Consider, as in Ex. 2.9, an observer with 4-velocity U(vector) who measures the properties of a particle
Question:
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
Step by Step Answer:
Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics
ISBN: 9780691159027
1st Edition
Authors: Kip S. Thorne, Roger D. Blandford