# A Robust Learning Rule for Soft-Bounded Memristive Synapses Competitive with
Supervised Learning in Standard Spiking Neural Networks

Thomas F. Tiotto14, Jelmer P. Borst2 and Niels A. Taatgen3  Groningen
Cognitive Systems and Materials Center
University of Groningen
Groningen, The Netherlands
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
4Corresponding author

###### Abstract

Memristive devices are a class of circuit elements that shows great promise as
future building block for brain-inspired computing. One influential view in
theoretical neuroscience sees the brain as a function-computing device: given
input signals, the brain applies a function in order to generate new internal
states and motor outputs. Therefore, being able to approximate functions is a
fundamental axiom to build upon for future brain research and to derive more
efficient computational machines. In this work we apply a novel supervised
learning algorithm - based on controlling niobium-doped strontium titanate
memristive synapses - to learning non-trivial multidimensional functions. By
implementing our method into the spiking neural network simulator Nengo, we
show that we are able to at least match the performance obtained when using
ideal, linear synapses and - in doing so - that this kind of memristive device
can be harnessed as computational substrate to move towards more efficient,
brain-inspired computing.

###### Index Terms:

Beyond CMOS, Neuromorphics, Memristors, Hebbian theory

##  I Introduction

Artificial neural networks (ANNs) are universal function approximators ([1]),
meaning that - given enough resources (neurons) - an ANN can potentially
approximate any and every function to an arbitrary degree of precision.
Philosophically, everything in Nature can be described as a function,
including the tasks the brain carries out; for example - alongside all others
- those relating to learning, memory, planning, emotions, and self-awareness.
Theoretical Neuroscience has also mostly adopted this view in regards to the
brain ([2]), defining and studying it as a device which applies functions to
its inputs in order to generate outcomes in the form of new internal states
and motor outputs.

There nowadays exist modelling tools that explicitly represent brain
functionality as mathematical functions with the most well-known and developed
probably being the spiking neural network simulator Nengo ([3]). Nengo
represents information using the principles of the Neural Engineering
Framework (NEF) ([4]), which prescribes how to encode and decode signals into
ensembles of spiking neurons and, based on the function being computed, sets
out an analytical method to determine the weights of optimal connection
matrices between these neuronal pools. Nengo also implements various learning
rules, with the MacNeil & Eliasmith general error-based learning rule
Prescribed Error Sensitivity (PES) ([5]) is commonly adopted for supervised
learning in classic neural network approaches. Another advantage of Nengo is
that is directly compiles to existing digital and mixed-signal neuromorphic
hardware, including Intel Loihi, SpiNNaker, FPGAs, and Braindrop.

Memristors are a novel class of device that has attracted great research
interest since its inception ([6]) and realisation ([7]) due to its capacity
to maintain a resistance state in absence of external stimuli. The fact that
the physical state of this device can be directly changed by applying voltage
pulses to the terminals allows its programming with a fraction of the power
needed for silicon transistors ([8]). Memristors have been applied to model
neurons, synapses, non-volatile random access memory, and many other more
specific applications ([9]).

One of the main reasons to consider a brain-inspired neuromorphic approach
that uses memristors for computation is because the efficiency improvement
that can be gained is prospected to go beyond anything achievable with
traditional algorithms running on silicon. Our present-day machines are
struggling to supply us with a sufficient balance between performance and
energy consumption ([10]). This imbalance is significant when considering the
environmental impact that our data centres have when confronted with ever more
massive amounts of big data, but is also crucial when the available energy
envelope is significantly limited, for example when computing at the extreme
edge. One reason for this relatively poor efficiency can be traced to the fact
that contemporary computers are based on the Von Neumann architecture, which
in itself entails inefficiency, given that memory and computation elements are
physically separate within the machine ([11]). Another reason - which follows
from the first - is that algorithms written for this computational
architecture seem to be inherently less capable of dealing with the kind of
uncertainty and massive quantities of data that result from interactions with
the real world ([12, 13]). In fact, one of the greatest strengths of ANNs is
that they do not need to be programmed in the traditional sense, but learn the
program/function directly from data.

When looking to advance beyond the Von Neumann architecture and Moore’s law,
it comes quite natural to look to the brain for inspiration as it is maybe the
most remarkable computational device we are aware of. Our brain consumes only
an estimated 20 W of power ([14]), compared to the more than 100 MW needed by
a single data centre. To put things into perspective: the difference in power
consumption is the same as if a golf ball were scaled to the size of a medium-
sized moon. Additionally, within this relatively tiny energy envelope the
brain is able to solve tasks that our best supercomputers still can’t approach
([15]). If we want to start to close the gap between our computers and the
brain, we need both new materials and novel computing paradigms: memristors
can give us the former, while we can start working towards the latter by
adopting a function-based approach as the brain’s and ANNs’.

We have already published a paper presenting a novel supervised learning
algorithm controlling the resistance of metal/niobium-doped strontium titanate
(Ni/Nb-doped SrTiO3) memristive synapses, which we named memristor PES (mPES),
and - as a proof of concept - showed that we could utilise it to learn two
very simple functions ($y=x$ and $y=x^{2}$) ([16]). In this work we extend our
methodology by applying mPES to learning non-trivial, non-linear functions of
varying dimensionality and also compare the results with the optimal analytic
implementation of the functions, as computed by Nengo using the NEF
principles. The functions we learn in this work are a lot more interesting
than the ones in our original mPES paper ([16]), as shall be made clear in
Section II. By implementing our method into Nengo, we show that we are able to
at least match the performance of PES; we do so by reproducing and extending
the benchmarks by Bekolay ([17]).

Artificial neural networks use perfectly linear synapses without considering
that this might not be the optimal choice, especially in a biologically-
plausible continuous learning context. In this context this kind of idealised
synapse might, as first noted by [18], actually lead to decreased learning
performance and memory capacity in the network. As shall become clear in the
main text, our memristors’ resistance dynamics follow a non-linear - power-law
- behaviour whose evolution slows as the resistance bounds are approached and
this matches the kind of device used by [19]. Thus, our current study aims at
exploring and confirming the important role that non-linear, soft-bounded
synaptic dynamics can play in learning and does so by using a model of a real
memristive device in a spiking neural network. One of the main values of our
study is to show that this kind of memristive device can be harnessed as
computational substrate to move towards more efficient, brain-inspired
computing.

The reader can now see how being able to approximate functions sets the
grounds for functional brain simulations and research, which will most
certainly be necessary to advance beyond our present-day computational
capabilities. Integrating analogue devices such as memristors into brain-
inspired models brings its own set of additional challenges but, given their
remarkable energy efficiency, this is quite certainly a valuable undertaking.

##  II Methods

In order to compare the performance of our mPES learning rule, to that of
standard PES, we reproduce and extend a methodology from existing literature
([17]).

The PES learning rule ([5]) is incorporated into Nengo and accomplishes online
error minimization by solving

$\Delta\omega_{ij}=\kappa\alpha_{j}e_{j}\boldsymbol{E}a_{i}$ (1)

where $\omega_{ij}$ is the weight of the connection between pre-synaptic
neuron $i$ and post-synaptic neuron $j$, $\kappa$ is the learning rate,
$\alpha_{j}$ and $e_{j}$ are NEF-specific parameters for neuron $j$,
$\boldsymbol{E}$ is the global $d$-dimensional error to minimize, and $a_{i}$
the activity of neuron $i$.

Memristive devices change their internal state and resistance in response to
voltages above a threshold. Here, we utilised simulated Ni/Nb-doped strontium
titanate (Ni/Nb-doped SrTiO3) devices, where resistive switching results from
changes occurring at the Ni/Nb:STO interface ([20]). To derive a model of the
device, we subjected it to a series of electrical measurements and found that
the power-law in Eq. (2) could explain the resistance change as a function of
the number $n$ of +0.1 V pulses applied to the device terminals

$\displaystyle R(n)$ $\displaystyle=R_{0}+R_{1}n^{c}$ (2)
$\displaystyle=200+2.3\times 10^{8}n^{-0.146}$

with $R(n)$ the device’s resistance in Ohm $\Omega$, $R_{0}$ and $R_{0}+R_{1}$
the minimum and maximum resistance, $n$ the number of voltage pulses supplied,
and $c$ a constant resulting from fitting. Fig. 1 shows measurements from the
real device and highlights how each successive +0.1 V voltage pulses brings
the device to a lower resistance state and how each one has a decreasing
effect. The devices can thus be characterised as soft-bound memristive
synapses, in line with those explored in existing literature ([19]).

Figure 1: Real memristor data showing the resistance $R(n)$ as a function of
the first ten +0.1 V pulses $n$ supplied to the device. Adapted from [16].

We do not model a decrease in the resistance due to a lack of experimental
data, but this does not impact our rule’s capacity to learn. It is important
to note that the initial resistance of each memristor is randomly initialised
around a resistance of $10^{8}\,\Omega$ and that each update resulting from a
pulse also presents a measure of stochasticity in order to better simulate the
physical uncertainties one has to deal with when using these kind of
materials. In order to simulate device-to-device variation and hardware noise
we introduce variation on both the initial resistance state of the devices and
on the magnitude of each update following a voltage pulse. To this end we
initialise the memristor resistances to a random high value in the range
$[10^{8}\,\Omega\pm 15\%]\,$ to reproduce the case (as measured in the real
devices) where the resistance is brought to a high value following the
application of multiple RESET voltage pulses. Secondly, we independently add
$15\%$ Gaussian noise to the $R_{0}$, $R_{1}$, and $c$ parameters in Eq. 2 at
the start of each simulation run. This has the effect of making every device
respond differently in response to the voltage pulses. We do not draw the
parameter values from a new distribution at each timestep because of
computational constraints. The full details on the device, measurements, and
modelling can be found in the Materials and Methods section of our original
mPES paper ([16]).

To be able to represent negative network weights - as resistance $R$ and its
reciprocal conductance $G$ are positive physical quantities - we define the
weight $\omega_{ij}$ of each synapse in the Nengo model as the scaled
difference between the normalised conductances of two memristors $M^{+}_{ij}$
and $M^{-}_{ij}$ as

$\displaystyle\omega_{ij}=\gamma\left[\left(\frac{G^{+}_{ij}-G^{+}_{0,ij}}{G^{+}_{1,ij}-G^{+}_{0,ij}}\right)-\left(\frac{G^{-}_{ij}-G^{-}_{0,ij}}{G^{-}_{1,ij}-G^{-}_{0,ij}}\right)\right]$
(3)

with $\gamma$ a gain factor (in all simulations in this work fixed at
$\gamma=10^{3}$), and $G^{\pm}_{ij}$, $G^{\pm}_{0,ij}$, and $G^{+}_{1,ij}$ the
present, minimum, and maximum conductance of $M^{\pm}$, respectively.

mPES is a novel learning rule that operates on the memristors’ conductances
$G^{\pm}_{ij}$ in Eq. (3) and is - essentially - a discretised version of PES.
Unlike PES, which operates on ideal weights, mPES needs to account for the
non-idealities and discrete changes - which decrease in magnitude with every
subsequent pulse - that result from applying voltage pulses to the memristors,
as can be seen in Fig. 1. Each memristor $M^{\pm}_{ij}$ receives a voltage
pulse to increase its conductance based on the sign of the
$V=e_{j}\boldsymbol{E}a_{i}$ term, defined analogously as in Eq. 1, which is a
matrix where every entry corresponds to a synapse in the Nengo model. The
local error $e_{j}\boldsymbol{E}$ represents how much the post-synaptic neuron
$j$ contributes to the global error $\boldsymbol{E}$, and $a_{i}$ is the
activity of the pre-synaptic neuron. PES also calculates this the local error,
but mPES skips the weight update if it is below a certain threshold because a
small local error has little to no effect in a real-valued weight update
setting but it can trigger a voltage pulse on a memristor and bring it to a
new discrete resistance state, thus having an outsized effect. If the sign of
$V_{ij}$ \- corresponding to the synapse between neurons $i$ and $j$ \- is
positive then the memristor $M^{+}_{ij}$ is pulsed. Each pulse increases the
conductance of $M^{+}_{ij}$ and this, as can be derived from Eq. 3,
strengthens the network weight $\omega_{ij}$ i.e., facilitates the synapse
between $i$ and $j$. A symmetric operation is applied to $M^{-}_{ij}$ when
$V_{ij}<0$ in order to weaken $\omega_{ij}$. The overall effect of this
procedure is that synapses whose neurons had a beneficial participation to the
error are facilitated, and those whose neurons had a negative effect are
depressed so as to decrease the probability of them playing a role in the
future. The full details can be found in the Materials and Methods section of
our original mPES paper ([16]).

In order to test the learning performance of mPES, we reproduced the
methodology described in “Learning nonlinear functions on vectors: examples
and predictions” ([17]) which was used to benchmark PES. This was done by
comparing the quality of multi-dimensional functions $f$ where the matrix
implementing the transformation was learned with PES to the case where the
matrix entries were computed analytically at the start of the simulation by
Nengo using the NEF (referred to simply as “NEF” from here onwards). In the
latter - NEF - case, the matrix implementing the function $f$ is not learned
and is not plastic, meaning that it does not change during a simulation run.
We extended this methodology by re-implementing it with the latest version of
Nengo at the time of writing (`Nengo 3.0`) and compared the outcome of using
our learning rule mPES to those when running PES and NEF.

The simple network topology shown in Fig. 2 is defined specifically to test
the learning rule: it consists of a noisy `input` signal $x$ projecting to a
`pre`-synaptic neuronal ensemble, a `post`-synaptic ensemble representing $y$
which is connected to `pre` via a plastic connection, a `ground truth`
ensemble representing the transformed input $f(x)$, and finally an `error`
ensemble which compares the activity in `post` and in `ground truth` as
$y-f(x)$. The input $x$ is an ideal white noise signal in multiple independent
dimensions, low-pass filtered by a 5 Hz cutoff. It is important to note the
the function $f$ is computed across the `input`-to-`ground truth` connection
analytically by the NEF; learning only happens on the `pre`-to-`post`
connection. The number of neurons in each neuronal ensemble and the simulation
run time are all altered depending on the specific function being tested, with
the specific values being reported in Table I. The neurons in each ensemble
are leaky integrate-and-fire whose parameters are unaltered from the Nengo
defaults.

We test learning by running mPES, PES, and NEF ten times each for each of the
functions $f$ in Table I, which are applied to the `input`-to-`ground truth`
connection. We limit ourselves to ten simulation runs because of computational
constraints and also to faithfully reproduce the methodology in ([17]). The
first function in Table I is interesting as an example of the simplest non-
linear function. The second tests if the linear combination of two non-
linearities can be learned with a one-dimensional error signal, corresponding
to a reinforcement learning setting. The third function’s inclusion is
motivated by it being a nearly minimal example of multidimensional-to-
multidimensional mapping. Circular convolution $\otimes$, of the kind of the
last two functions in Table I, is extremely important in many settings, but
especially so in Nengo as it is used to bind vectors together. One thing to
note is that an extra specialised neuronal ensemble `conv` is required between
`input` and `ground truth` in order to compute the circular convolution
between $x$’s components.

TABLE I: Functions $f$ under test. The central columns present the dimensionality $d$ of the signals represented in the main neuronal ensembles together with their number of neurons #. The penultimate reports the simulated time for each network run. The last column shows the error and 95 % confidence interval measured on the final testing block for mPES, PES, and NEF. Function $\boldsymbol{f}$ | Pre (d/#) | Error (d/#) | Post (d/#) | Sim. time | Error (mPES/PES/NEF)
---|---|---|---|---|---
$f\left(x_{1},x_{2}\right)=x_{1}\times x_{2}$ | 2-D / 200 | 1-D / 100 | 1-D / 200 | 50 s |  216 ($\pm 31$) / 227 ($\pm 40$) / 141 ($\pm 20$)
$f\left(x_{1},x_{2},x_{3},x_{4}\right)=x_{1}\times x_{2}+x_{3}\times x_{4}$ | 4-D / 400 | 1-D / 100 | 1-D / 400 | 100 s |  394 ($\pm 32$) / 446 ($\pm 60$) / 261 ($\pm 39$)
$f\left(x_{1},x_{2},x_{3}\right)=\left[x_{1}\times x_{2},x_{1}\times x_{3},x_{2}\times x_{3}\right]$ | 3-D / 300 | 3-D / 300 | 3-D / 300 | 100 s |  745 ($\pm 83$) / 773 ($\pm 92$) / 483 ($\pm 60$)
$f\left(x_{1},x_{2},x_{3},x_{4}\right)=\left[x_{1},x_{2}\right]\otimes\left[x_{3},x_{4}\right]$ | 4-D / 400 | 2-D / 200 | 2-D / 200 | 200 s |  771 ($\pm 70$) / 813 ($\pm 73$) / 710 ($\pm 63$)
$f\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\right)=\left[x_{1},x_{2},x_{3}\right]\otimes\left[x_{4},x_{5},x_{6}\right]$ | 6-D / 600 | 3-D / 300 | 3-D / 300 | 400 s |  1364 ($\pm 64$) / 1317 ($\pm 68$) / 1258 ($\pm 70$)

The mPES and PES learning rules use the information received from `error` to
act on the weights on the plastic connection between `pre` and `post`, with
the goal of making the value in `post` resemble that in `ground truth` as much
as possible. This results in the `pre`-to-`post` connection matrix being
progressively tuned to represent the transformation $f$ and thus for the value
$y$ in `post` to approximate that in `ground truth` i.e., $y\sim f(x)$. When
the NEF is benchmarked - in order to obtain a control “gold standard” to gauge
mPES and PES’s performance - the connection between `pre` and `post` is not
plastic but sees its weights pre-calculated in order to optimally represent
the function $f$. In this case, `error` is omitted from the network as $f(x)$
is already optimally being represented on the `pre`-to-`post` connection.

Compared to our original paper ([16]), we added the `ground truth` and `conv`
ensembles. This made the task more difficult as in the original model $f(x)$
was calculated analytically across the `input`-to-`error` connection while now
$f(x)$ is represented by the spiking in `ground truth`. This extra encoding
step leads to a loss of information, making the learning task somewhat harder
for mPES to solve.

Figure 2: Nengo network topology used to test the learning rule performance.
The signals $x$, $y$, and $f(x)$ are shown in correspondence to the neuronal
ensembles representing them. When $f=\otimes$ there is an extra conv ensemble
between input and ground truth. When the NEF is being benchmarked the error
ensemble is omitted.

The learning performance of mPES and PES are compared to the NEF baseline by
breaking up a simulation run into an equal number of 2.5 s learning and
testing blocks (for example, the first function in Table I sees the networks
being run for 50 s, giving a sequence of ten
$\|\textit{Learning}\|\textit{Testing}\|$ blocks). After each simulation run,
the values represented by `post` and `ground truth` during each testing block
are subtracted to give the absolute total error for that block. This choice
was made because the `error` ensemble is omitted when benchmarking the NEF,
but the calculated quantity would be analogous to that represented in `error`
during runtime in all cases.

##  III Results and Discussion

The results of our simulations comparing mPES to PES and NEF are shown in Fig.
3 and Fig. 4. In these plots the solid lines represent the accumulated error
at each testing block averaged over ten simulation runs and the shaded area
corresponds to the 95% confidence interval for this average value. The red
colour identifies the NEF control network, blue the control network with PES
learning, and green the network using our mPES learning rule. The $y$-axis
shows the average accumulated error during each testing block of the
simulation and is in arbitrary units; the $x$-axis shows the total length of
the simulation in seconds. Each data point is spaced 5 s apart, given that the
error is reported after a combination of 2.5 s learning and 2.5 s testing
blocks
($\|\textit{Learning}\|\textit{Testing}\|\ldots\|\textit{Learning}\|\textit{Testing}\|$).

Figure 3: Average accumulated learning error on each 2.5 s testing block for
variations on multiplication. Solid lines represent the accumulated error at
each testing block averaged over ten simulation runs and the shaded area
corresponds to the 95% confidence interval for this average value.
(a) $f\left(x_{1},x_{2}\right)=x_{1}\times x_{2}$
(b) $f\left(x_{1},x_{2},x_{3},x_{4}\right)=x_{1}\times x_{2}+x_{3}\times
x_{4}$
(c) $f\left(x_{1},x_{2},x_{3}\right)=\left[x_{1}\times x_{2},x_{1}\times
x_{3},x_{2}\times x_{3}\right]$

Figure 4: Average accumulated learning error on each 2.5 s testing block for
variations on circular convolution. Solid lines represent the accumulated
error at each testing block averaged over ten simulation runs and the shaded
area corresponds to the 95% confidence interval for this average value.
(a)
$f\left(x_{1},x_{2},x_{3},x_{4}\right)=\left[x_{1},x_{2}\right]\otimes\left[x_{3},x_{4}\right]$
(b)
$f\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\right)=\left[x_{1},x_{2},x_{3}\right]\otimes\left[x_{4},x_{5},x_{6}\right]$

One might expect the error in the NEF case to present no variation between
trials, given that the NEF analytically derives a matrix to optimally
represent the function computed across a connection. In reality, the actual
matrix entries in each simulation may differ based on the neuron parameters,
which are randomly initialised each time a Nengo simulation is run. The
fluctuations in NEF measured error within a simulation are due to the random
nature of the input data and to the transformation matrix better representing
some parts of the input space than others.

When learning the five different functions $f$ in Table I, mPES was able to
modulate the memristive synapses’ resistances which, when translated to
network weights, implemented $f$ as well or better than PES. In Fig. 3a, 3b,
3c, where the networks are learning variations on multiplication, mPES sees
the total error decrease slower than PES but the error measured on the final
testing block is within the confidence interval of PES in all three cases.
When the task is to learn 2-D and 3-D circular convolution of the input
components, as shown in Fig. 4a, 4b, mPES actually learns the function faster
than PES even though the final error is again comparable. That the performance
of mPES is in line to that of PES in all cases, can also be seen by looking at
the average error in the last testing block as reported in the rightmost
column of Table I. In the mPES and PES cases - where the connection weight
matrix is learned - the error at $t=0$ is due to the `pre`-to-`post` matrix
being randomly initialised and thus this data point does not reflect the
performance of the learning rules. As noted earlier, the NEF computes the
matrix entries analytically based on the specific - random - neuron parameters
of that simulation run; for this reason, the performance of the NEF network
also shows some variation between runs. Across the spectrum, as expected, the
performance of the NEF analytically-determined weight matrix is superior to
that of those obtained via online learning.

Thus, our mPES learning rule is able to match the learning performance of
standard PES, which is remarkable given the restriction imposed by having to
operate on non-ideal, stochastic items - as are the simulated memristors -
instead of real-valued, linear, continuous network weights. It is also notable
that mPES reaches this level of performance without having any information
about the magnitude of the updates happening on the underlying memristors;
given that the memristive devices’ resistance dynamics follow a power-law
subsequent voltage pulses have a monotonically decreasing effect, but mPES is
unaware of this. This leads us to speculate that mPES could also be applied to
systems based on different memristive synapses, not just to the Nb-doped
SrTiO3 ones utilised in this work, whose behaviour is depicted in Fig. 1.
Additionally the fact that we are still able to at least match the performance
of PES (and in the cases reported in Fig. 4 actually do better) supports the
finding by [19] that soft-bounded memristive synapses can enable improved
learning performance compared to linear synapses of similar resolution.

In this study we did not directly run ablation experiments to understand the
effect of failed devices on the network performance; however, in our original
paper ([16]) we experimented with increasing the variation in the parameters
and found that the learning performance of mPES degraded gracefully. Given
that the memristor model follows a power-law (Eq. 2) the effect of the
exponent $c$ is crucial to the behaviour of the simulated devices: higher
variation in $c$ leads an increasing number of devices becoming “stuck” in
either a high of a low resistance state, effectively rending them
unresponsive.

##  IV Conclusion

In this work we have modelled each weight in a spiking neural network by a
pair of simulated memristors and shown that a supervised learning rule we call
mPES, which operates on these devices’ resistances, can match the performance
of the standard general error-based learning rule PES ([5]). The performance
of our learning rule has been tested by reproducing and extending a
methodology from literature ([17]), where a Nengo spiking neural network was
used as basis for learning the five non-trivial multi-dimensional functions in
Table I.

Using memristors as basis for neural network weights (and for other ANN
elements), especially when paired with a biologically-plausible learning
algorithm, is one way of improving the energy efficiency of present-day
computing machines. Memristors are analogue devices and, as such, inherently
stochastic; this characteristic, if properly harnessed, could turn out to be
important to deal with the randomness present in all data resulting from real-
world interactions. The brain is continually learning from every input it
receives and soft-bounded memristive synapses of the kind used in this work
can help alleviate the issues of synapse hyper-specialisation and catastrophic
forgetting. Brittleness is one of the main downfalls of not just traditional
Von Neumann-based algorithms, but of machine learning models as a whole:
moving towards more robust, cognitive systems is taking one step further
towards the holy grail of artificial general intelligence.

While Nengo directly compiles to certain digital and mixed-signal neuromorphic
hardware, we are not currently able to directly implement our model on a
crossbar array of SrTiO3 memristors. Nonetheless, we believe that the main
value of this work is in showing that this kind of device can be harnessed as
computational substrate to move towards more efficient, brain-inspired
computing.

The brain is probably the most extraordinary computational device we know of,
but how it carries out its feats of intelligence is still mostly a mystery.
One way of understanding the brain is to view it as a function-computing
machine able - for example - to apply a function to the inputs received from
our retinae and decide that we are looking at a cat. Therefore, having a
memristor-based neuromorphic system which is able to learn to approximate non-
trivial functions - as the ones tested in this work - could prove to be a
valuable tool to start to functionally reproduce some of the tasks that the
brain seems to carry out so easily and that still elude our best computers and
learning models.

## Data Availability Statement

The experimental datasets generated during the current study are available
from the corresponding authors on request.

## Code Availability

A reproducible distribution of the code used and the experimental datasets
generated during in this study can be found on Code Ocean at
https://codeocean.com/capsule/4587296/tree.

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# Charged Particle Motion Near a Magnetized Black Hole: A Near-Horizon
Approximation

Noah P. Baker<EMAIL_ADDRESS>Theoretical Physics Institute, University of
Alberta, Edmonton, Alberta, Canada T6G 2E1  Valeri P. Frolov
<EMAIL_ADDRESS>Theoretical Physics Institute, University of Alberta,
Edmonton, Alberta, Canada T6G 2E1

###### Abstract

In this paper, the orbits of a charged particle near the event horizon of a
magnetized black hole are investigated. For a static black hole of mass $M$
immersed in a homogeneous magnetic field $B$, the dimensionless parameter
$b=eBGM/(mc^{4})$ controls the radius of the circular orbits and determines
the position of the innermost stable circular orbit (ISCO), where $m$ and $e$
are the mass and charge of the particle. For large values of the parameter
$b$, the ISCO radius can be very close to the gravitational radius. We
demonstrate that the properties of such orbits can be effectively and easily
found by using a properly constructed “near-horizon approximation”. In
particular, we show that the effective potential (which determines the
position of the orbit) can be written in a form which is invariant under
rescaling of the magnetic field, and as a result is universal in this sense.
We also demonstrate that in the near-horizon approximation, the particle
orbits are stationary worldlines in Minkowski spacetime. We use this property
to solve the equation describing slow changes in the distance of the particle
orbit from the horizon, which arise as a result of the electromagnetic field
radiated by the particle itself. This allows us to evaluate the life-time of
the particle before it reaches the ISCO and ultimately falls into the black
hole.

Alberta Thy 4-23

##  I Introduction

An isolated, non-rotating black hole does not have a magnetic field Ginzburg
and Ozernoi . This is a direct consequence of the famous no-hair theorem
Misner  _et al._ (1973). However, when a black hole is surrounded by an
accretion disk, a magnetic field generated by currents in the disk plasma may
exist. This field plays an important role in black hole astrophysics. For
example, the Blandford-Znejek mechanism, which provides an explanation behind
the formation of astrophysical jets around spinning supermassive black holes,
requires an accretion disk with a strong magnetic field around a rotating
black hole Blandford and Znajek (1977) (see also McKinney (2005) and
references therein).

In 2013, evidence of the existence of a magnetic field in the vicinity of the
black hole at the center of the Milky way was obtained Eatough  _et al._
(2013); Schwarzschild (2013). This was done by observing the Faraday rotation
of the linearly polarized radio waves emitted by the magnetar PSR J1745-2900,
which travelled close to the black hole on their way to the Earth. The angle
of Faraday rotation of a linearly polarized radio wave beam in magnetized
plasma is proportional to the integral of the product of the longitudinal
magnetic field and the density of free electrons over the ray’s path. It is
also proportional to the wavelength squared Thorne and Blandford (2017). This
makes it possible to obtain information about a regular magnetic field by
observing the Faraday rotation angle for different wavelengths. The radio
waves emitted by the magnetar allow one to measure the magnetic field at a
distance from the black hole about $10^{5}$ of its gravitational radius.
According to adopted models, the magnetic field increases inwardly, and the
estimated regular magnetic field near the black hole should be about $100G$
Eatough  _et al._ (2013); Schwarzschild (2013).

More recently, in 2021, polarization measurements by the Event Horizon
Telescope of the light emitted by the matter surrounding the black hole in the
galaxy M87 provided signatures of the magnetic field close to the edge of the
black hole Collaboration (2021). Based on these observations, the estimated
magnetic field strength was shown to be in the range $B\sim 1-30$G.
Estimations based on observations and theoretical models indicate that the
magnetic field near stellar mass black holes can in fact be much larger (see,
e.g., Piotrovich  _et al._ (2010, 2015)).

Let us emphasize that for realistic magnetic fields expected near black holes,
one can neglect their back-reaction on the spacetime geometry. On the other
hand, the effect of these fields on the orbits of charged particles near the
horizon might be very large. This happens because, in general, the
dimensionless quantity $e/(\sqrt{G}m)$ for a particle of charge $e$ and mass
$m$ is very large (e.g., for electrons it is of order $10^{16}$). This implies
that even close to the horizon of a magnetized black hole, the Lorentz force
acting on a charged particle can dramatically change its trajectory. We define
a dimensionless parameter which characterizes the strength of the magnetic
field, given by

$b={eBGM\over mc^{4}}\,.$ (1)

Here, $M$ is the mass of the black hole and $B$ is the characteristic value of
the magnetic field in its vicinity. The meaning of this parameter is as
follows. The surface gravity of a non-rotating black hole of mass $M$ is

$\kappa={c^{4}\over 4GM}\,,$ (2)

which is a red-shifted gravitational field strength calculated at the horizon.
Using $\kappa$, one can write the parameter $b$ in the form

$b={1\over 4}{eB\over m\kappa}\,.$ (3)

For a relativistic particle of charge $e$ and mass $m$ moving in a magnetic
field $B$, the quantity $eB$ is the value of the Lorentz force acting on the
particle, while $m\kappa$ characterizes the gravitational force. Relation (3)
shows that the dimensionless parameter $b$ is proportional to the ratio of the
Lorentz force and the gravitational one.

There exists a variety of publications discussing different aspects of
magnetized black holes and charged particle motion in their vicinity (see,
e.g., Prasanna and Vishveshwara (1978); Galtsov and Petukhov (1978); Sokolov
_et al._ (1978a, b); Aliev and Galtsov (1989); Aliev and Ozdemir (2002);
Komissarov (2004); Bičák  _et al._ (2006); Frolov and Shoom (2010); Al Zahrani
_et al._ (2011); Gralla  _et al._ (2016); Tursunov  _et al._ (2018); Zahrani
(2021); Chakraborty (2022); Zahrani (2022), along with the texts Frolov and
Novikov (1998); Frolov and Zelnikov (2011) and references therein).

One interesting observation is that the ISCO of a charged particle near a
magnetized black hole can be located much closer to the horizon than ISCO of a
neutral particle (see, e.g., Aliev and Ozdemir (2002); Frolov and Shoom
(2010)). In such a case, black holes (under special conditions) can “work” as
a powerful cosmic particle accelerators (see, e.g., Frolov (2012)).

The present paper is devoted to a detailed investigation of the near-horizon
orbits of charged particles in the vicinity of magnetized black holes. Our
starting point is a Schwarzschild black hole placed in a homogeneous magnetic
field. Such a magnetic field, deformed by the black hole’s gravity, can be
described by the electromagnetic vector potential, which is directly
proportional to the Killing vector generating rotations Wald (1974). We focus
on particle motion in the equatorial plane. Circular orbits of such a charged
particle are fixed points of the reduced Hamiltonian. For a particular
direction of motion, the Lorentz force acting on the particle is repulsive and
the particle trajectory may lie close to the horizon.

If the proper distance from the horizon $l$ is much smaller than the
gravitational radius $r_{g}$, the curvature effects and the orbit’s trajectory
bending may be neglected. Put simply, the idea of this paper is that, in such
a case, it is sufficient to perform calculations in flat spacetime for the
study of near-horizon trajectories. Namely, consider a near-horizon 3-volume
with proper length $L\ll r_{g}$ in both the tangent and orthogonal to the
horizon directions. The metric in this domain can be approximated by the
Rindler metric. We demonstrate that, by a proper rescaling, one may also
define a magnetic field which is static in the Rindler coordinates and which
generates the required near-horizon trajectories. In this near-horizon
approximation, the Hamiltonian describing the charged particle motion is
greatly simplified.

Rindler spacetime is nothing but Minkowski spacetime in specially chosen
(Rindler) coordinates. We demonstrate that in the near-horizon approximation,
circular orbits near magnetized black holes map onto so-called “stationary
orbits” in Minkowski spacetime. Such stationary curves are well studied and
classified (see, e.g., Letaw (1981)). Their characteristic property is that
their Frenet curvatures are constant. It should be emphasized that the
magnetic field which is static in the Rindler frame is time dependent in the
inertial Minkowski frame. This field is invariant under Rindler boosts and
translations in the direction parallel to the horizon. The charged particle’s
4-velocity coincided (up to a normalization constant) with a linear
combination of these two vectors.

It is of particular interest that in spite of the fact that we began with the
Killing magnetic field in the black hole geometry, the obtained magnetic field
close to the horizon and near the equatorial plane is in fact quite general in
the following sense: if the magnetic field respects the imposed symmetry and
is regular at the Rindler horizon, it is either constant (in Minkowski time)
and homogeneous, or it has the same leading asymptotic as the field obtained
in our procedure. This means that one may reasonably expect that our
conclusions regarding the properties of close to the horizon orbits may in
fact be valid for a more complicated and realistic structure of a regular
near-horizon magnetic field.

The paper is organized as follows. In sections II and III, we re-derive the
equations for circular orbits near static magnetized black holes and discuss
their properties. In section IV, we introduce the near-horizon approximation.
In section V, we apply this approximation in the of study near-horizon
trajectories of charged particles. In section VI, the dynamical evolution of
these orbits induced by the emission of electromagnetic radiation is studied.
We also evaluate the lifetime of such orbits. Section VII contains a brief
summary along with our discussion. In the appendix, we discuss the regular at
the Rindler horizon magnetic fields respecting the imposed symmetry
properties.

In this paper, we use the sign conventions adopted in Misner  _et al._ (1973)
and the Gaussian system of units.

##  II Charged particle motion near magnetized black holes

###  II.1 Charged Particle Motion

We express the worldline of a particle in the form $x^{\mu}=x^{\mu}(\lambda)$,
where $\lambda$ is some parameter specifying the position of the particle. To
describe charged particle motion in curved spacetime and in the presence of an
electromagnetic field, we begin with the following action111Let us note that
in our notation, the Lagrangian $L$ and Hamiltonian $H$ have dimensions
$[ML/T]$, which differs from the standard dimension of the energy,
$[ML^{2}/T^{2}]$. This is because the action $S$ is defined as the integral
over the proper length $d\lambda$ rather than the proper time $d\lambda/c$.
Since in what follows we shall be working with dimensionless quantities, this
difference in definitions of $L$ and $H$ is not important.

$\displaystyle S=\int d\lambda L\,,$ (4) $\displaystyle L={mc\over
2}\left[\eta^{-1}g_{\mu\nu}{dx^{\mu}\over d\lambda}{dx^{\nu}\over
d\lambda}-\eta\right]+{e\over c}A_{\mu}{dx^{\mu}\over d\lambda}\,.$

We denote by $m$ and $e$ the mass and electric charge of the particle, and by
$A_{\mu}$ the 4D electromagnetic field potential. The parameter
$\eta=\eta(\lambda)$ is a Lagrange multiplier which, under a change in
parametrization $\lambda\to\tilde{\lambda}(\lambda)$, transforms as

$\eta d\lambda=\tilde{\eta}d\tilde{\lambda}\,.$ (5)

The above relation ensures that the action $S$ is parametrization invariant. A
variation of the action with respect to $\eta$ yields the following constraint
equation

$g_{\mu\nu}{d{x}^{\mu}\over d\lambda}{d{x}^{\nu}\over d\lambda}+\eta^{2}=0\,.$
(6)

After variations of the action (LABEL:action), one can always set $\eta=1$.
For this gauge,

$g_{\mu\nu}u^{\mu}u^{\nu}=-1,\hskip 14.22636ptu^{\mu}={dx^{\mu}\over
d\lambda}\,.$ (7)

The corresponding parameter $\lambda$ has dimensions of length, and it
coincides with the proper time multiplied by $c$. From now on, we shall refer
to it as the proper time parameter and use this gauge.

Let us define the 4-momentum

$p_{\mu}={\partial L\over\partial{x}^{\mu}_{,\lambda}}=mcu_{\mu}+{e\over
c}A_{\mu}\,.$ (8)

Then the Hamiltonian $H=p_{\mu}u^{\mu}-L$ may be written in the form

$H={1\over 2mc}(\pi_{\mu}\pi^{\mu}+m^{2}c^{2})\,,$ (9)

where $\pi_{\mu}=p_{\mu}-{e\over c}A_{\mu}$. The constraint equation (7)
implies

$\pi_{\mu}\pi^{\mu}=-m^{2}c^{2}\,,$ (10)

and the Hamiltonian equations of motion are

${d{x}^{\mu}\over d\lambda}={\partial H\over\partial p_{\mu}},\hskip
14.22636pt{d{p}_{\mu}\over d\lambda}=-{\partial H\over\partial x^{\mu}}\,.$
(11)

One may check that these equations correctly reproduce the standard equations
for particle motion in curved spacetime and in the presence of an
electromagnetic field

$mc{Du^{\mu}\over D\lambda}={e\over c}F^{\mu\nu}u_{\nu}\,,$ (12)

where $F_{\mu\nu}=2A_{[\nu,\mu]}$ . We use the notation $D/D\lambda$ to denote
a covariant derivative along the particle worldline.

Let us note that the equations (11) are invariant under the following scale
trasformations

$\displaystyle{x}^{\mu}\to C_{1}{x}^{\mu},\hskip 5.69046pt\lambda\to
C_{1}\lambda\,,$ (13) $\displaystyle p_{\mu}\to C_{2}p_{\mu},\hskip
5.69046ptH\to C_{2}H\,,$

where $C_{1}$ and $C_{2}$ are constants.

###  II.2 Dimensionless Form of the Equations of Motion

It is more convenient to deal with equations written in dimensionless form.
The interval $ds^{2}$ has dimension of $[length]^{2}$. We choose $dx^{\mu}$ to
have dimensions of length, and the metric $g_{\mu\nu}$ to be dimensionless. We
will introduce a constant scale factor with dimensions of length and use it to
obtain dimensionless quantities. In what follows, we shall discuss particle
motion in the gravitational field of a static, spherically symmetric black
hole. In this case, a natural scale is its gravitational radius

$r_{g}={2GM\over c^{2}}\,.$ (14)

To distinguish the quantities before and after the rescaling, we use a tilde.
We thus have

$\displaystyle d\tilde{x}^{{\mu}}=r_{g}dx^{\mu},\hskip
5.69046pt\tilde{g}_{{\mu}{\nu}}=g_{\mu\nu}\,,$ (15) $\displaystyle
d\tilde{s}^{2}=\tilde{g}_{{\mu}{\nu}}d\tilde{x}^{{\mu}}d\tilde{x}^{{\nu}}=r_{g}^{2}ds^{2}\,,$
$\displaystyle ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,.$

We write the worldline of a particle in the form
$\tilde{x}^{{\mu}}=\tilde{x}^{{\mu}}(\tilde{\lambda})$, where
$\tilde{\lambda}$ is the “physical” (dimensional) proper-time parameter. Its
dimensionless form is given by $\lambda=\tilde{\lambda}/r_{g}$. For the
particle’s velocity ${\boldsymbol{u}}$, acceleration ${\boldsymbol{w}}$ and
“jerk” ${\boldsymbol{k}}$, one has

$\displaystyle\tilde{u}^{{\mu}}={d\tilde{x}^{{\mu}}\over
d\tilde{\lambda}}={dx^{{\mu}}\over d{\lambda}}=u^{\mu}\,,$ (16)
$\displaystyle\tilde{w}^{{\mu}}={D\tilde{u}^{{\mu}}\over
D\tilde{\lambda}}={1\over r_{g}}{Du^{{\mu}}\over D{\lambda}}={1\over
r_{g}}w^{\mu}\,,$ $\displaystyle\tilde{k}^{{\mu}}={D\tilde{w}^{{\mu}}\over
D\tilde{\lambda}}={1\over r_{g}^{2}}{Dw^{{\mu}}\over D{\lambda}}={1\over
r_{g}^{2}}k^{\mu}\,,$

and the equation of motion (12) may be written in the form

$w^{\mu}=f^{\mu},\hskip 14.22636ptf^{\mu}={er_{g}\over
mc^{2}}\tilde{F}^{{\mu}{\nu}}u_{\mu}\,.$ (17)

###  II.3 Motion Near Magnetized Black Holes

We write the dimensionless Schwarzschild metric in the form

$\displaystyle d{s}^{2}=-fd{t}^{2}+{d{r}^{2}\over f}+{r}^{2}d\omega^{2}\,,$
(18) $\displaystyle d\omega^{2}=d{\theta}^{2}+\sin^{2}{\theta}d{\phi}^{2}\,,$
$\displaystyle f=1-1/{r}\,,$

where $r=\tilde{r}/r_{g}$ is the dimensionless radius.

This metric has two commuting Killing vectors

${\boldsymbol{{\xi}}}={\partial\over\partial{t}},\hskip
14.22636pt{\boldsymbol{{\zeta}}}={\partial\over\partial\phi}$ (19)

which generate time translations and rotation about the symmetry axis,
respectively. The Killing equations

${\xi}_{(\mu;\nu)}={\zeta}_{(\mu;\nu)}=0$ (20)

imply that

${\xi}^{\mu}_{\ ;\mu}={\zeta}^{\mu}_{\ ;\mu}=0\,.$ (21)

Since the spacetime is Ricci flat, its Killing vectors obey the equations

${\xi}^{\mu\nu}_{\ \ ;\nu}={\zeta}^{\mu\nu}_{\ \ ;\nu}=0\,.$ (22)

Equations (21)–(22) reveal that the Killing vectors ${\boldsymbol{\xi}}$ and
${\boldsymbol{\zeta}}$ may be viewed as vector potentials of an
electromagnetic field ${\boldsymbol{{A}}}$ in the Lorentz gauge satisfying the
source-free Maxwell equations. For ${\boldsymbol{{\xi}}}$, one has a weakly
charged static black hole, while ${\boldsymbol{{\zeta}}}$ is the potential for
a weakly magnetized black hole.

In what follows, we focus on the latter case and write the corresponding
vector potential ${\boldsymbol{{A}}}$ in the form

${A}^{\mu}\partial_{\mu}={1\over 2}B\partial_{\phi}\,.$ (23)

One may check that the corresponding magnetic field at a distance far from the
black hole is homogeneous and directed along the axis of rotation
$\theta=0,\pi$. For this field, the equation of motion (12) may be written in
the following dimensionless form

$w^{\mu}=f^{\rho}$ (24)

where

$f_{\mu}=2b\zeta_{[\nu,\mu]}u^{\nu},\hskip 14.22636ptb={eBGM\over mc^{4}}\,.$
(25)

We denote

$a_{\mu}=b\zeta_{\mu}\,,$ (26)

so that one has $f_{\mu}=2a_{[\nu,\mu]}u^{\nu}$. Since the Killing vectors
commute, one has ${\cal L}_{{\xi}}a^{\mu}={\cal L}_{{\zeta}}a^{\mu}=0$. This
demonstrates that the magnetic field associated with the Killing vector
respects the spacetime symmetry.

As mentioned earlier, the magnetic field for the potential (23) is parallel to
the axis of symmetry $\theta=0,\pi$ at far distances, and its value is
constant and equal to $B$. We note that in a more realistic astrophysical
setup, the structure of the magnetic field interacting with the ionized plasma
of the accretion disk can be more complicated222 In the appendix, it is shown
that in the absence of a monopole magnetic charge, a regular at the horizon
magnetic field respecting the spacetime symmetry near the horizon and close to
the equatorial plane has the same leading form as the potential (26). In this
sense, using such a “toy model” is sufficient for the study of the motion of
charged particles near the horizon. .

To obtain the dimensionless Hamiltonian, we use the scaling transformations
(LABEL:CCCC) and set

$C_{1}=r_{g},\hskip 14.22636ptC_{2}=mc\,.$ (27)

After these transformations, the dimensionless Hamiltonian for a charged
particle moving in the equatorial plane $\theta=\pi/2$ in the magnetic field
(23) takes the form

${H}={1\over 2}\left[-{1\over f}p_{t}^{2}+fp_{r}^{2}+{1\over
r^{2}}(p_{\phi}-br^{2})^{2}+1\right]\,.$ (28)

One may check that the equations of motion for this Hamiltonian are identical
to (12). We write here only one of equations of motion, which we shall use
later

${d\phi\over d\lambda}={1\over r^{2}}(p_{\phi}-br^{2})\,.$ (29)

The constraint equation (10) gives ${H}=0$.

##  III Circular orbits in the equatorial plane

###  III.1 The Reduced Hamiltonian

Since the Hamiltonian $H$ does not depend on $t$ or $\phi$, the momenta
$p_{t}$ and $p_{\phi}$ are integrals of motion. Thus, one has

${d{p}_{t}\over d\lambda}=-\frac{\partial H}{\partial t}=0,\hskip
14.22636pt{d{p}_{\phi}\over d\lambda}=-\frac{\partial H}{\partial\phi}=0\,.$
(30)

In other words, if a particle begins its motion with some parameters $p_{t}$
and $p_{\phi}$, it will remain on the surface $p_{t}$=const and
$p_{\phi}$=const in the phase space. We denote these integrals of motion by
$p_{t}={\cal E}$ and $p_{\phi}=\ell$, and introduce a reduced Hamiltonian

$\displaystyle{\cal H}={1\over 2}fp_{r}^{2}+{\cal V}_{r}\,,$ (31)
$\displaystyle{\cal V}_{r}={1\over 2}\left[{1\over r^{2}}(\ell-
br^{2})^{2}+1-{{\cal E}^{2}\over f}\right]\,.$

The equations of motion become

$\displaystyle{dr\over d\lambda}={\partial{\cal H}\over dp_{r}}=fp_{r}\,,$
(32) $\displaystyle{dp_{r}\over d\lambda}=-{\partial{\cal H}\over dr}=-{1\over
2}f^{\prime}p_{r}^{2}-{\cal V}_{r}^{\prime}\,,$

where a prime denotes a derivative with respect to $r$.

In what follows, we focus solely on circular orbits. In the reduced 2D phase
space $(r,p_{r})$, they are represented by fixed points of the Hamiltonian
flux

${\partial{\cal H}\over\partial r}={\partial{\cal H}\over\partial p_{r}}=0\,.$
(33)

The constraint equation ${\cal H}=0$ at the fixed point implies ${\cal
V}_{r}=0$. This equation defines the energy ${\cal E}$. The second equation in
(LABEL:fixed) gives ${\cal V}_{r}^{\prime}=0$, an equation which determines
the radius $r$ of a circular orbit for given values of the integrals of
motion.

###  III.2 Stable and Innermost Stable Circular Orbits

Calculations give the following expressions for the second derivatives of the
Hamiltonian with respect to the canonical coordinates $r$ and $p_{r}$ at the
fixed point

${\partial^{2}{\cal H}\over\partial p_{r}^{2}}=f,\hskip
5.69046pt{\partial^{2}{\cal H}\over\partial r\partial p_{r}}=0,\hskip
5.69046pt{\partial^{2}{\cal H}\over\partial r^{2}}={\cal
V}_{r}^{\prime\prime}\,.$ (34)

The condition of stability for a circular orbit reads ${\cal
V}_{r}^{\prime\prime}>0$. For ${\cal V}_{r}^{\prime\prime}<0$, the orbits are
unstable. The “critical radius” where ${\cal V}_{r}^{\prime\prime}=0$
determines the position of the innermost stable circular orbit (ISCO).

One may express the equation ${\cal V}_{r}=0$ in the form

${\cal E}^{2}={\cal U},\hskip 14.22636pt{\cal U}=f\left[1+{1\over r^{2}}(\ell-
br^{2})^{2}\right]\,,$ (35)

and the two equations ${\cal V}_{r}^{\prime}=0$ and ${\cal
V}_{r}^{\prime\prime}=0$ then imply

${\cal U}^{\prime}=0,\hskip 14.22636pt{\cal U}^{\prime\prime}=0\,.$ (36)

The equation ${\cal U}^{\prime}=0$ defines the radius of the particle’s orbit
for given values of $\ell$ and $b$, and the equation ${\cal
U}^{\prime\prime}=0$ determines the radius of the ISCO. These equations
written in explicit form are

$\displaystyle 2b^{2}r^{5}-b^{2}r^{4}-2\ell
br^{2}+r^{2}-2\ell^{2}r+3\ell^{2}=0\,,$ (37) $\displaystyle
b^{2}r^{5}-r^{2}+2\ell br^{2}+3\ell^{2}r-6\ell^{2}=0\,.$ (38)

Using these equations, it is possible to show that for a large parameter $b$,
the ISCO for particles moving in the direction with positive $\ell$
(corresponding a repulsive Lorentz force) are close to the horizon. In the
leading order of $1/b$, one has Frolov and Shoom (2010)

$\displaystyle r_{ISCO}=1+{1\over\sqrt{3}b}\,,$ (39)
$\displaystyle\ell_{ISCO}=b+\sqrt{3}\,.$ (40)

###  III.3 Canonical Transformation

The kinetic part of the Hamiltonian (LABEL:RH) depends on the coordinate $r$.
It has the form

${1\over 2}{p_{r}^{2}\over\mu(r)}\,,$ (41)

where $\mu(r)=f^{-1}$ plays the role of mass and tends to infinity at the
horizon. It is possible to make a canonical transformation which transforms
the Hamiltonian to a more familiar form. Such new canonical variables are

$p=\sqrt{f}p_{r},\hskip 14.22636pt\rho=\int_{1}^{r}{dr\over\sqrt{f}}\,.$ (42)

In these variables, the Hamiltonian takes the form

$H={1\over 2}p^{2}+{\cal V}_{\rho},\hskip 14.22636pt{\cal V}(\rho)={\cal
V}_{r}(r(\rho))\,.$ (43)

We note that the new canonical coordinate $\rho$ is nothing but the
dimensionless proper distance to the horizon.

##  IV Near-horizon approximation

###  IV.1 Near-Horizon Geometry

Let us consider the motion of a charged particle close to the horizon of a
magnetized black hole with a large parameter $b$. The physical proper distance
$\tilde{l}$ of the ISCO orbit (39) to the horizon is $\sim r_{g}/\sqrt{b}$.
For sufficiently large $b$, its dimensionless version $l=\tilde{l}/r_{g}$,
which characterizes the bending of the orbit, is small. Near the horizon, the
curvature ${\cal R}$ is of the order of $r_{g}^{-2}$. Also in the regime of
large $b$, the quantity ${\cal R}l^{2}\sim 1/b$ characterizing the role of the
tidal forces is also small. This means that for our problem, one may neglect
the curvature and use a flat-spacetime approximation. Let us discuss this in
more detail.

We rewrite the metric (19) in the form

$ds^{2}=-fdt^{2}+{dr^{2}\over f}+{r^{2}\over\cosh^{2}z}(dy^{2}+dz^{2})\,,$
(44)

where we have made the following change of coordinates

$\phi=y,\hskip 14.22636pt\sin\theta={1\over\cosh z}\,.$ (45)

The second relation implies that

${dz\over d\theta}=-\cosh z\,.$ (46)

A point $p_{0}$ where $y=z=0$ corresponds to a point $\phi=0$ in the
equatorial plane. At this point, ${\boldsymbol{e}}_{y}=\partial_{y}$ is a unit
vector in the direction of $\phi$, while ${\boldsymbol{e}}_{z}=\partial_{z}$
is a unit vector in the direction of decreasing polar angle. The third unit
3-vector orthogonal to the vectors ${\boldsymbol{e}}_{y}$ and
${\boldsymbol{e}}_{z}$ is ${\boldsymbol{e}}_{r}=f^{1/2}\partial_{r}$.
Together, the vectors
$\\{{\boldsymbol{e}}_{y},{\boldsymbol{e}}_{z},{\boldsymbol{e}}_{r}\\}$ form a
right-handed orthonormal tetrad at $p_{0}$.

In the vicinity of $p_{0}$ and close to the horizon, one has

$r=1+q,\hskip 5.69046pt0<q\ll 1,\hskip 5.69046pt|y|\ll 1,\hskip
5.69046pt|z|\ll 1\,.$ (47)

In this approximation, one has

$ds^{2}=-qdt^{2}+{dq^{2}\over q}+dy^{2}+dz^{2}\,.$ (48)

One may check that this is indeed a metric of flat spacetime, where the
Rindler horizon is defined by the equation $q=0$. Let us now demonstrate how
equations (39)–(40) may be obtained in using this near-horizon approximation.

###  IV.2 Near-Horizon Orbits in the Equatorial Plane

We again consider the motion of a particle in the equatorial plane (where
$z=0$) and use a reduced 3D metric

$ds^{2}=-qdt^{2}+{dq^{2}\over q}+dy^{2}\,.$ (49)

We also use the following near-horizon approximation for the vector potential
of the magnetic field (26)

$a_{\mu}=b(1+2q)\delta_{\mu}^{y}\,.$ (50)

Let us note that we fix the values of the parameter $b$, which describes the
strength of the magnetic field.

It is convenient to introduce new variables

$\hat{q}=bq,\hskip 5.69046pt\hat{p}=p_{r}/b,\hskip 5.69046pt\hat{\cal
E}=\sqrt{b}{\cal E},\hskip 5.69046pt\hat{\ell}=\ell-b\,,$ (51)

in which the Hamiltonian (LABEL:RH) takes the form

$\displaystyle{\cal H}={b\over 2}\hat{q}\hat{p}^{2}+{1\over
2\hat{q}}\left[\hat{\cal U}-\hat{\cal E}^{2}\right]\,,$ (52)
$\displaystyle\hat{\cal U}=\hat{q}[(\hat{\ell}-2\hat{q})^{2}+1].$

At the fixed point of the Hamiltonian, where $\hat{p}=0$, the constraint
equation takes the form

$\hat{\cal E}^{2}=\hat{\cal U}\,.$ (53)

For a given $\hat{\ell}$, the coordinate $\hat{q}$ is determined by the
equation

${d\hat{\cal U}\over d\hat{q}}=0\,,$ (54)

while for the innermost stable orbit, one has

${d^{2}\hat{\cal U}\over d\hat{q}^{2}}=0\,.$ (55)

Explicit forms of equations (54) and (55) are given by

$\displaystyle 12\hat{q}^{2}-8\hat{q}\hat{\ell}+\hat{\ell}^{2}+1=0\,,$ (56)
$\displaystyle\hat{\ell}=3\hat{q}\,.$

Let us emphasize that both the potential $\hat{\cal U}$ and the equations
(LABEL:ppzz) written in the variables $(\hat{q},\hat{\ell})$ are invariant
under the rescaling of the magnetic field $b$, and in this sense they are
universal.

The first of these equations determines a coordinate distance $q=\hat{q}/b$ of
the trajectory of the charged particle from the horizon. One may also solve
this equation and find the corresponding integral of motion $\hat{\ell}$ for
the trajectory with a given value of $\hat{q}$. One has

$\hat{\ell}=4\hat{q}\pm\sqrt{4\hat{q}^{2}-1}\,.$ (57)

Figure 1: The parameter $\hat{\ell}$ as a function of $\hat{q}$ as defined by
equation (57). The lines $\gamma_{\pm}$ represent the two solutions with the
corresponding $\pm$ signs. These lines intersect at point “$1$”. The straight
line $\gamma_{0}$ is the separatice $\hat{\ell}=3\hat{q}$. Above this line,
$d^{2}\hat{\cal U}/d\hat{q}^{2}<0$, while below it one has $d^{2}\hat{\cal
U}/d\hat{q}^{2}>0$. Point “$0$” represents the innermost stable orbit. The
part of $\gamma_{-}$ to the right of point “$0$” represents stable orbits.

The two branches of this solution for both $\pm$ signs are plotted in Fig. 1,
given by the curves $\gamma_{\pm}$. These lines meet at point “$1$”, where
$(\hat{q}=1/2,\hat{\ell}=2)$. The line $\gamma_{0}$ is the separatice
determined by the equation $\hat{\ell}=3\hat{q}$. It divides the
$(\hat{q},\hat{\ell})$ plane into two parts. In the upper part of the plane
(above $\gamma_{0}$), $d^{2}\hat{\cal U}/d\hat{q}^{2}<0$, while in the lower
part of the plane (below $\gamma_{0}$), $d^{2}\hat{\cal U}/d\hat{q}^{2}>0$.
The critical point “$0$” represents the innermost stable orbit. The separatice
$\gamma_{0}$ intersects $\gamma_{-}$ at point “$0$”, where
$(\hat{q}=1/\sqrt{3},\hat{\ell}=\sqrt{3})$. The portion of $\gamma_{-}$ to the
right of this point represents stable orbits. At this point, the dimensionless
radius $r$ and the integral of motion $\ell$ are given by

$r=1+{1\over\sqrt{3}b},\hskip 14.22636pt\ell=b+\sqrt{3}\,.$ (58)

These relations correctly reproduce (39) in the regime of large $b$.

##  V Particle trajectories

###  V.1 Magnetic Field Near the Rindler Horizon

Let us make the following change of variables

$t=2\tau,\hskip 14.22636ptq={1\over 4}\rho^{2}\,.$ (59)

In these coordinates, the metric (48) takes the form

$ds^{2}=-\rho^{2}d\tau^{2}+d\rho^{2}+dy^{2}+dz^{2}\,,$ (60)

which is nothing but the standard Rindler form of a flat metric. A schematic
outlining the variables used in the near-horizon approximation is shown in
Fig. 2. If $(T,X,Y,Z)$ are Minkowski coordinates, then one has

$T=\rho\sinh(\tau),\hskip 5.69046ptX=\rho\cosh(\tau),\hskip
5.69046ptY=y,\hskip 5.69046ptZ=z\,.$ (61)

The 4-velocity of an observer at rest at $\rho$ in the Rindler frame is

$u_{0}^{\mu}={1\over\rho}\delta_{\rho}^{\mu}\,.$ (62)

The parameter $\tau$ is simply the dimensionless proper time as measured by an
observer at rest at $\rho=1$.

Figure 2: The near-horizon approximation. The 2D $(y,z)$-plane is tangent to a
unit sphere representing the event horizon of the black hole. The $\rho$-axis
is orthogonal to the $(y,z)$-coordinate plane. In the adopted near-horizon
approximation, the coordinate $\rho$ may be understood as the proper distance
from the horizon.

The dimensionless vector potential associated with the Killing vector
${\boldsymbol{\zeta}}$ written in $(\tau,r,y,z)$ coordinates is

$a_{\mu}=a_{y}\delta_{\mu}^{y},\hskip 14.22636pta_{y}=b{r^{2}\over(\cosh
z)^{2}}\,.$ (63)

Near the equatorial plane and close to horizon, where $\rho,y\ll 1$, the
leading term of the expansion of $a_{y}$ is of the form

$a_{y}=b+{b\over 2}(\rho^{2}-2z^{2})\,.$ (64)

The non-vanishing components of the field strength $F_{\mu\nu}=2a_{[\nu,\mu]}$
for the potential (64) are

$F_{\rho y}=b\rho,\hskip 14.22636ptF_{zy}=-2bz\,.$ (65)

Let us note that these expressions were obtained in using the Killing ansatz
(23) for the magnetic field. In the appendix we demonstrate that, in the
absence of a magnetic monopole contribution, expressions (64) and (65) are
nothing but the leading asymptotics for any regular static and $y-$independent
magnetic field in the near-horizon approximation.

On the plane at $z=0$, one has $F_{zy}=0$. The magnetic field vector in the
frame of an observer at rest in the Rindler frame is

$B_{\alpha}=-{1\over 2}e_{\alpha\beta\mu\nu}u_{0}^{\beta}F^{\mu\nu}\,,$ (66)

where $u_{0}^{\mu}$ is the 4-velocity of the observer (62) and
$e_{\alpha\beta\mu\nu}$ is the antisymmetric tensor, $e_{0123}=\rho$. The only
non-vanishing component of $B_{\mu}$ is

$B_{z}=b\rho\,.$ (67)

###  V.2 Stationary Worldlines

In Minkowski spacetime, there exists a special class of worldlines which are
called stationary Letaw (1981). Their characteristic property is that the
geodetic interval between any of its two points depends only on the proper
time interval. The stationary curves in Minkowski spacetime were described and
classified by Letaw Letaw (1981). He demonstrated that the stationary
worldlines are solutions of the Frenet equations when the curvature invariants
are constant. In this subsection, we demonstrate that the worldline of a
charged particle moving in the magnetic field (67) is a stationary curve.

For this purpose, let us consider a particle moving in $z=0$ plane and write
its worldline $X^{\mu}=X^{\mu}(\lambda)$ in Minkowski coordinates

$X^{\mu}=(\rho_{0}\sinh(\sigma\lambda),\rho_{0}\cosh(\sigma\lambda),V\lambda,0)\,.$
(68)

As earlier $\lambda$, is the dimensionless proper time parameter along the
trajectory. The particle’s 4-velocity is then

$u^{\mu}={dX^{\mu}\over
d\lambda}=(\rho_{0}\sigma\cosh(\sigma\lambda),\rho_{0}\sigma\sinh(\sigma\lambda),V,0)\,.$
(69)

The normalization condition ${\boldsymbol{u}}^{2}=-1$ implies

$\sigma={\sqrt{1+V^{2}}\over\rho_{0}}\,.$ (70)

The curve (68) is uniquely defined by two independent parameters: $\rho_{0}$
and $V$. One may check that the curve in the $(X-Y)$-plane representing the
trajectory of this particle is a catenary

$X=\rho_{0}\cosh\bigg{(}{\sqrt{1+V^{2}}\over\rho_{0}V}Y\bigg{)}.$ (71)

One may also easily find the four-acceleration ${w}^{\mu}=du^{\mu}/d\lambda$
and the “jerk” ${k}^{\mu}=dw^{\mu}/d\lambda$

$\displaystyle
w^{\mu}=\rho_{0}\sigma^{2}(\sinh(\sigma\lambda),\cosh(\sigma\lambda),0,0)\,,$
(72) $\displaystyle
k^{\mu}=\rho_{0}\sigma^{3}(\cosh(\sigma\lambda),\sinh(\sigma\lambda),0,0)\,,$

which obey the relations

$w^{\mu}w_{\mu}={(1+V^{2})^{2}\over\rho_{0}^{2}},\hskip
14.22636ptk^{\mu}k_{\mu}=-{(1+V^{2})^{3}\over\rho_{0}^{4}}\,.$ (73)

Let us note that both of these invariants are constant. This implies that the
Frenet coefficients calculated for these curves are constant as well, as they
should be in the stationary case.

The Rindler coordinates $(\tau,\rho,y,z)$ are related to the Minkowski
coordinates $(T,X,Y,Z)$ as follows

$\rho=\sqrt{X^{2}-T^{2}},\hskip 14.22636pt\tanh\tau={T\over X}\,.$ (74)

These relations give

$\displaystyle{\partial\tau\over\partial T}$
$\displaystyle={X\over\rho^{2}},\hskip 14.22636pt{\partial\tau\over\partial
X}=-{T\over\rho^{2}}\,,$ (75) $\displaystyle{\partial\rho\over\partial T}$
$\displaystyle=-{T\over\rho},\hskip 14.22636pt{\partial\rho\over\partial
X}={X\over\rho}\,.$

Using the relations

$u^{\tau}={\partial\tau\over\partial X^{\mu}}u^{\mu},\hskip
14.22636ptu^{\rho}={\partial\rho\over\partial X^{\mu}}u^{\mu},$ (76)

one may find $u^{\mu}$ in Rindler coordinates. One has

$u^{\mu}=(\sigma,0,V,0)\,.$ (77)

This relation shows that $d\tau/d\lambda=\sigma$.

Let us summarize. The particle with the worldline (68) in Minkowski
coordinates remains at a constant distance $\rho=\rho_{0}$ to the Rindler
horizon. For an observer at rest at this distance, the proper time $\tau_{0}$
is related to $\tau$ via $\tau_{0}=\rho_{0}\tau$. A velocity $v=dy/d\tau_{0}$
as measured by this observer is

$v={V\over\sqrt{1+V^{2}}}\,.$ (78)

The inverse relation determines $V$ as a function of the 3-velocity $v$

$V={v\over\sqrt{1-v^{2}}}\,.$ (79)

In a similar way, one finds the components of the vectors ${\boldsymbol{w}}$
and ${\boldsymbol{k}}$ in the Rindler frame

$\displaystyle w^{\mu}=(0,\rho_{0}\sigma^{2},0,0)\,,$ (80) $\displaystyle
k^{\mu}=(\sigma^{3},0,0,0)\,.$

The dimensionless Lorentz force (25) acting on the charged particle is

$f^{\mu}=bV\rho\delta_{\rho}^{\mu}\,.$ (81)

If the velocity $V$ is directed along the $y$ axis (such that $V$ is
positive), the Lorentz force is directed along the $\rho$ axis, i.e., away
from the horizon located at $\rho=0$. For motion with a fixed value
$\rho=\rho_{0}$, this repulsive force compensates the “fictitious” attractive
gravity force to the horizon in the Rindler space. This condition reads

$w^{\mu}=f^{\mu}\,.$ (82)

Substituting the expression for the 4-acceleration $w^{\mu}$ (LABEL:wwkk) into
(82), one finds

$\sigma^{2}=bV\,.$ (83)

Using the expression (70), one obtains

$b\rho_{0}^{2}={1+V^{2}\over V}\,.$ (84)

For a given value of $b$, this is the relation between the position of the
trajectory $\rho_{0}$ and the velocity $V$ of the particle. We have thus shown
that a stationary worldline given by (68) obeying (83) does indeed coincide
with the trajectory of a charged particle. In the classification of Letaw
Letaw (1981), such stationary curves belong to class (v).

Let us now show that (84) does in fact reproduce a solution of the first
equation in (LABEL:ppzz). Equation (29) allows one to write

$\hat{\ell}=V+2\hat{q}\,,$ (85)

where $\hat{q}=b\rho_{0}^{2}/4$. Substituting this expression into
(LABEL:ppzz) and solving the obtained equation, one finds

$\hat{q}={1+V^{2}\over 4V}\,.$ (86)

This equation is equivalent to (84). This means that the orbits which were
found by solving the dynamical equations of motion correctly reproduce the
results obtained by means of equation (LABEL:ppzz).

Relation (86) shows that $d\hat{q}/dV>0$ for $V>1$, where $\hat{q}>1/2$. This
means that when $\hat{q}$ decreases (and a point representing the state of the
system moves to the left along the lower branch of $\gamma_{-}$), the velocity
$V$ decreases as well. This behavior continues until the critical point
$\hat{q}=1/\sqrt{3}>1/2$, at which point the system becomes unstable. At the
critical point corresponding to the innermost stable orbit, one has

$V=V_{cr}={1\over\sqrt{3}},\hskip 5.69046ptv_{cr}={1\over 2},\hskip
5.69046pt\rho_{cr}={2\over 3^{1/4}\sqrt{b}}\,.$ (87)

###  V.3 Metric in a Co-Moving Frame

The equation of motion of the particle in the Rindler frame (60) frame is
given by

$\rho=\rho_{0},\hskip
14.22636pty=V\lambda={\rho_{0}V\over\sqrt{1+V^{2}}}\tau\,.$ (88)

To obtain the metric in a frame co-moving with the particle, it is sufficient
to make the following coordinate transformation (shifting to a reference frame
with velocity $V$ in the $y-$direction)

$y={\rho_{0}V\over\sqrt{1+V^{2}}}\tau+\bar{y}\,.$ (89)

In the coordinates $(\tau,\rho,\bar{y},z)$, the metric is

$\displaystyle ds^{2}$ $\displaystyle=-\left(\rho^{2}-\rho_{0}^{2}{V^{2}\over
1+V^{2}}\right)d\tau^{2}+{2\rho_{0}V\over\sqrt{1+V^{2}}}d\tau d\bar{y}$
$\displaystyle+d\rho^{2}+d\bar{y}^{2}+dz^{2}\,.$ (90)

We again emphasize that the coordinate $\tau$ in this expression is the proper
time for an observer at rest at $\rho=1$ in the Rindler frame.

For a charged particle moving in a magnetic field, the parameters $\rho_{0}$
and $V$ are not independent. Using the relation (84), one can express (89) as
follows

$y=\sqrt{V\over b}+\bar{y}\,,$ (91)

and the metric (V.3) in the co-moving frame takes the form

$\displaystyle ds^{2}=-(\rho^{2}-{V\over b})d\tau^{2}+2\sqrt{V\over b}d\tau
d\rho$ $\displaystyle+d\rho^{2}+d\bar{y}^{2}+dz^{2}\,.$ (92)

We note that this metric, associated with the moving particle, is stationary
and does not depend on $\tau$. This is a direct consequence of the fact that
its worldline is stationary.

###  V.4 Orbital Oscillations

So far, we have focused mainly on stationary orbits with fixed values of
$\rho=\rho_{0}$. Let us briefly discuss small perturbations of these orbits in
the radial direction. For this purpose, we use the Hamiltonian (43). In the
near-horizon approximation, it takes the form

$\displaystyle{\cal H}={1\over 2}p^{2}+{\cal V}_{\rho}\,,$ (93)
$\displaystyle{\cal V}_{\rho}={1\over 2}\left[(\hat{\ell}-{1\over
2}b\rho^{2})^{2}+1-{4{\cal E}^{2}\over\rho^{2}}\right]\,.$

Denote by $\rho_{0}$ the proper distance of the unperturbed orbit. Then
expanding the potential ${\cal V}_{\rho}$ near this point, one has

$\displaystyle{\cal H}={1\over 2}p^{2}+{\omega^{2}\over
2}(\rho-\rho_{0})^{2}\,,$ (94)
$\displaystyle\omega^{2}=b\left[2\sqrt{b^{2}\rho_{0}^{4}-4}-b\rho_{0}^{2}\right]\,.$

This expression demonstrates that a radial perturbation of the orbit results
in small oscillations around $\rho=\rho_{0}$. For a stable orbit, the
frequency $\omega$ is real, and it vanishes at the critical innermost stable
orbit. For $\rho_{0}<\rho_{cr}$, $\omega$ is imaginary, as expected for
unstable orbits.

##  VI Radiation friction

###  VI.1 Electromagnetic Radiation of a Charged Particle

An accelerated charged particle radiates electromagnetic waves, which reduces
its total energy and momentum. As a result, its motion is modified via an
effect called “radiation damping”. We assume that this effect is small, and
evaluate how it affects the particle’s motion near the horizon. For this
purpose, we use the approach developed in the previous sections. An
approximation of the radiation force when the effect is small is given by
Landau and Lifshitz (1980)

$\tilde{g}^{\mu}={2e^{2}\over
3c}\bigg{(}\tilde{k}^{\mu}+\tilde{u}^{\mu}\tilde{u}^{\nu}\tilde{k}_{\nu}\bigg{)}\,.$
(95)

In the presence of radiation, the force $\tilde{g}^{\mu}$ should be added to
the right-hand side of the equation of motion of the charged particle (12).
After making a transformation to dimensionless quantities, this equation takes
the form

$w^{\mu}=f^{\mu}+f_{R}^{\mu}\,,$ (96)

Here, $f^{\mu}$ is the dimensionless Lorentz force given by (81) and
$f_{R}^{\mu}$ is a radiation-friction force

$f_{R}^{\mu}={\epsilon\over b}g^{\mu},\hskip
14.22636ptg^{\mu}=k^{\mu}+u^{\mu}u^{\nu}k_{\nu}\,,$ (97)

where $\epsilon$ is a new dimensionless parameter

$\epsilon={e^{3}B\over 3m^{2}c^{4}}\,.$ (98)

In what follows, we assume that $\epsilon$ is small.

In order to study how the emitted radiation changes the distance $\rho$ of the
charged particle from the Rindler horizon, we consider the radiation force
$f_{R}^{\mu}$ as a perturbation. As such, we keep only terms up to first order
in $\epsilon$ in the equations of motion. Since this force is already of order
$\epsilon$, it is sufficient to substitute the unperturbed values of
${k}^{\mu}$ and $u^{\mu}$ into the expression for $g^{\mu}$. Using the
expression (LABEL:wwkk) for $k^{\mu}$ and (77) for $u^{\mu}$ in
$(\tau,\rho,y,z)$ coordinates, one obtains the following expression for
${f}_{R}^{\mu}$

${f}_{R}^{\mu}={\epsilon\over
b}(-{V^{2}(V^{2}+1)^{3/2}\over\rho^{3}},0,-{V(V^{2}+1)^{2}\over\rho^{2}},0)\,.$
(99)

One can see that the $\rho$ and $z$ components of this force vanish, while the
other components, $g^{\tau}$ and $g^{y}$, are negative. This implies that as a
result of the action of this force, the integrals of unperturbed motion$-$the
energy and the $y$-momentum$-$slowly decrease.

###  VI.2 Evolution of the Orbit

Let us now discuss how the the electromagnetic radiation of the charged
particle affects the orbit of a charged particle moving near the horizon. For
a given magnetic field $b$ and in the absence of radiation, the distance
$\rho$ of the particle from the horizon is a function of its velocity. If the
radiation is weak, the parameter $\rho$ as well as the velocity $V$ slowly
change in time. We choose the velocity $V$ as a parameter describing the state
of the system, and assume that $\rho$ is determined by it, i.e., that
$\rho=\rho(V)$. We also write

${dV\over d\lambda}=\epsilon f(V)\,,$ (100)

where $f(V)$ is a function which characterizes the rate of change of the
velocity due to the radiation. One has

${d\rho\over d\lambda}=\epsilon f{d\rho\over dV}\,.$ (101)

The 4-velocity of the particle moving along a worldline with a slowly changing
velocity $V$ is

$u^{\mu}=(\alpha,\epsilon f{d\rho\over dV},V,0)\,,$ (102)

and the standard normalization condition ${\boldsymbol{u}}^{2}=-1$ determines
the parameter $\alpha$

$\alpha={1\over\rho}\sqrt{1+V^{2}}\,.$ (103)

Here and later we neglect quantities of the second and higher order in
$\epsilon$.

Calculations give the 4-acceleration of the particle $w^{\mu}$ in Rindler
coordinates

$\displaystyle w^{\mu}=(w^{\tau},\alpha^{2}\rho,\epsilon f,0)\,,$ (104)
$\displaystyle w^{\tau}={\epsilon
f\over\rho^{2}\sqrt{1+V^{2}}}\left[V\rho+(1+V^{2}){d\rho\over dV}\right]\,.$

One may check that $u_{\mu}w^{\mu}=0$ as required. One also finds the
following expression for the dimensionless Lorentz force $f^{\mu}$

$f^{\mu}=(0,b\rho V,-\epsilon b\rho f(d\rho/dV),0)\,,$ (105)

while the friction force $f^{\mu}_{R}$ is given by (99).

Let

$J^{\mu}=w^{\mu}-f^{\mu}-f^{\mu}_{R}\,,$ (106)

then the equations of motion read $J^{\mu}=0$. The equation $J^{\rho}=0$
implies

$b\rho^{2}={1+V^{2}\over V}\,.$ (107)

This relation is identical to the relation (84). This means that, at least in
the leading order, the relation between $\rho$ and $V$ remains the same as in
the absence of radiation. The equation $J^{y}=0$ gives

$f=-{V(1+V^{2})^{2}\over b\rho^{2}[1+b\rho(d\rho/dV)]}\,.$ (108)

Substituting the expression for $\rho$ which follows from (107), one obtains

$f=-{2V^{4}(1+V^{2})\over 3V^{2}-1}\,.$ (109)

The equation $J^{z}=0$ is trivially satisfied. One can also check that the
last equation, $J^{\tau}=0$, is valid as a result of the relations (107) and
(109).

A solution of equation (100) for $V(\lambda)$ can be written in the following
parametric form

$\int{dV\over f}=\epsilon\lambda.$ (110)

Performing the integration, one gets

$\displaystyle\int{dV\over f}={\cal T}(V)\,,$ (111) $\displaystyle{\cal
T}(V)=2\arctan(V)-{1\over 6V^{3}}+{2\over V}\,.$

We define

${\cal T}_{c}={\cal T}(V_{c}=1/\sqrt{3})={\pi\over 3}+{3\sqrt{3}\over 2}\,,$
(112)

and denote

$T=T(V)={\cal T}_{c}-{\cal T}(V)\,.$ (113)

A plot of the function $T(V)$ is shown in Fig. 3. The quantity $T/\epsilon$ is
the dimensionless proper time parameter $\lambda$ required for a particle with
initial velocity $V$ to reach the innermost sable orbit.

Figure 3: The function $T=T(V)$.

Suppose a charged particle is initially in an orbit with a given parameter
$\rho$. Its velocity is then

$V=\frac{1}{2}\bigg{[}b\rho^{2}-\sqrt{b^{2}\rho^{4}-4}\bigg{]}\,.$ (114)

If one substitutes this expression into (113), one obtains the lifetime of the
charged particle moving near the horizon as a function of its initial
position. This function is shown in Fig. 4.

Figure 4: The lifetime $T$ as a function of
$\rho/\rho_{cr}=3^{1/4}\sqrt{b}\rho/2$.

###  VI.3 Motion at a Fixed Distance to the Horizon

As we saw in the previous section, a change in the velocity $V$ leads to a
change in the distance parameter $\rho$. Let us now briefly consider a
different problem. Namely, we assume that instead of a magnetic field
supporting the particle, there exists some other force which is independent of
the particle velocity $V$ yet fixes the distance of the particle to the
horizon, $\rho=\rho_{0}$. One then has (cf. Macdonald and Suen (1985))

${dV\over d\lambda}=-{2e^{2}\over 3mc^{2}r_{g}\rho_{0}^{2}}V(1+V^{2})^{2}\,.$
(115)

Using the physical proper time parameter $\tilde{\lambda}=r_{g}\lambda$ and
dimensional proper distance to the horizon $l=r_{g}\rho_{0}$, one may write
this relation in the form

${dV\over d\tilde{\lambda}}=-\gamma V(1+V^{2})^{2},\hskip
14.22636pt\gamma={2e^{2}\over 3mc^{2}l^{2}}\,.$ (116)

This equation can be easily integrated and its solution may be written in the
following parametric form

$\gamma\tilde{\lambda}={\cal F},\hskip 14.22636pt{\cal F}=-\ln V+{1\over
2}\ln(1+V^{2})-{1\over 2(1+V^{2})}\,.$ (117)

Using this relation, one can determine how the velocity $V$ changes with the
proper time parameter. The curve in Fig. 5 shows the dependence of the
velocity $V$ on the parameter $\gamma\tilde{\lambda}$.

Figure 5: The velocity $V$ as a function of $\gamma\tilde{\lambda}$ for
particle motion at a fixed $\rho=\rho_{0}$.

The function ${\cal F}(S)$ is monotonically decreasing. Its asymptotics,
$V=\infty$ at $\tilde{\lambda}=0$ and $V=0$ at $\tilde{\lambda}=\infty$, can
be easily obtained from (117). For small $V_{0}$, one has

$V=V_{0}\exp(-\gamma\tilde{\lambda})\,.$ (118)

Thus for a small initial velocity $V_{0}$, the velocity $V$ decays
exponentially with a characteristic time of decay given by $\gamma^{-1}$.

##  VII Discussion

In this paper, we have discussed the motion of charged particles near
magnetized, non-rotating black holes. In particular, we have studied particles
moving in the equatorial plane orthogonal to the magnetic field. We’ve focused
on orbits with a direction of motion corresponding to an outwardly directed
Lorentz force from the black hole horizon. When the magnetic field parameter
$b$ is large, such circular orbits are located close to the gravitational
radius $r_{g}=2GM/c^{2}$. In this regime curvature and trajectory bending
effects can be neglected and one can use the near-horizon approximation.

We considered a near-horizon spatial domain which extends from the horizon up
to the proper distance $L\ll r_{g}$, and assumed that its length in the
directions parallel to the horizon also have size $L$. In this domain, it is
shown that one can approximate the exact Schwarzschild geometry by the Rindler
metric. This defines a map from the near-horizon geometry to a corresponding
Minkowski spacetime. We demonstrated that it is possible to accompany this map
with a proper rescaling of the electromagnetic field, such that the near-
horizon trajectories of the charged particles map to stationary curves. These
timelike curves in Minkowski spacetime have remarkable properties. The
interval of a straight line connecting any two points on such a worldline
depends only on the proper time interval between these points. Stationary
curves are integral lines for a linear combination of the boost and rotation
Killing vectors, and their Frenet curvatures are constant Letaw (1981). In the
near-horizon approximation, the Hamiltonian of the charged particle is greatly
simplified.

As our model, we used a black hole immersed into a homogeneous at infinity
magnetic field. However, for the description of the near-horizon motion of
charged particles, only its near-horizon asymptotic is important. In fact, as
it is shown in the appendix, this near-horizon field is a general asymptotic
of _any_ regular at the horizon magnetic field respecting the imposed
symmetries.

Using the adopted near-horizon approximation, we solved the equations
describing the slow evolution of a circular orbit of the charged particle
induced by its electromagnetic radiation (see also Tursunov  _et al._ (2018)).
As a result of the radiation emitted by the particle, it loses both its
angular momentum and energy, and the radius of the orbit decreases until the
particle reaches the innermost stable orbit and falls into the horizon. We
calculated the corresponding lifetime for this process. Such a friction effect
on a particle moving near the horizon of a black hole and parallel to its
surface has a natural explanation in the framework of the membrane paradigm
Thorne  _et al._ (1986); Macdonald and Suen (1985). The motion of the particle
creates an electric current on the stretched horizon, and its Ohm’s power
dissipation is responsible for the friction force acting on the particle. It
is quite interesting that a similar effect is known to occur in “standard”
electrodynamics in flat spacetime. Namely, when a charged particle moves
parallel to the surface of a conducting material at some given distance, there
exists a force which decreases its velocity Boyer (1974).

The proper distance of the innermost stable orbit to the horizon depends on
the mass of the charged particle. Denote by $\rho_{e}$ and $\rho_{p}$ the
corresponding distances for the electron and proton then

$\rho_{p}-\rho_{e}=(\sqrt{m_{p}/m_{e}}-1)\rho_{e}\approx 42\rho_{e}\,.$ (119)

This implies that for a two component neutral gas of weakly interacting
protons and electrons there may exist a net electric charge associated with an
excess of electrons located in the ring of the equatorial plane between
$\rho_{p}$ and $\rho_{e}$. If one ejects such a neutral gas at some distance
from the horizon, the times required for the protons and electrons within it
to reach the innermost stable orbit are different. For steady accretion of the
neutral plasma into the black hole, this may result in the black hole becoming
charged.

For a non-rotating black hole, close to the horizon orbits for large values of
the parameter $b$ are “supported” by the magnetic field. For rapidly rotating
black holes, such orbits also exist as a result of the interaction of black
hole spin with the particle angular momentum. It would be interesting to
extend the near-horizon approximation adopted in this paper to the study the
near-horizon circular orbits in Kerr geometry.

## Acknowledgments

The authors thank the Natural Sciences and Engineering Research Council of
Canada and the Killam Trust for their financial support.

##  Appendix A Electromagnetic Field Near the Rindler Horizon

In this appendix, we discuss properties of the electromagnetic field near the
Rindler horizon. We write the Rindler metric in the form

$ds^{2}=-\rho^{2}d\tau^{2}+d\rho^{2}+dy^{2}+dz^{2}\,.$ (120)

The relations between the Rindler coordinates $(\tau,\rho,y,z)$ and the
Minkowski coordinates $(T,X,Y,Z)$ are given in (74). Relations (75) give the
partial derivatives of the Rindler coordinates $\tau$ and $\rho$ with respect
to $T$ and $X$.

We define two Killing vectors

$\xi^{\mu}\partial_{\mu}=\partial_{\tau},\hskip
14.22636pt\zeta^{\mu}\partial_{\mu}=\partial_{y}\,,$ (121)

and consider the electromagnetic field $A_{\mu}$ which respects these
symmetries

${\cal L}_{\xi}A_{\mu}={\cal L}_{\zeta}A_{\mu}=0\,.$ (122)

Here, ${\cal L}_{\xi}$ is the Lie derivative along a vector field $\xi$.
Relations (122) imply that

$A_{\mu}=A_{\mu}(\rho,z)\,.$ (123)

We impose the Lorenz gauge condition $A^{\mu}_{\ ;\mu}=0$, which for the
potential (123) in Rindler coordinates takes the form

${1\over\rho}\partial_{\rho}(\rho A_{\rho})+\partial_{z}A_{z}=0\,.$ (124)

Let us denote

$\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,$
(125) $\displaystyle J^{\mu}=F^{\mu\nu}_{\ \
;\nu}\equiv{1\over\sqrt{-g}}\partial_{\nu}\left(\sqrt{-g}F^{\mu\nu}\right)\,.$

Then the source-free Maxwell equations $J^{\rho}=J^{z}=0$ for the field (123)
are

$\partial_{z}F_{\rho z}=\partial_{\rho}(\rho F_{\rho z})=0\,.$ (126)

A solution of these equations is

$F_{\rho z}={C\over\rho}\,.$ (127)

The other two Maxwell equations, $J^{\tau}=J^{y}=0$, take the following form

$\displaystyle\rho\partial_{\rho}\left({1\over\rho}\partial_{\rho}A_{\tau}\right)+\partial^{2}_{z}A_{\tau}=0\,,$
(128)
$\displaystyle\partial^{2}_{\rho}A_{y}+{1\over\rho}\partial_{\rho}A_{y}+\partial^{2}_{z}A_{y}=0\,.$

Both of these equations allow a separation of variables and their general
solutions are

$\displaystyle A_{\tau}(\rho,z)=\int_{-\infty}^{\infty}d\omega e^{i\omega
z}\tilde{A}_{\tau}(\rho|\omega)\,,$ (129)
$\displaystyle\tilde{A}_{\tau}(\rho|\omega)=\rho\left[C^{1}_{\tau}(\omega)J_{1}(\omega\rho)+C^{2}_{\tau}(\omega)K_{1}(\omega\rho)\right]\,,$
$\displaystyle A_{y}(\rho,z)=\int_{-\infty}^{\infty}d\omega e^{i\omega
z}\tilde{A}_{y}(\rho|\omega)\,,$
$\displaystyle\tilde{A}_{y}(\rho|\omega)=C^{1}_{y}(\omega)J_{0}(\omega\rho)+C^{2}_{y}(\omega)K_{0}(\omega\rho)\,.$

Here, $J_{n}(x)$ and $K_{n}(x)$ are Bessel functions of the first and second
kind, respectively.

Near the Rindler horizon, i.e., for small $\rho\ll 1$, one has

$\displaystyle J_{0}(\omega\rho)\sim 1-{1\over 4}\omega^{2}\rho^{2},\hskip
14.22636ptJ_{0}(\omega\rho)\sim{1\over 2}\omega\rho\,,$ (130) $\displaystyle
K_{0}(\omega\rho)\sim-\ln(\omega\rho),\hskip
14.22636ptK_{1}(\omega\rho)\sim{1\over\omega\rho}\,.$

The electromagnetic field is regular at the Rindler horizon if its components
are finite and smooth functions of $(T,X,Y,Z)$ coordinates. For the field
(127), one has

$F_{TZ}=-{T\over\rho}F_{\rho z}=-{CT\over\rho^{2}},\hskip
5.69046ptF_{XZ}={CX\over\rho}F_{\rho z}={CX\over\rho^{2}}\,.$ (131)

Hence for a regular at the Rindler horizon solution (127), one should put
$C=0$ (such that $F_{\rho z}=0$).

For regularity of the field at the Rindler horizon, it is sufficient that the
components of its vector potential $A_{T}$, $A_{X}$, $A_{Y}$ and $A_{Z}$ are
finite and smooth functions of $(T,X,Y,Z)$. Using (LABEL:ASSIM), one can see
that for the solution $A_{y}$ given by (LABEL:SOL), one must put
$C^{2}_{y}(\omega)=0$. Using (75), one can also find the component $A_{\tau}$
of the vector potential in Cartesian $(T,X)$ coordinates

$A_{T}={X\over\rho^{2}}A_{\tau},\hskip
14.22636ptA_{X}=-{T\over\rho^{2}}A_{\tau}\,,$ (132)

and for a regular field one should put $C^{2}_{\tau}(\omega)=0$ in
(LABEL:SOL).

In the near-horizon approximation, we choose the coordinate $z$ such that
$z=0$ corresponds to the equatorial plane. One may use relations (LABEL:SOL)
to find the expression for the regular electromagnetic field in the domain
near the horizon and close to the equatorial plane, where $\rho\ll 1$ and
$z\ll 1$.

However, for $A_{y}$ this can be done directly. Let us denote

$P_{k}=P_{k}(\rho,z)\equiv\sum_{n=0}^{k}\sum_{i=0}^{n}C_{i,n-i}\rho^{i}z^{n-i}\,,$
(133)

and write the leading asymptotic of a regular near the horizon function
$A_{y}$ in the form $A_{y}=P_{2}(\rho,z)$. Substituting this expression in the
Maxwell equation $J^{y}=0$, one finds

$C_{10}=C_{11}=0,\hskip 14.22636ptC_{02}=-2C_{20}\,.$ (134)

Thus a regular at the horizon field $A_{y}$ has the form

$A_{y}={1\over 2}b(\rho^{2}-2z^{2})+b_{1}z={1\over
2}b(X^{2}-T^{2}-2Z^{2})+b_{1}Z\,.$ (135)

Here, we have set $C_{22}^{y}={1\over 2}b$ and $C_{10}=b_{1}$. We have also
set $C_{00}=0$ since this constant does not enter the expression for the field
strength. The final expression in (135) clearly shows that $A_{y}$ is regular
at the Rindler horizon. The field $F_{\mu\nu}$ for this potential has the
following components

$F_{\rho y}=b\rho,\hskip 14.22636ptF_{zy}=-2bz+b_{1}\,.$ (136)

The electric field in the Rindler frame for the potential $A_{y}$,

$E^{\mu}=F^{\mu\nu}u_{\nu}\equiv-\rho^{2}F^{\mu\tau},$ (137)

vanishes, while the magnetic field

$B^{\mu}={1\over 2}e^{\mu\nu\alpha\beta}u_{\nu}F_{\alpha\beta}\,,$ (138)

has the following non-vanishing components

$B^{\rho}=-2bz+b_{1},\hskip 14.22636ptB^{z}=b\rho\,.$ (139)

For $b=0$, this magnetic field is orthogonal to the horizon. Such a field in
the near-horizon region of the black hole exists if the black hole has a non-
vanishing magnetic monopole charge. In the absence of the monopole charge,
i.e., when $b_{1}=0$, the leading term of the near-horizon magnetic field has
a universal form which contains only one constant, $b$. In the case of a black
hole immersed in a homogeneous at infinity magnetic field, $b$ coincides with
its value at infinity.

A similar analysis can be done for the $A_{\tau}$ component of the potential,
which is responsible to the electric field. We write it in the form

$A_{\tau}=\rho^{2}{\cal A}(\rho,z)\,.$ (140)

The electromagnetic field for this potential is regular at the horizon when
${\cal A}$ is regular function of the Minkowski coordinates. The equation
$J^{\tau}=0$ implies that this function satisfies the equation

$\partial_{\rho}^{2}{\cal A}(\rho,z)+{3\over\rho}\partial_{\rho}{\cal
A}(\rho,z)+\partial_{z}^{2}{\cal A}(\rho,z)=0\,.$ (141)

We again use an asymptotic form of ${\cal A}$, namely ${\cal
A}=P_{2}(\rho,z)$, near the $z=0$ plane in the vicinity of the horizon. Then
equation (141) imposes the following conditions on the $C_{i,2-i}$

$C_{10}=C_{11}=0,\hskip 14.22636ptC_{02}=-4C_{20}\,.$ (142)

Thus

$A_{\tau}=\rho^{2}\left[C_{00}+C_{01}z+C_{20}\bigg{(}\rho^{2}-4z^{2}\bigg{)}\right],$
(143)

which is clearly regular with respect to Cartesian coordinates.

The resulting nonzero components of the field tensor are

$\displaystyle F_{\rho\tau}$
$\displaystyle=2\rho\bigg{[}C_{00}+C_{01}z+C_{20}\bigg{(}2\rho^{2}-4z^{2}\bigg{)}\bigg{]},$
(144) $\displaystyle F_{z\tau}$
$\displaystyle=\rho^{2}\bigg{[}C_{01}-8C_{20}z\bigg{]}\,.$

Using relations (75), one can show that

${\partial\tau\over\partial T}{\partial\rho\over\partial
X}-{\partial\tau\over\partial X}{\partial\rho\over\partial T}={1\over\rho}\,.$
(145)

Thus one has

$F_{TX}={1\over\rho}F_{\tau\rho}\,,$ (146)

and the leading term in the expression for $F_{\rho\tau}$ in (144) with
coefficient $C_{00}=E/2$ describes a constant electric field $\vec{E}$ in flat
spacetime directed along the $X$-axis.

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