zotero/storage/3T7LCGMD/.zotero-ft-cache

Page 12 APEIRON Nr. 19 June 1994
Stellar and Planetary Aberration
Thomas E. Phipps, Jr.
908 South Busey Avenue
Urbana, Illinois 61801
A way of looking at the topics of stellar and planetary aberration is suggested that enables them to
be viewed as closely related—both being conceived as dependent on source-detector relative velocity.
We show that the long-range limiting case, stellar aberration, has certain uniquely subtle aspects,
as well as some hypothesized characteristics that lend themselves to empirical testing.
Introduction
For some time the writer has been seeking improved
understanding of the several kinds of aberration—these
being rather superficially explained in most texts, particu
larly texts of relativity theory. Indeed, in some of the more
highly regarded of these (e.g., Møller 1972) the phenome
non of stellar aberration is so slightingly treated that use of
the formulas provided would literally yield wrong an
swers. During much of this investigation it seemed that
the fault lay with special relativity theory or with the
Lorentz transformation. This may still, or may not, be the
case. The reader will have to judge. Interest particularly in
stellar aberration has increased of late [cf., Ives 1950, Eisner
1967, Phipps 1989, Hayden 1993, etc.] With help from re
cent studies by Marmet (1994) and Sherwin (1993) it has
become apparent to the writer thatwith suitable interpretation
relativity theory can accommodate both principal types of
optical aberration, planetary and stellar. The purpose of
the present paper is to confirm this assertion, to exhibit
stellar aberration as (in a sense) a limiting case of planetary
aberration, to seek a physical model of the phenomenon,
to describe observations that could be made to test the
model, and to touch on residual aberration issues that re
main unsettled in the context of relativity theory.
1. Aberration via Lorentz Transformation
First, let us deal with the mathematics—which is the
easy part—so that we can get on to the real problem,
which (as so often happens in physics) is interpretation.
According to the special theory of relativity (Einstein
1905) both the Doppler effect and stellar aberration can be
treated by a Lorentz transformation of the four-vector of
light propagation,
k kkk k
x yz
i
c kkk c
μ xyz
ωω
= ==
FHG , , , IKJ , 2 + 2 + 2 .
Inertial system K' moves with velocity v relatively to sys
tem K along the direction of their common x-axes. We
consider both K and K' to be for the moment arbitrary
inertial systems. Propagation vectors (wave-normal or ray
path vectors) describing the same given light beam arek'
and k in K' and K, respectively. Applying a Lorentz trans
formation, we have
′ = + FHG IKJ
FHG IKJ
k kv
ic
i c
xx
γ ω , γ =1 1− 2
(v / c)
′= ′=
kk kk
yy zz
,,
i c
i c
iv
c kx
′= +
FH IK
ω γ ω , or ω ′ = γ (ω + vkx ).
From this it is verifiable that a directional turning of thek
vector, k'/k' ≠ k/k, is an unavoidable consequence of the
Lorentz transformation, except in the special case of paral
lelism of the v- and k-vectors.
To specialize for simplicity to a specific model, let the
x-direction (parallel to earth’s orbital motion), in our
“laboratory” (observatory) system K, lie in the horizontal
plane of the ecliptic; let the z-axis lie in the vertical, defin
ing the zenith direction; and (to eliminate diurnal effects)
consider our telescope to be located at the north pole of a
nonrotating earth, so that the pole of the ecliptic lies at the
zenith. To treat stellar aberration, consider a star emitting
light described by the k-vector to lie in the x–z plane at
angle α measured from the horizontal. Then kx = k cosα = ω c α
b gcos . In K' the corresponding angle α ′ obeys k'x = ′ ′
ωc α
b gcos . With these substitutions it follows from the first of the above Lorentz transformation equations that ω ω α
′′
b gcos = γ cosα + v c
c h and from the last of them that ω ′ ω = γ 1 + v c cosα
c h. On taking
the quotient of these equations we obtain the well-known
relation (Synge 1965)


APEIRON Nr. 19 June 1994 Page 13
cos
cos
cos
′=
+
+
α
α
α
v
c
v c
1
, (1a)
or its dual (symmetric with respect to interchange of K, K',
with sign change of relative velocity),
cos
cos '
cos '
α
α
α
=
−
−
v
c
v c
1
. (1b)
Subtracting cosα from both sides of (1a), using a standard
trigonometric identity (Peirce 1957), and introducing an
aberration angle ε = α − α ′ , we obtain
22 2 1
2
sin sin sin
cos
ε αε β α
βα
⋅ − =+
FH IK , (2)
where β = v c. Observing that ε = O(β), we can expand the left-hand side of (2) as a power series in ε ,
LHS = ε sinα − cosα ε − sinα ε + O β
26
2 34
d i.
Substituting ε = aβ + bβ + cβ + O β
23 4
( ), where a,b, c are unknown coefficients, and equating the resulting β power series to the series expansion of the right-hand side
of (2) in powers of β ,
RHS
O
=− ++
β αβ α α β αα β
sin cos sin sin cos ,
22 2
32 2 4
di
we obtain, upon equating coefficients of successive
powers of β , a sequence of equations that determine a,b, c , with the result (including the fourth-order term) that
εβ αβ α αβ α α
β αα α β
=− + +
− ++
sin sin cos sin cos
sin cos cos
2 32
4 25
1
2 12
1
8 12
1
6d i
d i di
O
(3a)
or ε = α +
v
c Ov
c
sin
2
2 . (3b)
The first-order term here [which verifies ε = O(β)] is the one usually given, and is the only order accessible to ob
servation with present-day telescopes. We show some of
the higher-order terms because they are seldom derived
and not always consistently represented (e.g., Aharoni 1965
shows an apparent γ -dependence). The dualism prescription, α ↔ α ', β → −β, with ε → −ε , works with the expansion, as with (1).
The above calculation loosely follows Einstein (1905),
who made the interpretation that K' is the rest system of
the light source and claimed thereby to describe stellar aber
ration. It is interesting to note that the above formal ma
nipulations apply to arbitrary inertial systems. But, if one
wants the formalism to relate to physics, close attention
must be paid to physical interpretation. Einstein’s 1905
interpretation, implying v vsd
= , where vsd is the sourcedetector relative velocity (rather, a transverse component
of it), applies formally to planetary aberration. It is incorrect
(i.e., gives the wrong numerical answer) if directly applied
to stellar aberration—as has been pointed out by Eisner
(1967) and by Hayden (1993). The error was noted by
Ives (1950) in connection with spectroscopic binary evi
dence and later by Phipps (1989) on related evidence. It is
trivially obvious, inasmuch as any dependence of stellar
aberration angle on vsd would imply dependence on stellar (source) velocities—which vary so tremendously as to
render description of the phenomenon by means of a sin
gle “constant of aberration” entirely unfeasible. Neverthe
less, and somewhat paradoxically, we shall presently show
that by suitable interpretation stellar aberration can be viewed
as a limiting case of planetary aberration.
2. Aberration via Velocity Composition
To finish with formal preliminaries we consider an
alternative relativistic description of aberration using the
general velocity composition law of special relativity the
ory. Again we consider arbitrary inertial systems K, K' with
light propagation vectors k, k', respectively, and with v the
velocity of K' with respect to K. This time, however, we
introduce photon velocity vectors u = ck/k = cn and
u' = ck'/k' = cn', where n, n' are unit vectors in the direction of propagation of a given light ray (“trajectories” of a
given photon in the two systems, for a particle model of
light). Møller (1972: Equation 2.55, page 51) gives the
general velocity composition formula
u 1 u uv v
1 uv
' ()
()
=
− +−− ⋅ −
−⋅
vc vc v
c
22 22 2
2
11 1
{} .
Introducing into this formula by v = cβm the unit
vector m in the direction of v , substituting for u,u' , and taking the scalar product of the resulting equation withn ,
we find
' ( )( )
()
nn 1 n m n m
nm
2
⋅=
−+−− ⋅− ⋅
−⋅
β ββ
β
11
1
2
{ } . (4)
Employing the angles α,α ' previously introduced and the “aberration angle” or angle between the photon trajecto
ries in the two systems as the difference ε = α − α ', and
noting that (n'⋅n) = cosε , (n ⋅ m) = −cosα , we obtain
cos cos cos
cos
ε β β αβ α
βα
=
− +− − +
+
1 11
1
2 22
e j . (5)
This can be shown by standard trigonometric identities to
agree with Equation (1) and thus with the expansion (3).


Page 14 APEIRON Nr. 19 June 1994
Alternatively, we can obtain the expansion directly from
(5) by representing ε , as before, as a power series in β with undetermined coefficients, expanding the left-hand
side of (5) as a Taylor series in β , equating this to the right-hand side similarly expanded, and evaluating the
undetermined coefficients to yield the same result as pre
viously obtained, Equation (3a). Thus the two methods,
(a) photon velocity composition and (b) Lorentz trans
formation of the propagation vectors, yield identical re
sults for aberration.
For describing stellar aberration both methods are
wrong if β is interpreted as vsd c , where vsd is sourcedetector relative velocity. Both are right if β is interpreted as vorb c , where vorb is orbital velocity of the earth in the
case of terrestrial observations, or ifcβ is interpreted more generally as the component transverse to line of sight to
the star of the relative velocity vdd of detector and detector (the same instrument at different times or different in
struments at the same time, etc.). The essential condition
for a viable physical interpretation of stellar aberration is
that only detector (not source) motion is involved.
3. Planetary Aberration
Planetary aberration is an effect of retarded propaga
tion of light. It obeys the aberration formula, Equation (3),
with v vsd
= . That is, planetary aberration increases with source-detector relative velocity (more exactly, with the compo
nent of such velocity transverse to the observer’s line of
sight—we sometimes omit this qualification for brevity).
During the time interval in which light emitted by the
source at position P is propagating to the detector, the
source moves to some new position P', which it occupies
at the instant of light detection (simultaneity being re
ferred to the rest system K of the detector). Thus there
may be a measurable angle between P and P' subtended at
the detector. This is called an aberration angle, though
other names might come to mind. (Eisner 1967 suggested
angle of “light time lag”—other possibilities include “lead
angle” of light aimed at the detector from the emitter,etc.).
The same is true of the “aberration” of Penrose (1959) and
Terrell (1959), which recycles the overworked ε of
Equation (3), with v vsd
= , as an apparent turning angle of a relatively-moving three-dimensional solid, as viewed by
a single point detector. We shall say nothing about that
cause célèbre.
There are two ways of interpreting the angle ε of
“planetary aberration,” both of which partake somewhat of
the metaphysical:
(1) It is the angle by which the direction (specified by an
gle α ) from the detector to the pointlike light-source’s
position at the event of emission differs from the di
rection (specified by α ') from detector to the light
source’s “present position” at detection time (time be
ing measured in the inertial system K of the detector).
(2) It is the “angle of lead” by which photons, fired like
bullets from the light source, must be aimed ahead of
the “present position” of the relatively-moving point
detector (at the event of emission, time being meas
ured in the inertial system K' of the source) in order to
intercept that detector at a later time.
Both interpretations assume linear extrapolations of
positions—that is, true inertiality of the motions of K and
K' during the entire interval of light propagation from
emission to detection (absorption). Thus the concept of
planetary aberration is physically useful primarily for
rather short propagation times and limited source-detector
distances. The linearity assumption breaks down com
pletely for light sources at stellar distances. In fact the very
concept of planetary aberration is useless for physics ex
cept in such cases (e.g., planets) as acquire meaning and
computability of “source trajectory” through the applica
tion of celestial mechanics. If initial conditions of source
motion are unknown or unknowable, trajectories are not
computable and position of the light source at detection time or of
the detector at emission time is an operationally meaningless
concept. Moreover, it is not a relativistically invariant con
cept, since it depends on distant simultaneity in one iner
tial system or the other.
Is there any invariant way of formulating the physical
definition of planetary aberration? Mathematically, this
may be accomplished without difficulty by combining the
symbols descriptive of aberration with those descriptive of
Doppler effect to form a 4-vector. But this evades the
question of physical definition . . . and we shall see pres
ently that any “fading to mere shadows” of the difference
between two operationally quite distinct physical phe
nomena creates another problem, which will be left as a
puzzle for the interested reader. Along with it we offer the
following thought questions: Do astronomers really need
the noninvariant concept of planetary aberration? Of what
use is it to them to quantify where a distant planet or the
Sun is at the present instant?
Unfortunately, it would seem that one cannot give the
quick and easy negative “relativistic” answers that spring to
mind . . . for in fact celestial mechanics, very accurately
governed on the scale of the solar system by Newton’s in
stantly-acting gravity, would appear to have good practical
uses, e.g., for the concept of “present position of the Sun.”
In fact, it may be hard to get along without it (since the
present position of the Sun is what our planet mainly de
pends on to guide its orbital progress from moment to
moment), and one rather hopes for the sake of future as
tronauts that astronautical science will, at whatever ideo
logical cost, avoid trying to do so. (Note that there is no
such known phenomenon as gravitational aberration
although for a century people such as Jefimenko (1994)
have persisted in trying to fly in the face of this fact by
representing gravity as retarded in its force action—and for


APEIRON Nr. 19 June 1994 Page 15
that very reason it is important to correct intra-solar
system observations for optical aberrations.)
4. Stellar Aberration
Stellar aberration in its physical interpretive aspects is
by a wide margin the subtlest scientific subject ever en
countered by this writer. New ideas about it seem to arise
almost daily. This paper has been rewritten half a dozen
times, and each time it has looked completely different.
The reader will have to bear with the present report as a
fallible one of tentative progress to date.
To gain a notion of the physical aspect of stellar aber
ration, consider the sketch in Figure 1. Inertial system K
comoves with our earthly telescope, designated #1. We
arbitrarily take our light source to be a star at the zenith
(α = π / 2 ). We suppose that the star is in fact at rest in
another inertial system K' moving with speed vsd relatively to K. If in the vicinity of the earth (for instance,
borne by an earth satellite) we have a second telescope,
designated #2, at rest in K', then that telescope must be
tilted at a different angle α ' such as to have a more
“forward” inclination in the direction of vsd , if it is to intercept photons comoving with those detectable on earth
(i.e., part of the same “ray”). The vertical downward pho
ton propagation direction in K is denoted by unit vector
n , the downward-slanted direction in K' by n'. (If m is a
unit vector in the direction of vsd , then m ⋅ n = − cosα , m ⋅ n' = − cosα ' .)
An important observation made by Marmet (1994),
and noted by others, is that in K (where the source moves
with speed vsd ) the path of the ray entering telescope #1 (negative of the apparent direction to the star) isfixed: The
photons have to come straight down from the zenith in K,
regardless of source motions. This means that in K' what
we have marked as “downward ray” will not enter tele
scope #2, nor will it be the same ray that enters #1. The
latter is instead the ray marked n', inclined at angle α ' in
K'. At first order we see from simple Galilean kinematics
that, in order to meet the condition of verticality in K, cn'
must deviate from verticality in K' by a “backward” or
“over the shoulder” velocity component equal to −vsd , just sufficient to cancel the effect of “forward” motion of K'
with respect to K. [At higher orders the relativistic velocity
composition law must replace the Galilean one, with the
same consequence of cancellation.] But this cancellation
condition, needed to ensure a fixed, rigorously vertical ray
in K, applies independently of the magnitude of vsd , given an omnidirectional source. In consequence, the parameter
vsd drops out of the discussion .. . so source speed plays no role in the description of stellar aberration—in agreement with
the nature of stellar aberration, known empirically since its
discovery by Bradley.
Thus an analysis of stellar aberration in terms ofsource
detector relative velocity—just as needed to describe planetary
aberration—nevertheless leads to elimination of the pa
rameter vsd from description of the phenomenon. In effect stellar aberration can be viewed as a limiting form of
planetary aberration, wherein both types share analysis in
terms of source-detector relative motion. We have repeat
edly emphasized in the mathematical analysis, and shown
in our formulas, a duality or reciprocity between K and K',
such that, by a simple swap of descriptors and change of
sign of relative velocity, the two systems are completely
interchangeable. There is also, as we know, a reciprocity of
Maxwell’s equations, having to do with their invariance
under time reversal, whereby the source and detector
functions (emission and absorption of radiation) can be
interchanged without altering the ray path. In a moment
we shall make use of this fact.
So far we have shown only that in inertial system K
what will be observed is unaffected by stellar proper ve
locities. But as long as K remains strictly inertial no aber
ration of stars will be observable in any case. The star im
age in Figure 1 must just sit forever at the zenith as viewed
in telescope #1. This is a well-known property of true
inertial systems, but it tells us nothing about the observ
able phenomenon of stellar aberration.
To make a quantitatively correct prediction of stellar
aberration employing the first-order formula derived
above, viz., ε = α − α ′ = (v / c)sinα (radians), it is necessary to interpret v in this formula as relative speed of the
detector in two different states of motion. [To emphasize this we may write v in the formula as vorb , denoting mean orbital speed of the Earth or detector (telescope).] This
has been recognized by some authors such as Synge
(1965) and not by numerous others such as Møller (1972).
The formalism gives of course no hint of how this parametric switch from vsd to vorb occurs. Let us assemble what we have learned so far and see if we have enough
logical ingredients to provide clues to this mystery. First,
consider the situation portrayed in Figure 1.
(A)For source motion arbitrary, detector motion inertial:
α in K retains a fixed, constant value.
α ' in K' adjusts itself automatically to cancel all ef
fects of source motion on detector aiming.
[The automaticity here has nothing to do with cybernetics
or precognition. It amounts to ray selection from an om
nidirectional source.] Now let us assert Maxwellian reci
procity of source and detector—with the ray reversing its
direction, while the telescopic detector (now in thought
acting as ray emitter) stays on earth in K and the stellar
source (now acting as absorber) stays at rest in K'—to hy
pothesize:
(B)For detector motion arbitrary, source motion inertial:
α ' in K' retains a fixed, constant value.
α in K adjusts itself automatically to cancel all ef
fects of detector motion on source aiming.


Page 16 APEIRON Nr. 19 June 1994
[By source aiming, we mean selection of a direction, from
an omnidirectional stellar source in K', coincident with a
ray capable of being detected in K.] From these hypothe
ses, assumed valid, we make a final extrapolation to the
general case:
(C) For arbitrary motions of both source and detector:
α in K adjusts itself automatically to cancel all ef
fects of detector motion on source aiming.
α ' in K' adjusts itself automatically to cancel all ef
fects of source motion on detector aiming.
Item (A) is simply a summary of empirically known
facts about stellar aberration. Item (B) is pure speculation
in need of observational confirmation (see below). Item
(C) is also speculative, but plausible if (B) is correct. Sup
posing that (C) is correct, we remark that despite all the
“automatic cancellation” hypothesized there remains an
effect of detector motion on detector aiming, and an effect of source
motion on source aiming. The former is well known; the lat
ter must remain speculative, since we have no control over
stellar source motions and cannot very easily experiment.
The most pressing need is to verify or refute item (B).
This we consider in the next section.
5. Observational Testing
Do the present considerations lend themselves to ob
servational testing? Yes, hypothesis (B) above is specula
tive but can probably be tested rather readily. A common
way to test an hypothesis is to assume its opposite and look
for a contradiction of fact or observation. Item (B) says
that for a star in inertial motion and our earthly telescope
in arbitrary noninertial motion (a requirement satisfied by
its actual orbital motion) α ', the angle of emission of the
ray seen by our telescope, will remain fixed regardless of
our detector motions. This is a very counter-intuitive no
tion. Much more intuitive is its contradiction, the “duck
hunter” model often used in explaining aberration:
Duck-hunter Model: The hunter’s gunbarrel (source emis
sion ray-path) has to be aimed to compensate not only for uniform
relative velocity of the duck (detector, telescope) and gun but also for
any noninertial maneuverings of the duck.
Our item (B) boldly contradicts this and thus stands in
need of observational testing. The most obvious concep
tion, based on the duck-hunter model, asserts that the
source directs its earth-detectable rays backwards “over its
shoulder” at just such a (variable) angle α ' as to cancel
(compensate) the effect of any source-detector relative velocity
component transverse to line-of-sight. Thus α ' varies
with detector proper motions. This can almost surely be
testeSid.nce we know in any case that nobody aboard a star is
aiming a searchlight in such a clever way as to compensate
variations in vsd , it is apparent that most stars are visible to us only because of the (approximate) omnidirectionality of
their light emissions. Testing of the duck-hunter model
[counter-hypothesis to (B)] should be feasible if stellar
sources can be found that emit their omnidirectional light
not uniformly but, e.g., in a high-intensity “beam” directed
almost toward us . . . or more generally if there exist stellar
light sources having strong local departures from direc
tional homogeneity of light emission on a scale of angular
variation comparable with (not vastly greater than) the 20
arc-seconds of stellar aberration due to earth motion.
In the case of a beam, if its “pointing” is a stable aspect
of the relative geometry (i.e., if the searchlight is fixed in
space, not variably “aimed”), so that the beam points in a
constant direction or in a direction that varies on a time
scale long compared with the earthly year, then we should
observe with any changes of either source velocity or de
tector velocity a modulation of light intensity having a pe
Figure 1. Stellar aberration in terms of source-detector relative velocity vsd . Telescope #1 at rest in
earth inertial system K, star at zenith. Telescope #2 and source at rest in K' (telescope distant from source). Horizontal component of cn' compensates vsd .


APEIRON Nr. 19 June 1994 Page 17
riod of one year for the detector motions and whatever
other period (if any) may govern source motions. To re
peat: Intensity modulations of one-year period must arise un
avoidably, according to the duck-hunter model, if the light
beam directed toward us is sharply directional. These
modulations reflect annual variations in the direction ofn'
or value of α ' , the emission angle (determined in the ge
ometry of Figure 2 by −n'⋅vsd = vsd cosα ') needed to intercept the earth in its annual motions. Figure 2 illustrates.
These directional variations, as we have said, are implied
by the duck-hunter model and are denied by our hy
pothesis (B).
Since a popular “relativistic” way of explaining ob
served apparent superluminal motions of sharply-beamed
astronomical “jets” is that these are directed very nearly to
ward us, the present considerations provide a double
barreled way of checking the consistency of theory: The
light from such nearly earth-pointing jets should exhibit
annual intensity changes additional to and unconnected
with the normal annual 20.5"direction aberrational changes,
if (B) is wrong This is the case because, as the earth orbits,
our telescope intercepts the light emitted from the source
at slightly different emission angles α '—off-angles of the
jet. If (B) is right, no such intensity changes of annual pe
riod are to be expected.
In effect, according to the duck-hunter model, as
sketched in Figure 2, the emitted photons have to be
“aimed” at us differently at different times of year in order
to enter our telescope, because we are a maneuvering tar
get. (Note that this effect, if it exists, is strictly a result of
our earth’s velocity changes—it has nothing to do with
parallax due to orbit diameter.) If no such annually peri
odic intensity modulations of “apparently superluminal”
sources are observed, this will constitute presumptive evi
dence in favor of our hypothesis (B). Should annual in
tensity modulations of any stellar light sources be ob
served, this will work in favor of the duck-hunter model
and necessitate retreat from the view of stellar aberration
here advocated.
According to the hypothesis (B) viewpoint, what is
wrong with the duck-hunter model? Nothing at all, to be
sure, if the detector were either a normal duck or a
mathematical point. But this is a duck with a difference, a
duck with complications. It is an armored duck that can be
killed only with a shot to the gullet, straight down its long,
tubular neck. It is also a singularly stupid or suicidal duck,
for although it is adept at dodging, with every change of its
velocity there is a correlated change of tilt of its tubular
neck. The rule is, according to (B), that the correlation
brings about precisely the tilt-angle change needed to can
cel the effect of the velocity change, so that a fixed aiming
direction on the part of the hunter suffices.
Thus our detector (telescope) is known to tilt annually
in correlation with its own noninertial motions, and by
the (B) view this tilting also effects a counter
compensation of the gun aimings at the source end de
signed to compensate for the duck’s dodgings—the result
being that the aiming angle α ' at the source end is unaf
fected by variable detector motions, just as the receiving
angle α at the detector end is unaffected by variable
source motions. If that is a trifle raffiniert, note that you
were warned.
6. Summary
We have arrived at item (C), above, which encapsu
lates our findings concerning the physical nature of stellar
aberration, on the basis of assumptions compatible with
special relativity theory. That is, our analysis has employed
as its sole velocity parameter thesource-detector relative velocity
vsd between inertial systems K, K', just as Einstein did in 1905. The other major ingredient in our reasonings—the
assumption of reciprocity or dualism between source and
detector, accompanying ray-direction reversal—is also
compatible with relativity and with a Maxwellian descrip
tion of light.
So, the present analysis and the morning stars all sing
together in celestial harmony with the known eternal
verities of physics, right? Any AIP journal would welcome
our submission, right? Wrong and wrong. There’s still
plenty that needs greasing among the squeaky joints that
connect relativity ideology to the real world. For instance,
we have not shown any mathematical path between the
vsd and vorb parametrizations. Consider the following twoparagraph essay entitled, “Where covariance fails:”
The Lorentz transformation of the kμ four-vector of light propagation is parametrized by the relative velocityv
Figure 2. Duck-hunter model, showing effect of “beaming” of star
light. At any six-month interval the two emission directions n′ in K',
produced by orbital detector motions affecting vsd , are separated by an
angle equal to twice the constant of aberration, ε 0 = 20.5". The light
intensities in these two directions are in the ratio of lengths SA:SB. An
nual modulation of the light intensity detected by an earthly detector
results.


Page 18 APEIRON Nr. 19 June 1994
of two inertial systems, K, K'. That transformation, ex
pressing the components of kμ ' in terms of those of kμ ,
describes both the Doppler effect on frequency and some
kind of optical aberration, purported to be stellar. (In fact
we have shown that there is some substance in that claim.)
The inextricably combined description of these two
physical phenomena, Doppler and aberration, reflects the
fact that they are physically symmetrical in exactly the
same way that “space” and “time” are physically symmet
rical—this being the meaning of covariance. Just as no correct
physicist can fail to speak of spacetime, and as our equally
politically aware Air Force brethren-in-propaganda take
care to speak of aerospace, we must twist up our forked
tongues and speak of “Dopplerstellaraberration,” all distinction
being faded to mere shadows. (This elocution will occur in
unison, commencing at 9 AM Monday, by Deans’ orders
posted on all bulletin boards.)
The Doppler effect depends for its quantitative de
scription on the source-detector relative velocity parame
ter vsd . Since there is one and only one v-parameter involved in the Lorentz transformation, it follows by ne
cessity that v vsd
= . Stellar aberration depends for its quantitative description on the earth’s orbital velocity vorb . Hence by equally stringent necessity v vorb
= . But in general v v
sd orb
≠ . Hence v ≠ v, a relationship that should trouble relativists, since it would trouble the mathemati
cians from whom as a class they are indistinguishable by
operational test.
We call attention to the need for some astronomical
data-gathering, or examination of existing data, in order to
determine if any examples exist of stellar objects with
known directional inhomogeneities of radiation emission
(particularly “relativistic jets”) that display intensity varia
tion periods exactly equal to one year. If this proves to be a
prevalent condition—or even if enough examples can be
found to exceed pure statistical chance —then our hy
pothesis (B), above, is empirically counter-indicated, the
simple “duck-hunter model” is supported, and the present
attempt to clarify the underlying physics of stellar aberra
tion will require agonizing reappraisal.
Finally, it might be mentioned that stellar aberration is
by no means uniquely dependent for its quantitative de
scription upon either Maxwell’s electrodynamics or Ein
stein’s relativity. Alternative theory (Phipps 1991) does
exist, although present-day instrumentation does not sup
port its crucial testing. In short, the mathematical part of
the description [particularly the higher-order corrections
in Equation (3)], which seems herein and to relativists so
secure, is actually the part most vulnerable to change;
whereas the physical or qualitative grasp of the situation,
once achieved, is forever. Come, fellow physicists .. .
surely something more to the point can be found than
Eddington’s umbrella.
References
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