Let S be the frame where the Sun is at rest. Imagine light from the North Star reaches the centre of the Sun, and let's define the equatorial plane as the plane that is perpendicular to this light and cuts the Sun into two hemisphere.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose a distant star A is on this equatorial plane and its light also reaches the centre of the Sun. Let the direction of the propagation of this light be our x axis. And let our y axis be pointing in the direction of the North Star.

In other words, the light from star A is travelling in the positive x direction, while that from the North Star is travelling in the negative y direction.

Let S' be the frame that is moving with respect to S in the positive x direction at velocity v.

By the transformation equation of velocity,

##u_x^\prime=\frac{u_x-v}{1-u_xv/c^2}## , ##u_y^\prime=\frac{u_y\sqrt{1-v^2/c^2}}{1-u_xv/c^2}## , ##u_z^\prime=\frac{u_z\sqrt{1-v^2/c^2}}{1-u_xv/c^2}##

the light from the North Star and that from star A are still travelling in the negative y direction (or y', to be precise) and in the positive x direction (or x'), respectively, to an observer in S'.

This contradicts the observation of stellar aberration, where the two velocities are not perpendicular in S'.

What did I miss?

Are two perpendicular vectors still perpendicular when we change reference frames?

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# Transformation equation of velocity contradicts stellar aberration?

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