Consider, as in Ex. 2.9, an observer with 4-velocity U(vector) who measures the properties of a particle with 4-momentum p(vector).(a) Show that the Euclidean metric of the observer’s 3-space, when thought of as a tensor in 4-dimensional spacetime, has the form

Show, further, that if A(vector) is an arbitrary vector in spacetime, then −A(vector) · U(vector) is the component of A(vector) along the observer’s 4-velocity U(vector), and

Data from Exercises 2.9

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

P=g+ . (2.31a)