Answer to Question #97659 in Mechanics | Relativity for AbdulRehman

Question

Use your knowledge of geometry (and/or trigonometry) to show that the
angle α between the worldline of an inertial observer and the lightcone
(drawn in an inertial frame using units in which light rays travel at 45
degrees) is the same as the angle β between that observer’s line of si-
multaneity and the lightcone. (Hint: Many people find it easier to use
trigonometry to solve this problem than to use geometry).

Answer 1

If we move our observer to the right, his or her worldline tilts to the right. To meet this worldline, the line of simultaniety also tilts up for the same angle. And since the lightcone is inclined at 45° from both the horizontal and vertical, the angles alpha and beta will be the same: they are equal 45° minus the angle between the time and the worldline, or the horizontal and the line of simultaneity. The same situation we'll have if the observer moves to the left.