shows the hyperplane of simultaneity defined in Section 16.16.1 when eˆ : (e0, e1, 0, 0)S. (a) Show.

shows the
hyperplane of simultaneity defined in Section 16.16.1 when eˆ : (e0, e1, 0,
0)S.

(a) Show that two
events in the hyperplane have a separation dr that has a zero dot product with
eˆ. Show, using that the plane and the orientation vector eˆ make equal angles
above and below the light cone, as shown.

(b) Show that if
a clock has a world line parallel to e and passing through the origin event,
then θ is the proper time measured on that clock as it moves from the
origin event to r.

(c) Show that if
S is a system in which eˆ : (1, 0, 0, 0)S then the coordinate time t in system
S is equal to θ.