A displacement four-vector has components metres

PHYSICS · Subjective · 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 displacement four-vector has components Aμ=(2,1,2,2)A^\mu = (2, 1, 2, 2) metres. Someone copies the metric from a particle-physics text that uses the signature (+,−,−,−)(+,-,-,-), lowers the index with it, and finds AμAμ=(2)(2)+(1)(−1)+(2)(−2)+(2)(−2)=−5 m2.A^\mu A_\mu = (2)(2) + (1)(-1) + (2)(-2) + (2)(-2) = -5\ \mathrm{m}^2. Then, remembering that in this book's signature a negative invariant square means timelike, they conclude that AμA^\mu is timelike. Find the error in this working, compute the invariant square in this book's signature (−,+,+,+)(-,+,+,+), and state whether AμA^\mu is timelike, spacelike or null.

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