# Jefimenko Made Easy: Electromagnetic Fields through Retardation

Shengchao Alfred Li  Amateur scientist in Potomac, Maryland, USA

###### Abstract

Oleg D. Jefimenko’s electrodynamics textbook is unique in its approaches to
deriving the electric and magnetic fields of arbitrary charge and current
distributions, and of an arbitrarily moving point charge. However, an uncommon
form of the inhomogeneous wave equations used in the early steps often poses
difficulty for readers right from the beginning. In this paper, we substitute
in a commonly used form, making his approaches readily accessible.

Oleg D. Jefimenko (1922-2009) was a Professor Emeritus in the Department of
Physics (renamed to the Department of Physics and Astronomy in 2012) at West
Virginia University. Persisting into his retirement were his studies in
theoretical physics, many of which are controversial. However, his work in
electromagnetic retardation, as collected in the first five chapters in his
book “Electromagnetic Retardation and Theory of Relativity: New Chapters in
the Classical Theory of Fields, Second Edition” [1], is compatible with other
textbooks (thus “correct”) [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14] [15] [16] [17] [18] [19] [20] but is unique enough to be worth reading.

Retardation refers to the phenomenon that it takes time for the effect of a
source to propagate to an observer. Starting from Maxwell’s Equations and the
concept of electromagnetic retardation, without having to rely on the concept
of retarded potentials, he derived Jefimenko’s Equations (named after him by
the popular textbook author David J. Griffiths [13] [10], but see [2] [21]
[22] ), which describe the electric and magnetic fields of an arbitrary charge
and current distribution. These equations are usually included in newer
textbooks, where they are found by differentiating retarded potentials [23]
[11] [13] [22] (but see [24] [15]). From these equations, he derived the
fields of several charge and current distributions. He derived the fields of
an arbitrarily moving point charge by reducing the size of a moving body of
charge, in contrast to the popular but non-straightforward approach of
differentiating the Lienard-Wiechert potentials in time domain [2] [3] [5] [6]
[7] [8] [9] [11] [13] [14] [16] [18] [19] [20] or in frequency domain [4].

However, in deriving Jefomenko’s Equations, he used an uncommon form of the
inhomogeneous wave equations, posing difficulty for readers from the very
beginning. In this paper, we replace the uncommon form of the inhomogeneous
wave equations in his work with a common one. We also add comments in places
as needed. Both are aimed at helping to understand.

This paper aligns mostly with the first two chapters of Jefimenko’s book.

##  1 Fields, Sources and Retardation

In classical electrodynamics, the concept of fields, invented by Faraday, is
used in describing electric and magnetic phenomena.

In a 3-dimensional space, a time-dependent scalar field is expressed as a
scalar function $V(x,y,z,t)$ defined at every position $(x,y,z)$ within a
region of space and at every time $t$ within a time interval. Similarly, a
time-dependent vector field (“vector wave field” by Jefimenko) is expressed as
a vector function ${\mathbf{V}}(x,y,z,t)$. We often omit the coordinates
$(x,y,z)$ and $t$ when there is no confusion.

The relation between a vector field ${\mathbf{V}(x,y,z,t)}$ and its source
field ${\mathbf{Z}(x,y,z,t)}$ is often expressed through a differential
equation or a set of differential equations. If ${\mathbf{Z}(x,y,z,t)}$ is
given, we can find ${\mathbf{V}(x,y,z,t)}$ by solving the equation or the set
of equations, subject to certain pre-determined conditions.

Since it takes time for the effect (e.g., field value at $(x,y,z)$ at time t)
to propagate from its cause (e.g., source at
$(x^{\prime},y^{\prime},z^{\prime})$ at time $t^{\prime}$)111Note that the
prime symbol “′” is only used to discriminate between observers and sources.
The not-primed and primed coordinates belong to the same coordinate system.
Those who have learned special relativity shall not confuse the primed
coordinates used here with those used in special relativity conventions.,
$t^{\prime}$ must be earlier than $t$. If the propagation speed is a fixed
speed $c$, we have $t^{\prime}=t-r/c$ where $r$ is the distance between
$(x^{\prime},y^{\prime},z^{\prime})$ and $(x,y,z)$, thus
$r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}$.

We define the retardation of ${\mathbf{Z}}$ as

$\displaystyle[\mathbf{Z}]$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}=t-r/c),$

where “$\operatorname{def}$” stands for “is defined as” and $[\ ]$ is called
the “retardation symbol”. When there is no confusion, we also call the value
$r/c$ retardation and $t^{\prime}=t-r/c$ retarded time.

If we utilize the delta function222It is often called Dirac’s delta function,
but Heaviside had intensely used it before Dirac. ($\delta$-function)
$\delta(t)$, the retardation of ${\mathbf{Z}}$ can be expressed as

$\displaystyle[{\mathbf{Z}}]$
$\displaystyle=\int{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime})\delta(t^{\prime}-(t-r/c))dt^{\prime}.$

For future use, we define333Note that, for the second expression, Jefimenko
used the symbol $\left[\frac{\partial{\mathbf{Z}}}{\partial t}\right]$, which
we think may cause confusion. This is because, for the sake of symmetry of
notations, we shall either use the pair $[\nabla^{\prime}{\mathbf{Z}}]$ and
$\left[\frac{\partial{\mathbf{Z}}}{\partial t^{\prime}}\right]$ or use the
pair $[\nabla{\mathbf{Z}}]$ and $\left[\frac{\partial{\mathbf{Z}}}{\partial
t}\right]$, but not the pair $[\nabla^{\prime}{\mathbf{Z}}]$ and
$\left[\frac{\partial{\mathbf{Z}}}{\partial t}\right]$ as Jefimenko did.

$\displaystyle[\nabla^{\prime}{\mathbf{Z}}]$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}(\nabla^{\prime}{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}))|_{t^{\prime}=t-r/c},$
$\displaystyle\left[\frac{\partial{\mathbf{Z}}}{\partial t^{\prime}}\right]$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}\left(\frac{\partial{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime})}{\partial
t^{\prime}}\right)\bigg{\rvert}_{t^{\prime}=t-r/c},$

where the differentiation $\nabla^{\prime}$ is applied to the primed
coordinates. Similarly, we define

$\displaystyle[\nabla^{\prime}\cdot{\mathbf{Z}}]$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}(\nabla^{\prime}\cdot{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}))|_{t^{\prime}=t-r/c},$
$\displaystyle[\nabla^{\prime}\times{\mathbf{Z}}]$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}(\nabla^{\prime}\times{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}))|_{t^{\prime}=t-r/c}.$

##  2 Maxwell’s Equations and the Inhomogeneous Wave Equations

Maxwell’s Equations in vacuum are,

$\displaystyle\nabla\cdot{\mathbf{E}}=\frac{1}{\epsilon_{0}}\rho,$ (1)
$\displaystyle\nabla\cdot{\mathbf{B}}=0,$ (2)
$\displaystyle\nabla\times{\mathbf{E}}=-\frac{\partial{\mathbf{B}}}{\partial
t},$ (3)
$\displaystyle\nabla\times{\mathbf{B}}=\mu_{0}{\mathbf{J}}+\mu_{0}\epsilon_{0}\frac{\partial{\mathbf{E}}}{\partial
t}=\mu_{0}{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial{\mathbf{E}}}{\partial
t},$ (4)

where $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field,
$\rho$ is the charge density, ${\mathbf{J}}$ is the current density,
$\epsilon_{0}$ is the permittivity of the vacuum, $\mu_{0}$ is the
permeability of the vacuum, and $\epsilon_{0}\mu_{0}=1/c^{2}$, where $c$ is
the speed of light in vacuum.

There is a continuity condition implicitly contained in Maxwell’s equations.
We make it explicit by applying the $\nabla\cdot$ operator to both sides of
Eq. (4),

$\displaystyle\nabla\cdot\nabla\times{\mathbf{B}}$
$\displaystyle=\mu_{0}\nabla\cdot{\mathbf{J}}+\mu_{0}\epsilon_{0}\frac{\partial\nabla\cdot{\mathbf{E}}}{\partial
t}$ (linearity property) $\displaystyle 0$
$\displaystyle=\nabla\cdot{\mathbf{J}}+\frac{\partial\rho}{\partial t}.$ (use
Eq. (A.47); use Eq. (1))

Given sources $\rho(x,y,z,t)$ and ${\mathbf{J}}(x,y,z,t)$, we can obtain
${\mathbf{E}}(x,y,z,t)$ and ${\mathbf{B}}(x,y,z,t)$ by solving these
equations. In this section we show that the four equations can be reduced to
two inhomogeneous wave equations, one relates ${\mathbf{E}}$ to its sources,
and the other relates ${\mathbf{B}}$ to its sources, as follows.

We apply the $\nabla\times$ operator to both sides of Eq. (3) and obtain,

$\displaystyle\nabla\times\nabla\times{\mathbf{E}}$
$\displaystyle=\nabla\times\left(-\frac{\partial{\mathbf{B}}}{\partial
t}\right)$ $\displaystyle=-\frac{\partial}{\partial
t}\nabla\times{\mathbf{B}}$ (linearity property)
$\displaystyle=-\frac{\partial}{\partial
t}\left(\mu_{0}{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial{\mathbf{E}}}{\partial
t}\right)$ (use Eq. (4))
$\displaystyle=-\mu_{0}\frac{\partial{\mathbf{J}}}{\partial
t}-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{E}}}{\partial t^{2}}.$ (linearity
property)

After rearrangement, we have

$\displaystyle\nabla\times\nabla\times{\mathbf{E}}+\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{E}}}{\partial
t^{2}}$ $\displaystyle=-\mu_{0}\frac{\partial{\mathbf{J}}}{\partial t}.$ (5)

Similarly, we apply the $\nabla\times$ operator to Eq. (4) and obtain,

$\displaystyle\nabla\times\nabla\times{\mathbf{B}}$
$\displaystyle=\nabla\times\left(\mu_{0}{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial{\mathbf{E}}}{\partial
t}\right)$
$\displaystyle=\mu_{0}\nabla\times{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\nabla\times{\mathbf{E}}$ (linearity property)
$\displaystyle=\mu_{0}\nabla\times{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\left(-\frac{\partial{\mathbf{B}}}{\partial t}\right)$ (use Eq. (3))
$\displaystyle=\mu_{0}\nabla\times{\mathbf{J}}-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{B}}}{\partial
t^{2}}.\ $ (linearity property)

After rearrangement, we have

$\displaystyle\nabla\times\nabla\times{\mathbf{B}}+\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{B}}}{\partial
t^{2}}$ $\displaystyle=\mu_{0}\nabla\times{\mathbf{J}}.$ (6)

Note that both Eq (5) and (6) are in the same general form

$\displaystyle\nabla\times\nabla\times{\mathbf{V}}(x,y,z,t)+\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{V}}(x,y,z,t)}{\partial
t^{2}}={\mathbf{K}}(x,y,z,t)$

where ${\mathbf{V}}$ is $\mathbf{E}$ or $\mathbf{B}$, and ${\mathbf{K}}$ is
$-\mu_{0}{\partial{\mathbf{J}}}/{\partial t}$ or
$\mu_{0}\nabla\times{\mathbf{J}}$, respectively. This is the inhomogeneous
wave equation Jefimenko dealt with.

However, if we plug the following equation

$\displaystyle\nabla\times(\nabla\times{\mathbf{E}})$
$\displaystyle=\nabla(\nabla\cdot{\mathbf{E}})-\nabla^{2}{\mathbf{E}}$ (use
Eq. (A.48))
$\displaystyle=\nabla\left(\frac{1}{\epsilon_{0}}\rho\right)-\nabla^{2}{\mathbf{E}}$
(use Eq. (1))
$\displaystyle=\frac{1}{\epsilon_{0}}\nabla\rho-\nabla^{2}{\mathbf{E}}$
(linearity property)

in Eq. (5), we get

$\displaystyle\frac{1}{\epsilon_{0}}\nabla\rho-\nabla^{2}{\mathbf{E}}+\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{E}}}{\partial
t^{2}}$ $\displaystyle=-\mu_{0}\frac{\partial{\mathbf{J}}}{\partial t},$

and after rearrangement,

$\displaystyle\nabla^{2}{\mathbf{E}}-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{E}}}{\partial
t^{2}}$
$\displaystyle=\frac{1}{\epsilon_{0}}\nabla\rho+\mu_{0}\frac{\partial{\mathbf{J}}}{\partial
t}.$ (7)

Similarly, if we plug the equation

$\displaystyle\nabla\times(\nabla\times{\mathbf{B}})$
$\displaystyle=\nabla(\nabla\cdot{\mathbf{B}})-\nabla^{2}{\mathbf{B}}$ (use
Eq. (A.48)) $\displaystyle=-\nabla^{2}{\mathbf{B}}$ (use Eq. (2))

in Eq. (6), we get

$\displaystyle-\nabla^{2}{\mathbf{B}}+\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{B}}}{\partial
t^{2}}$ $\displaystyle=\mu_{0}\nabla\times{\mathbf{J}},$

and after rearrangement,

$\displaystyle\nabla^{2}{\mathbf{B}}-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{B}}}{\partial
t^{2}}$ $\displaystyle=-\mu_{0}\nabla\times{\mathbf{J}}.$ (8)

Note that both Eq (7) and (8) are in the general form

$\displaystyle\nabla^{2}{\mathbf{V}}(x,y,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{V}}(x,y,z,t)}{\partial
t^{2}}={\mathbf{Z}}(x,y,z,t).$ (9)

where ${\mathbf{V}}$ is $\mathbf{E}$ or $\mathbf{B}$, ${\mathbf{Z}}$ is
${1}/{\epsilon_{0}}\nabla\rho+\mu_{0}{\partial{\mathbf{J}}}/{\partial t}$ or
$-\mu_{0}\nabla\times{\mathbf{J}}$, respectively, and $\nabla^{2}\mathbf{V}$
is called the Laplacian of $\mathbf{V}$. This is the inhomogeneous wave
equation we usually see in electrodynamics textbooks [25] [23] [26] [15] [22].
In this paper, we shall work with this form instead.

##  3 Retarded Integral as a Solution of the Inhomogeneous Wave Equation

The vector inhomogeneous wave equation is

$\displaystyle\nabla^{2}{\mathbf{V}}(x,y,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{V}}(x,y,z,t)}{\partial
t^{2}}$ $\displaystyle={\mathbf{Z}}(x,y,z,t),$ (Eq (9) revisited)

where ${\mathbf{V}}$ and ${\mathbf{Z}}$ are time-dependent vector fields. We
assume ${\mathbf{Z}}$ is given, and for simplicity, ${\mathbf{Z}}$ is zero
outside a finite region of space. Then ${\mathbf{V}}$ describes waves that
propagate through space and time.

######  Theorem 1.

The retarded integral

$\displaystyle{\mathbf{V}}(x,y,z,t)$
$\displaystyle=-\frac{1}{4\pi}\int\frac{{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}=t-r/c)}{r}dV^{\prime}$
$\displaystyle=-\frac{1}{4\pi}\int\frac{[{\mathbf{Z}}]}{r}dV^{\prime}$ (10)

is a solution of the inhomogeneous wave equation Eq. (9), subject to the
condition that $[{\mathbf{Z}}]$ is the cause and ${\mathbf{V}}$ is the effect,
and ${\mathbf{V}}$ is zero at positions far far away from the source, where
$r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}$, $c$ is
the speed of light, and $dV^{\prime}$ is the infinitesimal volume at position
$(x^{\prime},y^{\prime},z^{\prime})$ to be integrated with.

Note that due to linearity, Eq. (9) can be thought as having 3 components,
each is a scalar inhomogeneous differential equation corresponding to one of
the three coordinates, taking the form

$\nabla^{2}V(x,y,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}V(x,y,z,t)}{\partial
t^{2}}=Z(x,y,z,t).$ (11)

To prove this theorem, due to linearity, we only need to show that each
component of Eq. (10), taking the form

$\displaystyle{V}(x,y,z,t)=$
$\displaystyle-\frac{1}{4\pi}\int\frac{[{Z}]}{r}dV^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\frac{{Z}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}=t-r/c)}{r}dV^{\prime},$
(12)

satisfies the corresponding scalar inhomogeneous wave equation Eq. (11), where
$V$ and and $Z$ are scalars.

This theorem deserves extensive explanations before we prove it. We address
some questions as follows.

1\. Why is the wave equation linear?

Answer: The answer that it is because Maxwell’s Equations are linear is valid
but somewhat superficial. A deeper answer is that the linearity of the wave
equation and of Maxwell’s Equations merely reflects the experimental results
that the forces or fields produced by the sources are addable. ∎

2\. Why is the solution an integral?

Answer: This question is tightly related to the first one. Because the wave
equation is linear, by definition, we can divide the region occupied by the
source ${\mathbf{Z}}$ into a large number of small volumes. Then, the field
value ${\mathbf{V}}(x,y,z,t)$ becomes the sum of all the small field values,
each of which is caused by the sources (charge and current) in one of the
volumes. The sum becomes an integral when the sizes of the volumes are made
infinitesimal. ∎

3\. Why does retardation lead to a solution of the wave equation?

Answer: Retardation and wave propagation are essentially two different names
for one phenomenon. If a point source at $(x^{\prime},y^{\prime},z^{\prime})$
emits something444It helps to imagine that in 1-dimensional space, this
“something” can be stones, bullets; in 2-dimensional space, water surface
waves, expanding oil film on a surface; in 3-dimensional space, dust, colored
gas, sound waves, etc. Here we have found a common property of moving objects
and propagating waves: they move (propagate) in space over time. at time
$t^{\prime}$ which moves outwards with speed $c$, at time $t$ it will reach
position $(x,y,z)$ which lies on the surface of a sphere centered at the
source position with diameter $r=c(t-t^{\prime})$. ∎

4\. Why isn’t the distance $r$ placed in the retardation symbol $[\ ]$ as
$[{\mathbf{Z}}/r]$?

Answer: This is because $r$ is only related to the infinitesimal volumes to be
integrated with and does not depend on retardation or the retarded time
$t-r/c$. It is a property of space only555Readers who have learned the
Lienard-Wiechert potentials shall note that this is radically different from
$r$ there.. ∎

5\. What is special about ${\mathit{1}}/r$ so that it plays a role in the
solution?

Answer: The function of $1/r$ is very special because its Laplacian
$\nabla^{2}(1/r)$ is zero everywhere except at the point $r=0$, and at $r=0$
it is a $\delta$-function, whose volume integration is finite. Proof (follow
[17]):

$\displaystyle\nabla^{2}\frac{1}{r}$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}\nabla\cdot\nabla\frac{1}{r}=\nabla\cdot\left(-\frac{1}{r^{2}}\nabla
r\right)$ (chain rule)
$\displaystyle=\nabla\cdot\left(-\frac{1}{r^{2}}\nabla(r^{2})^{\frac{1}{2}}\right)$
(note it is easier to use $\nabla r={\mathbf{i}\frac{\partial r}{\partial
x}+\mathbf{j}\frac{\partial r}{\partial y}+\mathbf{k}\frac{\partial
r}{\partial z}}$)
$\displaystyle=\nabla\cdot\left(-\frac{1}{r^{2}}\frac{1}{2}(r^{2})^{-\frac{1}{2}}\nabla
r^{2}\right)$ (chain rule)
$\displaystyle=\nabla\cdot\left(-\frac{1}{r^{2}}\frac{1}{2r}\nabla
r^{2}\right)=\nabla\cdot\left(-\frac{1}{2r^{3}}\nabla({\mathbf{r}}\cdot{\mathbf{r}})\right)$
(a trick: $r^{2}={\mathbf{r}}\cdot{\mathbf{r}}$)
$\displaystyle=\nabla\cdot\left(-\frac{1}{2r^{3}}2(\nabla{\mathbf{r}})\cdot{\mathbf{r}}\right)=\nabla\cdot\left(-\frac{1}{r^{3}}{\mathbf{I}}{\mathbf{r}}\right)$
(chain rule; use Eq. (A.19))
$\displaystyle=\nabla\cdot\left(-\frac{\mathbf{r}}{r^{3}}\right)\ \mbox{(so
far we have proved }\nabla\frac{1}{r}=-\frac{\mathbf{r}}{r^{3}}\mbox{ and
}\nabla r=\frac{\mathbf{r}}{r}\mbox{ )}$ (13)
$\displaystyle=-\frac{1}{r^{3}}\nabla\cdot{\mathbf{r}}-{\mathbf{r}}\cdot\nabla\frac{1}{r^{3}}$
(chain rule, use Eq. (A.46))
$\displaystyle=-\frac{3}{r^{3}}-{\mathbf{r}}\cdot\nabla\frac{1}{({\mathbf{r}}\cdot{\mathbf{r}})^{\frac{3}{2}}}$
($\nabla\cdot{\mathbf{r}}=3$;
$r=({\mathbf{r}}\cdot{\mathbf{r}})^{\frac{1}{2}}$ from
$r^{2}={\mathbf{r}}\cdot{\mathbf{r}}$)
$\displaystyle=-\frac{3}{r^{3}}+{\mathbf{r}}\cdot\frac{3}{2}\frac{1}{({\mathbf{r}}\cdot{\mathbf{r}})^{\frac{5}{2}}}\nabla({\mathbf{r}}\cdot{\mathbf{r}})$
(chain rule)
$\displaystyle=-\frac{3}{r^{3}}+{\mathbf{r}}\cdot\frac{3}{2}\frac{2{\mathbf{r}}}{r^{5}}$
(for $\nabla({\mathbf{r}}\cdot{\mathbf{r}})$, see above or use Eq. (A.19))
$\displaystyle=-\frac{3}{r^{3}}+\frac{3r^{2}}{r^{5}}$
(${\mathbf{r}}\cdot{\mathbf{r}}=r^{2}$)
$\displaystyle=\begin{cases}0,&\text{if }r\neq 0,\\\ ?,&\text{if }r=0.\mbox{
(can't cancel them out as warned by
\cite[cite]{[\@@bibref{}{steane2012relativity}{}{}]})}\end{cases}$ (14)

At $r=0$ the value is ill-defined in our calculation, but its integration over
the volume of a small ball centered at the origin is well defined. We present
here the proof based on Gauss’ Law (but see [14] for a different one),

$\displaystyle\int_{\text{ball}}\nabla^{2}\frac{1}{r}dV$
$\displaystyle=\int_{\text{ball}}-\nabla\cdot\frac{\mathbf{r}}{r^{3}}dV$ (see
remarks in Eq. (13))
$\displaystyle=-\int_{\text{ball}}\nabla\cdot\frac{\hat{\mathbf{r}}}{r^{2}}dV$
($\hat{\mathbf{r}}={\mathbf{r}}/r$)
$\displaystyle=-\int_{\text{sphere}}\frac{\hat{\mathbf{r}}}{r^{2}}\cdot
d{\mathbf{S}}$ (to surface integration by Gauss’ Law)
$\displaystyle=-\frac{1}{r_{0}^{2}}\int_{\text{sphere}}\hat{\mathbf{r}}\cdot
d{\mathbf{S}}=-\frac{1}{r_{0}^{2}}\int_{\text{sphere}}dS=-\frac{1}{r_{0}^{2}}4\pi
r_{0}^{2}=-4\pi,$

where $r_{0}$ is the radius of the ball or sphere being integrated with. This
result is just the definition of a $\delta$-function,

$\displaystyle\nabla^{2}\frac{1}{r}=-4\pi\delta({\mathbf{r}}).$ (15)

We will use this equation when we prove the theorem. ∎

6\. What really is special about $1/r$?

Answer: The above mathematical answer to question 5 is valid but not very
intuitive. The answer may be in the Taylor expansion of a decreasing function
of $r$ that is infinity at $r=0$ and zero at $r=\infty$, with terms depending
on $1/r$, $1/r^{2}$, $1/r^{3}$, etc. The author of this paper has not yet come
up with an intuitive answer.∎

7\. Why is there a coefficient $-\frac{1}{4\pi}$ in the formula?

Answer: We inspect Eq. (15) and recognize that a coefficient $-1/(4\pi)$ can
normalize $\nabla^{2}(1/r)$ so that the result, a $\delta$-function, has
coefficient one. This, in turn, is related to the fact that the area of the
sphere is $4\pi r_{0}^{2}$. ∎

8\. How can the solution be found mathematically?

Answer: As we have argued previously, the solution of the inhomogeneous scalar
wave equation is an integral of the solutions of infinitesimal or point
sources.

We derive the solution of a point source by exploiting spherical
symmetry777Jefimenko criticized [9] and [10] for their assumption of the
source being spherically symmetrical. In hindsight, we can see the authors
were talking about a point source thus spherical symmetry assumption should
not cause problems. [10] was clearer than [9] on this issue but still did not
emphasize it enough to prevent misunderstanding.. In this case, the solution
is the Green function [2] [4] [8] [11] [14] [18]. It can be derived in the
time domain [4] [5] [27] [28] [17] [22] or in the frequency domain [2] [11]
[12] [28] [29] . We reproduce the result here in the time domain.

Suppose we want to solve the scalar inhomogeneous wave equation with a point
source at $(x^{\prime},y^{\prime},z^{\prime})=(0,0,0)$. The equation is
$\nabla^{2}V=0$ everywhere except at $(0,0,0)$. We first solve $\nabla^{2}V=0$
for the whole space except for $(0,0,0)$, then we apply the results from
question 5 to $(0,0,0)$.

This problem has spherical symmetry so we can simplify the equation by using
the spherical coordinate system. We assume that since a point source $Z$ is
independent of angles, so is $V$. Thus according to Eq. (A.17) we have

$\frac{1}{r}\frac{\partial^{2}}{\partial
r^{2}}(rV)-\frac{1}{c^{2}}\frac{\partial^{2}V}{\partial t^{2}}=0.$

let $u=rV$, we have $V=u/r$. Plug it into the equation we have

$\displaystyle\frac{1}{r}\frac{\partial^{2}u}{\partial
r^{2}}-\frac{1}{c^{2}r}\frac{\partial^{2}u}{\partial t^{2}}=0$

and for places $r\neq 0$ it reduces to

$\displaystyle\frac{\partial^{2}u}{\partial
r^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}u}{\partial t^{2}}=0$ (16)

This two-variable partial differentiation equation is readily solvable with
standard methods. We shall omit the derivation here and just write down the
general solution of Eq. (16) as follws,

$\displaystyle u$ $\displaystyle=g(t-r/c)+h(t+r/c),$

where $g$ and $h$ are arbitrary functions. Thus

$\displaystyle V=\frac{g(t-r/c)}{r}+\frac{h(t+r/c)}{r},$ (17)

where $g(t-r/c)/r$ is called the retarded solution while $h(t+r/c)/r$ is
called the advanced solution.

Combining the results with the work done in question 5, we find that we have
two particular solutions,

$\displaystyle{\mathbf{V}}_{\mbox{ret}}(x,y,z,t)=$
$\displaystyle-\frac{1}{4\pi}\int\frac{{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}=t-r/c)}{r}dV^{\prime},$
$\displaystyle{\mathbf{V}}_{\mbox{adv}}(x,y,z,t)=$
$\displaystyle-\frac{1}{4\pi}\int\frac{{\mathbf{Z}}(x^{\prime},y^{\prime},z^{\prime},t^{\prime}=t+r/c)}{r}dV^{\prime},$
(18)

where “ret” stands for “retarded”, and “adv” “advanced”. Of course, a weighted
mixture of the two is also a particular solution.

The retarded solution is attractive in physics because it is compatible with
the idea of causality (But see [30] for discussions about the advanced
solution). ∎

9\. We learn from the theories that to obtain the general solution of an
inhomogeneous partial differential equation, we superpose a particular
solution of the general solution of the corresponding homogeneous partial
differential equation. How does this reconcile with the fact that we have at
least two particular solutions in this case: the retarded integral and the
advanced integral?

Answer: Of cause, we immediately know from linearity [4] that the difference

$\displaystyle{\mathbf{V}}_{\mbox{ret}}(x,y,z,t)-{\mathbf{V}}_{\mbox{adv}}(x,y,z,t)$
(19)

is a solution of the homogeneous wave equation

$\displaystyle\nabla^{2}{\mathbf{V}}(x,y,z,t)-\frac{1}{c^{2}}\frac{\partial^{2}{\mathbf{V}}(x,y,z,t)}{\partial
t^{2}}$ $\displaystyle=0.$ (20)

This solution, when added to the advanced integral, gives us the retarded
integral.

We shall think about the physical meaning of this solution. It describes a
field, infinitesimally small at the distance of infinity, whose time-varying
ripples, or waves, propagate inwards with light speed c, toward where
${\mathbf{Z}}$ could have been. The waves are not absorbed by ${\mathbf{Z}}$
because it is missing. They propagate through where ${\mathbf{Z}}$ could have
been, and then propagate outwards, toward infinity just like the waves
generated by ${\mathbf{Z}}$, had ${\mathbf{Z}}$ existed.

Of cause, there are other solutions of the homogeneous equation Eq. (20) if
spherical symmetry is not required (e.g., a plane wave). These solutions, when
superposed with the particular solutions, give us other particular solutions.
∎

Now we are ready to prove this theorem. We follow Schwartz’s proof [6] (using
the $\delta$-function888Some authors tried to avoid using the
$\delta$-function. They divided the whole space to be integrated with into two
complementary volumes, infinitesimally small ball $V_{1}$ which contained the
observation position and $V_{2}$ which did not [9] [10] [1] [17], and argued
that in $V_{1}$ the wave equation became the Poisson’s equation
($\nabla^{2}V\neq 0$). We think this approach is less clear and merely
reproduces the work done by the $\delta$-function.) to show that the solution
satisfies the equation and the conditions. Though he proved the theorem for
potentials, the mathematics is similar.

###### Proof.

We demonstrate that Eq. (12) satisfies Eq. (11).

$\displaystyle\nabla^{2}V=$
$\displaystyle-\frac{1}{4\pi}\int\nabla^{2}\frac{[Z]}{r}dV^{\prime}$ (use Eq.
(12); linearity property) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\nabla\cdot\nabla\frac{[Z]}{r}dV^{\prime}$
(definition of $\nabla^{2}$) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\nabla\cdot\left(\frac{\nabla{[Z]}}{r}+{[Z]}\nabla\frac{1}{r}\right)dV^{\prime}$
(chain rule) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{\nabla^{2}[Z]}{r}+\nabla\frac{1}{r}\cdot\nabla[Z]+[Z]\nabla^{2}\frac{1}{r}+\nabla[Z]\cdot\nabla\frac{1}{r}\right)dV^{\prime}$
(chain rule) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{\nabla^{2}[Z]}{r}+2\nabla\frac{1}{r}\cdot\nabla[Z]+[Z]\nabla^{2}\frac{1}{r}\right)dV^{\prime}$
(combine terms) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{\nabla^{2}[Z]}{r}-2\frac{1}{r^{2}}\nabla
r\cdot\frac{\partial[Z]}{\partial t^{\prime}}\nabla
t^{\prime}+[Z]\nabla^{2}\frac{1}{r}\right)dV^{\prime}$ (chain rule; $[Z]$
depends on $(x,y,z)$ only through $t^{\prime}$) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{\nabla^{2}[Z]}{r}+2\frac{1}{r^{2}}\nabla
r\cdot\frac{\partial[Z]}{\partial t^{\prime}}\frac{1}{c}\nabla
r+[Z]\nabla^{2}\frac{1}{r}\right)dV^{\prime}$ (use $t^{\prime}=t-r/c$)
$\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{\nabla^{2}[Z]}{r}+\frac{2}{cr^{2}}\frac{\partial[Z]}{\partial
t^{\prime}}+[Z]\nabla^{2}\frac{1}{r}\right)dV^{\prime}.$ (see remarks in Eq.
(13)) (21)

The first term has $\nabla^{2}[Z]$ which we process as follows,

$\displaystyle\nabla^{2}[Z]=$ $\displaystyle\nabla\cdot(\nabla[Z])$
$\displaystyle=$ $\displaystyle\nabla\cdot\left(\frac{\partial[Z]}{\partial
t^{\prime}}\nabla t^{\prime}\right)$ ($[Z]$ depends on $(x,y,z)$ only through
$t^{\prime}$) $\displaystyle=$ $\displaystyle\nabla\frac{\partial[Z]}{\partial
t^{\prime}}\cdot\nabla t^{\prime}+\frac{\partial[Z]}{\partial
t^{\prime}}\nabla\cdot\nabla t^{\prime}$ (use Eq. (A.46)) $\displaystyle=$
$\displaystyle\frac{\partial{\nabla[Z]}}{\partial t^{\prime}}\cdot\nabla
t^{\prime}+\frac{\partial[Z]}{\partial t^{\prime}}\nabla\cdot\nabla
t^{\prime}$ (linearity property) $\displaystyle=$
$\displaystyle\frac{\partial^{2}[Z]}{\partial t^{\prime 2}}\nabla
t^{\prime}\cdot\nabla t^{\prime}+\frac{\partial[Z]}{\partial
t^{\prime}}\nabla\cdot\nabla t^{\prime}$ ($[Z]$ depends on $(x,y,z)$ only
through $t^{\prime}$) $\displaystyle=$
$\displaystyle\frac{\partial^{2}[Z]}{\partial t^{\prime
2}}\frac{1}{c^{2}}(-\nabla r)\cdot(-\nabla r)-\frac{\partial[Z]}{\partial
t^{\prime}}\frac{1}{c}\nabla\cdot\nabla r\ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{(use
$t^{\prime}=t-r/c$)}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial t^{\prime
2}}-\frac{1}{c}\frac{\partial[Z]}{\partial
t^{\prime}}\nabla\cdot\frac{\mathbf{r}}{r}\ \ \ \ \ \ \ \ \mbox{(see remarks
in Eq. (\ref{nablar}))}$ (22) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}-\frac{1}{c}\frac{\partial[Z]}{\partial
t^{\prime}}\nabla\cdot\frac{\mathbf{r}}{r}$ (use
$\frac{\partial^{2}[Z]}{\partial t^{\prime 2}}=\frac{\partial^{2}[Z]}{\partial
t^{2}}$) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}-\frac{1}{c}\frac{\partial[Z]}{\partial
t^{\prime}}\left(\frac{1}{r}\nabla\cdot{\mathbf{r}}+\nabla\frac{1}{r}\cdot{\mathbf{r}}\right)$
(use Eq. (A.46)) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}-\frac{1}{c}\frac{\partial[Z]}{\partial
t^{\prime}}\left(\frac{3}{r}-\frac{1}{r^{2}}\nabla r\cdot{\mathbf{r}}\right)$
($\nabla\cdot r=3$ and chain rule) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}-\frac{1}{c}\frac{\partial[Z]}{\partial
t^{\prime}}\left(\frac{3}{r}-\frac{1}{r}\right)$ (see remarks in Eq. (13))
$\displaystyle=$ $\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}-\frac{2}{cr}\frac{\partial[Z]}{\partial t^{\prime}}$ (see next footnote
for combining terms)

Plug this result back into Eq. (21) and get

$\displaystyle\nabla^{2}V=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{1}{r}\left(\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}-\frac{2}{cr}\frac{\partial[Z]}{\partial
t^{\prime}}\right)+\frac{2}{cr^{2}}\frac{\partial[Z]}{\partial
t^{\prime}}+[Z]\nabla^{2}\frac{1}{r}\right)dV^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\left(\frac{1}{r}\frac{1}{c^{2}}\frac{\partial^{2}[Z]}{\partial
t^{2}}+[Z]\nabla^{2}\frac{1}{r}\right)dV^{\prime}$ (cancel out
terms99footnotemark: 9) $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int\frac{[Z]}{r}dV^{\prime}-\frac{1}{4\pi}\int[Z]\nabla^{2}\frac{1}{r}dV^{\prime}$
(linearity property) $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}V-\frac{1}{4\pi}\int[Z]\nabla^{2}\frac{1}{r}dV^{\prime},$ (use Eq. (10)
for the first term)

and after rearrangement,

$\displaystyle\nabla^{2}V-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}V=$
$\displaystyle-\frac{1}{4\pi}\int[Z]\nabla^{2}\frac{1}{r}dV^{\prime}$
$\displaystyle=$ $\displaystyle\int
Z(x^{\prime},y^{\prime},z^{\prime},t^{\prime}=t-r/c)\delta({\mathbf{r}})dV^{\prime}$
(use Eq. (15)) $\displaystyle=$ $\displaystyle Z(x,y,z,t)$ (property of the
$\delta$-function; $t^{\prime}=t$ at $\mathbf{r}=0$)

∎

##  4 Jefimenko’s Equations

We apply Theorem 1 directly to the inhomogeneous wave equations Eq. (7) and
Eq. (8) and obtain

$\displaystyle{\mathbf{E}}=$
$\displaystyle-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\nabla^{\prime}\rho+\dfrac{1}{c^{2}}\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]}{r}dV^{\prime}$ (23) $\displaystyle{\mathbf{B}}=$
$\displaystyle\frac{\mu_{0}}{4\pi}\int\frac{[\nabla^{\prime}\times{\mathbf{J}}]}{r}dV^{\prime}$
(24)

These equations relate ${\mathbf{E}}$ and ${\mathbf{B}}$ to their causal
sources, and are the particular solutions of Maxwell’s Equations subject to
the conditions that the fields are zero at positions far, far away from the
sources, and that causality is assumed.

By manipulating these two equations, we can derive related formulas as
follows,

$\displaystyle{\mathbf{E}}$
$\displaystyle=-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\nabla^{\prime}\rho+\dfrac{1}{c^{2}}\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]}{r}dV^{\prime}$
$\displaystyle=-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\nabla^{\prime}\rho\right]}{r}dV^{\prime}-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\dfrac{1}{c^{2}}\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]}{r}dV^{\prime}$
$\displaystyle=-\frac{1}{4\pi\epsilon_{0}}\int\nabla\frac{\left[\rho\right]}{r}dV^{\prime}-\frac{1}{4\pi\epsilon_{0}}\int\nabla^{\prime}\frac{\left[\rho\right]}{r}dV^{\prime}-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\dfrac{1}{c^{2}}\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]}{r}dV^{\prime}$ (use Eq. (A.55))
$\displaystyle=-\frac{1}{4\pi\epsilon_{0}}\int\nabla\frac{\left[\rho\right]}{r}dV^{\prime}-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\dfrac{1}{c^{2}}\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]}{r}dV^{\prime}\ \ \ \ \ \ \ \ \mbox{(use Eq.
(\ref{eqn:intgradient})})$ (25)
$\displaystyle=\frac{1}{4\pi\epsilon_{0}}\int\left(\frac{[\rho]}{r^{3}}+\frac{1}{r^{2}c}\left[\frac{\partial\rho}{\partial
t^{\prime}}\right]\right){\mathbf{r}}dV^{\prime}-\frac{1}{4\pi\epsilon_{0}c^{2}}\int\frac{1}{r}\left[\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]dV^{\prime},\mbox{(use (\ref{eqn:gradientZr}))}$ (26)

and,

$\displaystyle{\mathbf{B}}$
$\displaystyle=\frac{\mu_{0}}{4\pi}\int\frac{[\nabla^{\prime}\times{\mathbf{J}}]}{r}dV^{\prime}$
$\displaystyle=\frac{\mu_{0}}{4\pi}\int\nabla\times\frac{[{\mathbf{J}}]}{r}dV^{\prime}+\frac{\mu_{0}}{4\pi}\int\nabla^{\prime}\times\frac{[{\mathbf{J}}]}{r}dV^{\prime}$
(use Eq. (A.56))
$\displaystyle=\frac{\mu_{0}}{4\pi}\int\nabla\times\frac{[{\mathbf{J}}]}{r}dV^{\prime}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\mbox{(use Eq. (\ref{eqn:intcurl}))}$ (27)
$\displaystyle=\frac{\mu_{0}}{4\pi}\int\left(\frac{[\mathbf{J}]}{r^{3}}+\frac{1}{r^{2}c}\left[\frac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]\right)\times{\mathbf{r}}\ dV^{\prime}\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \mbox{(use Eq. (\ref{eqn:curlZr}))}$ (28)

Eq. (26) and Eq. (28) are called Jefimenko’s Equations. There are other
tightly related equations [2] [21].

##  5 Retarded Potentials

Retarded potentials111111Here we treat potentials as tools that simplify
calculations. There are physicists who think potentials are real and have real
physical effects (e.g., the Aharonov–Bohm effect). Under different gauge
conditions, there are different potentials. can be introduced based on the
solutions of ${\mathbf{E}}$ and ${\mathbf{B}}$.

As to the magnetic field ${\mathbf{B}}$, we have

$\displaystyle{\mathbf{B}}=$
$\displaystyle\frac{\mu_{0}}{4\pi}\int\nabla\times\frac{[{\mathbf{J}}]}{r}dV^{\prime}$
(Eq. (27) revisited) $\displaystyle=$
$\displaystyle\nabla\times\left(\frac{\mu_{0}}{4\pi}\int\frac{[{\mathbf{J}}]}{r}dV^{\prime}\right)$
(linearity property)
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}$
$\displaystyle\nabla\times{\mathbf{A}},$

if we define retarded vector potential

$\displaystyle{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\int{\frac{[{\mathbf{J}}]}{r}}dV^{\prime}.$

As to the electric field ${\mathbf{E}}$, we have

$\displaystyle{\mathbf{E}}=$
$\displaystyle-\frac{1}{4\pi\epsilon_{0}}\int\nabla\frac{\left[\rho\right]}{r}dV^{\prime}-\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\dfrac{1}{c^{2}}\dfrac{\partial{\mathbf{J}}}{\partial
t^{\prime}}\right]}{r}dV^{\prime}$ (Eq. (25) revisited) $\displaystyle=$
$\displaystyle-\nabla\left(\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\rho\right]}{r}dV^{\prime}\right)-\frac{\partial}{\partial(t-r/c)}\left(\frac{\mu_{0}}{4\pi}\int\frac{[{\mathbf{J}}]}{r}dV^{\prime}\right)$
(linearity property; $\left[\frac{\partial\mathbf{J}}{\partial
t^{\prime}}\right]=\frac{\partial\mathbf{J}(x^{\prime},y^{\prime},z^{\prime},t^{\prime})}{\partial
t^{\prime}}|_{t^{\prime}=t-r/c}=\frac{\partial\mathbf{J}(x^{\prime},y^{\prime},z^{\prime},t-r/c)}{\partial(t-r/c)}$)
$\displaystyle=$
$\displaystyle-\nabla\left(\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\rho\right]}{r}dV^{\prime}\right)-\frac{\partial}{\partial
t}\left(\frac{\mu_{0}}{4\pi}\int\frac{[{\mathbf{J}}]}{r}dV^{\prime}\right)$
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}$
$\displaystyle-\nabla\varphi-\frac{\partial{\mathbf{A}}}{\partial t},$

again if we define retarded scalar potential $\varphi$ and borrow retarded
vector potential ${\mathbf{A}}$ from above such that

$\displaystyle\varphi=$
$\displaystyle\frac{1}{4\pi\epsilon_{0}}\int\frac{\left[\rho\right]}{r}dV^{\prime}$
(29) $\displaystyle{\mathbf{A}}=$
$\displaystyle\frac{\mu_{0}}{4\pi}\int\frac{[{\mathbf{J}}]}{r}dV^{\prime}.$
(30)

We collect the two equations relating fields to potentials as follows,

$\displaystyle{\mathbf{E}}=$
$\displaystyle-\nabla\varphi-\frac{\partial{\mathbf{A}}}{\partial t},$ (31)
$\displaystyle{\mathbf{B}}=$ $\displaystyle\nabla\times{\mathbf{A}}.$ (32)

##  6 Electromagnetic Induction

Jefimenko argued that since Eq. (26) and (28) show that ${\mathbf{E}}$ and
${\mathbf{B}}$ are generated by their corresponding sources, electric
induction (changing electric field induces magnetic field) and magnetic
induction (changing magnetic field induces electric field) are just illusions.

This argument sheds some light on electromagnetic phenomena. After all, in
electromagnetic waves, the electric field and magnetic field are in phase
[13], reaching peaks and troughs together, not alternatively.

He then argued that the physics world has long ignored the “electrokinetic
field” he defined, $-\frac{\mu_{0}}{4\pi}\int(\partial[{\mathbf{J}}]/\partial
t)/r\ dV^{\prime}$ (the second term of Eq. (26)), and spent two pages arguing
its significance. But since his results are the same as those of classical
electrodynamics, this argument may not be true.

##  7 The Fields of an Arbitrarily Moving Point Charge

Jefimenko spent chapters 3-5 of his book on finding electric field
${\mathbf{E}}$ and magnetic field ${\mathbf{B}}$ of a few case studies of
moving charge distributions. However, current distributions are not included,
except for current induced by the moving charges, making the case studies not
very general.

A special case about an arbitrarily moving point charge is interesting because
it is related to subatomic charged particles and thus is included in most
electrodynamics textbooks.

For this special case, a straightforward approach to obtaining $\mathbf{E}$
and $\mathbf{B}$ used by Jefimenko is to find the field expressions up to low-
order approximations of Eqs. (23) and (24), then let the size of the body
shrink infinitesimally while keeping the total charge constant.

The results are the same as those obtained in other textbooks with the
Lienard-Wiechert potential method, so are the results of such a charge moving
in a constant velocity, which is also identical to those obtained from special
relativity.

##  8 Discussion

Jefimenko’s electrodynamics textbook is a good complement to other
electrodynamics textbooks. The fact that the results regarding the electric
and magnetic fields are the same gives us confidence that they are what
Maxwell’s Equations mean to lead to.

As we move forward, we shall remind ourselves of the spectacular result that
the electric and magnetic fields of a point charge moving in constant velocity
obtained from retardation coincide with those obtained from special
relativity.

##  Appendix A Notations and Identities

In this appendix, we collect definitions and identities used in this paper.

### Definitions

We represent a vector in Cartesian coordinates as row vector $(x,y,z)$, or as
column vector $\left(\begin{array}[]{c}x\\\ y\\\
z\end{array}\right)=(x,y,z)^{T}$, where T stands for transpose, or as
$x{\mathbf{i}}+y{\mathbf{j}}+z{\mathbf{k}}$, or as
${\mathbf{i}}x+{\mathbf{j}}y+{\mathbf{k}}z$, where ${\mathbf{i}}$,
${\mathbf{j}}$, and ${\mathbf{k}}$ are unit vectors pointing to the positive
direction of the $x$, $y$, and $z$ axes, respectively, and $x$, $y$, and $z$
are coefficients. Note that we reuse $x,y,z$ symbols here both for the axes
and for the coefficients. Note that the unit vectors can be written as
${\mathbf{i}}=(1,0,0)^{T}$, ${\mathbf{j}}=(0,1,0)^{T}$,
${\mathbf{k}}=(0,0,1)^{T}$ thus

$\displaystyle x{\mathbf{i}}+y{\mathbf{j}}+z{\mathbf{k}}=$ $\displaystyle
x\left(\begin{array}[]{c}1\\\ 0\\\
0\end{array}\right)+y\left(\begin{array}[]{c}0\\\ 1\\\
0\end{array}\right)+k\left(\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right)=\left(\begin{array}[]{c}x\\\ y\\\ z\end{array}\right).$
(A.13)

We write down the space derivatives of a field in Cartesian coordinates,
Gradient:

$\displaystyle\nabla a={\mathbf{i}}\frac{\partial a}{\partial
x}+{\mathbf{j}}\frac{\partial a}{\partial y}+{\mathbf{k}}\frac{\partial
a}{\partial z},$

Divergence:

$\displaystyle\nabla\cdot{\mathbf{a}}=\frac{\partial a_{x}}{\partial
x}+\frac{\partial a_{y}}{\partial y}+\frac{\partial a_{z}}{\partial z},$

Curl:

$\displaystyle\nabla\times{\mathbf{a}}$
$\displaystyle=\left|\begin{array}[]{ccc}{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}}\\\
\frac{\partial}{\partial x}&\frac{\partial}{\partial
y}&\frac{\partial}{\partial z}\\\ a_{x}&a_{y}&a_{z}\end{array}\right|$ ($|\ |$
stands for determinant) $\displaystyle={\mathbf{i}}\left(\frac{\partial
a_{z}}{\partial y}-\frac{\partial a_{y}}{\partial
z}\right)+{\mathbf{j}}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial
a_{z}}{\partial x}\right)+{\mathbf{k}}\left(\frac{\partial a_{y}}{\partial
x}-\frac{\partial a_{x}}{\partial y}\right),$

Laplacian:

$\displaystyle\nabla^{2}a$ $\displaystyle=\nabla\cdot{\nabla
a}=\frac{\partial^{2}a}{\partial x^{2}}+\frac{\partial^{2}a}{\partial
y^{2}}+\frac{\partial^{2}a}{\partial z^{2}},$

Laplacian in spherical coordinates:

$\displaystyle\nabla^{2}V$
$\displaystyle=\frac{1}{r}\frac{\partial^{2}}{\partial
r^{2}}(rV)+\frac{1}{r^{2}\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
V}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}V}{\partial\phi^{2}}$
(A.17)

where $\phi$ is the azimuthal angle and $\theta$ is the zenith angle.

Laplacian of a vector:

$\displaystyle\nabla^{2}{\mathbf{a}}=$
$\displaystyle{\mathbf{i}}\nabla^{2}a_{x}+{\mathbf{j}}\nabla^{2}a_{y}+{\mathbf{k}}\nabla^{2}a_{z}$
$\displaystyle=$ $\displaystyle{\mathbf{i}}\nabla\cdot(\nabla
a_{x})+{\mathbf{j}}\nabla\cdot(\nabla a_{y})+{\mathbf{k}}\nabla\cdot(\nabla
a_{z})$ $\displaystyle=$
$\displaystyle{\mathbf{i}}\nabla\cdot\left({\mathbf{i}}\frac{\partial
a_{x}}{\partial x}+{\mathbf{j}}\frac{\partial a_{x}}{\partial
y}+{\mathbf{k}}\frac{\partial a_{x}}{\partial
z}\right)+{\mathbf{j}}\nabla\cdot\left({\mathbf{i}}\frac{\partial
a_{y}}{\partial x}+{\mathbf{j}}\frac{\partial a_{y}}{\partial
y}+{\mathbf{k}}\frac{\partial a_{y}}{\partial z}\right)$ $\displaystyle\ \ \ \
\ \ \ +{\mathbf{k}}\nabla\cdot\left({\mathbf{i}}\frac{\partial a_{z}}{\partial
x}+{\mathbf{j}}\frac{\partial a_{z}}{\partial y}+{\mathbf{k}}\frac{\partial
a_{z}}{\partial z}\right)$ $\displaystyle=$
$\displaystyle{\mathbf{i}}\left(\frac{\partial^{2}a_{x}}{\partial
x^{2}}+\frac{\partial^{2}a_{x}}{\partial
y^{2}}+\frac{\partial^{2}a_{x}}{\partial
z^{2}}\right)+{\mathbf{j}}\left(\frac{\partial^{2}a_{y}}{\partial
x^{2}}+\frac{\partial^{2}a_{y}}{\partial
y^{2}}+\frac{\partial^{2}a_{z}}{\partial z^{2}}\right)$ $\displaystyle\ \ \ \
\ \ \ +{\mathbf{k}}\left(\frac{\partial^{2}a_{z}}{\partial
x^{2}}+\frac{\partial^{2}a_{z}}{\partial
y^{2}}+\frac{\partial^{2}a_{z}}{\partial z^{2}}\right).$ (A.18)

### Identities

$\displaystyle\nabla({\mathbf{a}}\cdot{\mathbf{b}})=\nabla{\mathbf{a}}\cdot{\mathbf{b}}+\nabla{\mathbf{b}}\cdot{\mathbf{a}}$
$\displaystyle\mbox{(definition of }\nabla{\mathbf{a}}\mbox{ in proof)\ \ \ \
\ \ \ \ \ \ \ \ }$ (A.19)

###### Proof.

$\displaystyle\nabla({\mathbf{a}}\cdot{\mathbf{b}})=$
$\displaystyle\left({\mathbf{i}}\frac{\partial}{\partial
x}+{\mathbf{j}}\frac{\partial}{\partial
y}+{\mathbf{k}}\frac{\partial}{\partial
z}\right)(a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z})$ $\displaystyle=$
$\displaystyle{\mathbf{i}}\left(\frac{\partial a_{x}}{\partial
x}b_{x}+\frac{\partial a_{y}}{\partial x}b_{y}+\frac{\partial a_{z}}{\partial
x}b_{z}\right)+{\mathbf{j}}\left(\frac{\partial a_{x}}{\partial
y}b_{x}+\frac{\partial a_{y}}{\partial y}b_{y}+\frac{\partial a_{z}}{\partial
y}b_{z}\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+{\mathbf{k}}\left(\frac{\partial a_{x}}{\partial z}b_{x}+\frac{\partial
a_{y}}{\partial z}b_{y}+\frac{\partial a_{z}}{\partial z}b_{z}\right)$
$\displaystyle+{\mathbf{i}}\left(\frac{\partial b_{x}}{\partial
x}a_{x}+\frac{\partial b_{y}}{\partial x}a_{y}+\frac{\partial b_{z}}{\partial
x}a_{z}\right)+{\mathbf{j}}\left(\frac{\partial b_{x}}{\partial
y}a_{x}+\frac{\partial b_{y}}{\partial y}a_{y}+\frac{\partial b_{z}}{\partial
y}a_{z}\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+{\mathbf{k}}\left(\frac{\partial b_{x}}{\partial z}a_{x}+\frac{\partial
b_{y}}{\partial z}a_{y}+\frac{\partial b_{z}}{\partial z}a_{z}\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}\frac{\partial
a_{x}}{\partial x}&\frac{\partial a_{y}}{\partial x}&\frac{\partial
a_{z}}{\partial x}\\\ \frac{\partial a_{x}}{\partial y}&\frac{\partial
a_{y}}{\partial y}&\frac{\partial a_{z}}{\partial y}\\\ \frac{\partial
a_{x}}{\partial z}&\frac{\partial a_{y}}{\partial z}&\frac{\partial
a_{z}}{\partial z}\end{array}\right)\left(\begin{array}[]{c}b_{x}\\\ b_{y}\\\
b_{z}\end{array}\right)+\left(\begin{array}[]{ccc}\frac{\partial
b_{x}}{\partial x}&\frac{\partial b_{y}}{\partial x}&\frac{\partial
b_{z}}{\partial x}\\\ \frac{\partial b_{x}}{\partial y}&\frac{\partial
b_{y}}{\partial y}&\frac{\partial b_{z}}{\partial y}\\\ \frac{\partial
b_{x}}{\partial z}&\frac{\partial b_{y}}{\partial z}&\frac{\partial
b_{z}}{\partial z}\end{array}\right)\left(\begin{array}[]{c}a_{x}\\\ a_{y}\\\
a_{z}\end{array}\right)$ (A.32) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\partial a_{x}}{\partial
x}&\frac{\partial a_{x}}{\partial y}&\frac{\partial a_{x}}{\partial z}\\\
\frac{\partial a_{y}}{\partial x}&\frac{\partial a_{y}}{\partial
y}&\frac{\partial a_{y}}{\partial z}\\\ \frac{\partial a_{z}}{\partial
x}&\frac{\partial a_{z}}{\partial y}&\frac{\partial a_{z}}{\partial
z}\end{array}\right)^{T}\left(\begin{array}[]{c}b_{x}\\\ b_{y}\\\
b_{z}\end{array}\right)+\left(\begin{array}[]{ccc}\frac{\partial
b_{x}}{\partial x}&\frac{\partial b_{x}}{\partial y}&\frac{\partial
b_{x}}{\partial z}\\\ \frac{\partial b_{y}}{\partial x}&\frac{\partial
b_{y}}{\partial y}&\frac{\partial b_{y}}{\partial z}\\\ \frac{\partial
b_{z}}{\partial x}&\frac{\partial b_{z}}{\partial y}&\frac{\partial
b_{z}}{\partial z}\end{array}\right)^{T}\left(\begin{array}[]{c}a_{x}\\\
a_{y}\\\ a_{z}\end{array}\right)$ (A.45)
$\displaystyle\stackrel{{\scriptstyle\operatorname{def}}}{{=}}$
$\displaystyle(\nabla{\mathbf{a}})^{T}{\mathbf{b}}+(\nabla{\mathbf{b}})^{T}{\mathbf{a}}\stackrel{{\scriptstyle\operatorname{def}}}{{=}}\nabla{\mathbf{a}}\cdot{\mathbf{b}}+\nabla{\mathbf{b}}\cdot{\mathbf{a}}$

∎

Note that $\nabla{\mathbf{a}}$ (or $\nabla{\mathbf{b}}$) is a $3\times 3$
matrix. Its dot product with a vector is defined here as the product of the
transpose of the matrix and the vector. This definition is consistent with the
dot product of two vectors, i.e.,
${\mathbf{a}}\cdot{\mathbf{b}}={\mathbf{a}}^{T}{\mathbf{b}}$ (a scalar) where
${\mathbf{a}}$, ${\mathbf{b}}$ are $3\times 1$ vectors.

This formula when used in Electrodynamics is much better than another one used
in many textbooks [1] [6],

$\nabla({\mathbf{a}}\cdot{\mathbf{b}})=({\mathbf{a}}\cdot\nabla){\mathbf{b}}+({\mathbf{b}}\cdot\nabla){\mathbf{a}}+{\mathbf{a}}\times(\nabla\times{\mathbf{b}})+{\mathbf{b}}\times(\nabla\times{\mathbf{a}}),$

because the former formula exposes the physical intuition of the derivative of
a (dot) product of two factors, that the result is the sum of two terms, each
is the derivative of one factor multiplied by the other. It is a well-known
identity, but its first use in the differentiation of electromagnetic
potentials, as far as the author knows, is in Zucchini’s master’s thesis [31].

$\displaystyle\nabla\cdot(a{\mathbf{b}})$ $\displaystyle=\nabla
a\cdot{\mathbf{b}}+a\nabla\cdot{\mathbf{b}}$ (A.46)

###### Proof.

$\displaystyle\nabla\cdot(a{\mathbf{b}})=$
$\displaystyle\frac{\partial}{\partial x}(ab_{x})+\frac{\partial}{\partial
y}(ab_{y})+\frac{\partial}{\partial z}(ab_{z})$ $\displaystyle=$
$\displaystyle\frac{\partial a}{\partial x}b_{x}+\frac{\partial a}{\partial
y}b_{y}+\frac{\partial a}{\partial z}b_{z}+a\frac{\partial b_{x}}{\partial
x}+a\frac{\partial b_{y}}{\partial y}+a\frac{\partial b_{z}}{\partial z}$
$\displaystyle=$ $\displaystyle\left(\mathbf{i}\frac{\partial a}{\partial
x}+\mathbf{j}\frac{\partial a}{\partial y}+\mathbf{k}\frac{\partial
a}{\partial
z}\right)\cdot(\mathbf{i}b_{x}+\mathbf{j}b_{y}+\mathbf{k}b_{z})+a\nabla\cdot{\mathbf{b}}$
$\displaystyle=$ $\displaystyle\nabla
a\cdot{\mathbf{b}}+a\nabla\cdot{\mathbf{b}}$

∎

$\displaystyle\nabla\cdot\nabla\times{\mathbf{a}}$ $\displaystyle=0$ (A.47)

###### Proof.

$\displaystyle\nabla\cdot\nabla\times{\mathbf{a}}$
$\displaystyle=\nabla\cdot\left({\mathbf{i}}\left(\frac{\partial
a_{z}}{\partial y}-\frac{\partial a_{y}}{\partial
z}\right)+{\mathbf{j}}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial
a_{z}}{\partial x}\right)+{\mathbf{k}}\left(\frac{\partial a_{y}}{\partial
x}-\frac{\partial a_{x}}{\partial y}\right)\right)$
$\displaystyle=\frac{\partial}{\partial x}\left(\frac{\partial a_{z}}{\partial
y}-\frac{\partial a_{y}}{\partial z}\right)+\frac{\partial}{\partial
y}\left(\frac{\partial a_{x}}{\partial z}-\frac{\partial a_{z}}{\partial
x}\right)+\frac{\partial}{\partial z}\left(\frac{\partial a_{y}}{\partial
x}-\frac{\partial a_{x}}{\partial y}\right)$ $\displaystyle=0$

∎

$\displaystyle\nabla\times(\nabla\times{\mathbf{a}})=\nabla(\nabla\cdot{\mathbf{a}})-\nabla^{2}{\mathbf{a}}$
(A.48)

###### Proof.

$\displaystyle\nabla\times(\nabla\times{\mathbf{a}})=$
$\displaystyle\left|\begin{array}[]{ccc}{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}}\\\
\frac{\partial}{\partial x}&\frac{\partial}{\partial
y}&\frac{\partial}{\partial z}\\\ \frac{\partial a_{z}}{\partial
y}-\frac{\partial a_{y}}{\partial z}&\frac{\partial a_{x}}{\partial
z}-\frac{\partial a_{z}}{\partial x}&\frac{\partial a_{y}}{\partial
x}-\frac{\partial a_{x}}{\partial y}\end{array}\right|$ (A.52)
$\displaystyle=$
$\displaystyle{\mathbf{i}}\left(\frac{\partial^{2}a_{y}}{\partial x\partial
y}-\frac{\partial^{2}a_{x}}{\partial y^{2}}-\frac{\partial^{2}a_{x}}{\partial
z^{2}}+\frac{\partial^{2}a_{z}}{\partial x\partial z}\right)$
$\displaystyle+{\mathbf{j}}\left(\frac{\partial^{2}a_{z}}{\partial y\partial
z}-\frac{\partial^{2}a_{y}}{\partial z^{2}}-\frac{\partial^{2}a_{y}}{\partial
x^{2}}+\frac{\partial^{2}a_{x}}{\partial x\partial y}\right)$
$\displaystyle+{\mathbf{k}}\left(\frac{\partial^{2}a_{x}}{\partial x\partial
z}-\frac{\partial^{2}a_{z}}{\partial x^{2}}-\frac{\partial^{2}a_{z}}{\partial
y^{2}}+\frac{\partial^{2}a_{y}}{\partial y\partial z}\right),$
$\displaystyle\nabla(\nabla\cdot{\mathbf{a}})=$
$\displaystyle\nabla\left(\frac{\partial a_{x}}{\partial x}+\frac{\partial
a_{y}}{\partial y}+\frac{\partial a_{z}}{\partial z}\right)$ $\displaystyle=$
$\displaystyle{\mathbf{i}}\left(\frac{\partial^{2}a_{x}}{\partial
x^{2}}+\frac{\partial^{2}a_{y}}{\partial x\partial
y}+\frac{\partial^{2}a_{z}}{\partial x\partial z}\right)$
$\displaystyle+{\mathbf{j}}\left(\frac{\partial^{2}a_{x}}{\partial x\partial
y}+\frac{\partial^{2}a_{y}}{\partial y^{2}}+\frac{\partial^{2}a_{z}}{\partial
y\partial z}\right)$
$\displaystyle+{\mathbf{k}}\left(\frac{\partial^{2}a_{x}}{\partial x\partial
z}+\frac{\partial^{2}a_{y}}{\partial y\partial
z}+\frac{\partial^{2}a_{z}}{\partial z^{2}}\right),$

and compare
$\nabla(\nabla\cdot{\mathbf{a}})-\nabla\times(\nabla\times{\mathbf{a}})$ to
$\nabla^{2}\mathbf{a}$ (see Eq. (A.18)) we know that the identity is proved. ∎

### Manipulations of Retarded Quantities

In our derivations, we often need to perform mathematical manipulations of
expressions involving retarded quantities. Retarded quantities often emerge
with the “source” and are with the primed coordinates. Retarded quantities are
often represented with the retardation operator $[\ ]$.

Recall that we let $\nabla$ denote derivatives with respect to the ordinary
coordinates and $\nabla^{\prime}$ with respect to the primed coordinates. We
use $\mathbf{r}$ to represent the vector distance from the source at
$(x^{\prime},y^{\prime},z^{\prime})$ to the field at $(x,y,z)$ as
$\mathbf{r}=\mathbf{i}(x-x^{\prime})+\mathbf{j}(y-y^{\prime})+\mathbf{k}(z-z^{\prime})$
and $r$ the scalar distance
$r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}$.

$\displaystyle\frac{\partial}{\partial
x}f({\mathbf{r}})+\frac{\partial}{\partial x^{\prime}}f({\mathbf{r}})=0,$
$\displaystyle\frac{\partial}{\partial x}f(r)+\frac{\partial}{\partial
x^{\prime}}f(r)=0,$ $\displaystyle\frac{\partial}{\partial
x}{\mathbf{f}}({\mathbf{r}})+\frac{\partial}{\partial
x^{\prime}}{\mathbf{f}}({\mathbf{r}})=0,$
$\displaystyle\frac{\partial}{\partial
x}{\mathbf{f}}(r)+\frac{\partial}{\partial x^{\prime}}{\mathbf{f}}(r)=0,$
(A.53)

where $f({\mathbf{r}})$, $f(r)$, ${\mathbf{f}}({\mathbf{r}})$ and
${\mathbf{f}}(r)$ are any scalar or vector functions of the vector distance
${\mathbf{r}}$ or the scalar distance $r$ between the field position and the
source position.

The intuition of these equations is that, if the field and its source are
related only through the distance vector, moving the observation position is
equivalent to moving the source position in the opposite direction. This is
the familiar Galilean relativity.

###### Proof.

It suffices to prove the first equation.

$\displaystyle\frac{\partial}{\partial
x}f({\mathbf{r}})+\frac{\partial}{\partial x^{\prime}}f({\mathbf{r}})=$
$\displaystyle\frac{\partial
f({\mathbf{r}})}{\partial(x-x^{\prime})}\frac{\partial(x-x^{\prime})}{\partial
x}+\frac{\partial
f({\mathbf{r}})}{\partial(x-x^{\prime})}\frac{\partial(x-x^{\prime})}{\partial
x^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{\partial
f({\mathbf{r}})}{\partial(x-x^{\prime})}(\frac{\partial(x-x^{\prime})}{\partial
x}+\frac{\partial(x-x^{\prime})}{\partial x^{\prime}})$ $\displaystyle=$
$\displaystyle 0$

∎

$\displaystyle\frac{[\partial Z/\partial
x^{\prime}]}{r}=\frac{\partial}{\partial
x}\frac{[Z]}{r}+\frac{\partial}{\partial x^{\prime}}\frac{[Z]}{r},$
$\displaystyle\frac{[\partial Z/\partial
y^{\prime}]}{r}=\frac{\partial}{\partial
y}\frac{[Z]}{r}+\frac{\partial}{\partial y^{\prime}}\frac{[Z]}{r},$
$\displaystyle\frac{[\partial Z/\partial
z^{\prime}]}{r}=\frac{\partial}{\partial
z}\frac{[Z]}{r}+\frac{\partial}{\partial z^{\prime}}\frac{[Z]}{r}.$ (A.54)

###### Proof.

Because of symmetry we only need to prove the first equation.

$\displaystyle\frac{\partial}{\partial
x}\frac{[Z]}{r}+\frac{\partial}{\partial x^{\prime}}\frac{[Z]}{r}$
$\displaystyle=$ $\displaystyle[Z]\frac{\partial}{\partial
x}\frac{1}{r}+\frac{1}{r}\frac{\partial}{\partial
x}[Z]+[Z]\frac{\partial}{\partial
x^{\prime}}\frac{1}{r}+\frac{1}{r}\frac{\partial}{\partial x^{\prime}}[Z]$
(chain rule) $\displaystyle=$ $\displaystyle[Z]\left(\frac{\partial}{\partial
x}\frac{1}{r}+\frac{\partial}{\partial
x^{\prime}}\frac{1}{r}\right)+\frac{1}{r}\left(\frac{\partial}{\partial
x}[Z]+\frac{\partial}{\partial x^{\prime}}[Z]\right)$ (combine terms)
$\displaystyle=$ $\displaystyle\frac{1}{r}\left(\frac{\partial}{\partial
x}\bigg{\rvert}_{t^{\prime}}[Z]+\frac{\partial}{\partial
t^{\prime}}[Z]\frac{\partial t^{\prime}}{\partial
x}\right.+\left.\frac{\partial}{\partial
x^{\prime}}\bigg{\rvert}_{t^{\prime}}[Z]+\frac{\partial}{\partial
t^{\prime}}[Z]\frac{\partial t^{\prime}}{\partial x^{\prime}}\right)$ (use Eq.
(A.53)) $\displaystyle=$
$\displaystyle\frac{1}{r}\left(\frac{\partial}{\partial
t^{\prime}}[Z](\frac{\partial t^{\prime}}{\partial x}+\frac{\partial
t^{\prime}}{\partial x^{\prime}})+\frac{\partial}{\partial
x^{\prime}}\bigg{\rvert}_{t^{\prime}}[Z]\right)$ ($[Z]$ depends on $x$ only
through $t^{\prime}$ ) $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
x^{\prime}}\bigg{\rvert}_{t^{\prime}}[Z]$ (use Eq. (A.53), $t^{\prime}=t-r/c$
is a function of $r$) $\displaystyle=$ $\displaystyle\frac{[\partial
Z/\partial x^{\prime}]}{r},$

where $\frac{\partial}{\partial x^{\prime}}|_{t^{\prime}}$ means when
calculating the derivatives, we do not go into $t^{\prime}$ with the chain
rule. ∎

We notice that if we replace the factor $1/r$ with any function $f(r)$, the
equations still hold.

The intuition of these equations is, if we co-move $x$ and $x^{\prime}$, $y$
and $y^{\prime}$, or $z$ and $z^{\prime}$, the net change of the right hand of
the equations is caused by the change in $Z$ value due to change of the primed
coordinates.

$\displaystyle\frac{[\nabla^{\prime}Z]}{r}=\nabla\frac{[Z]}{r}+\nabla^{\prime}\frac{[Z]}{r}$
(A.55)
$\displaystyle\frac{[\nabla^{\prime}\times{\mathbf{Z}}]}{r}=\nabla\times\frac{[{\mathbf{Z}}]}{r}+\nabla^{\prime}\times\frac{[{\mathbf{Z}}]}{r}$
(A.56)

###### Proof.

We notice that these are simple extensions of Eq. (A.54). ∎

$\displaystyle\int\partial/\partial x^{\prime}$ $\displaystyle
f(x^{\prime},y^{\prime},z^{\prime})dx^{\prime}=0,$
$\displaystyle\int\partial/\partial y^{\prime}$ $\displaystyle
f(x^{\prime},y^{\prime},z^{\prime})dy^{\prime}=0,$
$\displaystyle\int\partial/\partial z^{\prime}$ $\displaystyle
f(x^{\prime},y^{\prime},z^{\prime})dz^{\prime}=0,$ (A.57)

if $f(x^{\prime},y^{\prime},z^{\prime})$ is zero outside of a finite region of
space.

###### Proof.

Because of symmetry we only need to prove the first equation.

$\displaystyle\int\partial/\partial
x^{\prime}f(x^{\prime},y^{\prime},z^{\prime})dx^{\prime}$ $\displaystyle=\int
d_{x^{\prime}}(f(x^{\prime},y^{\prime},z^{\prime})$
$\displaystyle=f(\infty,y^{\prime},z^{\prime})-f(-\infty,y^{\prime},z^{\prime})=0$

∎

The intuition is that, if we start from the foot on the left side of a hill,
climb to the peak, descend to the right, and reach the foot of the hill on the
right side, the net height we cover is zero.

$\displaystyle\int\nabla^{\prime}f(x^{\prime},y^{\prime},z^{\prime})=0,$
(A.58) $\displaystyle\int\nabla^{\prime}\times
f(x^{\prime},y^{\prime},z^{\prime})=0,$ (A.59)

if $f(x^{\prime},y^{\prime},z^{\prime})$ is zero outside of a finite region of
space.

###### Proof.

We notice that these are simple extensions of Eq. (A.57). ∎

This proof is simple because we have limited
$f(x^{\prime},y^{\prime},z^{\prime})$ to be zero outside of a finite region of
space. A more general version of the condition requires the value of $f$ to
vanish at infinity with a rate sufficiently fast, regarding distance $r$.

$\displaystyle\nabla\frac{[Z]}{r}$
$\displaystyle=-\frac{\mathbf{r}}{r^{3}}[Z]-\frac{\mathbf{r}}{r^{2}c}\left[\frac{\partial{Z}}{\partial
t^{\prime}}\right]$ (A.60)

###### Proof.

$\displaystyle\nabla\frac{[Z]}{r}$ $\displaystyle=$
$\displaystyle[{Z}]\nabla\frac{1}{r}+\frac{1}{r}\nabla[Z]$ (chain rule)
$\displaystyle=$
$\displaystyle[{Z}]\left(-\frac{\mathbf{r}}{r^{3}}\right)+\frac{1}{r}\frac{\partial[Z]}{\partial
t^{\prime}}\nabla t^{\prime}$ (see remarks in Eq. (13)) $\displaystyle=$
$\displaystyle[{Z}]\left(-\frac{\mathbf{r}}{r^{3}}\right)+\frac{1}{r}\left[\frac{\partial{Z}}{\partial
t^{\prime}}\right]\nabla(t-\frac{r}{c})$ $\displaystyle=$
$\displaystyle[{Z}]\left(-\frac{\mathbf{r}}{r^{3}}\right)-\frac{1}{rc}\left[\frac{\partial{Z}}{\partial
t^{\prime}}\right]\nabla r$ $\displaystyle=$
$\displaystyle-\frac{\mathbf{r}}{r^{3}}[{Z}]-\frac{\mathbf{r}}{r^{2}c}\left[\frac{\partial{Z}}{\partial
t^{\prime}}\right].$ (see remarks in Eq. (13))

∎

$\displaystyle\nabla\times\frac{[{\mathbf{Z}}]}{r}=\left(\frac{1}{r^{3}}[{\mathbf{Z}}]+\frac{1}{r^{2}c}\left[\frac{\partial{\mathbf{Z}}}{\partial
t^{\prime}}\right]\right)\times{\mathbf{r}}$ (A.61)

###### Proof.

$\displaystyle\nabla\times\frac{[{\mathbf{Z}}]}{r}$ $\displaystyle=$
$\displaystyle\nabla\frac{1}{r}\times[{\mathbf{Z}}]+\frac{1}{r}\nabla\times[{\mathbf{Z}}]$
(chain rule) $\displaystyle=$
$\displaystyle-\frac{\mathbf{r}}{r^{3}}\times[{\mathbf{Z}}]+\frac{1}{r}\left|\begin{array}[]{ccc}{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}}\\\
\frac{\partial}{\partial x}&\frac{\partial}{\partial
y}&\frac{\partial}{\partial z}\\\ {[}Z_{x}]&[Z_{y}]&[Z_{z}]\\\
\end{array}\right|$ (see remarks in Eq. (13)) $\displaystyle=$
$\displaystyle\frac{1}{r^{3}}[{\mathbf{Z}}]\times{\mathbf{r}}+\frac{1}{r}\left|\begin{array}[]{ccc}{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}}\\\
\frac{\partial t^{\prime}}{\partial x}&\frac{\partial t^{\prime}}{\partial
y}&\frac{\partial t^{\prime}}{\partial z}\\\ \frac{\partial[Z_{x}]}{\partial
t^{\prime}}&\frac{\partial[Z_{y}]}{\partial
t^{\prime}}&\frac{\partial[Z_{z}]}{\partial t^{\prime}}\end{array}\right|$
(A.68) $\displaystyle=$
$\displaystyle\frac{1}{r^{3}}[{\mathbf{Z}}]\times{\mathbf{r}}+\frac{1}{r}\left|\begin{array}[]{ccc}{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}}\\\
-\frac{1}{c}\frac{\partial r}{\partial x}&-\frac{1}{c}\frac{\partial
r}{\partial y}&-\frac{1}{c}\frac{\partial r}{\partial z}\\\
\frac{\partial[Z_{x}]}{\partial t^{\prime}}&\frac{\partial[Z_{y}]}{\partial
t^{\prime}}&\frac{\partial[Z_{z}]}{\partial t^{\prime}}\end{array}\right|$
(because $t^{\prime}=t-r/c$) $\displaystyle=$
$\displaystyle\frac{1}{r^{3}}[{\mathbf{Z}}]\times{\mathbf{r}}+\frac{1}{r}\left|\begin{array}[]{ccc}{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}}\\\
-\frac{1}{rc}r_{x}&-\frac{1}{rc}r_{y}&-\frac{1}{rc}r_{z}\\\
\frac{\partial[Z_{x}]}{\partial t^{\prime}}&\frac{\partial[Z_{y}]}{\partial
t^{\prime}}&\frac{\partial[Z_{z}]}{\partial t^{\prime}}\end{array}\right|$
(A.75) $\displaystyle=$
$\displaystyle\frac{1}{r^{3}}[{\mathbf{Z}}]\times{\mathbf{r}}-\frac{1}{r^{2}c}{\mathbf{r}}\times\frac{\partial[{\mathbf{Z}}]}{\partial
t^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{3}}[{\mathbf{Z}}]\times{\mathbf{r}}-\frac{1}{r^{2}c}{\mathbf{r}}\times\left[\frac{\partial{\mathbf{Z}}}{\partial
t^{\prime}}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{1}{r^{3}}[{\mathbf{Z}}]+\frac{1}{r^{2}c}\left[\frac{\partial{\mathbf{Z}}}{\partial
t^{\prime}}\right]\right)\times{\mathbf{r}}$

∎

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Robert Flynn, Anton Ragni

# Leveraging Cross-Utterance Context For ASR Decoding

###### Abstract

While external language models (LMs) are often incorporated into the decoding
stage of automated speech recognition systems, these models usually operate
with limited context. Cross utterance information has been shown to be
beneficial during second pass re-scoring, however this limits the hypothesis
space based on the local information available to the first pass LM. In this
work, we investigate the incorporation of long-context transformer LMs for
cross-utterance decoding of acoustic models via beam search, and compare
against results from n-best rescoring. Results demonstrate that beam search
allows for an improved use of cross-utterance context. When evaluating on the
long-format dataset AMI, results show a 0.7% and 0.3% absolute reduction on
dev and test sets compared to the single-utterance setting, with improvements
when including up to 500 tokens of prior context. Evaluations are also
provided for Tedlium-1 with less significant improvements of around 0.1%
absolute.

Index Terms: speech recognition, language modelling, cross-utterance, beam-
search, rescoring

##  1 Introduction

The decoding stage of end-to-end automated speech recognition (ASR) systems
often benefits from the use of an external language model. For Connectionist
temporal classification (CTC) [1] based acoustic models (AM) this is often
crucial for good performance due to their conditional independence assumption.
Traditionally, $n$-gram models have been used for this task, although due to
data scarcity these models are usually restricted to a limited context of 3-4
preceding words. Consequently, neural models are often applied during second
pass [2, 3, 4, 5, 6, 7, 8, 9] or first pass decoding [10, 11, 12, 13, 14]
stages for further reductions in word error rate (WER).

The advent of the transformer architecture [15] has lead increased performance
on many tasks, including ASR. In part, this can be attributed to its ability
to effectively propagate gradients across long-distances, enabling the
learning of much longer-term dependencies. The use of external transformer
language models (TLMs) to improve ASR has been investigated in prior work [3,
16, 2, 4], with results demonstrating an advantage over recurrent-based
architectures.

While many realistic use cases for ASR will involve long and continuous
conversations or talks, ASR systems are mainly trained over very short
utterances with a (often invalid) $i.i.d$ assumption. Likewise decoding/re-
scoring is typically performed over individual utterances, which can lead to
context fragmentation reductions in performance.

The contributions made as part of this work are as follows: 1. We assess the
benefit of cross-utterance information for decoding, including how much
context is useful and the impact of errors in the history. 2. We demonstrate
that external language model integration via beam search improves the ability
to utilise cross-utterance information compared to re-scoring. 3. A set of
adaptions from prior work [17, 18] are proposed for use in this task making
shallow fusion with TLMs more feasible.

The remainder of the paper is organized in the following manner: Section 2
provides a brief overview of related work, Section 3 details our method
including adaptions made for efficient decoding (3.3) and language modelling
in a conversational setting (3.4). Experimental details such as model and
decoding configuration, and datasets is provided in section 4. Results and key
findings are given in Section 5 with our conclusion in Section 6.

##  2 Related work

Long-range linguistic context has been previously been exploited in ASR
through the use of external neural language models. In [2, 5, 8, 9] cross-
utterance information is found to be beneficial for perplexity (PPL) and WER
during second-pass re-scoring with LSTM and transformer type models. Similar
findings are presented in [7] where LSTMs are adapted for a conversational
setting and effectively used for re-scoring with an extended history. However
it is not clear from these works how much context is useful for re-scoring.
First pass cross-utterance decoding using LSTM models has previously been
explored [10] using on-the-fly composition with a WSFT [19] based decoder.
This work shows some performance benefit for using the LSTM during the initial
pass compared to re-scoring. No benefit is found to using greater than 4
sentences of prior context, which may be due limitations of LSTMs use of
context [20].

##  3 Transformer Decoding

###  3.1 Transformer Language Modelling

While typically a transformer [15] may consist of a bi-directional encoder,
followed by a causal decoder, for language modelling we use the decoder-only
variant transformer. This consists of alternating multi-headed self-attention
with a causal mask, and feed-forward modules.

Given a word sequence $\bm{w}=(w_{1},..,w_{T})$ causal language models are
trained to estimate the conditional probability of $P(w_{t}|\bm{w}_{<t})$.
Word sequence probabilities can then be obtained by an expansion resulting in
$P(\bm{w})$, when working with log-likelihoods this equates to:
$\sum_{t=1}^{T}\log P(w_{t}|\bm{w}_{<t})$. For the purpose of decoding these
likelihoods can be treated as scores and combined with the AM through a log-
linear interpolation.

###  3.2 Self-Attention

Self attention, which is one of the key components of TLMs, is a mixing
operation between input tokens that enables the learning of local and long-
range statistical patterns. The input $X$ is first transformed to obtain the
query, key and value matrices.

$Q,K,V=XW^{Q},XW^{K},XW^{V}$ (1)

In this work we use the key query normalisation variant of attention [21].

$\hat{q}_{i}=\frac{q_{i}}{\lVert q_{i}\rVert},\hat{k}_{i}=\frac{k_{i}}{\lVert
k_{i}\rVert}$ (2)

This involves applying $L_{2}$ normalisation to the keys and queries,
obtaining their dot product, and scaling the result by a learnt parameter $g$.
To encode positional information information an additional matrix $P$ is added
as a bias to the similarity matrix, alongside a causal mask $M$ to prevent
interactions with future tokens, before the softmax is applied. The resultant
distribution is used to obtain a weighting of the value vectors, which is used
to produce inputs to the following layer.

$\mathrm{Attention}(\hat{Q},\hat{K},V)=\mathrm{Softmax}(g\cdot\hat{Q}\hat{K}^{T}+P+M)V$
(3)

###  3.3 Adaptions for Efficient Decoding

####  3.3.1 Key-Value Caching

Unlike recurrent-based networks TLMs do not condense the history into a single
representation, and instead process their input in parallel. For each token or
utterance passed to the model we want to avoid re-computing the previous
history as this would be extremely costly. As the language model is causal,
and prior states are not updated based on future information, it is possible
to cache keys and values from the attention sub-layers to avoid re-
computation. This is not an approximation, and can be viewed as computing the
attention in a sequential manner which is explored in prior work [18, 22].
This enables efficient cross-utterance decoding using TLMs with a linear
increase in memory for each token in the cache.

To achieve this we simply concatenate keys $\hat{K}$ and values $V$ from
previous timesteps $t_{0:t-1}$ that are cached during each forward pass with
the data from the current timestep.

$\hat{K}_{0:t},V_{0:t}=(\hat{K}_{0:t-1},\hat{K}_{t}),(V_{0:t-1},V_{t})$ (4)

With $\hat{Q}_{t},\hat{K}_{0:t},V_{0:t}$ acting as input to equation 3 for
this method. To limit the sequence length at inference time, the cache can be
truncated, when doing this we find it necessary to maintain the beginning of
sentence token at the start of the history.

####  3.3.2 Multi-Query Attention

Additionally, we employ Multi-Query Attention [17], which uses only 1 head for
both the keys and the values. This reduces slowdown due to memory bandwidth,
with negligible performance degradation, and was recently validated in large-
scale training of PaLM [23]. When decoding incrementally with a batch size of
25 and cache/history size of 500 tokens, this improves our iteration time by
74%.

###  3.4 Conversational Language Modelling

Figure 1: Modifications for language modelling in a conversational setting.

As the language model is operating over series of utterances
$U=(u_{0},...u_{t})$, where utterance boundaries may indicate speaker changes,
or changes in topic we choose to model this with a few modifications to the
architecture. These modifications are illustrated in figure 1.

A separator token $SEP$ is introduced to denote utterance boundaries, this is
passed to the model alongside the input at the end of each utterance. There
are no targets for this token and it is simply used to denote a boundary in
the history.

To predict the first token in a following utterance, we introduce initial
token prediction, where learnt per dimension scalar $\gamma$ and offset
$\beta$ transformations are applied to the logit predictions from the last
token in the current utterance. Specifically, given a sequence of logits
$L^{t}=(l^{t}_{1}...l^{t}_{n})$ from utterance $u_{t}$, probabilities for the
first token in the following utterance $y_{0}^{t+1}$ are obtained as follows:

$y^{t+1}_{1}=\mathrm{Softmax}(l^{t}_{n}\odot\gamma+\beta)$ (5)

Hence the model can learn to modulate probabilities for tokens that may be
more, or less likely at the beginning of an utterance. The inclusion of eq. 5
was motivated by the frequent speaker changes in AMI [24], with new utterances
often beginning with a hesitation such as hmm.

###  3.5 Beam Search

Language models can be combined with the output probabilities of an AM using
beam search. Ideally, we would like to find the most probable path across all
alignments using all models. However, this is infeasible so we approximate by
restricting the search to a set of top paths or beams. For each time-step $t$
we use the AM probabilities and look-ahead probabilities from the TLM to score
the vocabulary indices at that time-step $i_{t}$.

$\mathrm{Score}(i_{t})=\log P_{AM}(i_{t})+\mathrm{Score}_{LM}(i_{t})$ (6)

Our AM uses the CTC decoding algorithm [1] which features blank tokens
$\theta$ as part of the vocabulary, and allows for repetitions which are
collapsed as part of decoding. As the TLM does not provide probabilities for
repeats $i_{t}=i_{t-1}$ or blanks $\theta$ an insertion bonus $\beta$ is used
to prevent the search favouring these tokens. Additionally, a scaler $\alpha$
is used to weight the TLMs score. Hence the TLMs score is as follows [12]:

$\mathrm{Score}_{LM}(i_{t})=\begin{cases}0,&\text{if $i_{t}=\theta$ or
$i_{t}=i_{t-1}$}\\\ \alpha\log
P_{LM}(i_{t})+\beta,&\text{otherwise}\end{cases}$ (7)

##  4 Experimental Setup

###  4.1 Model and Training Specifications

The TLMs all have 12 layers with a hidden dimension of 256. For the
feedforward blocks, SwiGLU layers [25] are used with an expansion factor of 4.
Key query normalised attention is used in-place of dot product attention as in
[21]. 8 heads are used for the queries during attention with 1 head for the
keys and values [17]. For positional encoding, the dynamic position bias
proposed in [26, 27] is used. TLMs are trained without dropout, as they under-
fit the pre-training corpus. In total the TLMs amount to around 11M
parameters.

During training of the TLM windows of up to 25 prior utterances are used as
context. A TLM is pre-trained and fine-tuned for each dataset, in order to
match the AMs vocabulary and training takes around 5 days on a GTX 3060 GPU.

For the AM, we use an encoder-only Conformer architecture [28] trained with
self-conditioned CTC and intermediate losses [29]. Batch normalisation [30] is
swapped out in favour of batch renormalisation [31]. Key query normalised
attention and dynamic position biases are also used with the AM. For both AMs
a byte pair vocabulary of size 128 is used. For the Tedlium dataset, the model
has 12 layers, a hidden dimension of 256, 8 attention heads, and a convolution
kernel width of 31. For AMI a smaller hidden dimension of 176 is used with 4
attention heads, 16 layers, and a convolution kernel width of 15. SpecAugment
[32] is used during training alongside dropout ($0.1$ on feedforwards and
$0.3$ post-attention) for regularisation. Training of the AMs takes around 2
days on AMI and 4 days on Tedlium using a V100 GPU.

Models are trained using the Madgrad optimizer [33] with weight decay of
$1e-6$. An EMA of model parameters set to 0.9999 is used for the AM, and
during finetuning of the TLM.

###  4.2 Beam Search

For beam search a beam width of 25 is used. The search is constrained at each
time-step to indices within a given threshold of the argmax of the AM
probabilities (cut-off threshold). The insertion bonus $\beta$, TLM weighting
$\alpha$ and the cut-off threshold were trained through a random search. The
ranges explored in the search are as follows: For $\alpha$ $[0.0,1.0]$,
$\beta$ $[-0.1,0.8]$, and $[-4,-12]$ for the cutt-off threshold. Only the top
beam is maintained when transferring across utterance boundaries, this helps
reduce the propagation of errors found in low probability beams from previous
utterances. The beam search is parallelised over each utterance in the single
utterance setting, or over each meeting/talk for cross-utterance evaluation
using the ray library [34]. Single-utterance decoding takes around 10 minutes
and 40 minutes on Tedlium and AMI, and 35-55 minutes and 2.5-3.5 hours in the
cross-utterance setting, however our implementation is not optimized and is
implemented in python.

###  4.3 N-best Rescoring

To obtain the N-best list, beam search with a width of 1000 is used with a
n-gram trained with the same BPE vocabulary as the AM, this is then re-ranked
through a word-level n-gram model and pruned to an n-best list of 100
hypotheses. The TLM is combined with first-pass models through weighted
addition of the log probabilities. Additionally, a length penalty is included.
These weights/hyperparameters are trained through a random search. For best
performance it was necessary to standardise the log probabilities of the TLM
using mean and standard deviation statistics from the top hypotheses.

###  4.4 Datasets

####  4.4.1 Tedlium 1

Tedlium [35] is an ASR corpus consisting of single-speaker TED talks. This
data is selected as it is an ideal use-case for long-range ASR, with many
unique speakers using varied language, talking continuously for periods of up
to 18 minutes. Despite this there is, too our knowledge, no work that attempts
to fully utilise this longer time-frame for ASR. For this work we use the
smaller $1^{st}$ release of Tedlium. For pre-processing, spaces between the
apostrophe in contraction are removed. In this data there are instances of
large un-annotated gaps between utterances (this is alot less prevalent in the
test and dev data). To address this, when training cross-utterance TLMs the
cache is not propagated over gaps larger than ten seconds. Statistics for this
corpus are provided in table 1.

| Total Words | Duration (h) | Recordings
---|---|---|---
Train | 764,088 / 1,315,797 | 80.1 / 118.1 | 132 / 774
Dev | 95,374 / 17,733 | 9.8 / 1.6 | 18 / 8
Test | 89,978 / 27,500 | 9.4 / 2.6 | 16 / 11
Table 1: Corpus statistics for AMI / Tedlium-1 Dataset | 0 | 50 | 100 | 250 | 500 | 1000
---|---|---|---|---|---|---
AMI | 89.77 / 75.49 | 74.37 / 67.82 | 71.43 / 66.12 | 68.18 / 64.47 | 66.53 / 63.80 |  64.89 / 62.54
Tedlium | 148.72 / 136.21 | 132.47 / 114.60 | 126.79 / 118.96 | 119.70 / 108.10 | 117.86 / 104.63 |  116.33 / 101.91
Table 2: Perplexity (PPL) for TLMs on dev / splits at (0, 50, 100, 250, 500) tokens of context from preceding utterances Dataset | Decoding Method | 0 | 50 | 100 | 250 | 500
---|---|---|---|---|---|---
AMI | Rescoring | 20.96 / 19.41 |  20.88 / 19.31 | 20.88 / 19.31 | 20.89 / 19.30 | 20.88 / 19.30
Rescoring (GTH) | 20.96 / 19.41 | 20.84 / 19.26 |  20.83 / 19.26 | 20.83 / 19.23 | 20.83 / 19.21
Beam Search | 20.52 / 18.98 | 19.82 / 18.75 |  19.78 / 18.72 | 19.78 / 18.67 | 19.80 / 18.65
Beam Search (GTH) | 20.52 / 18.98 | 19.71 / 18.68 | 19.70 / 18.65 | 18.67 / 18.62 |  19.66 / 18.57
Tedlium | Rescoring | 9.51 / 8.48 | 9.52 / 8.44 | 9.52 / 8.45 |  9.50 / 8.44 | 9.52 / 8.45
Rescoring (GTH) | 9.51 / 8.48 | 9.48 / 8.42 | 9.47 / 8.41 |  9.44 / 8.42 | 9.50 / 8.42
Beam Search | 9.73 / 8.61 | 9.69 / 8.54 | 9.69 / 8.48 |  9.67 / 8.47 | 9.72 / 8.48
Beam Search (GTH) | 9.73 / 8.61 | 9.62 / 8.52 | 9.59 / 8.48 |  9.56 / 8.35 | 9.61 / 8.41
Table 3: Results (WER) for each decoding method on AMI and Tedlium with (0, 50, 100, 250, 500) tokens of context from preceding utterances. Results are given for dev / test sets. GTH denotes the use of the Ground Truth transcripts to form the History. Dataset | Dev | Test
---|---|---
Tedlium | 11.73 / 9.89 | 10.37 / 8.80
AMI | 22.94 / 21.09 | 21.49 / 19.58
Table 4: Baselines WERs for each dataset, results are given without/with an
n-gram language model

####  4.4.2 AMI

AMI [24] consists of multi-speaker meetings. The individual headphone
microphone (IHM) version is used in this work. This data was selected due to
it's long-format conversational setting and it's use in similar, prior work.
However, features of this dataset such as, very frequent speaker changes and
speaker overlaps, may present difficulties for measuring the long-range
performance of ASR systems in isolation.

Utterances in the training data longer than 16 seconds are re-segmented using
the provided time-stamps, while dev and test splits are kept intact.
Statistics are provided in table 1

####  4.4.3 OpenSubtitles

As a pre-training corpus for the TLMs
OpenSubtitles111http://www.opensubtitles.org/ [36], is used. We pretrain on a
subset of the corpus for containing 408,985,555 words for one epoch. Numbers
and monetary values are converted to their orthographic form, and all text is
set to lower case and punctuation excluding apostrophes is removed.

##  5 Experimental Results

Perplexities for the TLMs are provided in table 2. Here the models show
continued decreases in PPL which begins to plateau at 1000 tokens of prior
context.

Table 4 presents our baseline results for both greedy decoding, and the
initial first-pass which is used in re-scoring. Notably, our baselines achieve
good results, performing better on both datasets than other single-domain AMs
in the literature which do not incorporate i-vectors [2, 9, 37].

Results for re-scoring and beam search are given in table 3. The presentation
of results for AMI is split into sub-sections that are based on the key
components of this work, lastly we overview and discuss performance on Tedlium
separately.

###  5.1 Re-scoring vs Beam Search

For decoding via beam search on AMI we see sizeable improvement over re-
scoring, both in the utilisation of context, and more generally in the single-
utterance setting. When zero prior context is available we see a 2.4% and 2.5%
(dev and test) absolute decrease over the greedy decoding baseline for beam
search, compared to 2.0% and 2.1% for re-scoring.

With additional context we find much more consistent improvements with beam
search. For example, with the full 500 tokens of prior context beam search
shows an absolute decrease over the single-utterance setting of 0.7% and 0.3%
for dev and test respectively. For re-scoring 0.1% and 0.1% absolute decreases
are attained with the additional context. Hence we find that re-scoring is
limiting the use of the context with improvements from beam search becoming
more pronounced in the cross-utterance setting.

###  5.2 How much context is useful?

Unsurprisingly, on the AMI corpus, decoding benefits most from including the
recent history of the previous 50 tokens, with gains of 0.7% and 0.2% (dev and
test) absolute over the single-utterance setting using beam search. On average
there is 23.7 tokens per utterance on AMI, hence this amounts to around 2
prior utterances of context. For the test set we can see continued
improvements with up to 500 tokens of prior context with a further 0.1%
absolute decrease when over 50 tokens of context. This is on average 21
utterances of context that we are able to benefit from, increasing beyond 500
tokens brought no further improvements. On the dev set while we see
significant gains from the cross-utterance context of 0.7% absolute there is
no benefit from increasing the context beyond 100 tokens.

###  5.3 Impact of errors in the context

During the training the model is provided with ground truth transcripts, while
during decoding, only the decoded history is available. To examine the effect
of this mismatch on the use of the context we present additional results where
during decoding the transcripts from previous utterances are used as the
context. This is denoted as Ground Truth Histoy (GTH) in table 3.

For both dev and test dataset on AMI when decoding via beams search we see
around an additional 0.1% absolute decrease when using the GTH across all
context lengths. Additionally, with the GTH we see continued improvements on
the dev set until 500 tokens of context, while otherwise this plateaus at 100
tokens. Similar findings are seen for re-scoring. As such, we find that test-
time errors have a consistent impact on the use of the history. This does not
account for the impact of errors within the current utterance, which is likely
more severe.

###  5.4 Performance on Tedlium

On the Tedlium data re-scoring provides an improvement over beam search of
0.2% and 0.1% absolute in the single utterance setting. During re-scoring the
TLM benefits from interpolation with the first-pass models. Consequently, this
is likely due to the TLMs much higher perplexity on this corpora, causing
greater reliance on the n-gram models.

As with AMI we find that beam search enables better utilisation of the context
with around a 0.1% decrease over the single-utterance setting on the test
data, compared to fairly insignificant decrease of less than 0.1% for re-
scoring. For both decoding methods we see an increase in WER when using 500
tokens of context. Additionally, there is no benefit from the context on the
development set when using the model outputs as history. However, for the GTH
evaluations there is around a 0.2% absolute decrease over single-utterance
when using beam search suggesting that the poor performance on this split may
be due to misleading errors in the history.

##  6 Conclusion

In this work, we examined the benefit of context from prior utterances during
the decoding stage of ASR. Up to 500 tokens were found to be helpful when
decoding on AMI with reductions in a similar range to prior work. From Tedlium
there are much smaller gains from the context. However, it is difficult to
tell to what degree this is a result of our method or the prevalence of useful
long-range dependencies in this dataset.

It was found that decoding directly via beam search allowed for greater use of
the context, with increasing improvements compared to re-scoring as more
context was made available. As we find better results on AMI where the TLMs
show a much lower PPL, it is reasonable to assume that increasing the scale of
our models would provide improved results on both datasets. Additionally, due
to our findings that the use of context is limited during re-scoring due to
the first-pass models, it is likely also limited by the AMs implicit language
model. Ideally, cross-utterance information would be incorporated during both
encoding and decoding stages, which can be investigated as part of future
work.

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