Special Relativity | Part IB, 2001

A particle of rest mass mm and four-momentum P=(E/c,p)P=(E / c, \mathbf{p}) is detected by an observer with four-velocity UU, satisfying UU=c2U \cdot U=c^{2}, where the product of two four-vectors P=(p0,p)P=\left(p^{0}, \mathbf{p}\right) and Q=(q0,q)Q=\left(q^{0}, \mathbf{q}\right) is PQ=p0q0pqP \cdot Q=p^{0} q^{0}-\mathbf{p} \cdot \mathbf{q}.

Show that the speed of the detected particle in the observer's rest frame is

v=c1PPc2(PU)2v=c \sqrt{1-\frac{P \cdot P c^{2}}{(P \cdot U)^{2}}}

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