xi(r)$ and $\psi(r)$. These differences come from the different mathematical
treatment. In [3, 10] the geometry of the magnetic field configuration was
prescribed a priori, assuming that $B_{\theta}(r,\theta)=\tan\theta
B_{r}(r,\theta)/2$. This type of assumption regarding the $B_{\theta}$ and
$B_{r}$ is based on the configuration satisfying the second Maxwell equation
on flat spacetime, viz. $\nabla\times\mathbf{B}=0$. The modifications of the
magnetic field were then inspected using Eq. (7), while no further reference
was made with respect to the other Maxwell equation, Eq. (8), since it is not
modified by the presence of the non-minimal coupling. In this work, however,
no a priori assumption regarding the field configuration, and thus the
relationship between $B_{\theta}$ and $B_{r}$, has been made, and both Maxwell
equations are solved simultaneously, making the treatment complete and self-
consistent, thus improving the mathematical analysis. The predicted
enhancement/suppression for a given $\tilde{q}$ in [3, 10] is larger than what
the present work predicts about the component $B_{r}$, whereas for
$B_{\theta}$ component it is, in fact, smaller.

Figure 3:  The radial component of the uniform magnetic field in Schwarzschild
spacetime scaled by the same quantity in flat spacetime as a function of $r$
for different values of $\tilde{q}$. For $\tilde{q}=0.5$ the enhancement of
the magnetic field component near the horizon with respect to the case of
minimal coupling ($\tilde{q}=0$) can be seen, while for $\tilde{q}=-0.5$ we
observe the change in sign and magnitude of the field component near the
horizon. Figure 4: The azimuthal component of the uniform magnetic field in
Schwarzschild spacetime scaled by the same quantity in flat spacetime as a
function of $r$ for different values of $\tilde{q}$. For $\tilde{q}=-0.5$ the
enhancement of the magnetic field component near the horizon with respect to
the case of minimal coupling ($\tilde{q}=0$) can be seen, while for
$\tilde{q}=0.5$ we observe the change in sign and magnitude of the field
component near the horizon. Figure 5: An asymptotically uniform magnetic field
near the Schwarzschild horizon for the minimal coupling scenario. Figure 6: An
asymptotically uniform magnetic field near the Schwarzschild horizon for the
non-minimal coupling parameter $\tilde{q}=0.5$. Figure 7: An asymptotically
uniform magnetic field near the Schwarzschild horizon for the non-minimal
coupling parameter $\tilde{q}=-0.5$.

The modification of an asymptotically uniform magnetic field near a
Schwarzschild black hole in a minimal coupling scenario has been widely
studied in the literature (see [40] and references therein). As expected, the
components of asymptotically uniform magnetic field get enhanced/suppressed
near the event horizon depending on the sign of the coupling constant
$\tilde{q}$ (see Figs. 3 and 4). There are two points to note here: (i) in
minimal coupling scenario the radial component, $B_{r}$, remains unaffected,
whereas in the case of non-minimal coupling both the components, $B_{r}$ and
$B_{\theta}$, are affected; (ii) for $\tilde{q}\neq 0$ the direction of
magnetic field changes near the horizon – for positive values of $\tilde{q}$
the direction switches to the opposite direction near the equatorial plane,
whereas for negative values of $\tilde{q}$ the direction changes near the two
poles (see Figs. 6 and 7; to contrast them from the case of minimal coupling
scenario, see Fig. 5). Therefore, in contrast to the case of a dipole field,
an asymptotically uniform magnetic field is also modified qualitatively, apart
from the quantitative enhancement/suppression both cases have in common. The
manifestations of this qualitative change on the motion of a charged particle
would be worth further exploration. But since in the following we study the
motion in the equatorial plain, the effect of the change of direction near the
poles (for $\tilde{q}>0$) is irrelevant.

As can be seen in Fig. 7, the asymptotically uniform magnetic field is being
repelled by the black hole in the vicinity of the Schwarzschild horizon, i.e.
similar to the Meissner effect in superconductors. This Meissner-like effect
near the black hole horizon vis-a-vis its undermining effects for the
efficiency of the Blandford-Znajek mechanism for jet formations in active
galactic nuclei (AGNs), gamma-ray bursts (GRBs), galactic black hole binaries,
etc., has been widely studied (see, for example, [41, 42, 43] and references
therein). In this work we are only considering the vacuum solution, such that
our results are not opposed to the Blandford-Znajek mechanism which requires a
plasma-filled magnetosphere around the black hole, where this Meissner-like
effect have been found to be absent by many researchers.

##  4 Motion in the Equatorial Plane: Effective Potential

Due to the symmetry of the problem we are investigating, the 4-potential can
be assumed to have the form [44]

$A_{\mu}=\left(0,0,0,A_{\phi}(r,\theta)\right)\,,$ (16)

while the Lagrangian for the motion of a charged test particle of charge $Q$
and mass $m$ can be written as [45]

$\mathscr{L}=\frac{1}{2}m\textnormal{{g}}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+QA_{\phi}\dot{\phi}\,,$
(17)

where $\textnormal{{g}}_{\mu\nu}$ denotes the metric components and
$\dot{x}^{\mu}$ is the 4-velocity. From this Lagrangian, apart from the
equation of motion

$\frac{{\rm d}^{2}x^{\mu}}{{\rm
d}\tau^{2}}+\Gamma^{\mu}{}_{\rho\sigma}\frac{{\rm d}x^{\rho}}{{\rm
d}\tau}\frac{{\rm d}x^{\sigma}}{{\rm
d}\tau}=\frac{Q}{m}F^{\mu}{}_{\nu\,\rm}\,\frac{{\rm d}x^{\nu}}{{\rm
d}\tau}\,,$ (18)

the two conserved quantities

$\frac{\partial{\mathscr{L}}}{\partial{\dot{t}}}=-m\left(1-\frac{2M}{r}\right)\frac{{\rm
d}t}{{\rm d}\tau}\equiv-E$ (19)

and

$\frac{\partial{\mathscr{L}}}{\partial{\dot{\phi}}}=mr^{2}\sin^{2}\theta\frac{{\rm
d}\phi}{{\rm d}\tau}+QA_{\phi}\equiv l$ (20)

may also be obtained. Here we stress that $F^{\mu\nu}$ represents the total
electromagnetic tensor which already includes the corrections to magnetic
fields coming from the non-minimal coupling effect (and not only the
“background” part of the field) because it is the solution of the non-
minimally coupled Maxwell equation, as can already be seen from the Eqs. (6)
and (7). The potential $A_{\mu}$ discussed here is defined with respect to
this total field in the usual manner, i.e.
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. $E$ and $l$ can be
interpreted as the (effective) energy and angular momentum, respectively.
Therefore, although in the considered equations the terms responsible for the
non-minimal coupling are not explicitly present, they are de facto implicitly
contained in the field terms such as $A_{\mu}$. We define $\tilde{E}\equiv
E/m$, $\tilde{l}\equiv l/m$ and $q\equiv Q/m$. From the equation of motion
(18) and the conservation equations (19) and (20), we get

$\tilde{E}^{2}=\left(\frac{{\rm d}r}{{\rm d}\tau}\right)^{2}+V_{\rm eff}^{2}$
(21)

for the motion in the equatorial plane, where

$V_{\rm
eff}^{2}=\left(1-\frac{2M}{r}\right)\left[1+\frac{1}{r^{2}}\left(\tilde{l}-qA_{\phi}(r)\right)^{2}\right]\,,$
(22)

and, furthermore, $A_{\phi}(r)$ is subject to the ordinary differential
equation

$\frac{{\rm d}A_{\phi}}{{\rm
d}r}=-\mu\left(1-\frac{2M}{r}\right)^{-1/2}\frac{\psi}{r^{2}}\,,$ (23)

where the non-minimal coupling enters through the modifications of the
function $\psi$.

###  4.1 Dipole Magnetic Field

We should mention the available analytical expression for $A_{\phi}(r,\theta)$
for the case of minimal coupling scenario [38],

$A_{\phi}=-\frac{3\mu\sin^{2}\theta}{8M^{3}}r^{2}\left[\ln\left(1-\frac{2M}{r}\right)+\frac{2M}{r}\left(1+\frac{M}{r}\right)\right]\,.$
(24)

For the non-minimal coupling scenario $V_{\rm eff}^{2}$ can be obtained
numerically by solving the system of Eqs. (9), (10) and (23). The proper
initial/boundary conditions to be used is $A_{\phi}(r_{0})\simeq 0$ when
$r_{0}\rightarrow\infty$ (which is compatible with Eq. (24)) and, again, those
for $\xi(r)$ and $\psi(r)$ are as discussed earlier. We introduce two
dimensionless parameters, $\lambda\equiv\mu{q}/M^{2}$ and
$H\equiv\tilde{l}/M$, as an aid to study the effective potential.

####  4.1.1 Case 1: The Angular Momentum is Parallel to Magnetic Moment

Let us first review some basic features of this setting in the minimal
coupling case. In general, the feature-wise richest type of the effective
potential has two maxima and two minima [38]. Between these two maxima a
potential well (minimum) is present. The maxima nearest to the horizon are
shown in the respective figure insets, while the minima farthest from the
horizon are not shown, as they are not of significant physical interest. For a
given value of $H$, increasing $\lambda$ amounts to a flattening-out of the
potential well, raising the level of the first maximum, lowering the level of
the outer maximum and shifting the first minimum outwards – in case it exists.
For a given $\lambda$, increasing $H$ amounts to narrowing the potential well,
raising the level of the outer maxima, lowering the level of the inner maxima
and shifting the first minimum inwards (see Figs. 8 and 11). So, if we focus
on the case for which no outer maximum (and thus no potential well) initially
exists for a given $\lambda$ and $H$, then with increasing $H$ at one point
the outer maximum appears. Furthermore, its level increases with $H$, and thus
the potential well becomes narrower and deeper. Finally, it reaches a maximum
depth when both the maxima attain equal height. If $H$ is then further
increased, the depth of the potential well starts decreasing as the inner
maximum level decreases. Hence, increasing the value of $H$ further, the
potential well will at some point again vanish as the inner maximum is
vanishing (e.g. $\lambda=50$, $H\sim 20-150$).

In case of non-minimal coupling, a positive value of the coupling constant
$\tilde{q}$ shifts the potential well outwards, does not affect the depth of
the potential well significantly (just slightly reducing it) and raises the
level of the first maximum significantly (see the inset of Fig. 8). A negative
value of $\tilde{q}$ shifts the potential well inwards and lowers the first
minimum significantly (so it can affect the potential well, see Fig. 11).
Depending on the particle energy $\tilde{E}$, it will be subjected to
scattering, a stable/unstable circular orbit, a bounded orbit with Larmor
motion, or it may plunge into the black hole.

Scattering: A particle coming from infinity and having an energy smaller than
either of the maxima will be scattered away back to infinity. When the inner
maximum is the absolute maximum, depending on the sign of $\tilde{q}$, the
spectrum (i.e. the energy dependence) of the scattered particles will differ.
For a positive $\tilde{q}$ particles with higher energies will be scattered
away, while for a negative $\tilde{q}$ the scattered particles will have
smaller energies than in the case of minimal coupling (see Figs. 8 and 11).
When the outer maximum is the absolute maximum (e.g. $\lambda=30$, $H=70.7816$
[38]), a nonzero $\tilde{q}$ will not affect the scattering significantly, as
the outer maximum is not significantly influenced by the effect of vacuum
polarization.

Plunging into the black hole: A particle coming from infinity with an energy
bigger than both of the maxima will plunge into the black hole. The effects of
a nonzero $\tilde{q}$ on the spectrum of particles plunging into the black
hole will be the opposite to what has been discussed regarding scattering.

Circular motion: A particle coming from infinity with energy equal to either
of the maxima will have a unstable circular orbit. Therefore, when $\tilde{q}$
is positive (negative), it is possible for particles of respectively higher
(lower) energies (in comparison to the case of minimal coupling) to have an
unstable circular orbit. On the other hand, a particle having the energy which
is equal to the first minimum and of local origin (i.e. not coming from
infinity) will have a stable circular orbit. This kind of circular orbit goes
by the popular term “innermost stable circular orbit” (isco). In the absence
of a magnetic field a charged particle cannot have a stable/unstable circular
orbit below a radius of $6M$, while a magnetic field allows circular orbits
well below this radius. Furthermore, in the non-minimal coupling case, with
nonzero $\tilde{q}$, this radius gets additionally affected (it
increases/decreases for positive/negative values of $\tilde{q}$, see Figs. 8
and 11). This will have implications for the modeling of accretion disks, as
will be discussed later.

Particle trapping: A particle of local origin which has an energy smaller than
both of the maxima will see two turning points and will be trapped in a
gyrating (Larmor motion, see, for example, Figs. 5 to 9 of [38] and/or Figs. 1
to 3 of [3]) bound orbit. Depending on the values of $\lambda$ and $H$ the two
maxima can have the same height, such that the depth of the corresponding
potential well will be maximum (e.g. $\lambda=50$, $H=68.5249$). A positive
value of $\tilde{q}$, apart from shifting the potential well outwards, hardly
makes any difference in comparison with the minimal coupling scenario (see
Fig. 10), while a negative $\tilde{q}$ significantly affects the potential
well by modifying the first maximum and thus the trapping of particles (see
Fig. 9). A more negative value of $\tilde{q}$ can bring down the first maximum
even lower with respect to the outer maximum and can hence make the potential
well shallower (or can even wash it out completely (see Fig. 10). On the other
hand, in the case when a negative $\tilde{q}$ does not make the potential well
shallower (for certain combinations of values of $\lambda$, $H$, see e.g. Fig.
9), then – as the potential well is in this case positioned closer to the
event horizon – this well is capable of holding more energetic particles
trapped in itself in comparison to the minimal coupling scenario.

In all the settings discussed above the study of the spectrum of the scattered
particles stands out as the most distinguished case for manifesting the
difference between minimal and non-minimal coupling scenarios. With respect to
the spectrum of the scattered particles, all other aspects of motion represent
a less significant kind of probe that could serve as a signature for the
existence of the effect of non-minimal coupling.

Figure 8: The effective potential for the motion of a charged particle in the
equatorial plane of the Schwarzschild spacetime in an asymptotically dipolar
magnetic field, for $\lambda=50$. The colors blue, red and black correspond to
the angular momentum parameter $H$ values 17.6954, 24.7736 and 34.4136,
respectively. The solid, dashed, dashdotted curves are for $\tilde{q}=0.0$ and
dotted, dashdotdotted, densely dashdotted curves are for $\tilde{q}=0.5$,
respectively. Due to a lack of space we have presented these line-styles in
the figure legend of Fig. 11 (the only differences are the signs of $H$ and
$\tilde{q}$, while the magnitudes are the same). The values for $\lambda$ and
$H$ are taken from [38]. In order to overcome the scaling difficulties, the
peaks of the effective potentials are plotted in the inset where the left
labeling of vertical axis is for $\tilde{q}=0.5$ and right labeling for
$\tilde{q}=0$. Figure 9: Same as Fig. 8, but for $\tilde{q}=-0.5$. The peaks
of effective potentials for $\tilde{q}=0.0$ are plotted in the inset. The
decrease in the heights of the peaks for $\tilde{q}=-0.5$ in relation to the
case of the minimal coupling can be seen. Figure 10: The effective potential
for the purpose of illustrating maximum depth potential (and hence maximal
particle trapping). The used parameter values are $\lambda=50.0,\,H=68.5249$
and the values of $\tilde{q}$ are shown in the legends. As can be seen, a
negative coupling constant ($\tilde{q}$) significantly reduces the capacity of
trapping charged particles near the horizon, while a positive $\tilde{q}$ does
not affect it so much.

####  4.1.2 Case 2: The Angular Momentum is Anti-Parallel to the Magnetic
Moment

For a particle having its angular momentum oriented anti-parallel to magnetic
dipole, the effective potential has only one maximum (very close to the
horizon) and a minimum far away from the horizon. Hence, such a particle will
effectively not see any potential well. If the energy $\tilde{E}$ of the
particle is greater than the maximum, it will plunge into the black hole,
while if it is smaller, it will be scattered away. There also exists a
possibility of an unstable circular orbit very near the event horizon (if the
particle’s energy is equal to the maximum), and a stable circular orbit far
away from the event horizon (for less energetic particles). The effects of
nonzero $\tilde{q}$ on the scattering of particles are similar to what has
been discussed for the case $H>0$ (see Figs. 11 and 12).

Figure 11: Same as Fig. 8, but here for the negative values of the angular
momentum parameter ($H$) used there. The peaks for $\tilde{q}=0.5$ are shown
in the inset. An increase in the heights of the peaks for $\tilde{q}=0.5$ with
respect to the case of minimal coupling may be seen here. Figure 12: Same as
Fig. 11, but here for $\tilde{q}=-0.5$. The decrease of the heights of the
peaks for a negative coupling compared to the case of minimal coupling may be
seen here.

###  4.2 Uniform Magnetic Field

Again, we point out the existence of an analytical expression for
$A_{\phi}(r,\theta)$ for the case of the minimal-coupling scenario [40],

$A_{\phi}=\frac{B_{0}}{2}r^{2}\sin^{2}\theta\,,$ (25)

where $B_{0}$ is the strength of magnetic field at infinity. To obtain the
effective potential numerically for the case of non-minimal coupling, we
follow the same set of procedures as described earlier for the case of a
dipole field. We define one more dimensionless quantity, $\beta\equiv
qB_{0}M/m$, and will make use of $H$ as defined earlier. For the radial
distance we will use a dimensionless parameter, $\rho=r/M$ , for which
$\rho_{+}$ and $\rho_{-}$ will be used to denote the radius of the isco when
the angular momentum is parallel and anti-parallel to the magnetic field,
respectively. Unlike in the case of a dipole magnetic field, the uniform
magnetic field at large distances from the horizon plays an important role as
a barrier for the particles coming from infinity (see Figs. 13 and 14).

Figure 13: The effective potential for the motion of a charged particle in the
equatorial plane of the Schwarzschild spacetime in an asymptotically uniform
magnetic field for the magnetic field parameter $\beta=0.5$, for the case of
the magnetic field and the angular momentum being parallel. The colors blue,
red and black corresponds to the angular momentum parameter ($H$) values
$2.36$, $4.14$ and $9.2$, respectively. The solid, dashed, dashdotted curves
are for $\tilde{q}=0.0$ and dotted, dashdotdotted, densely dashdotted curves
are for $\tilde{q}=0.5$, respectively. Because of a lack of space in this
figure we have presented these line styles in the legend of Fig. 14 (the only
difference being the sign $\tilde{q}$, while its magnitude is the same). To
highlight the two interesting features (one extra maximum and one additional
minimum near the horizon) in the effective potential due to positive coupling,
the inset plot is included. Only the right-side labeling of the vertical axis
corresponds to the inset plots. Figure 14: Same as Fig. 13, but now for
$\tilde{q}=-0.5$. Unlike for the case of positive $\tilde{q}$, here for
$\tilde{q}=-0.5$ the effective potentials do not differ much from the case of
the minimal-coupling scenario. Figure 15: The isco radii($\rho_{+}$) as a
function of $\beta$ when the angular momentum of the particle is parallel to
the uniform magnetic field (asymptotically). The colors black, blue and red
are for the values of $\tilde{q}=$0.125, 0.0 and -0.125, respectively. As
expected in all three (different values of $\tilde{q}$) cases, the isco radius
should be $6M$ when there is no magnetic field (i.e. when $\beta$=0). For the
case of minimal coupling with an increase in the strength of the magnetic
field parameter $\beta$, the isco radius saturates for $r_{isco}\rightarrow
2M$ to $\beta\rightarrow\infty$. Figure 16: The angular momentum parameter
($H_{+}$) for isco as a function of $\beta$. The colors black, blue and red
are for the values of $\tilde{q}=0.125$, $0.0$ and $-0.125$, respectively. As
expected, in absence of any magnetic field the angular momentum here is
$+\sqrt{12}M$. Figure 17: Location of the extrema of the effective potential
$V_{\rm eff}^{2}$ for $\beta=0.5$ in the $r$-$H$ parameter space (upper
branch) for different values of $\tilde{q}$. For $\tilde{q}=0.5$ two
minima/maxima (denoted as Min A/B and Max A/B, respectively) exist. Note that
while this upper branch has strictly $H>0$ in the case of non-minimal
coupling, for $\tilde{q}=0.5$ it obtains values with $H<0$.

####  4.2.1 Case 1: The Angular Momentum is Parallel to the Magnetic Field

For our investigation of non-minimal coupling of an asymptotically uniform
magnetic field with Schwarzschild spacetime, we have used the values of
parameters $\beta$ and $H$ from the caption of Fig. 3 from [40]. We also note
that the quoted $l\simeq 3.22$ in that figure caption is a typographical error
and we find it to be $l\simeq 4.6$. The parameters $\rho$, $\beta$ and $H$
defined here corresponds to $2\rho$, $b$ and $2l$ in [40], respectively. We
will focus our discussion mainly on the case for $\tilde{q}\neq 0$, as the
$\tilde{q}=0$ case is nicely presented in [40].

In the case of a minimally coupled magnetic field, for lower values of $H$
there are, at first, no extrema of the effective potential present, while for
an increase in $H$ we see one extremum (inflection point), and increasing $H$
even further results in one maximum near and one minimum away from the horizon
(as can be seen in Fig. 13 for $H\simeq 2.36,\,4.14,\,9.2$). For a further
increase in $H$, both of these extrema shift outwards from the horizon with
the increased values of the effective potential. A positive $\tilde{q}$
introduces none, one or two more extrema near the horizon, depending on $H$.
So, for those values of $\beta$ and $H$, for which there are no extrema at all
in the case of minimal coupling, a positive $\tilde{q}$ can introduce an
inflection point or a minimum and hence lead to the existence of an isco or a
potential well. For those values of $\beta$ and $H$, for which there is an
inflection point, a positive $\tilde{q}$ can introduce a minimum and hence a
potential well. Finally, for those higher values of $\beta$ and $H$, for which
there is a maximum and a minimum, a positive $\tilde{q}$ can introduce one
more maximum and one more minimum and hence one more potential well.
Effectively, a positive $\tilde{q}$ introduces bound gyrating trajectories
(and stable/unstable circular orbits) very near the horizon, separated from
the same type of trajectories away from the horizon. This observation can have
its implications in modeling of accretion disks and while studying any
emission (possibly synchrotron emission, see below) from particles gyrating in
the bounded orbits.

A negative $\tilde{q}$ introduces at most one inflection point or minimum near
the horizon if no such features are there in case of minimal coupling. For an
already existing potential well, a negative $\tilde{q}$ increases the depth
(see Fig. 14). Unlike the case of $\tilde{q}>0$, here we do not see two sets
of bounded orbits, but with an increased depth of the potential well, trapped
particle will have higher energies compared to the case of minimal coupling.

In Fig. 15 we have plotted the dependence of the radius of the isco on the
magnetic field parameter $\beta$. In the figure we clearly can see the isco
radius at $6M$ for zero magnetic field, then, for an increasing magnetic
field, it shifts towards the horizon. As expected, for a positive $\tilde{q}$
and in the case of smaller values of magnetic field (i.e. values up to which
the non-minimal coupling does not introduce any inner minima, but rather just
shifts the extrema of the minimal coupling scenario outwards, see Fig. 13) the
isco radii are larger than those for the minimal-coupling case. Above a
certain strength of the magnetic field, the non-minimal coupling starts
introducing additional extrema near the horizon (this aspect can be seen in
Fig. 15 around $\beta\simeq 1.3$. We choose $|\tilde{q}|=0.125$ instead of
$0.5$ to show this transitory feature clearly, since for $\tilde{q}=0.5$ this
feature shows near $\beta\sim 0.1$). Therefore, we see that beyond certain
values of $\beta$ there are two sets of stable circular orbits (well within
$6M$ radius) separated from each other. The termination of plot for negative
$\tilde{q}$ should be interpreted as isco grazing the horizon and being
occupied with angular momentum of the particles $H\rightarrow 0$. Because of
limited resolution ($\Delta H$) used in our code, this curve is terminating
there. This feature is not surprising, as we carefully observe the plot of the
effective potential in the Fig. 14 for $H=2.36$. Like in the case of minimal
coupling for $\beta\rightarrow\infty$ [40], we expect the radius of isco to
saturate at some value (depending on $\tilde{q}$) near $2M$ (the Schwarzschild
radius).

In Fig. 16 we have plotted the angular momentum for isco as a function of
magnetic field parameter $\beta$. We can see that, starting from the angular
momentum of $\sqrt{12}M$ for $\beta=0$, the particles with lower angular
momentum initially occupy the isco, while with the increase in the strength of
the magnetic field particles of higher angular momentum orbit in the isco as
well. Again, the transitory feature near $\beta\simeq 1.3$ for
$\tilde{q}=0.125$ reflects the introduction of an inflection point or a
shallow potential well near the horizon because of non-minimal coupling (see
the inset of Fig. 13). The termination of the curve for $\tilde{q}=-0.125$
near $\beta\simeq 2.2$ (with $H\simeq 0.4$) reflects the fact that for the
continuation of this curve one needs a progressively finer and finer
resolution ($\Delta H$) requiring heavy computation. As the gained information
is not justifying such heavy computations, we have terminated our calculations
beyond these limits.

As a concluding remark and summary to this section we present the change of
the position of the minima and maxima of the effective potential in the
$r$-$H$ phase space in Fig. 17. As already described, the introduction of
$\tilde{q}\neq 0$ indeed greatly modifies the form of the effective potential,
in particular by creating additional extrema and also shifting this “upper
branch” (i.e. the branch of $H>0$ for $\tilde{q}=0$) to negative values of
$H$. In fact, this modification can alter observational signatures which, in
turn, may be used in the future to derive limits on $\tilde{q}$, as described
in Sec. 6.1.

####  4.2.2 Case 2: The Angular Momentum is Anti-Parallel to the Magnetic
Field

Figure 18: The effective potential for the motion of a charged particle in the
equatorial plane of the Schwarzschild spacetime in an asymptotically uniform
magnetic field with magnetic field parameter $\beta=0.3$, for the case of the
magnetic field being anti-parallel to angular momentum. Figure 19: Same as
Fig. 18, but now for $\tilde{q}=-0.5$. Figure 20: The isco radii ($\rho_{-}$)
as a function of $\beta$ when angular momentum of the particle is anti-
parallel to the uniform magnetic field (asymptotically). The colors black,
blue and red are for the values of $\tilde{q}=0.5$, $0.0$ and $-0.5$,
respectively. As expected, in the absence of a magnetic field the radius
should be $6M$. For the case of minimal coupling, for an increase of the
strength of the magnetic field parameter $\beta$, the isco radius saturates as
$r_{isco}\rightarrow 4.3M$ for $\beta\rightarrow\infty$ [40]. Figure 21:
Location of the extrema of the effective potential $V_{\rm eff}^{2}$ for
$\beta=0.3$ in the $r$-$H$ parameter space (lower branch) for different values
of $\tilde{q}$. In contrast to the upper branch (cf. Fig. 17), the change with
$\tilde{q}$ of the overall structure here is much smaller. Figure 22: The
angular momentum parameter ($H_{-}$) for the isco as a function of $\beta$.
The colors black, blue and red are for the values of $\tilde{q}=0.5$, $0.0$
and $-0.5$, respectively. As expected, in the absence of a magnetic field the
angular momentum is $-\sqrt{12}M$.

When the angular momentum of a particle is anti-parallel to the magnetic field
we hardly see any interesting features in the effective potential in both
coupling scenarios. Results in the case of non-minimal coupling are just
slightly enhanced/suppressed quantitatively. We can speculate about the
possible reason for such a situation to be the saturation of the isco radius
($\rho_{-}$) comparatively (with the case of parallel angular momentum) far
from the horizon, $\rho_{-}\rightarrow 4.3M$ as $\beta\rightarrow\infty$ [40].

In Figs. 18 and 19 we have presented a few illustrative plots for the
effective potential. Depending on $\beta$ and $H$, there exist a maximum (and
hence an unstable circular orbit) near the horizon, followed by a minimum (and
hence a stable circular orbit and/or potential well for particle trappings)
away from the horizon. A positive/negative $\tilde{q}$ raises/lowers the
height of the maxima and shifts inward/outwards. The effects of
positive/negative $\tilde{q}$ on the minima is to shift it inwards/outwards
and hardly makes any change in its height.

In Fig. 20 we have plotted the radii ($\rho_{-}$) of the isco as a function of
the parameter $\beta$. A positive/negative value of the coupling constant
$\tilde{q}$ allows the isco to be closer/farther to/from the event horizon
when compared with the case of minimal coupling. Correspondingly, the energy
of particles trapped in unstable/stable circular orbits or in gyrating bound
orbits will be different for the two kinds of coupling. In these figures, for
a zero magnetic field ($\beta=0$) we again can see the isco radius $6M$ and
the corresponding angular momentum $-\sqrt{12}\,M$.

Finally, in a similar way as for the previous section, in Fig. 21 we present
the behavior of the extrema of the effective potential in the $r$-$H$
parameter space (for the lower branch, i.e. $H<0$ with respect to $\tilde{q}$.
While a modification is clearly visible, in contrast to the upper branch
described before the basic shape remains the same and no additional extrema
are introduced.

####  4.2.3 Energy of a Charged Particle in a Marginally Stable Circular Orbit

As a prequel to the study of the collision of charged particles, in this
section we briefly discuss the energetics of charged particles moving in a
marginally stable circular orbit. In the case of the angular momentum being
parallel to the asymptotically uniform magnetic field, a non-minimal coupling
makes it possible to have the isco closer to the horizon compared to the case
of minimal coupling (of course above a certain value of magnetic field
strength this feature is common to both the positive and the negative charge,
see Figs. 15 and Fig. 23). The break/gap in the energy plot for
$\tilde{q}=0.125$ is understandable if we observe Fig. 15, since for $\beta<5$
there are no isco radii in the range $\sim 2.12-2.47$, while with the increase
in $\beta$ this gap will fill.

Figure 23: Energy of a charged particle in an isco as function of the isco
radius when its angular momentum is parallel to the uniform magnetic field.
Figure 24: Energy of a charged particle in an isco as function of the isco
radius, when its angular momentum is anti-parallel to the uniform magnetic
field.

When the angular momentum is anti-parallel to the magnetic field, similar to
the results in the case of minimal coupling, the non-minimal coupling also
enables trapping of the particles with higher energies ($\tilde{E}>1$, see
Fig. 24). For a positive/negative value of the coupling constant $\tilde{q}$
and any given $\beta$ these high energy particles can move in closer/farther
iscos in comparison to the case of minimal coupling.

##  5 Particle Collision/Acceleration

Taking the case of an asymptotically uniform magnetic field, we consider the
collision of two charged particles of equal mass ($m_{0}$), opposite charge
and moving in opposite directions on an isco (therefore having the same
angular momentum). For this we calculate the center of mass energy of the two
colliding particles, given by

$E_{\rm cm}^{2}=2m_{0}^{2}\left(1-g_{\mu\nu}u^{\mu}_{1}u^{\nu}_{2}\right)\,,$
(26)

where $u^{\mu}_{1}$ and $u^{\mu}_{2}$ are their respective 4-velocities. For
this configuration of the collision the 4-velocities of the particles take the
form

$u^{\alpha}_{\pm}=\left(-\frac{{\rm d}t}{{\rm d}\tau},\,0,\,0,\,\pm\frac{{\rm
d}\phi}{{\rm d}\tau}\right)$ (27)

and are subject to the normalization $u^{\alpha}u_{\alpha}=-1$. Using Eqs.
(19), (20) and (22), we get

$\frac{E_{\rm
cm}}{m_{0}}=\sqrt{2}\left(1-\frac{2M}{r}\right)^{-1/2}\sqrt{\tilde{E}^{2}+V^{2}_{\rm
eff}}\,.$ (28)

In Fig. 25 we have plotted the center of mass energy over the rest energy of a
charged particle as function of the isco radius when the angular momentum of
the particles is parallel to the magnetic field. For the case of minimal
coupling we do not see much gain in energy [46]. In the case of non-minimal
coupling, beyond some values of magnetic field strength, near the horizon, the
ratio $E_{\rm cm}/m_{0}$ dominates when compared to the case of minimal
coupling. The energy of a charged particle in an isco near the horizon in case
of a nonzero $\tilde{q}$ is not much larger comparatively (see Fig. 23).
However, because of the relative proximity to the horizon (when compared to
the case of minimal coupling), the term $\sqrt{1-2M/r}$ in Eq. (28) enhances
this ratio $E_{\rm cm}/m_{0}$.

Figure 25: The center of mass energy for a head-on collision of two charged
particles of equal mass $m_{0}$, equal and opposite charge, moving in opposite
directions, therefore having equal angular momenta (when the angular momenta
of these charged particles are parallel to the magnetic field) as a function
of the isco radii. Figure 26: The center of mass energy for a head-on
collision of two charged particles of equal mass $m_{0}$, equal and opposite
charge, moving in opposite directions, therefore having equal angular momenta
(when the angular momenta of these charged particles are anti-parallel to the
magnetic field) as a function of the isco radius.

The center of mass energy over the rest energy of a charged particle as
function of the isco radius when the angular momentum of the particles is
anti-parallel to the magnetic field is shown in Fig. 26. Unlike the case of
parallel angular momentum here the isco radii are away from the horizon, the
effect of which can be seen if we observe the numbers on vertical axis of Fig.
26. The seemingly blowing-off of $E_{\rm cm}/m_{0}$ near the minimum isco
radius is not to be interpreted as a gain of energy due to the black hole, but
coming from the fact that these particles are already energetic (see Fig. 24).
In other words, they have high energies when measured by an observer at
infinity, i.e. the black hole is not functioning as an accelerator here [46].

##  6 Possible Applications in Astrophysical and Cosmological Scenarios

Due to the required amount of work and space, it is not possible to present
any detailed numerical and quantitative predictions regarding the systems in
which the non-minimal coupling could have its astrophysical application. While
leaving such more detailed and extensive investigations for the future, in
this section we will explore several possible implications of the non-minimal
coupling qualitatively. Recently the non-minimal coupling of photon to the
Weyl tensor was considered in [19]. The observational constraint on coupling
parameter thus derived has been estimated to be $\sim 0.563$. Though in this
treatment a different metric was used for the description of black hole, and
not the Schwarzschild metric, their estimate is not too different from the
order of $\tilde{q}\sim 0.5-1$, used in our work for example. Whereas in the
context of the Sun and pulsar(of mass $1.33M_{\odot}$), for observable effects
of non-minimal coupling the constraint on $\tilde{q}$ obtained in [22] are
$\sim 3.3\times 10^{9}$ and $\sim 1.5\times 10^{2}$. Clearly for the later
case we can not claim the validity of our present work Applications of this
effect for objects like a neutron star, the sun and Earth are discussed in
[3], such that we will not discussion them here. In [3] the application to
primordial black holes is also discussed, but nevertheless here we would like
to present it in a different light.

One of the central issues which appears when discussing the potential
observational effects of the non-minimal coupling is the value of the coupling
parameter. Broadly speaking, one approach to this question is to consider the
coupling constant to be a free parameter that has to be obtained or
constrained phenomenologically, i.e. from observations of astrophysical
sources, as have been considered in the literature [4, 5]. On the other hand,
Drummond & Hathrell [2] have presented a derivation of the non-minimal
coupling from first principles, demonstrating it to be the consequence of the
QED vacuum polarization on the curved spacetime. In their derivation they have
found that the coupling constant is determined to be
$q_{3}=-\alpha{\lambda_{e}^{2}}/90\pi\,$ $\sim-10^{-28}\,{\rm m}^{2}$, where
$\alpha$ is fine structure constant and $\lambda_{e}$ is the Compton
wavelength of the electron. It is, however, important to take into account
that this value is determined for a very specific and elementary setting of a
photon propagating in vacuum on the curved spacetime. It is not at all simple
to determine what is the connection between this very simple case of one
photon and a very complex setting of photons leading to macroscopic magnetic
field distributions, such as the dipole field. Therefore, it is not possible
to simply claim that the coupling parameter for macroscopic configurations of
magnetic fields – i.e. the ones encountered in astrophysics – is related to
the value calculated in [2]. In principle, the value of the coupling parameter
could even vary for different field strengths and configurations. It therefore
seems that the value of the coupling parameter for the macroscopic magnetic
field configurations of interest is basically unknown and should be
constrained from the phenomenological considerations. However, the
significance of the value determined in [2] comes from the fact that it can
serve at least as a conservative lower limit for the value of $\tilde{q}$.
Therefore, when discussing the potential observational implications we can
take it as a reasonable assumption that the value of $\tilde{q}$ should be at
least of the order of magnitude predicted in [2] or higher. In the following
paragraph we will discuss the cosmological consequences of non-minimal
coupling focusing on the case of this limit. Taking this conservative lower
limit, $\tilde{q}=q_{3}/r_{0}^{2}$, where $r_{0}\sim 2M$, to be of order 1,
the mass $M$ has to be $\sim 10^{13}$ kg, such that for most of the
astrophysical sources of mass comparable to or higher than the solar mass, it
is $\tilde{q}\ll 1$. Therefore, it follows that in such a case the observable
effects will be negligible.

However, an interesting candidate relevant even for this conservative limit is
given by hypothetical astrophysical objects called Primordial Black Holes
(PBHs) [47, 48], which have a mass range conjectured (depending on the model,
i.e. on the time when they formed after the Big Bang) to be between $10^{-8}$
kg and many solar masses. Another point to note is that Hawking’s theory of
black holes evaporation [49] predicts that any PBH of a mass less than $\sim
10^{11}$ kg would have evaporated by now. Needles to say, it is of course very
questionable to expect that this approximation – which assumes a weak field
and the one-loop approximation – can be applicable for a PBH of mass $\sim
10^{-8}$ kg ($\tilde{q}\sim 10^{41}$). On the other hand, if we take the
Drummond & Hathrell [2] theory to be applicable for $\tilde{q}\sim 1-10$ (i.e.
a mass of $\sim 10^{13}-10^{11}$ kg, since $\tilde{q}\propto 1/\sqrt{\rm
mass}$), then it can be utilized for both kinds of PBHs – the ones which are
already evaporated and the ones which still exist. With regard to evaporated
PBHs it is in principle quite likely that the local distribution of magnetic
fields previously associated with them would still be present at their
location. Thus, an astrophysical observation of a local suppression of the
magnetic field can be further analyzed as a possible candidate for the
detection of PBH. The investigation of PBHs in relation to non-minimal
coupling can in general take two directions. In the case of a significant
localized suppression of magnetic field being observed (while previously not
being recognized as a PHB candidate), we can take non-minimal coupling
considerations into account and claim this localized suppression to be a
signature of a PBH. On the other hand, if some location is already claimed to
be a PBH candidate by some other model or consideration, then – if a
significant suppression of magnetic field around this point is established –
it can serve as a way to establish a constraint on the value of $\tilde{q}$.
For the PBHs which exist until the present day ($10^{11}$ kg $\lesssim$ mass
$\lesssim 10^{13}$ kg) the same arguments as for the usual black holes in [3,
10] may be made. We note that the discussed question of PBHs is also
interesting because it connects the question of non-minimal coupling with the
question of the evolution of cosmological magnetic fields, which could also be
connected with the question of the origin of the Universe [50].

In any realistic modeling of an accretion disk around a black hole, the study
of factors which affect the isco is important. The implications of the correct
modeling of an accretion disk is also important for the analysis of the black
hole images made by the Event Horizon Telescope [51, 52, 53] and for similar
future observations. In the present work we saw that a non-minimal coupling
affects isco not only quantitatively but also qualitatively. One of the means
to distinguish between a black hole and hypothetical/exotic objects (like a
naked singularity, a non-singular black hole, etc.) has been the investigation
of the properties of isco in absence of and in presence of magnetic field (in
the minimal coupling scenario). Including a non-minimal coupling
considerations in these studies can answer the questions like – can a black
hole, when its gravity is non-minimally coupled with a magnetic field, have
same/distinguishable observational signatures as those of exotic objects? For
instance, the $\gamma$ \- metric – one of the candidates for naked singularity
– shows two well separated regions of the isco for $\gamma\in(1/\sqrt{5},1/2)$
(in absence of [54] and in presence of magnetic fields [55]). We can see from
Fig. 15 that this particular feature is present for a black hole too when the
non-minimal coupling is considered. So it would be interesting to further
investigate how a black hole and exotic objects differ in their observational
signature when non-minimal coupling of electromagnetic field with strong
gravity is considered. Very often the estimation of spin of a black hole
candidate has been derived from the theoretical relationship between spin and
the isco radius [56, 57]. Though in the present work we have not considered a
rotating black hole (i.e. a Kerr black hole), from previous studies in minimal
coupling we can expect that there too the properties of the isco would be
different if non-minimal coupling is taken into consideration.

From observational astronomy it is well established that many astrophysical
objects (active galactic nuclei, radio galaxies, both stellar mass and super-
massive black holes, etc.) often have jets associated with them. Many proposed
mechanisms for the formation of jets assumes the presence of a magnetic field
in the system comprising a central object (often a neutron star or black
hole), an accretion disk and a jet. The transfer of energy and material from
accretion disk to jet is facilitated by magnetic fields in many models [58].
So far only the minimal coupling of magnetic field with gravity have been
considered in the models of jet formation. A study of non-minimal coupling of
magnetic fields with gravity can possibly shed some new light on the nature of
the phenomenon.

Figure 27: The location of minima (denoted as “Min”) and maxima (denoted as
“Max”) in the $r$-$H$ parameter space for different values of $\tilde{q}$. As
discussed above, while for the case of minimal coupling and for $\tilde{q}<0$
only one minimum/maximum exists, for $\tilde{q}>0$ there are two
minima/maxima. In this latter case “Min A”/“Max A” denotes the minimum/maximum
which is closer to the event horizon, while “Min B”/“Max B” denotes the
minimum/maximum farther away from the event horizon which converges with the
one for $\tilde{q}\leq 0$ for large $r$ or $H$. Figure 28: Upper limit
$\omega_{\rm max}$ for the frequency of the emitted synchrotron radiation by a
proton orbiting a stellar black hole for different values of $\tilde{q}$
(according to Eq. (33)) as a function of the radius $r_{\rm min}$ at which the
minimum of the effective potential (and hence the stable circular orbit which
the proton eventually reaches) is located. Note the deviation at small $r_{\rm
min}$ for the case $\tilde{q}=0.5$ which is due to the second minimum/maximum
of the effective potential.

###  6.1 Synchrotron Radiation

Finally, as a concrete astrophysical application and in order to give a
quantitative estimate how it may be changed by non-minimal coupling, we
consider the possibility of synchrotron radiation from the vicinity of a black
hole. Doing so requires the trapping of high energy particles near the horizon
[45]. In absence of a magnetic field the possibility of particle trappings
within a radius of $6M$ in unbound and unstable orbits [59, 60, 61] has
limitations. In the presence of a magnetic field, even in a minimal coupling
scenario, highly energetic particles can be trapped in stable circular orbits
within a radius of $6M$ and near the horizon. We saw here that non-minimal
coupling further facilitates the trapping of high energy particles near the
horizon, such that in any study of synchrotron radiation from the vicinity of
a black hole the inclusion of non-minimal considerations can be useful.

In order to demonstrate a concrete possibility for such an investigation, we
start with the concepts developed for synchrotron radiation from a weakly
magnetized Schwarzschild black hole with minimal coupling in [62] and extend
them to the non-minimal case. In particular, we consider the case of a proton
orbiting a stellar Schwarzschild black hole in a uniform magnetic field (cf.
Sec. 4.2) which translates to a value for $\beta$ of $\beta\simeq 4.718\times
10^{7}$.

In [62], for $\beta\gg 1$ and minimal coupling, the following relations for
$H_{\rm max}(\beta,r)$ and $H_{\rm min}(\beta,r)$, the values for the location
of the minimum and maximum of the effective potential in the parameter space,
respectively, has been derived:

$\displaystyle H_{\rm max}$ $\displaystyle=\frac{\beta r_{\rm max}^{2}(r_{\rm
max}-M)}{4M^{2}(3M-r_{\rm max})}+\mathcal{O}(\beta^{-1})\,,$ (29)
$\displaystyle H_{\rm min}$ $\displaystyle=\frac{\beta r_{\rm
min}^{2}}{4M^{2}}+\frac{M}{2\beta(r_{\rm min}-2M)}+\mathcal{O}(\beta^{-3})\,,$
(30)

where $r_{\rm max}$ and $r_{\rm min}$ are the corresponding normalized radii.
From this the respective energies $E_{\rm max}$ and $E_{\rm min}$ for a
constant value of $H$ may be approximated by the expressions

$\displaystyle E_{\rm max}^{2}$ $\displaystyle\simeq\frac{\beta^{2}r_{\rm
max}(r_{\rm max}-2M)}{(3M-r_{\rm max})^{2}}\,,$ (31) $\displaystyle E_{\rm
min}^{2}$ $\displaystyle\simeq 1-\frac{2M}{r_{\rm
min}}+\frac{M^{4}}{\beta^{2}r_{\rm min}^{3}(r_{\rm min}-2M)}\,.$ (32)

Taking into account the form of the effective potential, this results in a
helical trajectory around the circular orbit with radius $r_{\rm min}$.
Without synchrotron radiation this helical movement would be stable, while due
to the energy loss the particle ”spirals down” to $r=r_{\rm min}$, such that
in the end the trajectory is nearly circular (i.e. it has descended to the
minimum of the effective potential).

For this situation [62] presents a well-motivated estimate for the upper limit
$\omega_{\rm max}$ of the frequency of the synchrotron radiation, namely

$\omega_{\rm max}\simeq 0.87\beta E_{\rm max}.$ (33)

When considering the non-minimal case, i.e. $\tilde{q}\neq 0$, one first has
to consider the structure of the effective potential, which can be done using
Fig. 27: For $\tilde{q}\leq 0$ we have a maximum and a minimum, however non-
minimal coupling considerably shifts them in the $r$-$H$ parameter space and,
furthermore, also regarding their actual value. This, in turn, results in a
upwards shift of $\omega_{\rm max}$, as can be seen in Fig. 28. This means
that there is, in principle, a possibility to limit $\tilde{q}$ from below by
measuring the synchrotron frequency emitted in the vicinity of a black hole.

A more interesting situation develops for $\tilde{q}>0$. As described above,
in this parameter region, depending on the values of $r$ and $H$, up to two
minima and maxima can exist. For this case we name the maximum closer to the
event horizon, at $r_{\rm max}^{\rm A}$, “maximum A” and the one farther from
it, at $r_{\rm max}^{\rm B}$, “maximum B”, i.e. $r_{\rm max}^{\rm A}<r_{\rm
max}^{\rm B}$ (and analogous to that for the minima at $r_{\rm min}^{\rm A}$
and $r_{\rm min}^{\rm B}$). When calculating $\omega_{\rm max}$ in this case,
one has to consider two different scenarios (as before, for a fixed $H$):
$V_{\rm eff}^{2}(r_{\rm max}^{\rm A})<V_{\rm eff}^{2}(r_{\rm max}^{\rm B})$
and $V_{\rm eff}^{2}(r_{\rm max}^{\rm A})\geq V_{\rm eff}^{2}(r_{\rm max}^{\rm
B})$. In the former scenario, in order to have a conservative limit, a
particle which eventually ends up in the minimum A can initially at most have
an energy corresponding to the maximum A, while a particle arriving at the
minimum B can start with an energy corresponding to the maximum B. On the
other hand, in the second scenario, particles finishing at the trajectories
corresponding to either of the two minima may start at the higher energy, i.e.
at the one corresponding to maximum A. This, together with the particular
values of the effective potential at the maxima, results in the rather
different behavior of $\omega_{\rm max}$ with $r_{\rm min}$ for
$\tilde{q}=0.5$ in Fig. 28, as for very small values of $r_{\rm min}$ close to
the event horizon the maximum frequency is dramatically higher than for the
case of $\tilde{q}\leq 0$, which might result in either a clear observational
feature or, in case it is not observed, a rather strict upper limit on
$\tilde{q}$.

##  7 Concluding Remarks

New astrophysical observations, such as the results obtained by the Event
Horizon Telescope and the study of super-massive black holes, make it possible
to investigate the structure of magnetic fields near the event horizon more
precisely [63, 64, 65, 66]. This topic is also of fundamental theoretical
interest since black hole horizons are associated with strongest gravitational
fields currently accessible to observations. It is precisely in this strong
gravity setting that one could expect to find some signatures of new effects
going beyond the standard description of relationship between gravity and
electromagnetism. As we have discussed, one of such effects – that can be
motivated from the fundamental field-theory considerations – is the non-
minimal coupling of gravity and electromagnetism. For instance, this type of
non-minimal coupling naturally comes as a consequence of the effect of QED
vacuum polarization on curved spacetime. We note that in the setting of
Schwarzschild spacetime near the horizon one could also expect other effects
beyond standard electrodynamics to manifest, such as some non-linear effects
[67]. With an aim to study this effect in a setting which is relevant for the
astrophysical application, we have considered the case of non-minimal coupling
between magnetic fields and gravity on Schwarzschild spacetime in the weak
magnetic field limit. We have thus followed, improved and considered some new
applications of the ideas recently proposed in [3, 10], with the primary
motivation to elaborate the physical consequences of the non-minimal coupling
between gravity and electromagnetism and to help bringing them closer to the
possibility of observational verification.

In the following we are going to summarize our findings and the conclusions
which may be drawn from them.

  * •

We have first reviewed the already known solutions for the case of minimally
coupled magnetic fields on Schwarzschild spacetime and then considered the
non-minimal coupling effects for both the asymptotically dipole and uniform
magnetic fields in detail, while discussing the changes induced by the non-
minimal coupling with respect to the minimal coupling case. We have also
provided a detailed discussion of the changes that come as a result of non-
minimal coupling in both of the mentioned magnetic field configurations. In
all of the discussed settings we have found that even the modest values of the
coupling parameter $\tilde{q}$ – which do not seem to appear to be in
contradiction with other observations – can cause a significant change in the
magnetic fields near the event horizon. These changes can have the character
of an amplification or a suppression, depending on the sign of the non-minimal
coupling parameter.

  * •

Apart from this quantitative change (which appears for both the asymptotically
dipole and the asymptotically uniform case), in the case of an asymptotically
uniform field a further qualitative change also appears: the direction of
magnetic fields changes near the horizon – for the positive value of
$\tilde{q}$ the direction alters near the equatorial plane, while for the
positive values the direction alters near the two poles.

  * •

We have also studied in detail how non-minimal coupling modifies the features
of the effective potential in the equatorial plane. The non-minimal coupling
can increase/decrease the effective potential, change the position of the
potential well and raise or lower the level of the maxima and minima. Due to
these reasons, under proper conditions non-minimal coupling can significantly
change the spectrum of the scattered particles, affect the innermost stable
circular orbit (isco), influence the energy spectrum of trapped particles and
change the center of mass energy during the acceleration of charged particles.
All these processes were discussed in detail qualitatively and quantitatively.
The most promising effect for manifesting the difference between minimal and
non-minimal coupling seems to be the study of the spectrum of scattered
particles for the case of a dipole field configuration.

  * •

We have also discussed various physical scenarios of observational interest in
which the non-minimal coupling effects could play a significant role. Such
scenarios and settings include primordial magnetic fields and primordial black
holes, studies of the consequences for the isco (which, for instance, are of
interest for discrimination between black holes and potential exotic objects
with similar properties like non-singular black holes, naked singularities,
etc.), potential effects of non-minimal coupling in jet formation and
synchrotron radiation.

  * •

For the latter we analyzed the potential change of the observational
signatures due to non-minimal coupling in more detail, such that in the future
it might be used to set limits on the value of $\tilde{q}$. In particular, for
non-minimal coupling the frequency of the radiation emitted at an intermediate
distance from the black hole might be moderately shifted up or down for
negative and positive values of $\tilde{q}$, respectively, compared to the
minimal case. On the other hand, close to the Schwarzschild radius a positive
$\tilde{q}$ might result in a dramatic increase of the (maximal) synchrotron
frequency, hence potentially giving the most significant observational
signature.

We believe that future theoretical research done in these topics – combined
with further improvements in experimental techniques for the study of magnetic
fields near the black hole horizon – can lead to the potential observation of
non-minimal coupling, or at least strongly constraining the value of the
coupling parameter. Such progress could be a significant step forward in our
understanding of strong gravitational fields and quantum effects on them, as
well as the connection between electromagnetism and gravity. In fact, if
observed, non-minimal coupling between electromagnetism and gravity would be
the first detection of a curved spacetime quantum process.

## Acknowledgments

The work of AS is supported by the Russian Science Foundation under grant no.
22-11-00063.

## Data Availability Statement

The data generated and/or analyzed during the current study are not publicly
available for legal/ethical reasons but are available from the corresponding
author on reasonable request. The data that support the findings of this study
are available upon reasonable request from the authors.

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