# Relativistic Aberration:
The Transformation of a Propagating Field

Aran O’Hare<EMAIL_ADDRESS>
###### Abstract

The fields used to describe the influence of masses and electric charges are
generally accepted to emanate at the speed of light from their sources. To
obtain these fields for a moving particle which are consistent with special
relativity, transformations should be applied to the emanating field in the
particle’s rest frame to acquire the field propagating in the particle’s
primed inertial frame. These transformations are of its coordinates, and also
the velocity vectors of the field’s propagation, which consequently lead to a
change in the field’s direction and as a result its density and strength. Yet,
an electric charge’s field propagation velocities are neglected and not
currently transformed. This omission has consequences when it comes to
transforming the field from the source’s rest frame to the source’s primed
frame, as this primed frame’s field would then be inconsistent with the
calculated retarded field’s directions and consequently its strengths in this
primed frame. Here the retarded field of a moving point particle will be
derived and from this the requirement of the velocity transforms and the
aberrational effects will be shown. Hence, the full consistent inertial frame
transform of the emanating field will be given. This work shows the general
theoretical basis and legitimacy of the aberrational and time dilation effects
on the field’s density and thus strength and acts as an argument for their
implementation on all luminally emanating fields.

Relativistic Aberration, relativistic beaming, Emanating Field Transformation,
Propagating Field Transformation, Retarded Field strength transformations

##  I Introduction

In astrophysics, when describing the observed luminosity from a relativistic
plasma jet, the transformation of the radiated light’s velocities, known as
relativistic aberration or relativistic beaming, is used rybicki2008radiative
; lind1985semidynamical . These aberrational effects are required to explain
the luminosity of the radiated light (flux of photons) from different points
in the plasma, as depending on the plasma’s relative velocity at a position
there is a different weighting on the intensity of the number of photons
received by an observer. However, coordinate transformations are not included
in derivations, hence the requirement to apply the effects of aberration on
the flux and a derivation of the frame transforms have not been shown.

In general, it is accepted that the electric field and its information
propagate from a particle at the speed of light. However, when it comes to the
relativistic transformations of electric fields, the literature applies the
Lorentz transforms to the field’s coordinates without transforming the field’s
propagation velocity EFGriffiths:1492149 ; EFgron2012electrodynamics ;
EFpurcell2013electricity ; EFresnick1968introduction . If the electric field
propagates at the speed of light then relativity requires the velocity vector
of this propagation at each coordinate to also be transformed. This
transformation does not change the luminal speed of the propagation but does
change the propagation direction.

Here, the retarded field of a moving particle will be derived. From this, we
will show the requirement of applying both the aberrational effects (due to
velocity transforms) on the field’s propagation, which result in a change of
the field’s direction and density, and the time dilation effects on the
field’s density. This work shows the theoretical basis and legitimacy of both
the aberrational and time dilation effects and acts as an argument for their
implementation on all luminally emanating fields.

##  II The Retarded Field

In the primed frame of a particle $q$, it is taken that the particle is moving
in the positive Z-Axis only with velocity $\mathbf{V_{q}}=(0,0,v_{q})$, with
it currently (primed time $t^{\prime}=0$) positioned at the origin. If the
field emanates from the particle at luminal speed $c$. Then the field that is
currently propagating through a general primed coordinate
$\vec{R^{\prime}}=(x^{\prime},y^{\prime},z^{\prime})$, is the field that was
emanated from the particle when it was at a previous time and position, which
can be calculated from its trajectory. These corresponding coordinates are
referred to as the retarded time $T^{\prime}_{ret}$ (the time when the field
had emanated from the particle to reach $\vec{R^{\prime}}$ at time
$t^{\prime}=0$) and the particle’s retarded position
$\mathbf{P}^{\prime}_{ret}=(0,0,v_{q}T_{ret}^{\prime})$, corresponding to this
retarded time. Hence the retarded field displacement that the field has
propagated along is

$\vec{R^{\prime}}_{ret}=\vec{R^{\prime}}-\mathbf{P}^{\prime}_{ret}=\begin{pmatrix}x^{\prime}\\\
y^{\prime}\\\ z^{\prime}-v_{q}T_{ret}^{\prime}\end{pmatrix}.$ (1)

A visual of these values are shown in figure (1) below.

$y^{\prime}$$z^{\prime}$$x^{\prime}$$\mathbf{P}^{\prime}_{ret}$$\vec{R}^{\prime}_{ret}$$\vec{R^{\prime}}$$v_{q}$
Figure 1: Diagram of a particle’s primed frame, showing a particle positioned
at the origin at time $t^{\prime}=0$, moving in the positive Z-direction with
velocity $v_{q}$. The field at a general primed coordinate
$\mathbf{R}^{\prime}$, corresponds to the field that was emitted from the
particle when it was at its retarded position on the Z-Axis given by
$\mathbf{P}^{\prime}_{ret}=(0,0,v_{q}T_{ret}^{\prime})$, where
$T_{ret}^{\prime}$ is the corresponding retarded time. The field has
propagated the retarded field displacement $\mathbf{R}^{\prime}_{ret}$, to get
to this general primed coordinate.

The magnitude of the retarded field displacement is equal to the distance the
field, propagating at the speed of light, travels in the corresponding
retarded time. This gives

$\begin{split}\left(cT_{ret}^{\prime}\right)^{2}&=\|\vec{R^{\prime}}_{ret}\|^{2}\\\
&=(x^{\prime 2}+y^{\prime 2}+z^{\prime 2})+v_{q}^{2}T_{ret}^{\prime
2}-2v_{q}z^{\prime}T_{ret}^{\prime},\end{split}$ (2)

rearranging this, we get the quadratic

$T_{ret}^{\prime
2}+\left(2\gamma^{2}\frac{v_{q}}{c^{2}}z^{\prime}\right)T_{ret}^{\prime}-\frac{\gamma^{2}}{c^{2}}(x^{\prime
2}+y^{\prime 2}+z^{\prime 2})=0,$ (3)

where $\gamma=(1-v_{q}^{2}/c^{2})^{-1/2}$. Now taking the solution for the
past time, and making use of the identity

$\gamma^{2}=1+\gamma^{2}\frac{v_{q}^{2}}{c^{2}},$ (4)

we get the result

$\begin{split}T_{ret}^{\prime}&=-\gamma^{2}\frac{v_{q}}{c^{2}}z^{\prime}-\sqrt{\left(-\gamma^{2}\frac{v_{q}}{c^{2}}z^{\prime}\right)^{2}+\frac{\gamma^{2}}{c^{2}}(x^{\prime
2}+y^{\prime 2}+z^{\prime 2})}\\\
&=-\gamma^{2}\frac{v_{q}}{c^{2}}z^{\prime}-\frac{\gamma}{c}\sqrt{x^{\prime
2}+y^{\prime 2}+\left(1+\gamma^{2}\frac{v_{q}^{2}}{c^{2}}\right)z^{\prime
2}}\\\
&=-\gamma^{2}\frac{v_{q}}{c^{2}}z^{\prime}-\frac{\gamma}{c}\sqrt{x^{\prime
2}+y^{\prime 2}+\gamma^{2}z^{\prime 2}}\\\
&=-\gamma^{2}\frac{v_{q}}{c^{2}}z^{\prime}-\frac{\gamma}{c}\|\mathbf{R}\|.\end{split}$
(5)

In the final step, we required $t^{\prime}=0$ for all coordinates, and hence
used the Lorentz transform of the spatial coordinates;
$\mathbf{R}=(x,y,z)=(x^{\prime},y^{\prime},\gamma z^{\prime})$, to get
$\|\mathbf{R}\|$. With this we can rewrite the Z-component of equation (1) as

$z^{\prime}-v_{q}T_{ret}^{\prime}=\gamma\left(z+\frac{v_{q}\cdot\|\mathbf{R}\|}{c}\right),$
(6)

giving the retarded field displacement vector to be

$\vec{R}^{\prime}_{ret}=\begin{pmatrix}x\\\ y\\\
\gamma\left(z+\dfrac{v_{q}\cdot\|\mathbf{R}\|}{c}\right)\end{pmatrix}=\|\mathbf{R}\|\begin{pmatrix}\frac{x}{\|\mathbf{R}\|}\\\
\frac{y}{\|\mathbf{R}\|}\\\
\gamma\left(\frac{z}{\|\mathbf{R}\|}+\dfrac{v_{q}}{c}\right)\\\
\end{pmatrix}.$ (7)

Since the field propagates along this in the primed frame, the unit vector of
the primed propagation velocity at any general primed coordinate can be worked
out to be

$\mathbf{\hat{\mathbf{\text{$\mathbf{U}^{\prime}$}}}}=\mathbf{\hat{\mathbf{\text{$\mathbf{R}^{\prime}$}}}}_{ret}=\dfrac{1}{\text{\AA}}\begin{pmatrix}\frac{x}{\|\mathbf{R}\|}\\\
\frac{y}{\|\mathbf{R}\|}\\\
\gamma\left(\frac{z}{\|\mathbf{R}\|}+\dfrac{v_{q}}{c}\right)\end{pmatrix},$
(8)

where the factor

$\text{\AA}=\gamma\left(1+\frac{v_{q}}{c}\frac{z}{\|\mathbf{R}\|}\right).$ (9)

Now taking the magnitude of the retarded field displacement from equation (7)
and using equation (4), we have

$\begin{split}\|\vec{R}^{\prime}_{ret}\|^{2}&=x^{2}+y^{2}+\gamma^{2}\left(z^{2}+\frac{v_{q}^{2}}{c^{2}}\|\mathbf{R}\|^{2}+2\frac{v_{q}}{c}z\|\mathbf{R}\|\right)\\\
&=\gamma^{2}\|\mathbf{R}\|^{2}+\frac{v_{q}^{2}}{c^{2}}\gamma^{2}z^{2}+2\frac{v_{q}}{c}\gamma^{2}z\|\mathbf{R}\|\\\
&=\gamma^{2}\left(\|\mathbf{R}\|+\frac{v_{q}}{c}z\right)^{2}\\\
&=\gamma^{2}\left(1+\frac{v_{q}}{c}\frac{z}{\|\mathbf{R}\|}\right)^{2}\|\mathbf{R}\|^{2}\\\
&=\text{\AA}^{2}\|\mathbf{R}\|^{2}.\end{split}$ (10)

Since the luminal speed of the propagation will be same in both frames we have

$\begin{split}c=\frac{\|\vec{R}\|}{T_{R}}&=\frac{\|\vec{R}^{\prime}_{ret}\|}{T_{ret}^{\prime}}\\\
&=\frac{\text{\AA}\|\vec{R}\|}{T_{ret}^{\prime}}\\\
T_{ret}^{\prime}&=\text{\AA}T_{R},\end{split}$ (11)

where $T_{R}$ is the proper (particles rest frame) time the field takes to
propagate to $\mathbf{R}$ from the particle. Time and length are stretched in
the primed frame along $\mathbf{R}^{\prime}_{ret}$ relative to the proper
frame along $\mathbf{R}$ by a factor Å, leading to the relative radial density
of the field in the primed frame to that of the proper frame, which we will
refer to as the radial field strength weighting, given as

$W_{\rho}=\frac{1}{\text{\AA}}.$ (12)

##  III Aberration

Figure 2: Graphs of the field propagation velocities from a particle in its
proper (left) and primed (right) frames, with the same evenly distributed
proper angles. The black arrows show the magnitude and direction in the yz-
plane of the propagation velocity in each frame for the corresponding proper
angle, and the central red arrow shows the velocity of the particle in the
primed frame $v_{q}=0.9$.

The propagation of the field in the proper frame, with the particle at rest,
is taken as evenly distributed with its strength spherically symmetrical, and
with the magnitude of its velocity equal to the speed of light $c$, in all
directions, as shown in the left graph of figure (2) below. If the field
propagates, then a velocity transform of this propagation at all coordinates
should also be performed in order to change inertial frame. The luminal
velocity of a field emanating from a particle in its own proper frame, can be
written in spherical polar coordinates as

$\mathbf{U}=\begin{pmatrix}U_{x}\\\ U_{y}\\\
U_{z}\end{pmatrix}=c\begin{pmatrix}\sin\theta\cos\phi\\\ \sin\theta\sin\phi\\\
\cos\theta\end{pmatrix},$ (13)

where $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$ are the angles from the Z-Axis
and X-axis respectively. It is taken that the primed frame is moving as it did
in the previous chapter, with velocity $\mathbf{V}=(0,0,-v_{q})$ relative to
the particles proper frame. Therefore we only require a transformation of the
Z-component to get the primed propagation velocity. Using the equation for the
relativistic transform of a general velocity vector between two inertial
frames, we have

$\begin{split}\mathbf{U}^{\prime}&=\dfrac{1}{\gamma}\dfrac{\mathbf{U}+\Big{[}\dfrac{\gamma-1}{\|\mathbf{V}\|^{2}}(\mathbf{U}\cdot\mathbf{V})-\gamma\Big{]}\mathbf{V}}{1-\dfrac{\mathbf{U}\cdot\mathbf{V}}{c^{2}}}\\\
&=\dfrac{c}{\text{\AA}}\begin{pmatrix}\sin\theta\cos\phi\\\
\sin\theta\sin\phi\\\
\gamma\left(\cos\theta+\dfrac{v_{q}}{c}\right)\end{pmatrix}=\dfrac{c}{\text{\AA}}\begin{pmatrix}\frac{x}{\|\mathbf{R}\|}\\\
\frac{y}{\|\mathbf{R}\|}\\\
\gamma\left(\frac{z}{\|\mathbf{R}\|}+\dfrac{v_{q}}{c}\right)\end{pmatrix},\end{split}$
(14)

with

$\text{\AA}=\gamma\left(1+\dfrac{v_{q}}{c}\cos\theta\right)=\gamma\left(1+\frac{v_{q}}{c}\frac{z}{\|\mathbf{R}\|}\right),$
(15)

where the propagation velocity is given in spherical polar and Cartesian
coordinates respectively. It can be seen from equation (8) that this
propagation velocity’s direction is the same as what is given by the retarded
field. The field’s propagation direction has been aberrated, while its
propagation speed remains equal to $c$, which can be shown by taking the
magnitude of the primed velocity vector. This aberration leads to a higher
concentration of the field in the direction of the particle’s motion, as shown
in the right graph of figure (2) above. The propagation directions of the
field as seen in figure (2), are transformed such that the angles between them
and the Z-Axis, have the aberrational relationship

$\cos\theta^{\prime}=\frac{\mathbf{U}^{\prime}_{z}}{\|\mathbf{U}^{\prime}\|}\equiv\frac{\mathbf{R}^{\prime}_{ret_{z}}}{\|\mathbf{R}^{\prime}_{ret}\|}=\dfrac{\cos\theta+\dfrac{v_{q}}{c}}{1+\dfrac{v_{q}}{c}\cos\theta}.$
(16)

This is the relativistic aberration formula einstein1905electrodynamics , it
shows how the field’s propagation direction transforms, it can be used to give
Å in primed terms, by rearranging this for $\cos\theta$ and substituting into
equation (15), leading to

$\text{\AA}=\frac{1}{\gamma\left(1-\dfrac{v_{q}}{c}\cos\theta^{{}^{\prime}}\right)}=\frac{1}{\gamma\left(1-\dfrac{v_{q}}{c}\dfrac{\mathbf{R}^{\prime}_{ret_{z}}}{\|\mathbf{R}^{\prime}_{ret}\|}\right)}.$
(17)

###  III.1 The Field Strength’s Dependence on Angular Density

The proper frame’s differential solid angle element

$d\Omega=\sin{\theta}d\theta d\phi,$ (18)

encompasses a certain amount of the field, this is the same amount of the
field that is encompassed by the coinciding aberrated differential solid angle

$d\Omega^{\prime}=\sin{\theta^{\prime}}d\theta^{\prime}d\phi^{\prime}.$ (19)

We can calculate this element by differentiating both sides of equation (16)
with respect to $\theta$ hogg1997special , which gives

$\sin{\theta^{\prime}}d\theta^{\prime}=\dfrac{1-\dfrac{v_{q}^{2}}{c^{2}}}{\left(1+\dfrac{v_{q}}{c}\cos{\theta}\right)^{2}}\sin{\theta}d\theta=\frac{1}{\text{\AA}^{2}}\sin{\theta}d\theta.$
(20)

Using this and $d\phi^{\prime}=d\phi$ (as the angle $\phi$ is always
perpendicular to the motion of the particle and hence unaffected by
transformation) we have the solid angle in the primed frame given as

$d\Omega^{\prime}=\frac{1}{\text{\AA}^{2}}\sin{\theta}d\theta d\phi.$ (21)

The relative primed field strength at a given angle is taken as being
proportional to the amount of the field per solid angle in the primed frame
relative to that in the proper frame, referred to here as the aberrational
field strength weighting, given as

$W_{\Omega}=\frac{d\Omega}{d\Omega^{\prime}}=\text{\AA}^{2}.$ (22)

##  IV The Field Transformation

In the particle’s proper frame the field strength’s magnitude $f$, obeys the
inverse square law, Giving

$f=\frac{k}{\|\mathbf{R}\|^{2}},$ (23)

where $k$ is a constant for the particle. In the primed frame the field
strength at a coordinate will be equal to the inverse square of the retarded
field displacement with constant $k$ and the field strength weightings from
equations (12) and (22) applied

$f^{\prime}=W_{\rho}W_{\Omega}\frac{k}{\|\vec{R}^{\prime}_{ret}\|^{2}}=\text{\AA}\frac{k}{\|\vec{R}^{\prime}_{ret}\|^{2}}.$
(24)

It can be noted that both field strength weightings together give Å, which is
the same as the generalised Doppler factor. Now using this equation of the
field strength and the direction of the field’s propagation which is the unit
vector for the retarded field displacement, we have the primed frame’s field
described by the vector

$\mathbf{f}^{\prime}=f^{\prime}\mathbf{\hat{\mathbf{\text{$\mathbf{R}^{\prime}$}}}}_{ret}=\text{\AA}\dfrac{k\vec{R}^{\prime}_{ret}}{\|\vec{R}^{\prime}_{ret}\|^{3}}.$
(25)

##  V Discussion

If the electric field propagates at the speed of light and special relativity
is applied, then the aberrational and time dilation effects are required, and
hence the electric field will be described differently than previous
relativistic transforms, described by equation (25).
One consequence of aberration is that if a particle approaches the speed of
light, then every proper propagation direction of the field from the particle
would, in the primed frame, approach the direction of the particle’s motion,
the retarded position of the particle would also tend to negative infinity.
Given this and that the particle’s motion is approaching the speed of the
luminal propagation, then there would be no field outside of where the
particle is located. Therefore if a photon did have a emanating field itself,
it would be expected that a field would not exist outside the space that the
photon occupies.
The aberrational weighting from the transformations of a field in this paper
can also be applied to the average flux of photons in all directions from a
moving radiating source (instead of the field strength) and might provide
another way of explaining the distribution of synchrotron radiation, where the
Liénard – Wiechert field equations which include similar factors have been
used to explain the phenomena wiedemann2015theory .

###### Acknowledgements.

The many discussions with Patrick Mullan helped pave the way to this work.

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