A particle moves at along

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics

The Geometry of Gravity, Ch Minkowski spacetime and relativistic dynamics, completed

Problem

A particle moves at v=0.60cv=0.60c along xx, so γ=1.25\gamma=1.25 and, in this book's signature (−,+,+,+)(-,+,+,+), its four-velocity is uμ=(54c,34c,0,0)u^\mu=(\tfrac54c,\tfrac34c,0,0). ((a)) Lower the index in this book's signature to find uμu_\mu, and confirm uμuμ=−c2u^\mu u_\mu=-c^2 by direct computation. ((b)) Using the signature-conversion rule, find uμu_\mu in the opposite convention (+,−,−,−)(+,-,-,-) without changing uμu^\mu, and confirm uμuμ=+c2u^\mu u_\mu=+c^2 there. ((c)) Explain, in one sentence, why uμu^\mu itself did not change between (a) and (b), even though uμu_\mu and uμuμu^\mu u_\mu both did.

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