D5 Transverse Velocities and Aberration
How is it for black, when in the red system an object moves with a velocity of u' transverse to the direction of relative motion v? Up to now we have considered only movements along the x-axis.
For the derivation of the transformation of such ‘transverse velocity’ we do not need the Lorentz transformations. Knowledge of the basic phenomena is completely sufficient. Since u' is a velocity e.g. in the y'-direction of red, it follows
u' = ∆y' / ∆t' = ∆y / ∆t' = ∆y / (∆t • √) = (∆y / ∆t) / √ = u / √
Black thus measures the smaller lateral velocity u = u' • √
We will need this result in E1. We use it here for the derivation of the correct formula for aberration. By aberration (lat. aberrare ~ wander, deviate) we understand the change of direction of velocities, which arise as a result of the fact that the viewer likewise moves. James Bradley recognized in 1728 that the tiny annual-periodic position shifts of fixed stars are to be understood as the consequence of the movement of the earth around the sun. According to legend he got the idea, as he rode in his coach in windless rainy English weather and thereby observed that the rain seemed to fall diagonally, the faster the coach moved the more diagonal the rain.
Consider a telescope, pointing in a direction perpendicular to the momentary direction of the earth’s movement in its orbit:
In the time the light of a star needs to arrive from the objective to the eyepiece, the earth has already advanced on its course. We must therefore tip the telescope through an angle α, in order for the star to be presented in the center of the visual field. This angle defines the amount ‘the ray of light wanders off’ due to the movement of the earth. The resulting formula for this aberration is tan (α) = v/c, where v is the velocity of the earth in its orbit (approximately 30 km/s). The angle has a size of approximately 20 arc seconds.
Einstein had already in 1905 drawn attention to the fact that this traditional formula is only approximately correct. Lateral velocities should be transformed according to the above formula, resulting in the correct formula tan (α) = v/(u' • √) = v/(c • √) One also obtains this formula, if one assigns (correctly) the distance travelled by the light to the hypotenuse of the triangle instead of the leg in the above figure. For the lateral velocity u of the light, the Pythagorean Theorem gives u = c • √, which again yields for tan (α) the value v/u = v/(c • √). Thus for light the new accurate aberration formula is sin (α) = v/c. This correction is astronomically insignificant, since the values of the sine and the tangent functions hardly differ for small angles.
By the way: The angle of aberration is independent of the speed of light in the telescope tube. The angle of aberration does not change if you fill your telescope with water! A more in depth argument would consider the direction of the optical wave planes.
We could now consider the general case, where in the red system an object moves with any velocity w’ + u’ in any direction. Einstein already handled this case in his original publication of 1905 [09-140ff] and presents beautiful symmetrical formulas for the resulting velocities and angles in the black system. Likewise the aberration is treated completely generally [09-146ff]. The appropriate calculations should now be well comprehensible to the reader. We will however not need these results in what follows. K3 makes some references to it.