An observer with 4-velocity U(vector) measures the properties of a particle with 4-momentum p(vector). The energy she measures is E = −p(vector) · U(vector) [Eq. (2.29)].

(a) Show that the particle’s rest mass can be expressed in terms of p(vector) as

(b) Show that the momentum the observer measures has the magnitude

(c) Show that the ordinary velocity the observer measures has the magnitude

where |p| and E are given by the above frame-independent expressions.

(d) Show that the ordinary velocity v, thought of as a 4-vector that happens to lie in the observer’s slice of simultaneity, is given by

E = - p. . (2.29)