11institutetext: CERN, Geneva, Switzerland

# RF measurements I: signal receiving techniques

F. Caspers

###### Abstract

For the characterization of components, systems and signals in the RF and
microwave range, several dedicated instruments are in use. In this paper the
fundamentals of the RF-signal sampling technique, which has found widespread
applications in ‘digital’ oscilloscopes and sampling scopes, are discussed.
The key element in these front-ends is the Schottky diode which can be used
either as an RF mixer or as a single sampler. The spectrum analyser has become
an absolutely indispensable tool for RF signal analysis. Here the front-end is
the RF mixer as the RF section of modern spectrum analysers has a rather
complex architecture. The reasons for this complexity and certain working
principles as well as limitations are discussed. In addition, an overview of
the development of scalar and vector signal analysers is given. For the
determination of the noise temperature of a one-port and the noise figure of a
two-port, basic concepts and relations are shown. A brief discussion of
commonly used noise measurement techniques concludes the paper.

##  0.1 Introduction

In the early days of RF engineering the available instrumentation for
measurements was rather limited. Besides elements acting on the heat developed
by RF power (bimetal contacts and resistors with very high temperature
coefficient) only point/contact diodes, and to some extent vacuum tubes, were
available as signal detectors. For several decades the slotted measurement
line [1] was the most used instrument for measuring impedances and complex
reflection coefficients. Around 1960 the tedious work with such coaxial and
waveguide measurement lines became considerably simplified with the
availability of the vector network analyser. At the same time the first
sampling oscilloscopes with 1 GHz bandwidth arrived on the market. This was
possible due to progress in solid-state (semiconductor) technology and
advances in microwave elements (microstrip lines). Reliable, stable, and
easily controllable microwave sources are the backbone of spectrum and network
analysers as well as sensitive (low noise) receivers. This paper will only
treat signal receiving devices such as spectrum analysers and oscilloscopes.
For an overview of network analysis tools see RF measurements II: network
analysis.

##  0.2 Basic elements and concepts

Before discussing several measurement devices, a brief overview of the most
important components in such devices and some basic concepts are presented.

###  0.2.1 Decibel

Since the unit dB is frequently used in RF engineering a short introduction
and definition of terms is given here. The decibel is the unit used to express
relative differences in signal power. It is expressed as the base 10 logarithm
of the ratio of the powers of two signals:

$P\text{[dB]}=10\cdot\text{log}(P/P_{0})\hskip 5.69046pt.$ (1)

It is also common to express the signal amplitude in dB. Since power is
proportional to the square of a signal’s amplitude, the voltage in dB is
expressed as follows:

$V\text{[dB]}=20\cdot\text{log}(V/V_{0})\hskip 5.69046pt.$ (2)

In Eqs. (1) and (2) $P_{0}$ and $V_{0}$ are the reference power and voltage,
respectively. A given value in dB is the same for power ratios as for voltage
ratios. Please note that there are no ‘power dB’ or ‘voltage dB’ as dB values
always express a ratio.

Conversely, the absolute power and voltage can be obtained from dB values by

$\displaystyle P=P_{0}\cdot 10^{\frac{P\text{[dB]}}{10}}\hskip 5.69046pt,$ (3)
$\displaystyle V=V_{0}\cdot 10^{\frac{V\text{[dB]}}{20}}\hskip 5.69046pt.$ (4)

Logarithms are useful as the unit of measurement because

  1. 1.

signal power tends to span several orders of magnitude and

  2. 2.

signal attenuation losses and gains can be expressed in terms of subtraction
and addition.

Table 1 helps to indicate the order of magnitude associated with dB.

Table 1: Overview of dB key values and their conversion into power and voltage ratios. | Power ratio | Voltage ratio
---|---|---
$-$20 dB | 0.01 | 0.1
$-$10 dB | 0.1 | 0.32
$-$3 dB | 0.50 | 0.71
$-$1 dB | 0.74 | 0.89
0 dB | 1 | 1
1 dB | 1.26 | 1.12
3 dB | 2.00 | 1.41
10 dB | 10 | 3.16
20 dB | 100 | 10
$n\cdot$10 dB | 10n | 10${}^{n\text{/2}}$

Frequently dB values are expressed using a special reference level and not SI
units. Strictly speaking, the reference value should be included in
parentheses when giving a dB value, e.g., +3 dB (1 W) indicates 3 dB at
$P_{0}$ = 1 watt, thus 2 W. However, it is more common to add some typical
reference values as letters after the unit, for instance, dBm defines dB using
a reference level of $P_{0}$ = 1 mW. Thus, 0 dBm correspond to $-$30 dBW,
where dBW indicates a reference level of $P_{0}$ = 1 W. Often a reference
impedance of 50 $\Omega$ is assumed. Other common units are

  * •

dBmV for the small voltages with $V_{0}$ = 1 mV and

  * •

dBmV/m for the electric field strength radiated from an antenna with reference
field strength $E_{0}$ = 1 mV/m

###  0.2.2 The RF diode

One of the most important elements inside all sophisticated measurement
devices is the fast RF diode or Schottky diode. The basic metal–semiconductor
junction has an intrinsically very fast switching time of well below a
picosecond, provided that the geometric size and hence the junction
capacitance of the diode is small enough. However, this unavoidable and
voltage dependent junction capacity will lead to limitations of the maximum
operating frequency.

The equivalent circuit of such a diode is depicted in Fig. 1 and

VRF inVideo outRF bypasscapacitordiode impedance50 Figure 1: The equivalent
circuit of a diode

an example of a commonly used Schottky diode can be seen in Fig. 2.

Figure 2: A commonly used Schottky diode. The RF input of this detector diode
is on the left and the video output on the right (figure courtesy Agilent)

One of the most important properties of any diode is its characteristic which
is the relation of current as a function of voltage. This relation is
described by the Richardson equation [2]:

$I=AA_{\text{RC}}T^{2}\text{exp}\left(-\frac{q\phi_{\text{B}}}{kT}\right)\left[\text{exp}\left(\frac{qV_{\text{J}}}{kT}\right)-\text{M}\right]\hskip
5.69046pt,$ (5)

where $A$ is the area in cm2, $A_{\text{RC}}$ the modified Richardson
constant, $k$ Boltzmann’s constant, $T$ the absolute temperature,
$\phi_{\text{B}}$ the barrier height in volts, $V_{\text{J}}$ the external
Voltage across the depletion layer, M the avalanche multiplication factor and
$I$ the diode current.

This relation is depicted graphically for two diodes in Fig. 3.

I50 $\mu$A/divTypicalLBSDTypicalSchottkyDiodeV50 mV/div Figure 3: Current as a
function of voltage for different diode types (LBSD = low barrier Schottky
diode)

As can be seen, the diode is not an ideal commutator (Fig. 4) for small
signals. Note that it is not possible to apply big signals, since this kind of
diode would burn out.

Threshold voltage VoltageCurrent Figure 4: The current–voltage relation of an
ideal commutator with threshold voltage

However, there exist rather large power versions of Schottky diodes which can
stand more than 9 kV and several 10 A but they are not suitable in microwave
applications due to their large junction capacity.

The Richardson equation can be roughly approximated by a simpler equation [2]:

$I=I_{\text{s}}\left[\text{exp}\left(\frac{V_{\text{J}}}{0.028}\right)-1\right]\hskip
5.69046pt.$ (6)

This approximation can be used to show that the RF rectification is linked to
the second derivation (curvature) of the diode characteristic.

If the DC current is held constant by a current regulator or a large resistor
assuming external DC bias111Most diodes do not need an external bias, since
they have a DC return self-bias., then the total junction current, including
RF is

$I=I_{0}=\text{i}_{0}\text{\,cos\,}\omega t$ (7)

and hence the current–voltage relation can be written as

$V_{\text{J}}=0.028\text{ln}\left(\frac{I_{\text{S}}+I_{0}+\text{icos\,}\omega
t}{I_{\text{S}}}\right)=0.028\text{ln}\left(\frac{I_{0}+I_{\text{S}}}{I_{\text{S}}}\right)+0.028\text{ln}\left(\frac{\text{icos\,}\omega
t}{I_{0}+I_{\text{S}}}\right)\hskip 5.69046pt.$ (8)

If the RF current $I$ is small enough, the second term can be approximated by
Taylor expansion:

$V_{\text{J}}\approx
0.028\text{ln}\left(\frac{I_{0}+I_{\text{S}}}{I_{\text{S}}}\right)+0.028\left[\frac{\text{icos\,}\omega
t}{I_{0}+I_{\text{S}}}-\frac{\text{i}^{2}\text{cos}^{2}\omega
t}{2(I_{0}+I_{\text{S}})^{2}}+\ldots\right]=V_{\text{DC}}+V_{\text{J}}\text{\,cos\,}\omega
t+\text{higherorderterms}$ (9)

With the identity cos${}^{2}=0.5$, the DC and the RF voltages are given by

$V_{\text{J}}=\frac{0.028}{I_{0}+I_{\text{S}}}\text{i}=R_{\text{S}}\text{i\hskip
5.69046ptand\hskip
5.69046pt}V_{\text{DC}}=0.028\text{ln}\left(1+\frac{I_{0}}{I_{\text{S}}}\right)-\frac{0.028^{2}}{4(I_{0}+I_{\text{S}})^{2}}=V_{0}-\frac{V_{\text{J}}^{2}}{0.112}\hskip
5.69046pt.$ (10)

The region where the output voltage is proportional to the input power is
called the square-law region (Fig. 5).

[algebraic=true,linecolor=olive]911-0.15*x*x+3.25*x-10.6[algebraic=true,linecolor=olive]911-0.15*x*x+3.25*x-10.6without
loadsquare law loadedLBSDLBSD0-10-20-30-40-500.0050.050.55.050500Input power
[dBm]Output power [mV] Figure 5: Relation between input power and output
voltage

In this region the input power is proportional to the square of the input
voltage and the output signal is proportional to the input power, hence the
name square-law region.

The transition between the linear region and the square-law region is
typically between $-$10 and $-$20 dB (Fig. 5).

There are fundamental limitations when using diodes as detectors. The output
signal of a diode (essentially DC or modulated DC if the RF is amplitude
modulated) does not contain a phase information. In addition, the sensitivity
of a diode restricts the input level range to about $-$60 dBm at best which is
not sufficient for many applications.

The minimum detectable power level of an RF diode is specified by the
‘tangential sensitivity’ which typically amounts to $-$50 to $-$55 dBm for 10
MHz video bandwidth at the detector output [3].

To avoid these limitations, another method of operating such diodes is needed.

###  0.2.3 Mixer

For the detection of very small RF signals a device that has a linear response
over the full range (from 0 dBm ( = 1mW) down to thermal noise = $-$174 dBm/Hz
= 4$\cdot$10-21 W/Hz) is preferred. An RF mixer provides these features using
1, 2, or 4 diodes in different configurations (Fig. 6). A mixer is essentially
a multiplier with a very high dynamic range implementing the function

$f_{1}(t)f_{2}(t)\text{\hskip 5.69046ptwith}f_{1}(t)=\text{RFsignal\hskip
5.69046ptand}f_{2}(t)=\text{LOsignal}\hskip 5.69046pt,$ (11)

or more explicitly for two signals with amplitude $a_{i}$ and frequency
$f_{i}$ ($i=1,2$):

$a_{1}\text{\,cos}(2\pi f_{1}t+\varphi)\cdot a_{2}\text{\,cos}(2\pi
f_{2}t)=\frac{1}{2}a_{1}a_{2}\left[\text{cos}((f_{1}+f_{2})t+\varphi)+\text{cos}((f_{1}-f_{2})t+\varphi)\right]\hskip
5.69046pt.$ (12)

Thus we obtain a response at the IF (intermediate frequency) port that is at
the sum and difference frequency of the LO (local oscillator $=f_{1}$) and RF
($=f_{2}$) signals.

Examples of different mixer configurations are shown in Fig. 6.

Figure 6: Examples of different mixer configurations

As can be seen from Fig. 6, the mixer uses diodes to multiply the two ingoing
signals. These diodes function as a switch, opening different circuits with
the frequency of the LO signal (Fig. 7).

[algebraic=true]00.60.1*sin(10.4719755*x)LO[algebraic=true]00.60.1*sin(10.4719755*x)
Figure 7: Two circuit configurations interchanging with the frequency of the
LO where the switches represent the diodes

The response of a mixer in time domain is depicted in Fig. 8.

[algebraic=true,linecolor=myred]06.283185sin(2*x)RFLO[algebraic=true,linewidth=0.01,linestyle=dashed]06.283185sin(2*x)[algebraic=true,linewidth=0.01,linestyle=dashed]06.283185sin(2*x
+
3.1415926)[algebraic=true,linecolor=orange]01sin(2*x)[algebraic=true,linecolor=orange]23sin(2*x)[algebraic=true,linecolor=orange]45sin(2*x)[algebraic=true,linecolor=orange]66.283185sin(2*x)[algebraic=true,linecolor=orange]12sin(2*x
+ 3.1415926)[algebraic=true,linecolor=orange]34sin(2*x +
3.1415926)[algebraic=true,linecolor=orange]56sin(2*x + 3.1415926)IF Figure 8:
Time domain response of a mixer

The output signal is always in the ‘linear range’ provided that the mixer is
not in saturation with respect to the RF input signal. Note that for the LO
signal the mixer should always be in saturation to make sure that the diodes
work as a nearly ideal switch. The phase of the RF signal is conserved in the
output signal available form the RF output.

###  0.2.4 Amplifier

A linear amplifier augments the input signal by a factor which is usually
indicated in decibel. The ratio between the output and the input signal is
called the transfer function and its magnitude—the voltage gain $G$—is
measured in dB and given as

$G[\text{dB}]=20\cdot\frac{\text{V}_{\text{RFout}}}{\text{V}_{\text{RFin}}}\text{\hskip
5.69046ptor\hskip
5.69046pt}\frac{\text{V}_{\text{RFout}}}{\text{V}_{\text{RFin}}}=20\cdot\text{log}G[\text{lin}]\hskip
5.69046pt.$ (13)

The circuit symbol of an amplifier is shown in Fig. 9 together with its
S-matrix.

12S = $\left(\begin{array}[]{cc}0&0\\\ G&0\end{array}\right)$ Figure 9:
Circuit symbol an S-matrix of an ideal amplifier

The bandwidth of an amplifier specifies the frequency range where it is
usually operated. This frequency range is defined by the $-$3 dB points222The
$-$3 dB points are the points left and right of a reference value (e.g., a
local maximum of a curve) that are 3 dB lower than the reference. with respect
to its maximum or nominal transmission gain.

In an ideal amplifier the output signal would be proportional to the input
signal. However, a real amplifier is nonlinear, such that for larger signals
the transfer characteristic deviates from its linear properties valid for
small signal amplification. When increasing the output power of an amplifier,
a point is reached where the small signal gain becomes reduced by 1 dB (Fig.
10).

Figure 10: Example for the 1dB compression point [4]

This output power level defines the 1 dB compression point, which is an
important measure of quality for any amplifier (low level as well as high
power).

The transfer characteristic of an amplifier can be described in terms which
are commonly used for RF engineering, i.e., the S-matrix (for further details
see the paper on S-matrices of this School). As implicitly contained in the
S-matrix, the amplitude and phase information of any spectral component are
preserved when passing through an ideal amplifier. For a real amplifier the
element $G=\text{S}_{21}$ (transmission from port 1 to port 2) is not a
constant but a complex function of frequency. Also the elements S11 and S22
are not 0 in reality.

###  0.2.5 Interception points of nonlinear devices

Important characteristics of nonlinear devices are the interception points.
Here only a brief overview will be given. For further information the reader
is referred to Ref. [4].

One of the most relevant interception points is the interception point of 3rd
order (IP3 point). Its importance derives from its straightforward
determination, plotting the input versus the output power in logarithmic scale
(Fig. 10). The IP3 point is usually not measured directly, but extrapolated
from measurement data at much smaller power levels in order to avoid overload
and damage of the device under test (DUT). If two signals
$(f_{1},f_{2}>f_{1})$ which are closely spaced by $\Delta f$ in frequency are
simultaneously applied to the DUT, the intermodulation products appear at +
$\Delta f$ above $f_{2}$ and at $-$ $\Delta f$ below $f_{1}$.

The transfer functions or weakly nonlinear devices can be approximated by
Taylor expansion. Using $n$ higher order terms on one hand and plotting them
together with an ideal linear device in logarithmic scale leads to two lines
with different slopes
($x^{n}\stackrel{{\scriptstyle\text{log}}}{{\rightarrow}}n\cdot\text{log}x$).
Their intersection point is the intercept point of $n$th order. These points
provide important information concerning the quality of nonlinear devices.

In this context, the aforementioned 1 dB compression point of an amplifier is
the intercept point of first order.

Similar characterization techniques can also be applied with mixers which with
respect to the LO signal cannot be considered a weakly nonlinear device.

###  0.2.6 The superheterodyne concept

The word superheterodyne is composed of three parts: super (Latin: over),
$\epsilon\tau\epsilon\rho\omega$ (hetero, Greek: different) and
$\delta\upsilon\nu\alpha\mu\iota\sigma$ (dynamis, Greek: force) and can be
translated as two forces superimposed333The direct translation (roughly) would
be: Another force becomes superimposed.. Different abbrevations exist for the
superheterodyne concept. In the US it is often referred to by the simple word
‘heterodyne’ and in Germany one can find the terms ‘super’ or ‘superhet’. The
‘weak’ incident signal is subjected to nonlinear superposition (i.e., mixing
or multiplication) with a ‘strong’ sine wave from a local oscillator. At the
mixer output we then get the sum and difference frequencies of the signal and
local oscillator. The LO signal can be tuned such that the output signal is
always at the same frequency or in a very narrow frequency band. Therefore a
fixed frequency bandpass with excellent transfer characteristics can be used
which is cheaper and easier than a variable bandpass with the same
performance. A well-known application of this principle is any simple radio
receiver (Fig. 11).

BPRF amplifierMixerLocal oscillator (often locked to a quarz)Bandpass filterIF
amplifierDemodulatorAudio amplifier Figure 11: Schematic drawing of a
superheterodyne receiver

##  0.3 Oscilloscope

An oscilloscope is typically used for acquisition, display, and measurement of
signals in time domain. The bandwidth of real-time oscilloscopes is limited in
most cases to 10 GHz. For higher bandwidth on repetitive signals the sampling
technique has been in use since about 1960. One of the many interesting
features of modern oscilloscopes is that they can change the sampling rate
through the sweep in a programmed manner. This can be very helpful for
detailed analysis in certain time windows. Typical sampling rates are between
a factor 2.5 and 4 of the maximum frequency (according to the Nyquist theorem
a real-time minimum sampling rate of twice the maximum frequency
$f_{\text{max}}$ is required).

Sequential sampling (Fig. 12) requires a pre-trigger (required to open the
sampling gate) and permits a non-real-time bandwidth of more than 100 GHz with
modern scopes.

[algebraic=true,linestyle=dashed]2.53.5-7.2*x*x*x+64.8*x*x-189*x+182
[algebraic=true,linestyle=dashed]2.53.57.2*x*x*x-64.8*x*x+189*x-182

Figure 12: Illustration of sequential sampling

Random sampling (rarely used these days, Fig. 13) was developed about 40 years
ago (around 1970) for the case where no pre-trigger was available and relying
on a strictly periodic signal to predict a pre-trigger from the measured
periodicity.

[algebraic=true,linestyle=dashed]2.53.5-7.2*x*x*x+64.8*x*x-189*x+182
[algebraic=true,linestyle=dashed]2.53.57.2*x*x*x-64.8*x*x+189*x-182

Figure 13: Illustration of random sampling

Sampling is discussed in more detail in the following. Consider a bandwidth-
limited time function s($t$) and its Fourier transform S($f$). The signal
s($t$) is sampled (multiplied) by a series of equidistant $\delta$-pulses
p($t$) [5]:

$\text{p}(t)=\sum^{+\infty}_{n=-\infty}{\delta(t-nT_{\text{s}})=III(t/T_{\text{s}})}$
(14)

where the symbol $III$ is derived from the Russian letter III and is
pronounced ‘sha’. It represents a series of $\delta$-pulses.

The sampled time functions s${}_{\text{s}}(t)$ is

$\displaystyle\text{s}_{\text{s}}(t)=\text{s}(t)\text{p}(t)=\text{s}(t)III(t/T_{\text{s}})$
$\displaystyle\text{S}_{\text{s}}(f)=\text{S}(f)*\frac{1}{T_{\text{s}}}III(T_{\text{s}}f)$
(15)
$\displaystyle\text{S}_{\text{s}}(f)=\frac{1}{T_{\text{s}}}\sum^{+\infty}_{n=-\infty}{\text{S}(f-mF)}\text{with}F=\frac{1}{T_{\text{s}}}\hskip
5.69046pt.$ (16)

Note that the spectrum is repeated periodically by the sampling process. For
proper reconstruction, one ensures that overlapping as in Fig. 14 does not
occur.

$\left|\text{S}(f)\right|$$\left|\text{S}_{\text{s}}(f)\right|$$f_{\text{g}}$–$f_{\text{g}}$+$f_{\text{g}}$–$\frac{1}{T_{\text{s}}}$+$\frac{1}{T_{\text{s}}}$$\frac{\text{s}(f)}{T_{\text{s}}}$
Figure 14: Periodically repeated component of the Fourier Transform of
s${}_{\text{s}}(t)$ [5]

If the spectra overlap as in Fig. 14 we have undersampling, the sampling rate
is too low. If big gaps occur between the spectra (Fig. 15) we have
oversampling, the sampling rate is too high.

$f$LOW PASS
H${}_{\text{LP}}(f)$$\left|\text{S}_{\text{s}}(f)\right|$–$f_{\text{g}}$+$f_{\text{g}}$–$1/T_{\text{s}}$+$1/T_{\text{s}}$
Figure 15: Reconstruction of S$(f)$ via ideal lowpass from S${}_{\text{s}}(f)$
(slightly oversampled)

But this scheme applies in most cases. In the limit we arrive at a Nyquist
rate of $1/T_{\text{s}}=2f_{\text{g}}=F$.

The rules mentioned above are of great importance for all ‘digital’
oscilloscopes. The performance (conversion time, resolution) of the input ADC
(analog–digital converter) is the key element for single-shot rise time. With
several ADCs in time-multiplex one obtains these days 8-bit vertical
resolution at 20 GSa/s = 10 GHz bandwidth.

Another way to look at the sampling theorem (Nyquist) is to consider the
sampling gate as a harmonic mixer (Fig. 16).

[algebraic=true]00.60.1*sin(10.4719755*x)s($t$)$R_{i}$q($t$)$C_{\text{S}}$$R_{\text{L}}$

Figure 16: Sampling gate as harmonic mixer; C${}_{\text{s}}$ = sampling
capacitor [6]

This is basically a nonlinear element (e.g., a diode) that gives product terms
of two signals superimposed on its nonlinear characteristics.

The switch in Fig. 16 may be considered as a periodically varying resistor
R($t$) actuated by q($t$). If q($t$) is not exactly a $\delta$-function then
the higher harmonics decrease with $f$ and the spectral density becomes
smaller at high frequencies.

For periodic signals one may apply a special sampling scheme. With each signal
event the sampling time is moved by a small fraction $\Delta t$ along the
signal to be measured (Fig. 17).

[algebraic=true]0.451.5sin(6*x + 3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true,linestyle=dashed]4.5150.6*sin(0.6*x + 3.14192)$\Delta
t$2 $\Delta t$3 $\Delta t$4 $\Delta t$ Figure 17: Signal reconstruction with
sampling shift by $\Delta t$ per pulse [7]

The highest possible signal frequency for this sequential sampling is linked
to the width of the sampling pulse. This sampling or gating pulse should be as
short as possible otherwise signal averaging during the ‘gate-open’ period
would take place.

The sampling pulse is often generated by step-recovery diodes (snap-off
diodes) which change their conductivity very rapidly between the conducting
and non-conducting state. The actual switch (Schottky diode) becomes
conductive during the gate pulse and charges a capacitor (sample and hold
circuit) but not to the full signal voltage. Assuming a time constant
$R_{i}C_{\text{s}}$ much bigger than the ‘open’ time of the sampling gate, we
obtain approximately (Fig. 18)

$i_{\text{c}}(t)=\frac{\text{s}(t)}{R_{i}+R_{\text{d}}(t)}\hskip 5.69046pt.$
(17)

[algebraic=true]00.60.1*sin(10.4719755*x)s($t$)$R_{i}$q($t$)$C_{\text{S}}$u${}_{\text{C}}(t)$[algebraic=true]00.60.1*sin(10.4719755*x)s($t$)$R_{\text{d}}(t)$$C_{\text{S}}$$R_{i}$

Figure 18: Equivalent circuit for the sample-and-hold element [6]

After the sampling process we have [6]

$u_{\text{c}}(t)=\frac{1}{C_{\text{s}}}\int^{+\infty}_{-\infty}i_{\text{c}}(t)\text{d}t=\int^{+\infty}_{-\infty}\text{s}(t)\frac{1}{C_{\text{s}}(R_{i}+R_{\text{d}}(t))}\hskip
5.69046pt,$ (18)

with the control signal for the Schottky diode being

$\text{q}(t)=\frac{1}{C_{\text{s}}(R_{i}+R_{\text{d}}(t))}\hskip 5.69046pt.$
(19)

The control or switching signal is moved by $\tau$ or n$\Delta t$ (Fig. 21)
with respect to the signal to be sampled s($t$):

$u_{\text{c}}(\tau)=\int^{+\infty}_{-\infty}{\text{s}(t)\text{q}(t-\tau)\text{d}t}\hskip
5.69046pt.$ (20)

Note that the time constant $R_{i}C_{\text{s}}$ is much bigger than the length
of q$(t)$. $C_{\text{s}}$ is only charged to a fraction of s$(t)$ (Fig. 19).

tA [V]s($t$)q($t-\tau$) Figure 19: Sampling with finite-width sampling pulse

The sampling efficiency $\eta$ is defined as

$\eta=\frac{u_{\text{c}}(\tau)}{\text{s}(\tau)}\hskip 5.69046pt.$ (21)

In order to circumvent the problem of poor sampling efficiency a feedback loop
technique (integrator) can be used. This integrator amplifies the voltage step
on the sampling capacitor, after the sampling gate is closed, exactly by a
factor $1/\eta$. If the sampling gate has not moved with respect to the
trigger, the sampling capacitor is already charged to the correct voltage
u${}_{c}(\tau)$ and there is no change. Otherwise the change in uc just
amounts to the change in signal voltage.

The sampling gate is interesting from a technological point of view. As
aperture times (Fig. 19) may be of the order of 10 ps, MIC (Microwave
Integrated Circuit) technology has been used for many years. Today, the latest
generation of sampling heads (50 GHz) is even one step further with MMIC
(Monolithic Microwave Integrated Circuits) technology.

In MIC technology the sampling pulse is applied to a slotline in the ground-
plane metallization of a microstrip substrate (Fig. 20).

Figure 20: Sampling circuits [7]

This slot line has a length of some 10 mm and is shorted at both ends. With a
voltage across the slotline the fast Schottky diodes open and connect the
microstrip line via a through hole to the sampling capacitor $C_{\text{s}}$.
Owing to the particular topology of the circuit the signal line (microstrip)
is decoupled from the sampling pulse line over a wide frequency range (Fig.
20).

To move the sampling pulse by $\Delta t$ for each event requires a pre-trigger
(several 10 ns ahead), to start a fast-ramp generator. The intersection
(comparator) of the ramp generator output with a staircase-like reference
voltage defines the sampling time and $\Delta t$ (Fig. 21).

$\Delta t$2 $\Delta t$3 $\Delta t$4 $\Delta t$5 $\Delta t$FASTSWEEPREF Figure
21: Timing of sampling pulses [7]

The delay required for the pre-trigger has been a significant problem for many
applications, since it may be as large as 70 ns on certain (older)
instruments. A 70 ns delay-line leads to considerable signal distortions
especially for the high-frequency components.

To avoid the delay for the pre-trigger a technique named ‘random sampling’ was
developed about 45 years ago. It requires a strictly periodic signal rather
than just the repetitive one for sequential sampling (Fig. 21). By measuring
the (constant) repetition frequency of this strictly periodic signal, a
prediction of the next pulse arrival time can be given in order to generate a
trigger. Today there is little interest in random sampling, as pre-trigger
delays are drastically reduced (12 ns). There are also problems with jitter,
and random sampling needs repetition rates of serveral kHz [7].

Features of modern sampling scopes are summarized in Table 2.

Table 2: Features of modern sampling scopes Rise time: 7 ps |  $\approx$ 50 GHz
---|---
Jitter |  $\approx$ 1.5 ps
Static operation possible, no minimum repetition rate required. |
Optical sampling (mode-locked laser), 1 ps rise time |  $\approx$ 350 GHz

##  0.4 Spectrum analyser

Radio-frequency spectrum analysers can be found in virtually every control-
room of a modern particle accelerator. They are used for many aspects of beam
diagnostics including Schottky signal acquisition and RF observation. A
spectrum analyser is in principle very similar to a common superheterodyne
broadcast receiver, except for the requirements of choice of functions and
change of parameters. It sweeps automatically through a specified frequency
range which corresponds to an automatic turning of the nob on a radio. The
signal is then displayed in the amplitude/frequency plane. Thirty years ago,
instruments were set manually and had some sort of analog or CRT (cathode ray
tube) display. Nowadays, with the availability of cheap and powerful digital
electronics for control and data processing, nearly all instruments can be
remotely controlled. The microprocessor permits fast and reliable setting of
the instrument and reading of the measured values. Extensive data treatment
for error correction, complex calibration routines, and self tests are a great
improvement. However, the user of such a sophisticated system may not always
be aware what is really going on in the analog section before all data are
digitized. The basis of these analog sections are discussed now.

In general there are two types of spectrum analyser:

  * •

Scalar spectrum analysers (SA) and

  * •

Vector spectrum analysers (VSA).

The SA provides only information of the amplitude of an ingoing signal, while
the VSA provides the phase as well.

###  0.4.1 Scalar spectrum analysers

A common oscilloscope displays a signal in the amplitude-time plane (time
domain). The SA follows another approach and displays it in frequency domain.

One of the major advantages of the frequency-domain display is the sensitivity
to periodic perturbations. For example, 5% distortion is already difficult to
see in the time domain but in the frequency domain the sensitivity to such
‘sidelines’ (Fig. 22) is very high ($-$120 dB below the main line).

[algebraic=true,linecolor=orange]040.06*sin(5*x)[algebraic=true,linecolor=orange]040.06*sin(5*x)A
[V]A [dB]tt2% AMPLITUDE MODULATION2% AM IN FREQUENCY DOMAIN– 400 Figure 22:
Example of amplitude modulation in time and frequency domain

A very faint amplitude modulation (AM) of 10${}^{-\text{12}}$ (power) on some
sinusoidal signals would be completely invisible on the time trace, but can be
displayed as two sidelines 120 dB below the carrier in the frequency domain
[8].

We will now consider only serial processing or swept tuned analysers (Fig.
23).

SIGNALTUNABLEBANDPASSAMPLITUDEDETECTORDISPLAY Figure 23: A tunable bandpass as
a simple spectrum analyser (SA)

The easiest way to design a swept tuned spectrum analyser is by using a
tunable bandpass. This may be an LC circuit, or a YIG filter (YIG = Yttrium-
Iron-Garnet) beyond 1 GHz. The LC filter exhibits poor tuning, stability and
resolution. YIG filters are used in the microwave range (as preselector) and
for YIG oscillators. Their tuning range is about one decade, with Q-values
exceeding 1000.

For much better performance the superheterodyne principle can be applied (Fig.
11).

SIGNALINPUTSWITCHABLEATTENUATORLOW PASSMIXERIF FILTERSAW
TOOTHGENERATORTUNABLEOSCILLATORLOIFAMPLIFIERDISPLAYVIDEOAMPLIFIERVIDEO
FILTERLOW PASSAMPLITUDEDETECTOR Figure 24: Block diagram of a spectrum
analyser

As already mentioned, the nonlinear element (four-diode mixer, double-balanced
mixer) delivers mixing products as

$f_{\text{s}}=f_{\text{LO}}\pm f_{\text{IF}}\hskip 5.69046pt.$ (22)

Assuming a signal range from 0 to 1 GHz for the spectrum analyser depicted in
Fig. 24 and $f_{\text{LO}}$ between 2 and 3 GHz we get the frequency chart
shown in Fig. 25.

1234523 GHZf${}_{\text{LO}}$fS(+)(–) Figure 25: Frequency chart of the SA of
Fig. 24, intermediate frequency = 2 GHz

Obviously, for a wide input frequency range without image response we need a
sufficiently high intermediate frequency. A similar situation occurs for AM-
and FM-broadcast receivers (AM-IF = 455 kHz, FM-IF = 10.7 MHz). But for a high
intermediate frequency (e.g., 2 GHz) a stable narrow-band IF filter is
difficult to construct which is why most SAs and high quality receivers use
more than one IF. Certain SAs have four different LOs, some fixed, some
tunable. For a large tuning range the first, and for a fine tune (e.g., 20
kHz) the third LO is tuned.

Multiple mixing is necessary when going to a lower intermediate frequency
(required when using high-Q quartz filters) for good image response
suppression of the mixers.

It can be shown that the frequency of the $n$-th LO must be higher than the
(say) 80 dB bandwidth of the ($n$ $-$ 1)th IF-band filter. A disadvantage of
multiple mixing is the possible generation of intermodulation lines if
amplitude levels in the conversion chain are not carefully controlled.

The requirements of a modern SA with respect to frequency are

  * •

high resolution

  * •

high stability (drift, phase noise)

  * •

wide tuning range

  * •

no ambiguities,

and with respect to amplitude response are

  * •

large dynamic range (100 dB)

  * •

calibrated, stable amplitude response

  * •

low internal distortions.

It should be mentioned that the size of the smallest IF-bandpass filter width
$\Delta f$ has an important influence on the maximum sweep rate (or step-width
and -rate when using a synthesizer)

$\frac{\text{d}f}{\text{d}t}<(\Delta f)^{2}\hskip 5.69046pt.$ (23)

In other words, the signal frequency has to remain at $\Delta T=1/\Delta f$
within the bandwidth $\Delta f$.

On many instruments the proper relation between $\Delta f$ and the sweep rate
is automatically set to the optimum value for the highest possible sweep
speed, but it can always be altered manually (setting of the resolution
bandwidth).

Certain SAs do not use a sinusoidal LO signal but, rather, periodic short
pulses or a comb spectrum (harmonic mixer). This is very closely related to a
sampling scope, except that the spacing of the comb lines is different

$f_{\text{s}}=Nf_{\text{LO}}\pm f_{\text{IF}}\hskip 14.22636ptn=1,2,3,...$
(24)

A single, constant input-frequency line may appear several times on the
display. This difficulty (multiple response) was a particular problem with
older instruments. Certain modulation and sweep modes permit the
identification and rejection of these ‘ghost’ signals. On modern spectrum
analysers the problem does not occur, except at frequencies beyond 60 GHz,
when a tracking YIG filter may need to be installed.

Caution is advised when applying, but not necessarily displaying, two or more
strong (> 10 dBm) signals to the input. Intermodulation 3rd-order products may
appear (from the first mixer or amplifier) and could lead to misinterpretation
of the signals to be analysed.

SAs usually have a rather poor noise figure of 20–40 dB as they often do not
use preamplifiers in front of the first mixer (dynamic range, linearity). But
with a good preamplifier the noise figure can be reduced to almost that of the
preamplifier. This configuration permits amplifier noise figure measurements
to be made with reasonable precision of about 0.5 dB. The input of the
amplifier to be tested is connected to a hot and a cold termination and the
corresponding two traces on the SA display are evaluated [9, 10, 11, 12, 13].

Spectrum analysers can also be used to directly measure the phase noise of an
oscillator provided that the LO phase noise in the SA is much lower than that
of the device under test [9]. For higher resolution, set-ups with delay lines
and additional mixers (SA at low frequencies or FFT) are advised.

##  0.5 Vector spectrum and FFT analyser

The modern vector spectrum analyser (VSA) is essentially a combination of a
two-channel digital oscilloscope and a spectrum analyser FFT display. The
incoming signal gets down-mixed, bandpass (BP) filtered and passes an ADC
(generalized Nyquist for BP signals; $f_{\text{sample}}$ = 2 BW). A schematic
drawing of a modern VSA can be seen in Fig. 26.

Figure 26: Block diagram of a vector spectrum analyser

The digitized time trace then is split into an I (in phase) and Q (quadrature,
90 degree offset) component with respect to the phase of some reference
oscillator. Without this reference, the term vector is meaningless for a
spectral component.

One of the great advantages is that a VSA can easily separate AM and FM
components.

An example of vector spectrum analyser display and performance is given in
Fig. 27 and Fig. 28. Both figures were obtained during measurements of the
electron cloud in the CERN SPS.

Figure 27: Single-sweep FFT display similar to a very slow scan on a swept
spectrum analyser Figure 28: Spectrogram display containing about 200 traces
as shown on the left side in colour coding. Time runs from top to bottom

##  0.6 Noise basics

The concept of ‘noise’ was applied originally to the type of audible sound
caused by statistical variations of the air pressure with a wide flat spectrum
(white noise). It is now also applied to electrical signals, the noise ‘floor’
determining the lower limit of signal transmission. Typical noise sources are:
Brownian movement of charges (thermal noise), variations of the number of
charges involved in the conduction (flicker noise), and quantum effects
(Schottky noise, shot noise). Thermal noise is only emitted by structures with
electromagnetic losses which, by reciprocity, also absorb power. Pure
reactances do not emit noise (emissivity = 0).

Different categories of noise can be defined:

  * •

white, which has a flat spectrum,

  * •

pink, being low-pass filtered, and

  * •

blue, being high-pass filtered.

In addition to the spectral distribution, the amplitude density distribution
is also required in order to characterize a stochastic signal. For signals
coming from very many independent sources, the amplitude density has a
Gaussian distribution. The noise power density delivered to a load by a black
body is given by Planck’s formula:

$\frac{N_{\text{L}}}{\Delta f}=hf\left(\text{e}^{hf/kT}-1\right)^{-1}\hskip
5.69046pt,$ (25)

where $N_{\text{L}}$ is the noisepower delivered to a load, $h=6.625\cdot
10^{-34}$ Js the Planck constant and $k=1.38056\cdot 10^{-23}$ J/K Boltzmann’s
constant.

Equation (25) indicates constant noise power density up to about 120 GHz (at
290 K) with 1% error. Beyond, the power density decays and there is no
‘ultraviolet catastrophe’, i. e., the total noise power is finite.

The radiated power density of a black body is given as

$W_{\text{r}}(f,T)=\frac{hf^{3}}{c^{2}\left[\text{e}^{hf/kT}-1\right]}\hskip
5.69046pt.$ (26)

For $hf<<kT$ the Rayleigh–Jeans approximation of Eq. (25) holds:

$N_{\text{L}}=kT\Delta f\hskip 5.69046pt,$ (27)

where in this case $N_{\text{L}}$ is the power delivered to a matched load.
The no-load noise voltage $u(t)$ of a resistor $R$ is given as

$\overline{u^{2}(t)}=4kTR\Delta f$ (28)

and the short-circuit current $i(t)$ by

$\overline{i^{2}(t)}=4\frac{kT\Delta f}{R}=4kTG\Delta f\hskip 5.69046pt,$ (29)

where $u(t)$ and $i(t)$ are stochastic signals and $G$ is $1/R$. The linear
average $\overline{u(t)},\overline{i(t)}$ vanishes. Of special importance is
the quadratic average $\overline{u^{2}(t)},\overline{i^{2}(t)}$.

The available power (which is independent of $R$) is given by (Fig. 29)

$\frac{\overline{u^{2}(t)}}{4R}=kT\Delta f\hskip 5.69046pt.$ (30)

[algebraic=true]00.60.1*sin(10.4719755*x)R1 = noiseless resistorR2 = noiseless
loadW${}_{\text{u}}=4kT\text{R}_{1}$ Figure 29: Equivalent circuit for a noisy
resistor $R_{1}$ terminated by a noisless load $R_{2}$

We define a spectral density function [9]

$\displaystyle W_{\text{u}}(f)$ $\displaystyle=$ $\displaystyle 4kTR$
$\displaystyle W_{\text{i}}(f)$ $\displaystyle=$ $\displaystyle 4kTG$ (31)
$\displaystyle\overline{u^{2}(t)}$ $\displaystyle=$
$\displaystyle\int_{f_{1}}^{f_{2}}W_{\text{u}}(f)\text{d}f\hskip 5.69046pt.$

A noisy resistor may be composed of many elements (resistive network). In
general, it is made from many carbon grains which have homogeneous
temperatures. But if we consider a network of resistors with different
temperatures and hence with an inhomogeneous temperature distribution (Fig.
30)

$R_{i},W_{\text{u}}$$R_{1},T_{1}$$W^{\prime}_{\text{u}1}$$W^{\prime}_{\text{u}2}$$W^{\prime}_{\text{u}3}$$R_{2},T_{2}$$R_{2},T_{2}$
Figure 30: Noisy one-port with resistors at different temperatures [14, 9]

the spectral density function changes to

$\displaystyle
W_{\text{u}}=\sum_{j}W_{\text{u}j}=4kT_{\text{n}}R_{\text{i}}\hskip
5.69046pt,$ (32) $\displaystyle T_{\text{n}}=\sum_{j}\beta_{j}T_{j}\hskip
5.69046pt,$ (33)

where $W_{\text{u}j}$ are the noise sources (Fig. 31), $T_{\text{n}}$ is the
total noise temperature, $R_{\text{i}}$ the total input impedance, and
$\beta_{j}$ are coefficients indicating the fractional part of the input power
dissipated in the resistor $R_{j}$. It is assumed that the $W_{\text{u}j}$ are
uncorrelated for reasons of simplicity.

(2,2.4)(7,2.4)$W_{\text{u}}$$R_{i}$$R_{i}$$W_{\text{u}}$$W_{\text{u}1}$$W_{\text{u}2}$$W_{\text{u}i}$
Figure 31: Equivalent sources for the circuit of Fig. 30.

The relative contribution ($\beta_{j}$) of a lossy element to the total noise
temperature is equal to the relative dissipated power multiplied by its
temperature:

$T_{\text{n}}=\beta_{1}T_{1}+\beta_{2}T_{2}+\beta_{3}T_{3}+\cdots$ (34)

A nice example is the noise temperature of a satellite receiver, which is
nothing else than a directional antenna. The noise temperature of free space
amounts roughly to 3 K. The losses in the atmosphere, which is an air layer of
10 to 20 km length, cause a noise temperature at the antenna output of about
10 to 50 K. This is well below room temperature of 290 K.

So far only pure resistors have been considered. Looking at complex
impedances, it can be seen that losses from dissipation occur in $Re$(Z) only.
The available noise power is independent of the magnitude of Re($Z$) with
Re($Z$) > 0\. For Figs. 30 and 31, Eq. (33) still applies, except that
$R_{\text{i}}$ is replaced by $Re$(Z${}_{\text{i}}$). However, it must be
remembered that in complex impedance networks the spectral power density
$W_{\text{u}}$ becomes frequency dependent [14].

The rules mentioned above apply to passive structures. A forward-biased
Schottky diode (external power supply) has a noise temperature of about
$T_{0}$/2 + 10%. A biased Schottky diode is not in thermodynamic equilibrium
and only half of the carriers contribute to the noise [9]. But it represents a
real 50 $\Omega$ resistor when properly forward biased. For transistors and in
particular field-effect transistors (FETs), the physical mechanisms are
somewhat more complicated. Noise temperatures of 50 K have been observed on a
FET transistor at 290 K physical temperature.

##  0.7 Noise-figure measurement with the spectrum analyser

Consider an ideal amplifier (noiseless) terminated at its input (and output)
with a load at 290 K with an available power gain ($G_{\text{a}}$). We measure
at the output [10, 15]:

$P_{\text{a}}=kT_{0}\Delta fG_{\text{a}}\hskip 5.69046pt.$ (35)

For $T_{0}$ = 290 K (or often 300 K) we obtain $kT_{0}$ = $-$174 dBm/Hz
($-$dBm = decibel below 1 mW). At the input we have for some signal Si a
certain signal/noise ratio S${}_{i}/\text{N}_{i}$ and at the output
S${}_{0}/\text{N}_{0}$. For an ideal (= noiseless) amplifier
S${}_{i}/\text{N}_{i}$ is equal to S${}_{0}/\text{N}_{0}$, i. e., the signal
and noise levels are both shifted by the same amount. This gives the
definition of the noise figure $F$:

$F=\frac{\text{S}_{i}/\text{N}_{i}}{\text{S}_{0}/\text{N}_{0}}\hskip
5.69046pt.$ (36)

The ideal amplifier has $F$ = 1 or $F$ = 0 dB and the noise temperature of
this amplifier is 0 K. The real amplifier adds some noise which leads to a
decrease in $\text{S}_{0}/\text{N}_{0}$ due to the noise added (=
N${}_{\text{ad}}$):

$F=\frac{\text{N}_{\text{ad}}+kT_{0}\Delta fG_{\text{a}}}{kT_{0}\Delta
fG_{\text{a}}}\hskip 5.69046pt.$ (37)

For a linear system the minimum noise figure amounts to $F_{\text{min}}$ = 1
or 0 dB. However, for nonlinear systems one may define noise figures $F$ < 1\.
Now assume a source with variable noise temperature connected to the input and
measure the linear relation between amplifier output power and input
termination noise temperature ($T_{\text{s}}$ = $T_{\text{source}}$).

In a similar way a factor $Y$ can be defined:

$\displaystyle Y=\frac{T_{\text{e}}+T_{\text{H}}}{T_{\text{e}}+T_{\text{C}}}$
$\displaystyle T_{\text{e}}=\frac{T_{\text{H}}-YT_{\text{C}}}{Y-1}$
$\displaystyle
F=\frac{\left[T_{(\text{H}}/290)-1\right]-Y\left[(T_{\text{C}}/290)-1\right]}{Y-1}\hskip
5.69046pt,$ (38)

where $T_{\text{e}}$ is the effective input noise temperature (see Fig. 32)
and $T_{\text{H}}$ and $T_{\text{C}}$ are the noise temperatures of a hot or
cold input termination.

P${}_{\text{OUT}}$TS [K]Source temperatureNOISE FREE++Slope $\approx
k\text{G}_{\text{a}}\Delta$fNoise freeT0 = 300 KT${}_{\text{hot}}$ = 12000 K
(Drawing not to scale) Figure 32: Relation between source noise temperature
$T_{\text{s}}$ and output power $P_{\text{out}}$ for an ideal (noise free) and
a real amplifier [10, 15]

To find the two points on the straight line of Fig. 32 one may switch between
two input terminations at 373 K (100∘ C) and 77 K. For precise reading of RF
power, calibrated piston attenuators in the IF path (Intermediate Frequency
Superheterodyne Receiver) are in use. This is the hot/cold method. The
difference between the $Y$-factor and the hot/cold method is that for the
latter the input of the amplifier becomes physically connected to resistors at
different temperatures (77, 373 K). For the $Y$-factor, the noise temperature
of the input termination is varied by electronic means between 300 K and 12
000 K (physical temperature always around 300 K).

As a variant of the 3-dB method with a controllable noise source, the excess
noise temperature definition ($T_{\text{ex}}=T_{\text{H}}-T_{\text{C}}$) is
often applied. A switchable 3-dB attenuator at the output of the amplifier
just cancels the increase in noise power from $T_{\text{H}}-T_{\text{C}}$.
Thus the influence of nonlinearities of the power meter is eliminated. To
measure the noise of one port one may also use a calibrated spectrum analyser.
However, spectrum analysers have high noise figures (20–40 dB) and the use of
a low-noise preamplifier is recommended. This ‘total power radiometer’ [11] is
not very sensitive but often sufficient, e. g., for observation of the
Schottky noise of a charged particle beam. Note that the spectrum analyser may
also be used for two-port noise figure measurements. An improvement of this
‘total power radiometer’ is the ‘Dicke Radiometer’ [11]. It uses a 1 kHz
switch between the unknown one port and a controllable reference source. The
reference source is made equal to the unknown via a feedback loop, and one
obtains a resolution of about 0.2 K. Unfortunately, switch spikes sometimes
appear. Nowadays, switch-free correlation radiometers with the same
performance are available [6].

The noise figure of a cascade of amplifiers is [14, 9, 10, 15, 6]

$F_{\text{total}}=F_{1}+\frac{F_{2}-1}{G_{\text{a}1}}+\frac{F_{3}-1}{G_{\text{a}1}G_{\text{a}2}}+\cdots$
(39)

As can be seen from Eq. (39) the first amplifier in a cascade has a very
important effect on the total noise figure, provided $G_{\text{a}1}$ is not
too small and F2 is not too large. In order to select the best amplifier from
a number of different units to be cascaded, one can use the noise measure $M$:

$M=\frac{F-1}{1-(1/G_{\text{a}})}\hskip 5.69046pt.$ (40)

The amplifier with the smallest M has to be the first in the cascade [15].

## References

  * [1] F. Caspers, RF engineering basic concepts: the Smith chart, CAS proceedings, 2010.
  * [2] G. D. Vendelin, A. M. Pavio and U. L. Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques, second edition, Wiley-Interscience, 2005, ISBN-10 0-471-41479-4.
  * [3] M. Thumm, W. Wiesbeck and S. Kern, Hochfrequenzmesstechnik, B.G. Teubner Stuttgart $\cdot$ Leipzig, 1998, ISBN 3-519-16360-8
  * [4] R. A. Witte, Spectrum and Network Measurements, Prentice-Hall Inc., 1991, ISBN 0-13-826959-9
  * [5] H. D. Lueke, Signaluebertragung, Springer, 1975, ISBN 3-540-07125-3.
  * [6] B. Schiek, Messysteme der Hochfrequenztechnik, Hiithig 1984, ISBN 3-7785-1045-2.
  * [7] K. Lipinski, Moderne Oszillographen, VDE, 1974.
  * [8] W. O. Schleifer, Hochfrequenz und Mikrowellenmesstechnik in der Praxis, Hiithig, 1981, ISBN 3-7785-0675-7.
  * [9] B. Schiek and H. J. Sieveris, Rauschen im Hochfrequenzschaltungen, Hiithig, 1984, ISBN 3-7785-2007-5.
  * [10] P. C. L. Yip, High Frequency Circuit Design and Measurement, Chapman and Hall, 1990, ISBN 0-412-34160-3.
  * [11] G. Evans and C. W. McLeisch, RF-Radiometer Handbook, Artech, 1977, ISBN 0-89006-055-X.
  * [12] F. R. Connor, Noise, Edward Arnold, 1973, ISBN 0-7131-3306-6.
  * [13] F. Landstorfer and H. Graf, Rauschprobleme der Nachrichtentechnik, Oldenbourg, 1981, ISBN 3-486-24681-X.
  * [14] O. Zinke and H. Brunswig, Lehrbuch der Hochfrequenztechnik, Zweiter Band, Springer, 1974, ISBN 3-540-06245-9.
  * [15] Fundamentals of RF and microwave noise figure measurements, Hewlett-Packard Application Note 57-1.


*mps*

# Charging up Boosted Black Holes

Prakruth Adari  Department of Physics and Astronomy, Stony Brook University,
Stony Brook, NY 11794 Roman Berens  Department of Physics, Columbia
University, New York, NY 10027 Janna Levin  Department of Physics and
Astronomy, Barnard College of Columbia University, New York, NY 10027

###### Abstract

Contrary to a prevailing assumption that black holes would swiftly discharge,
we argue that black holes can charge preferentially when boosted through an
ambient magnetic field. Though the details are very different, the preference
for charge is related to the precipitation of the Wald charge on a spinning
black hole in an ambient magnetic field. The gravito-electrodynamics upstages
naive arguments about screening electric fields in determining the value of
the charge accrued. Charged test particles, which build up the black hole
charge, exhibit chaotic behavior as evidenced by fractal basin boundaries
between dynamical regions. Charged, boosted black holes will generate their
own electromagnetic fields and thereby their own luminous signatures, even if
they are initially bare. We therefore add boosted black holes to the growing
list of potentially observable black hole signatures, alongside black hole
batteries and black hole pulsars. The implications should be relevant for
supermassive black holes that are boosted relative to a galactic magnetic
field as well as black holes merging with magnetized neutron stars.

Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
    1. 1.1 Spinning Black Holes and Screened Electric Fields
  2. 2 Maxwell Equations around a Boosted Black Hole
    1. 2.1 Screening of the electric field
  3. 3 The dynamics of charging up a boosted black hole
    1. 3.1 Orbits seeded on the $x$-axis
      1. 3.1.1 Triangles
    2. 3.2 The Equatorial Plane and Fractal Basin Boundaries
  4. 4 Summary
  5. A Solving the Maxwell Equations
  6. B Initial Conditions

##  1 Introduction

Detailed direct observations of black holes this century motivate a renewed
interest in the electromagnetic potentiality of black holes. Even bare,
initially dark black holes cleverly exploit ambient magnetic fields—whether
seeded by cohabitating neutron stars or anchored in a diffuse background—to
power novel luminous channels. For instance, black hole batteries turn on
during merger with a highly-magnetized neutron star, leading to faint
electromagnetic counterparts to gravitational wave observations [1, 2, 3, 4].
Relatedly, a spinning black hole in an external magnetic field will
preferentially acquire charge, as shown in an oft overlooked observation
attributable to Wald [5]. A spinning charge generates a magnetic dipole
inducing a black hole pulsar [6]. Even the Penrose process is enormously
amplified around charged and magnetized black holes [7]. All of these
phenomena—black hole batteries, black hole pulsars, the electromagnetic
Penrose process—are distinct yet interrelated. They are each consequences of
the motion of a black hole through a magnetic field.

We extend the range of luminous phenomena by considering black holes that are
relativistically boosted through a uniform magnetic field. Like spinning black
holes, these boosted black holes will acquire charge, although the details are
rather different. The boosted black hole perceives both a magnetic field
$B_{0}$ and an electric field $E_{0}=\beta B_{0}$ due to the Lorentz
transformation of the ambient magnetic field $\gamma^{-1}B_{0}$. Nearby
charges will respond to the boosted black hole resulting in a charged boosted
black hole, as we show.

The gravito-electrodynamics are far more complicated for a boosted black hole
than a spinning black hole. The boost breaks axial symmetry so a constant of
geodesic motion is lost, namely the angular momentum. The Carter constant is
also lost [8, 9, 6]. The only generic constants of motion retained for
particles around the black hole are the energy and the timelike constraint on
the four-velocity. With the loss of constants of motion, dynamics in phase
space are no longer tightly constrained and the window to chaotic dynamics is
opened. We therefore anticipate, and observe, extremely intricate orbits,
reflecting the underlying chaos.

The complicated dynamics also highlight another general conclusion: dynamics
upstage screening. By this we mean that while charges may be inclined to
distribute themselves so as to screen an electric field, the actual dynamics
may not comply with this inclination. Consequently, the charge acquired
depends sensitively on the initial data and distribution of the test charges,
as we show. Some distributions lead to a charged black hole and others do not.
Qualitatively, the magnitude of the black hole charge will be set by the boost
$\beta$, the ambient magnetic field $B_{0}$, the mass of the black hole $M$.
Based on units, this would indicate, again only qualitatively, $Q\propto\beta
B_{0}M^{2}$. Due to the extreme sensitivity to initial conditions of chaotic
dynamics, there is no simple quantitative formula to quote for the anticipated
magnitude of the charge. In general, we do see an anti-symmetry about the
$y$-axis: if charge $Q$ precipitates from charges on one side of the $y$-axis,
then $-Q$ will precipitate from charges on the other side. For completely
symmetric distributions, this would lead to an average of zero black hole
charge, although test charges continue to flux along the field lines.
Asymmetric distributions of ambient charges will lead to charged boosted black
holes.

Although an assessment of realistic luminosities is beyond the scope of this
work, it is important to note that a boosted charge creates its own magnetic
field, which in turn leads to its own electromagnetic signatures [10].
Ultimately, a charged boosted black hole could have observable implications
for supermassive black holes like M87* or Sagittarius A* as well as black
hole/neutron star merges detectable by the LIGO-VIRGO-KAGRA network of
gravitational wave observatories [11, 12, 13].

Below we consider bare black holes, so black holes in vacuum except for the
test charges. We are not imagining black holes immersed in a magnetosphere,
which would lead to the force-free condition $\textbf{E}\cdot\textbf{B}=0$, as
in [14]. We review the Wald argument for spinning black holes in order to show
that dynamics upstage screening. We then dive into the dynamics around boosted
black holes to demonstrate charge acquisition as well as chaos.

###  1.1 Spinning Black Holes and Screened Electric Fields

In a classic paper from 1974, Wald argued that to screen the electric field
generated by the spin $a$ of a black hole in a uniform magnetic field, black
holes would acquire a charge $Q_{W}=2aMB_{0}$, hereafter called the Wald
charge [5]. A spinning black hole of charge $Q_{W}$ will be screened in the
sense that charges will not experience a drop in the electromagnetic potential
between infinity and the event horizon. Wald’s argument was formulated on the
spin axis. In a recent article, we showed that the argument could be extended
off-axis, although charges will still flux and radiate around the black hole
[6].

The gravito-electrodynamics can lead to charge acquisition that deviates from
the value anticipated from screening arguments, $Q_{W}$. For this purpose it
is sufficient to consider particles along the spin axis. We take the spin of
the black hole and the magnetic fields to be aligned with the $z$-axis.

A particle with charge per unit mass $\overline{q}$ will couple with the
electromagnetic potential through

$\overline{q}A_{t}=-\chi_{Q}(g_{tt}+1).$ (1.1)

The final term in the potential is a gauge term we have subtracted to ensure
that the electrostatic potential is zero at infinity, as in [7], and we have
defined

$\chi_{Q}=\overline{q}\left(\frac{Q}{2M}-aB_{0}\right)$ (1.2)

as in [7]. On the $z$-axis,

$g_{tt}=-\frac{1}{g_{rr}}=-\frac{\Delta}{\Sigma},$ (1.3)

with

$\displaystyle\Delta$ $\displaystyle=z^{2}+a^{2}-2Mz$ $\displaystyle\Sigma$
$\displaystyle=z^{2}+a^{2}.$ (1.4)

Thus the conserved energy per unit mass for the particle is given by

$e=-\frac{1}{m}\pi_{\mu}\eta^{\mu}=-(u_{t}+\overline{q}A_{t}),$ (1.5)

where $\pi^{\mu}=mu^{\mu}+qA^{\mu}$ is the canonical momentum and $\eta^{\mu}$
is the timelike Killing vector. The change in the electromagnetic potential
for a particle falling from infinity to cross the black hole event horizon at
$r_{+}$ is then

$\Delta\epsilon=\left.-\overline{q}A_{t}\right|_{z=r_{+}}+\left.\overline{q}A_{t}\right|_{z=\infty}=\chi_{Q}$
(1.6)

which vanishes at the Wald charge $Q_{W}=2aMB_{0}$. Consequently, the electric
field is screened at the Wald charge [5, 4, 15, 16].

However, the dynamics can override the desire to screen the electric field. If
we seed particles initially at rest along the $z$-axis stretching from the
event horizon out to infinity, the value of the charge that precipitates onto
the black hole depends on the initial data which we will now show.

On the positive $z$-axis, $\theta=0$, and the equations of motion simplify
greatly with only two coordinates, $(t,z)$. We can use an effective potential
formulation since there exist enough constants of motion, $e$ and $u\cdot u$,
to reduce the ODEs to first order equations. From $e$ we have

$\dot{t}=\frac{\Sigma}{\Delta}(e-\chi_{Q})+\chi_{Q}.$ (1.7)

Now we leverage $u\cdot u=-1$, using (1.7) to eliminate $\dot{t}$ to write

$\frac{1}{2}\dot{z}^{2}+V_{\rm eff}=\frac{e^{2}-1}{2}$ (1.8)

with

$V_{\rm
eff}=\frac{2Mz}{\Sigma}\chi_{Q}\left(e-\chi_{Q}\left(\frac{Mz}{\Sigma}\right)\right)-\frac{Mz}{\Sigma}.$
(1.9)

At $z\to\infty$, $V_{\rm eff}\to 0$ and therefore for $\dot{z}_{0}=0$ we have
$e=1$ in this limit. Notice for this value that $\dot{t}_{0}=1$.

Letting $\Sigma_{0}$ and $\Delta_{0}$ denote the values of $\Sigma$ and
$\Delta$ at $z=z_{0}$, we set $\dot{z}_{0}=0$ so that the particles are
initially at rest, finding $\dot{t}_{0}=\sqrt{\frac{\Sigma_{0}}{\Delta_{0}}}$,
which constrains

$e=\frac{\Delta_{0}}{\Sigma_{0}}\left(\sqrt{\frac{\Sigma_{0}}{\Delta_{0}}}-\chi_{Q}\right)+\chi_{Q}.$
(1.10)

As required, $e$ approaches 1 as $z_{0}\to\infty$. We can use this value for
$e$ in $V_{\rm eff}$ as shown in Fig. 1.

| $\left|\chi_{Q}\right|=0.5$ | $\left|\chi_{Q}\right|=1.5$
---|---|---
| $z_{0}=2.5M$
---
|
---
|
---
| $z_{0}=5M$
---
|
---
|
---
Figure 1: Sample effective potentials with $a=0.5M$ with varying $z_{0}$ and
$Q$. Solid lines correspond to effective potential and dashed lines to the
energy of a charged particle starting at rest at the listed $z_{0}$. The blue
plots have $\chi_{Q}>0$ and the orange have $\chi_{Q}<0$. Note that for
$z_{0}=2.5M$, the transition value of $\chi_{Q}$ is 1.4, and for $z_{0}=5M$,
it is 0.64.

As the potential diagrams show, the dynamics can favor one sign of charge over
another even away from $\chi_{Q}=0$. Since the slope of the potential
determines if a particle rolls out to infinity or rolls in towards the black
hole, we suggest as a rough guide setting the slope to zero to find a band
around the Wald charge which can lead to stable accretion of charge.

Setting

$\left.\frac{\partial V}{\partial z}\right|_{z=z_{0}}=0$ (1.11)

and solving for $\chi_{Q}$ gives

$\chi_{Q}=\frac{1}{2}\sqrt{\frac{\Sigma_{0}}{\Delta_{0}}},$ (1.12)

which goes to $1/2$ as $z_{0}\to\infty$. In other words, above the Wald
charge, positive charges are still favored, and below the Wald charge,
negative charges are still favored in a band defined by

$Q_{W}-\frac{M}{\left|\overline{q}\right|}<Q<Q_{W}+\frac{M}{\left|\overline{q}\right|}$
(1.13)

for charges that begin far from the black hole.

To examine this behavior, we simulated the trajectories of a series of
particles. We seeded positive and negative particles at $z_{0}$ and let them
evolve. If both particles escaped or both fell into the black hole, the
simulation ended. If one fell in and one escaped, the charge of the black hole
was increased or decreased and the point was seeded again, repeating until the
charge of the black hole stabilized. For the first set of simulations, we took
the initial charge to obey
$Q_{0}>Q_{W}+\frac{M}{|\overline{q}|}\sqrt{\frac{\Sigma_{0}}{\Delta_{0}}}$,
and took
$Q_{0}<Q_{W}-\frac{M}{|\overline{q}|}\sqrt{\frac{\Sigma_{0}}{\Delta_{0}}}$ for
the second set. Fig. 2 shows the final charge in units of $Q_{W}$ as a
function of $z_{0}$ in units of $M$.

Figure 2: $Q/Q_{W}$ vs. $r_{0}/M$. The blue curves are
$Q_{W}\pm\frac{M}{|\overline{q}|}\sqrt{\frac{\Sigma_{0}}{\Delta_{0}}}$ in
units of $Q_{W}$. The straight orange lines are the asymptotic values
$Q_{W}\pm\frac{M}{|\overline{q}|}$. The red dots are the results of the
simulations. The initial charge of the black hole was taken to be above the
blue curve for the simulations yielding dots above the band, and vice versa
for the dots below the band. The parameters are $\overline{q}B_{0}=10\,M^{-1}$
and $a=0.5M$, so $Q_{W}=0.1M$.

The nearer you are to the black hole, the larger the deviation as shown in
Fig. 2. This result is in no way generalizable. Some parameter values will
lead to oscillatory orbits and significant deviations from the figure. We
stress that we are merely pressing the point that the specific initial seeding
of charges and the gravito-electrodynamics can favor accretion of charges of a
particular sign, even if this does not lead to the full screening effect that
Wald originally described.

We take that lesson with us as we leave spinning black holes behind and turn
our attention to boosted black holes.

##  2 Maxwell Equations around a Boosted Black Hole

We take the black hole to be immersed in an external magnetic field that is
asymptotically uniform in the positive $z$ direction. We also take the black
hole to be moving with constant speed $\beta$ in the negative $y$ direction.
In the frame of the black hole, the magnitude of the asymptotic field is
$B_{0}$ and there is an asymptotic electric field in the negative $x$
direction, $\textbf{E}_{0}=\bm{\beta}\times\textbf{B}_{0}$. In Appendix A, we
find the value of the electromagnetic fields everywhere.

The Schwarzschild metric is given by

$ds^{2}=-N^{2}c^{2}\,dt^{2}+N^{-2}\,dr^{2}+r^{2}\,d\Omega^{2},$ (2.1)

with $N^{2}=1-\dfrac{2M}{r}$. The Maxwell equations in vacuum are

$\partial_{\nu}\left(\sqrt{-g}F^{\mu\nu}\right)=0,\quad\partial_{\nu}\left(\sqrt{-g}(\star
F)^{\mu\nu}\right)=0,$ (2.2)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and $(\star
F)^{\mu\nu}=\frac{1}{2}\sqrt{-g}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$. We
neglect the contribution of the electromagnetic fields to the geometry of the
spacetime. Allowing for charge on the black hole, we solve the Maxwell
equations, finding

$\displaystyle A_{t}$
$\displaystyle=-\frac{Q}{r}-E_{0}\left(1-\frac{2M}{r}\right)r\sin\theta\cos\phi$
(2.3) $\displaystyle A_{\phi}$
$\displaystyle=\frac{1}{2}B_{0}r^{2}\sin^{2}\theta.$ (2.4)

as did the authors of [17] previously.

$Q=0$ | $Q=0.5M$ | $Q=M$
---|---|---
|
---
|
---
|
---
Figure 3: Plots of $\textbf{E}\cdot\textbf{B}$ in the $y$-$z$ plane in units
of $M^{-2}$ for $\overline{q}B_{0}=0.5M^{-1}$, $\beta=0.5$, and varying $Q$.

For reference, we also report the invariant,

$\displaystyle\textbf{E}\cdot\textbf{B}$
$\displaystyle=\frac{1}{4}F^{\mu\nu}(\star F)_{\mu\nu}$
$\displaystyle=\frac{QB_{0}}{r^{2}}\cos\theta-
E_{0}B_{0}\frac{2M}{r}\cos\theta\sin\theta\cos\phi.$ (2.5)

Derivations of (2.5) and (2.4) can be found in Appendix A. Contour plots of
$\textbf{E}\cdot\textbf{B}$ in the $y$-$z$ plane for various values of $Q$ are
shown in Fig. 3. We note that this vanishes in the equatorial plane. Even with
a force-free magnetosphere, the best charges can do in attempts to screen the
electric field is to achieve $\textbf{E}\cdot\textbf{B}=0$. In our vacuum
scenario, there is no value of $Q$ that leads to the force-free condition
$\textbf{E}\cdot\textbf{B}=0$ everywhere. In the recent Ref. [10], the authors
considered the electromagnetic field structure to motivate charged boosted
black holes from asymmetric distributions.

###  2.1 Screening of the electric field

Although we expect charge acquisition to deviate substantially from a
prediction based on screening of the electric field, it is worth considering
the effect of $Q$ on the electric potential, in analogy with section 1.1 for a
spinning black hole.

We consider a test particle of charge per unit mass $\overline{q}$ moved from
$r=r_{0}$ to the horizon. As before, the conserved energy per mass of the
particle is

$e=-(u_{\mu}+\overline{q}A_{\mu})\eta^{\mu}$ (2.6)

Following Wald, we see that the change in electrostatic energy per unit mass
is

$\Delta\epsilon=-\overline{q}A_{\mu}\eta^{\mu}\big{|}_{r=2M}+\overline{q}A_{\mu}\eta^{\mu}\big{|}_{r=r_{0}}$
(2.7)

or

$\Delta\epsilon=\frac{\overline{q}Q}{2M}-\frac{\overline{q}Q}{r_{0}}-\overline{q}E_{0}N(r_{0})^{2}r_{0}\sin\theta\cos\phi.$
(2.8)

We can rewrite this as

$\Delta\epsilon=\frac{\overline{q}}{2M}\left(Q-Q_{T}(\theta,\phi)\right)\left(1-\frac{2M}{r_{0}}\right)$
(2.9)

with the threshold charge

$Q_{T}(\theta,\phi)=2MEr_{0}\sin\theta\cos\phi.$ (2.10)

Notice that $Q_{T}$ averages to zero. Clearly, and not surprisingly, there is
no value of $Q$ that screens the electric field in the frame of the black
hole.

The electric field does disappear in the frame of test charges that are at
rest with respect to $\gamma^{-1}B_{0}$ (for an uncharged black hole).
However, as the black hole whizzes by, these charges will automatically begin
to move and thereby will begin to feel an electric field. The gravito-
electrodynamics then determine the acquisition of charge on the black hole and
not the aspiration to screen the electric field.

##  3 The dynamics of charging up a boosted black hole

For concreteness, we set particles at rest with respect to each other until
the black hole flies by. In the frame of the black hole, the charges have
initial velocity $\textbf{v}_{0}=\beta\hat{\textbf{y}}$. More precisely, this
is the velocity measured by a local observer at the initial location of the
particle. In Appendix B, we define a proper orthonormal basis to explicitly
set the initial data with both relativistic effects included.

Figure 4: Plot of charged particle trajectories for
$\overline{q}B_{0}=0.1M^{-1}$, $\beta=0.05$, and $r_{0}=9M$. The top plots
have positive charges and the bottom plots have negative charges. Each column
is at a fixed $Q$, with $Q$ decreasing from left to right by $0.1M$ until the
final column where both the positive and negative test charge fall in,
stabilizing the black hole charge at $Q=-0.21M$. The $x$ and $y$ coordinates
are in units of $M$.

Intuitively, we expect that test particles extremely close to the event
horizon will indiscriminately fall in, unless charges are absurdly high.
Similarly, far enough away from the black hole, the dynamics of test charges
will be dominated by electromagnetic forces. In the intermediate region within
tens of Schwarzschild radii, the gravito-electrodynamics are quite
complicated. In fact, the dynamics are most assuredly chaotic as we have lost
two constants of motion, both the angular momentum and the Carter constant.

The magnetic field selects a preferred equatorial plane around the black hole.
Only trajectories in the equatorial plane remain two-dimensional. To explore
the dynamics of charged particles and the influence on charge acquisition, we
will pay particular attention to these equatorial orbits. Furthermore, the
electric field picks out the $x$-axis as special. We begin by exploring orbits
that are seeded on the $x$-axis with the initial data set as described above.
We then consider orbits seeded anywhere in the equatorial plane.

Figure 5:  Plot of charged particle trajectories for
$\overline{q}B_{0}=0.1M^{-1}$, $\beta=0.05$, and $r_{0}=13M$. The top plots
have positive charges and the bottom plots have negative charges. Each column
is at a fixed $Q$, with $Q$ decreasing from left to right by $0.1M$ until the
final column where both the positive and negative test charge escape,
stabilizing the black hole charge at $Q=-0.18M$. $x$ and $y$ are in units of
$M$.

###  3.1 Orbits seeded on the $x$-axis

$r_{0}<r_{I}$ | Both charges fall in.
---|---
$r_{I}<r_{0}<r_{II}$ | Negative charges fall in until a positive charge falls in.
$r_{II}<r_{0}<r_{III}$ | Negative charges fall in until they are repelled. Positive charges always escape.
$r_{III}<r_{0}$ | Both charges escape.
Table 1: Summary of behavior of charges in each interval.

We plant charges on the positive $x$-axis at varying $x=r_{0}$ and observe
four different kinds of behavior with ascending radius as summarized in the
chart above. The exact values of the radii at which we transition between the
four behaviors depends on the parameters $\beta$ and $\overline{q}B_{0}$.

For radii $r_{0}<r_{I}$, both charges fall in and the black hole remains
uncharged. For radii $r_{I}<r_{0}<r_{II}$, the black hole acquires negative
charge. Although positive charges are pulled to the left by the electric
field, they overshoot the black hole and so do not contribute to the charge.
Consequently, positive charges tend to escape while negative charges fall in.
As each particle falls in we update the charge of the black hole, which
becomes increasingly negative until a positive charge falls in. This is shown
in Fig 4.

As we move to larger radii, $r_{II}<r_{0}<r_{III}$, the behavior changes.
Initially, positive charges still escape and negative charges fall in. But now
the charge stabilizes when a negative charge is repelled, not when a positive
charge is pulled in. In addition, the negative charges undergo many tight
spirals at the beginning of their trajectories. This behavior is shown in Fig
5. Lastly, at radii $r_{0}>r_{III}$, both charges escape and the black hole
remains uncharged.

####  3.1.1 Triangles

| $|\overline{q}|B_{0}=0.01M^{-1}$ | $|\overline{q}|B_{0}=0.05M^{-1}$ | $|\overline{q}|B_{0}=0.1M^{-1}$
---|---|---|---
| $\beta=0.05$
---
|
---
|
---
|
---
| $\beta=0.15$
---
|
---
|
---
|
---
| $\beta=0.25$
---
|
---
|
---
|
---
Figure 6: Plots of $Q/M$ vs. $r_{0}/M$ for particles seeded on the $x$-axis
with varying $\overline{q}B_{0}$ and $\beta$. The red circled point and the
orange boxed point in the upper right plot correspond to the trajectories in
Figs. 4 and 5, respectively.

To get a sense of the magnitude of the charge acquired in the four different
intervals, we consider the following set up: We seed positive and negative
charges at a distance $r_{0}$ on the positive $x$-axis. All the charges have
an initial velocity of $\beta$ in the positive $y$ direction, with initial
conditions set as described above. We then increment or decrement the charge
of the black hole if either the positive or negative particle (but not both)
falls in and then we seed two more charges at the same spot. We repeat this
process until the charge of the black hole stabilizes. We then increment the
starting position and repeat the process. This generates a plot of the final
charge of the black hole $Q$ as a function of the starting position $r_{0}$.
We assume the mass of the black hole is effectively unperturbed for particles
with a large charge and small mass.

Intriguingly, we consistently see triangles as shown in Fig. 6 wherein the
charge acquisition as a function of radius shows the behavior described above
across the four intervals. We show nine such triangles for different values of
$\beta$ and $\overline{q}B_{0}$. The specific values of $r_{I}$, $r_{II},$ and
$r_{III}$ depend on these parameters but the qualitative behavior is the same
across Fig. 6. The trajectories shown in Figs. 4 and 5 correspond to red and
orange points highlighted in the upper right plot of Fig. 6. The behaviors
shown in these figures are consistent along each leg of the triangle.

If instead we were to seed particles on the negative $x$-axis, the triangle
would flip vertically around the $Q=0$ axis. If the distribution of charge is
symmetric about $y$, then the net charge is zero, although charges will
continue to flux around the black hole since $\mathbf{E}\cdot\mathbf{B}\neq
0$.

Figure 7: The triangle from the upper left of Fig. 6 with higher resolution.
The red circle, blue square, and green triangle correspond to the trajectories
in Fig. 8.

The boundaries $r_{I}$, $r_{II}$, $r_{III}$ appear to be zero-dimensional on
the $x$-axis for most of the triangles in Fig. 6. However, the top left
triangle is broken up in a complex way. We probe three details of the broken
triangle shown at higher resolution in Fig. 7. The three results indicated by
a red circle, blue square, and green triangle are all very near each other in
the space of initial conditions with values of $r_{0}$ within $0.3M$ of each
other, yet with greatly disparate outcomes. The specific trajectories
corresponding to these outcomes are shown in Fig. 8. In each panel we are
seeing in-falling trajectories of negative charges transition to escape
trajectories at the corresponding value of $Q$ highlighted in Fig. 7. The
extreme delicacy of the outcome is typical of chaotic systems.

|
---
|
---
|
---
Figure 8: Trajectories corresponding to the red circle, blue square, and green
triangle in Fig. 7, respectively. All have $\overline{q}B_{0}=0.01M^{-1}$ and
$\beta=0.05$, and have $r_{0}=31.92M$, $32.08M$, and $32.22M$, respectively.
All trajectories are for negative particles, and the two trajectories in the
same plot correspond to slightly different values of $Q$, separated by
increments of $\Delta Q=0.005M$. The blue trajectory has the greater (less
negative) $Q$, and the red trajectory’s escape is what stabilizes the charge
at its final value. In the first plot, the negative particles orbited the
black hole many times in a figure-eight trajectory until falling in, or, for
sufficiently negative $Q$, escaping. This stabilized the charge at
$Q=-0.285M$, far above the points lying on the triangle. In the second plot,
the two trajectories overlap until the charges approach the black hole,
wherein one falls in and one escapes, stabilizing the charge at $Q=-0.420M$,
still above the other points on the triangle but forming a new line in the
plot. In the third plot, the usual behavior is observed, wherein the charges
undergo many spirals before falling in or escaping, as in Fig. 5. The final
charge here was $Q=-0.650M$, along the leg of the triangle.

As we move into the rest of the equatorial plane, one might expect the
boundaries to be one-dimensional, that is, lines that define the basin
boundaries between the types of dynamical behavior. However, given that we
anticipate chaotic behavior, we should rather expect these boundaries to be
fractal [18, 19, 20, 21, 22, 23, 24], reflecting the extreme sensitivity to
initial conditions and the mixing and folding of trajectories that
characterize chaos. The first triangle of Fig. 6, blown up in Fig. 7, is
already demonstrating such behavior. Below we show that this is pervasive in
the equatorial plane: the boundaries between zero charge and charge are
fractal.

###  3.2 The Equatorial Plane and Fractal Basin Boundaries

| $|\overline{q}|B_{0}=0.01M^{-1}$ | $|\overline{q}|B_{0}=0.05M^{-1}$ | $|\overline{q}|B_{0}=0.1M^{-1}$
---|---|---|---
| $\beta=0.05$
---
|
---
|
---
|
---
| $\beta=0.15$
---
|
---
|
---
|
---
| $\beta=0.25$
---
|
---
|
---
|
---
|
Figure 9: Plots of $Q$ in the $x$-$y$ plane for varying $|\overline{q}|B_{0}$
and $\beta$. All plots have $-20M\leq x,y\leq 20M$. If both positive and
negative particles seeded at the $x$ and $y$ for that cell fall into the black
hole, then the cell is colored light blue. If both particles escape, the cell
is colored light red. If one falls in and one escapes, the charge of the black
hole is increased or decreased until it stabilizes. The spectrum from dark
blue to dark red represents the final charge in units of $M$. The range of the
triangles on the $x$-axis are visible in most of these panels, although the
restricted standardized range of these figures doesn’t cover the same ranges
as those in Fig. 6. The box in the corner of the upper right plot corresponds
to the region examined in the first plot of Fig. 10.

Extending our analysis to the entire equatorial plane, the triangles are
represented in the basin boundaries of Fig. 9. The simulation proceeds much as
in the previous section. We divide the region $-20M\leq x,y\leq 20M$ into
cells of size $0.1M$, seed positive and negative particles at the top left
corner of each cell, and color the cell based on the behavior of the charges.
As before, if one particle falls in and the other escapes, the black hole’s
charge is increased or decreased and the cell is seeded again, repeating until
the charge has stabilized. The cell is then colored to represent the final
charge of the black hole as shown in the legend. Extracting the data from the
positive $x$ axis alone would give Fig. 6. Reversing the signs of $x_{0}$,
$\overline{q}$, and $Q$ reflects the force about the $y$-axis, leading to the
antisymmetry about $y$ in the figures. Again, symmetric distributions lead to
$Q=0$, asymmetric distributions to black hole charge, and in both cases,
charged particles continue to flux since $\mathbf{E}\cdot\mathbf{B}\neq 0$.

As anticipated, the boundaries show evidence of chaotic behavior. For a
dynamical system with $N$ coordinates and $N$ canonical momenta, the motion in
phase space will be restricted to tori if there are $N$ constants of motion.
The argument is simply that a set of coordinates can always be found such that
each canonical momentum is set equal to a constant, creating a set of constant
frequencies. The corresponding coordinates are then the angular variables on
the $N$-torus. All integrable systems, that is, non-chaotic systems, have this
feature.

As constants of motion are lost, the system becomes non-integrable with no
closed-form analytic solution as a function of initial conditions. The system
is still deterministic, however the loss of constants of motion allows orbits
to exist off of torii in phase space. The orbits show extreme sensitivity to
initial conditions and a mixing and folding of trajectories, as expected with
chaotic systems.

A spinning black hole has four constants of motion: energy, angular momentum
about the spin axis, $u\cdot u$, and the Carter constant. Therefore the orbits
are integrable. In boosting the black hole through the magnetic field, we have
lost constants of motion. We expect the most general orbits to be non-
integrable and chaotic. A nice demonstration of chaotic behavior is the
presence of fractal basin boundaries [18, 19, 20, 21, 22, 23, 24], which we
happened to be scanning in our generalization to the equatorial plane of the
triangles seen in one dimension.

|  |
---|---|---
Figure 10: Basin boundaries at various scales. All have
$\overline{q}B_{0}=0.1M^{-1}$ and $\beta=0.05$. The first plot has $10M\leq
x\leq 20M$, $-20M\leq y\leq-10M$, and a spacing of $0.1M$. The second has
$14.84M\leq x\leq 15.54M$, $-15.10M\leq y\leq-14.40M$, and a spacing of
$0.01M$. The third has $14.96M\leq x\leq 15.06M$, $-15.10M\leq y\leq-15.00M$,
and a spacing of $0.001M$.

The substructure is already apparent in Fig. 9. To see that the fractal
structure persists, we zoom in on the region shown in the box in the upper
right plot in Fig. 9. The results are show in Fig. 10, where we continue to
zoom in on smaller regions. The repetition of structure on smaller and smaller
scales is the definition of a fractal. The mixing and folding of trajectories
is evident in the mixed colors at the boundary. And the sensitivity to initial
conditions is evident as we zoom in and still see that a minuscule deviation
in the initial data leads to very different orbits.

To close this section, we show a pair of nearby non-equatorial trajectories in
Fig. 11. As is evident, the small difference in initial conditions yields very
different behavior, as one particle escapes and one ends in the black hole.
This demonstrates that the chaotic behavior exists outside the equatorial
plane as well, and we expect fractal basin boundaries to appear throughout
space, though investigating the entire dynamics is beyond the scope of this
paper.

Figure 11: A pair of non-equatorial trajectories. Both particles are negative,
and have $x_{0}=14.6M$ and $y_{0}=-15M$. The red trajectory has $z_{0}=2.8M$,
and the blue trajectory has $z_{0}=2.9M$. The system parameters are
$\overline{q}B_{0}=0.1M^{-1}$, $\beta=0.1$, and $Q=0$.

##  4 Summary

As anticipated, a black hole boosted through an ambient magnetic field can
preferentially acquire charge. The charge acquired depends on the details of
the initial data and the distribution of particles around the black hole.
Symmetric distributions lead to $Q=0$; asymmetric distributions to black hole
charge. Regardless, charged particles continue to flux around the black hole.
Our analysis is not in any way intended to be a model of a realistic
astrophysical set up. We simply intend to demonstrate the general principle:
All motion through magnetic fields will contribute to the charge of a black
hole in different ways. Another general conclusion is that the gravito-
electrodynamics determine the actual outcome of charge accretion, not the
screening of the electric field. In our case, the gravito-electrodynamics are
chaotic as evidenced by the fractal basin boundaries between outcomes.

Once charged, the black hole can support its own electromagnetic field leading
to its own intrinsic luminosity. Possible astrophysical settings for boosted
black holes include supermassive black holes that are boosted relative to the
central galactic magnetic field, black holes with magnetized and offset
accretion disks, and black holes orbiting through the magnetosphere of a
companion neutron star. We are excited about the prospects for observing
charged bare black holes during this prolific black hole century through
surveys, the EHT project, and the multi-messenger networks around LIGO-VIRGO-
KAGRA.

## Acknowledgements

We thank Albert Law and Kshitij Gupta for collaborative conversations. RB is
supported in part by the U.S. Department of Energy grant DE-SC0011941 and
Simons Foundation Award Number 555117. JL is supported in part by the Tow
Foundation.

##  Appendix A Solving the Maxwell Equations

The Maxwell equations are given in (2.2), but we repeat them here:

$\partial_{\nu}\left(\sqrt{-g}F^{\mu\nu}\right)=0,\quad\partial_{\nu}\left(\sqrt{-g}(\star
F)^{\mu\nu}\right)=0.$ (A.1)

Following [17], we take the nonvanishing components of the vector potential to
be $A_{t}\left(r,\theta^{\prime}\right)$ and $A_{\phi}(r,\theta)$, where
$\theta^{\prime}$ is the “polar” angle down from the positive $x$ axis. We
have chosen this unusual angle because we have an electric field aligned with
the negative $x$ axis at infinity. With this assumption, the dual Maxwell
equations are automatically satisfied.

Taking $\mu=t$ and $\mu=\phi$ in the first equation yields

$\displaystyle N^{2}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial
A_{t}}{\partial
r}\right)+\frac{1}{\sin\theta^{\prime}}\frac{\partial}{\partial\theta^{\prime}}\left(\sin\theta^{\prime}\frac{\partial
A_{t}}{\partial\theta^{\prime}}\right)$ $\displaystyle=0$ (A.2) $\displaystyle
r^{2}\frac{\partial}{\partial r}\left(N^{2}\frac{\partial A_{\phi}}{\partial
r}\right)+\sin\theta\frac{\partial}{\partial\theta}\left(\frac{1}{\sin\theta}\frac{\partial
A_{\phi}}{\partial\theta}\right)$ $\displaystyle=0,$ (A.3)

where as before, $N^{2}=1-\frac{2M}{r}.$ The electric field is uniform and in
the negative $x$ direction at infinity. In spherical coordinates, this becomes
the boundary condition that for large $r$,

$\left(E^{r},E^{\theta},E^{\phi}\right)\to-
E_{0}\left(\sin\theta\cos\phi,\frac{1}{r}\cos\theta\cos\phi,-\frac{1}{r}\frac{\sin\phi}{\sin\theta}\right).$
(A.4)

Note that $\cos\theta^{\prime}=\sin\theta\cos\phi$. This gives the boundary
condition for large $r$

$A_{t}\to-E_{0}r\cos\theta^{\prime}.$ (A.5)

We shall also impose that $A_{t}$ vanish at the horizon. Since we also have a
uniform magnetic field in the $z$ direction at infinity, we also have the
boundary condition

$\left(B^{r},B^{\theta},B^{\phi}\right)\to
B_{0}\left(\cos\theta\cos\phi,-\frac{1}{r}\sin\theta,0\right).$ (A.6)

This implies

$A_{\phi}\to\frac{1}{2}B_{0}r^{2}\sin^{2}\theta.$ (A.7)

We also impose that $A_{\phi}$ vanish at the origin.

There is one additional boundary condition that we impose, that of regularity
of the observed energy density $\rho_{\text{obs}}=T_{\mu\nu}u^{\mu}u^{\nu}$ at
the horizon, where

$u^{\mu}=\left(\left(1-\frac{2M}{r}\right)^{-1},-\sqrt{\frac{2M}{r}},0,0\right)^{\mu}$
(A.8)

is the four-velocity of a freely falling observer. It can be shown that this
forces $A_{t}$ to vanish at the horizon at least linearly, while $A_{\phi}$
simply cannot diverge at the horizon.

We now take $A_{t}=f(r)P\left(\theta^{\prime}\right)$ in the first equation
and separate variables, yielding

$\displaystyle\left(1-\frac{2M}{r}\right)\left(2rf^{\prime}(r)+r^{2}f^{\prime\prime}(r)\right)-\ell(\ell+1)f(r)$
$\displaystyle=0$ (A.9)
$\displaystyle\frac{1}{\sin\theta^{\prime}}\frac{d}{d\theta^{\prime}}\left(\sin\theta^{\prime}\frac{dP\left(\theta^{\prime}\right)}{d\theta^{\prime}}\right)+\ell(\ell+1)P\left(\theta^{\prime}\right)$
$\displaystyle=0.$ (A.10)

We have written the separation constant as $\ell(\ell+1)$ for later
convenience. We shall take $\ell$ to be real, as complex $\ell$ yields non-
regular solutions. The angular equation is Legendre’s differential equation,
solved by the Legendre functions of the first and second kind, i.e.,
$P_{\ell}(\cos\theta^{\prime})$, and $Q_{\ell}(\cos\theta^{\prime})$. The
Legendre functions of the second kind are singular at $\theta=0$ and $\pi$, as
are the Legendre functions of the first kind for non-integer $\ell$, so we
take $P(\theta)=P_{\ell}(\cos\theta^{\prime})$ and restrict to integer $\ell$.
We note that the separation constant has the same value for $\ell=-n$ and
$\ell=n-1$ for non-negative $n$, so we can take $\ell\geq 0$ as well.

We can rewrite the radial equation as

$x(x-1)f^{\prime\prime}(x)+2(x-1)f^{\prime}(x)-\ell(\ell+1)f(x)=0,$ (A.11)

where $x=r/2M$. This is a hypergeometric differential equation with parameters
$a=-\ell$, $b=\ell+1$, and $c=2$, so one solution is

$f_{\ell}^{(1)}(x)={}_{2}F_{1}(-\ell,\ell+1,2;x).$ (A.12)

We note that this is a polynomial of degree $\ell$ since $\ell$ is a non-
negative integer. It also vanishes at $x=1$. The second solution, which can be
found using the method of Frobenius, has a logarithmic contribution that
diverges at the horizon for $\ell>0$, so it can be ignored. We shall
separately handle the $\ell=0$ case, where the solutions are $1$ and $1/x$. We
thus have

$A_{t}\left(r,\theta^{\prime}\right)=C_{0}+D_{0}\frac{2M}{r}+\sum_{\ell=1}^{\infty}C_{\ell}f_{\ell}^{(1)}\left(r/2M\right)P_{m}\left(\cos\theta^{\prime}\right)$
(A.13)

with arbitrary constants $C_{0}$, $D_{0}$, and $C_{\ell}$. To make $A_{t}$
vanish at the horizon, we impose the condition $C_{0}+D_{0}=0$. Matching with
$-E_{0}r\cos\theta$ at infinity yields $C_{1}=2ME_{0}$ and $C_{\ell}=0$ for
$\ell>1$. Thus our solution becomes

$A_{t}\left(r,\theta^{\prime}\right)=C_{0}\left(1-\frac{2M}{r}\right)-E_{0}\left(1-\frac{2M}{r}\right)r\cos\theta^{\prime}.$
(A.14)

It turns out that the coefficient of the $1/r$ piece will contribute to the
charge of the black hole. Since this black hole is uncharged, we set
$C_{0}=0$, so we have

$A_{t}\left(r,\theta^{\prime}\right)=-E_{0}N^{2}r\cos\theta^{\prime},\quad
A^{t}=E_{0}r\cos\theta^{\prime}.$ (A.15)

Moving on to the $A_{\phi}$ equation, we take $A_{\phi}=g(r)Q(\theta)$ and
separate variables, finding

$\displaystyle\left(1-\frac{2M}{r}\right)r^{2}g^{\prime\prime}(r)+2Mg^{\prime}(r)-\ell(\ell+1)g(r)$
$\displaystyle=0$ (A.16)
$\displaystyle\sin\theta\frac{d}{d\theta}\left(\frac{1}{\sin\theta}\frac{dQ(\theta)}{d\theta}\right)+\ell(\ell+1)Q(\theta)$
$\displaystyle=0.$ (A.17)

We can rewrite the radial equation as

$x(x-1)g^{\prime\prime}(x)+g^{\prime}(x)-\ell(\ell+1)g(x)=0,$ (A.18)

where again $x=r/2M$. This can be transformed into another hypergeometric by
taking $g(x)=x^{2}h(x)$. The first solution is

$g_{\ell}^{(1)}(x)=x^{2}\,{}_{2}F_{1}(\ell+2,1-\ell,3;x).$ (A.19)

This only has integers powers of $x$ in its Taylor series if $\ell$ is an
integer. Since this must eventually be matched to $r^{2}$ for large $r$, we
thus take $\ell$ to be an integer, in which case $g_{\ell}^{(1)}(x)$ goes as
$x^{\ell+1}$ for large $x$. Again, we can take $\ell\geq 0$. Lastly, for
$\ell=0$, the first solution has a $\ln(1-x)$ contribution that diverges at
the horizon, so we can ignore it. The second solution, as before, involves a
logarithmic contribution for $\ell\neq 0$ that diverges at the horizon and can
thus be ignored. For $\ell=0$, the second solution is a constant and must be
included.

The angular equation can be rewritten in terms of $u=\cos\theta$:

$(1-u^{2})Q^{\prime\prime}(u)+\ell(\ell+1)Q(u)=0.$ (A.20)

This can also be transformed into a hypergeometric equation by taking
$z=u^{2}$. The two solutions are

$Q_{\ell}^{(1)}(u)={}_{2}F_{1}\left(\frac{\ell}{2},-\frac{\ell+1}{2},\frac{1}{2},u^{2}\right)\quad\text{
and }\quad
Q_{\ell}^{(2)}(u)=u\,{}_{2}F_{1}\left(\frac{\ell+1}{2},-\frac{\ell}{2},\frac{3}{2},u^{2}\right).$
(A.21)

The second solution is not smooth at $\theta=0$ or $\pi$ for $\ell>0$, so we
can ignore it for those values of $\ell$. For $\ell=0$, the second solution is
simply $u$. We thus have

$A_{\phi}(r,\theta)=G_{0}+G^{\prime}_{0}\cos\theta+\sum_{\ell=1}^{\infty}F_{\ell}g_{\ell}^{(1)}\left(r/2M\right)Q_{\ell}^{(1)}(\theta)$
(A.22)

with arbitrary constants $G_{0}$, $G_{0}^{\prime}$, and $F_{\ell}$. Clearly
$g_{\ell}^{(1)}(0)=0$, so we require $G_{0}=G_{0}^{\prime}=0$ to make
$A_{\phi}$ vanish at the origin. Since $g_{\ell}^{(1)}(r)$ goes as
$r^{\ell+1}$ for large $r$, the boundary condition at infinity sets
$F_{1}=2M^{2}B_{0}$ and $F_{\ell}=0$ for $\ell>1$. Thus we have

$A_{\phi}=\frac{1}{2}B_{0}r^{2}\sin^{2}\theta,\quad
A^{\phi}=\frac{1}{2}B_{0}.$ (A.23)

Since the Maxwell equations are linear, we can incorporate a charged black
hole by simply adding the potential for a point charge, finding

$\displaystyle A_{t}\left(r,\theta^{\prime}\right)$
$\displaystyle=-\frac{Q}{r}-E_{0}\left(1-\frac{2M}{r}\right)r\sin\theta\cos\phi$
$\displaystyle A_{\phi}(r,\theta)$
$\displaystyle=\frac{1}{2}r^{2}B_{0}\sin^{2}\theta.$

We now compute the electric and magnetic fields of this solution. Recalling
that

$E^{i}=F^{0i},\quad B^{i}=(\star
F)^{i0}=-\frac{1}{2}\widetilde{\epsilon}^{0ijk}F_{jk},$ (A.24)

where $\widetilde{\epsilon}_{0ijk}=\sqrt{|g|}\epsilon_{0ijk}$, we find

$\displaystyle E^{r}$
$\displaystyle=\frac{Q}{r^{2}}-E_{0}\sin\theta\cos\phi\quad$ $\displaystyle
B^{r}$ $\displaystyle=B_{0}\cos\theta$ $\displaystyle E^{\theta}$
$\displaystyle=-\frac{1}{r}E_{0}\cos\theta\cos\phi\quad$ $\displaystyle
B^{\theta}$ $\displaystyle=-\frac{1}{r}B_{0}\sin\theta$ $\displaystyle
E^{\phi}$ $\displaystyle=E_{0}\frac{\sin\phi}{r\sin\theta}\quad$
$\displaystyle B^{\phi}$ $\displaystyle=0.$ (A.25)

We see that the $\theta$ and $\phi$ components are not affected by the black
hole, and at first glance it appears that the $r$ components are not either.
However, the coordinate basis vector $\textbf{e}_{r}=\partial_{r}$ in
Schwarzschild spacetime time only becomes the standard spherical unit vector
in the $r\to\infty$ limit. This can be made more clear by normalizing the
basis vector. We find

$\hat{\textbf{e}}_{r}=N\textbf{e}_{r},$ (A.26)

so for the electric field

$E^{r}\textbf{e}_{r}=\left(\frac{Q}{r^{2}}-E_{0}\sin\theta\cos\phi\right)N^{-1}\hat{\textbf{e}}_{r}.$
(A.27)

This yields the expected electric field in flat space only in the $r\to\infty$
limit. The same is true for the magnetic field.

We can compute the charge using Gauss’s law. In the language of differential
forms, we have

$4\pi Q_{G}=\int_{V}\star J=\int_{V}\text{d}(\star F)=\oint_{\partial V}\star
F.$ (A.28)

Taking $V$ to be a sphere of radius $R$, we find

$\displaystyle 4\pi Q_{G}$ $\displaystyle=\oint_{r=R}\left(\star
F\right)_{\theta\phi}d\theta\wedge d\phi$
$\displaystyle=\int_{S^{2}}R^{2}\sin\theta\,\partial_{r}A_{t}|_{r=R}\,d\theta\,d\phi$
$\displaystyle=\int_{S^{2}}R^{2}\sin\theta\,\left(\frac{Q}{R^{2}}-E_{0}\sin\theta\cos\phi\right)\,d\theta\,d\phi$
$\displaystyle=Q\int_{S^{2}}\sin\theta\,d\theta\,d\phi-
E_{0}R^{2}\int_{S^{2}}\sin^{2}\theta\cos\phi\,d\theta\,d\phi$
$\displaystyle=4\pi Q,$ (A.29)

as we should. We also compute the invariant

$\displaystyle\textbf{E}\cdot\textbf{B}$
$\displaystyle=\frac{1}{4}F^{\mu\nu}(\star F)_{\mu\nu}$
$\displaystyle=\frac{QB_{0}}{r^{2}}\cos\theta-
E_{0}B_{0}\frac{2M}{r}\cos\theta\sin\theta\cos\phi.$ (A.30)

##  Appendix B Initial Conditions

We want to set the initial conditions in such a way that an observer at rest
at a given point measures the velocity of the test charge at that point to be
$\textbf{v}=\beta\hat{y}$. To this end, we build an orthonormal basis for the
observer. If $e^{\mu}_{a}=\left(\partial_{a}\right)^{\mu}$ denotes the $\mu$th
component of basis vector $a$, an orthonormal basis for such an observer is

$\hat{e}^{\mu}_{t}=\frac{1}{\sqrt{1-\frac{2M}{r_{0}}}}e^{\mu}_{t},\quad\hat{e}^{\mu}_{r}=\sqrt{1-\frac{2M}{r_{0}}}e^{\mu}_{r},\quad\hat{e}^{\mu}_{\theta}=\frac{1}{r_{0}}e^{\mu}_{\theta},\quad\hat{e}^{\mu}_{\phi}=\frac{1}{r_{0}\sin\theta_{0}}e^{\mu}_{\phi},$
(B.1)

which obeys $g_{\mu\nu}\hat{e}^{\mu}_{a}\hat{e}^{\mu}_{b}=\eta_{ab}.$ The
components of the four-velocity of the particle in the orthonormal basis
should be $\gamma(1,\textbf{v})$ with $\textbf{v}=\beta\hat{\textbf{y}}$. Thus
the initial conditions are

$u_{\mu}\hat{e}^{\mu}_{t}=\frac{1}{\sqrt{1-\beta^{2}}},\quad
u_{\mu}\hat{e}^{\mu}_{r}=\frac{\beta\sin\theta_{0}\sin\phi_{0}}{\sqrt{1-\beta^{2}}},\quad
u_{\mu}\hat{e}^{\mu}_{\theta}=\frac{\beta\cos\theta_{0}\sin\phi_{0}}{\sqrt{1-\beta^{2}}},\quad
u_{\mu}\hat{e}^{\mu}_{\phi}=\frac{\beta\cos\phi_{0}}{\sqrt{1-\beta^{2}}},$

which we can write as

$\dot{t}(0)=\frac{1}{\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{2M}{r_{0}}\right)}},\quad\dot{r}(0)=\frac{\beta\sin\theta_{0}\sin\phi_{0}}{\sqrt{1-\beta^{2}}}\sqrt{1-\frac{2M}{r_{0}}},$
(B.2)
$\dot{\theta}(0)=\frac{\beta\cos\theta_{0}\sin\phi_{0}}{r_{0}\sqrt{1-\beta^{2}}},\quad\dot{\phi}(0)=\frac{\beta\cos\phi_{0}}{r_{0}\sin\theta_{0}\sqrt{1-\beta^{2}}}.$

These are the initial conditions used in all the simulations above.

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