## K3    General Formulas for Velocity Addition, the Doppler Effect and Aberration

In sections D4 and D6 we considered how to add velocities in x direction in the STR. We also determined the correct formula for the Doppler Effect when the source moves in a direct line either toward or away from the observer. In Section D5, we saw how ‘transverse’ velocities transform and we also derived the angle of aberration one must take into account when the observer moves perpendicular to the segment connecting the source and observer.

These results are (important) special cases of more general results, which we could easily derive given our previous work. We will forego doing so and simply present references for further reading. Perhaps you may want to find your own derivation?

General formulas for the addition of velocities:

• by Einstein himself: [9-140ff]
• by Horst Melcher: [27-37ff]by Michael Fowler: [24-69f]
• by Roman Sexl and Herbert Schmidt: [25-100ff]
• by Jürgen Freund: [26-73ff]

General formulas for the Doppler Effect:

• by Einstein without calculations: [9-146ff]
• by Horst Melcher: [27-72ff]
• by Jürgen Freund: [26-117ff]
• by Edwin Taylor und John Archibald Wheeler with 4-dimensional vectors: [11-263]

General formulas for Aberration:

• by Einstein without calculations: [9-146ff]
• by Horst Melcher: [27-74ff]
• by Jürgen Freund: [26-89ff]