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Why the Bradley aberration cannot be used to measure absolute speeds. A comment
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EUROPHYSICS LETTERS
Europhys. Lett., 58 (4), pp. 637–638 (2002) Comment

15 May 2002

Why the Bradley aberration cannot be used to measure absolute speeds. A comment

K. Kassner
Institut fu¨r Theoretische Physik, Otto-von-Guericke-Universita¨t Magdeburg Postfach 4120, D-39016 Magdeburg, Germany

(received 3 September 2001; accepted 25 February 2002)

PACS. 06.30.Gv – Velocity, acceleration, and rotation. PACS. 07.87.+v – Spaceborne and space research instruments, apparatus, and components
(satellites, space vehicles, etc.). PACS. 95.55.-n – Astronomical and space-research instrumentation.

Abstract. – In a recent article in this journal (Sardin G., Europhys. Lett., 53 (2001) 310), Sardin proposed to use the Bradley aberration of light for the construction of a speedometer capable of measuring absolute speeds. The purpose of this comment is to show that the device would not work.

Stellar aberration appears to be independent of the velocity of the star observed. This fact, even though explained long ago, has remained a source of continuing confusion. After all, it seems difficult to reconcile it with the relativity principle. Indeed, if the arguments of [1] could be upheld, they would lead to a rejection of this principle.
Roughly speaking, aberration is the difference between the observed and “true” angular positions of a star, caused by the motion of either the observer or the star. In order to get a workable definition of true aberration, a frame must be given in which to measure the true position of a star. Let us choose this frame at rest with respect to the star, because then the latter will always be seen at the same angle. It is easy to calculate the so-defined true aberration for various situations. For simplicity, the calculations will be done with the velocities of the star and the observer parallel to a prescribed line taken as x-axis.
First, let us assume the star to move at velocity −V (in a certain frame Σ ) and the observer to be at rest. Let a light ray emitted at an angle θ with respect to the x-axis in the star’s frame Σ hit the observer’s eye. What is the observed angle θ ? The answer is provided by the relativistic addition theorem for velocities:

u =u

+

u

⊥

=

u −V 1 − V u/c2

+

γ(1

u⊥ − V u/c2)

,

(1)

where u is the velocity in Σ, u ≡ V (V u)/V 2 and u⊥ ≡ u − u are its components parallel and perpendicular to V , respectively, u , u , u ⊥ are the corresponding quantities in Σ and γ = (1−V 2/c2)−1/2. Note that while the speed of light is the same in the two frames, its velocity
(vector) is not. In Σ, we have u = −c(cos θex + sin θey), so we obtain u = −c(cos θ ex + sin θ ey), with cos θ = (cos θ + V /c)/[1 + (V /c) cos θ] and sin θ = sin θ/[γ(1 + (V /c) cos θ)], which can be simplified, using tan x/2 = sin x/(1 + cos x), to yield the well-known result

θ

θ

tan = tan

2

2

c−V c+V

.

(2)

c EDP Sciences

638

EUROPHYSICS LETTERS

Consider now the situation where the observer moves at V in some system Σ˜ , while the star is at rest (hence Σ˜ = Σ). If the angle observed in Σ is θ again, what will its value be in Σ˜ ? One finds, using the addition theorem with V instead of −V : tan θ˜/2 = tan θ /2 (c + V )/(c − V ). Hence θ˜ = θ, and the true aberration is the same for the two situations. It depends on the
relative velocities of star and observer only.
However, with an a priori unknown velocity of the star, an observer at rest with respect
to it is not normally handy. Therefore, what is measured is not the true aberration, in
general. Let us then define as relative aberration the difference in angles observed by two
arbitrary observers moving at different velocities V1 and V2 and looking at the same star (the moment they meet). From (2), we have tan θ1/2 = tan θ/2 (c − V1s)/(c + V1s) and tan θ2/2 = tan θ/2 (c − V2s)/(c + V2s), where V1s and V2s are the velocities of the observers relative to the star. θ may be eliminated to obtain tan θ2/2 = tan θ1/2 (c − V2s)(c + V1s)/(c + V2s)(c − V1s). Using the velocity addition theorem to express V1s/2 by V1/2 and the unknown velocity W of the star, V1s/2 = (V1/2 − W )/(1 − V1/2W/c2), we find:

tan θ2 = tan θ1

2

2

(c (c

− +

W )(c W )(c

+ −

V1) V1)

(c (c

+ −

W )(c W )(c

− +

V2) V2)

=

tan

θ1 2

c c

− +

V21 V21

,

(3)

where V21 = (V2 − V1)/(1 − V1V2/c2) is the relative velocity between the two observers. The speed W of the star cancels out of the formula! Of course, we could have obtained this result directly by using the same reasoning as in the derivation of (2). But eq. (3) is much more instructive, as it gives us also conditions under which the velocity of the star drops out.
For what is measured as Bradley aberration is not precisely the relative aberration. We do not simultaneously have two observers handy. What is measured is the difference in angles obtained by one observer in different states of motion, at different times. But this is the same thing as the relative aberration provided the velocity of the star is constant between measurements. Then the only velocity that counts is V21, a relative velocity again. If the velocity of the star changed, we would obtain, instead of (3),

tan θ2 = tan θ1

2

2

(c (c

− +

W1)(c W1)(c

+ −

V1) V1)

(c (c

+ −

W2)(c W2)(c

− +

V2) V2)

,

(4)

where now W1 and W2 are the velocities of the star, when it emitted the light rays reaching the observer when she had the velocities V1 and V2, respectively.
Sardin’s device to measure “absolute speeds” replaces the star by a light source moving along with the observer. Thus the velocity of the source changes between successive observations at different observer speeds. Hence, the independence of Bradley aberration from the source velocity does not hold. But this independence is the central argument on which the working principle of the device is based. Therefore, the device will not work. In fact, we can calculate the aberration measured by the device from (4), because the velocities of the light source are known here. In any state of uniform motion we have W1 = V1, W2 = V2. The result tan θ2/2 = tan θ1/2 indicates that there will be no observable aberration. There is no need to abandon the relativity principle.
A more complete version of this comment, including figures, is given at the e-Print archive (http://arXiv.org) as astro-ph/0203056.

REFERENCES [1] Sardin G., Europhys. Lett., 53 (2001) 310 .