A particle of rest mass is at rest in frame

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

A particle of rest mass mm is at rest in frame SS, so its four-velocity there is uμ=(c,0,0,0)u^\mu = (c, 0, 0, 0). Someone proposes a candidate covector CμC_\mu and writes, in SS, the fixed numbers Cμ=(mc, 0, 0, 0)C_\mu = (mc,\, 0,\, 0,\, 0). They argue: 'Contracting gives Cμuμ=mc⋅c=mc2C_\mu u^\mu = mc\cdot c = mc^2, a single definite number. Since CμuμC_\mu u^\mu is invariant, the quotient rule tells us CμC_\mu is a genuine type (0,1)(0,1) tensor, and its components are (mc,0,0,0)(mc,0,0,0) in every inertial frame.' ((a)) State precisely what hypothesis of the quotient rule this argument has failed to check. ((b)) Take frame S′S', moving at v=0.6cv = 0.6c along xx relative to SS (so γ=1.25\gamma = 1.25), in which the particle's correctly boosted four-velocity is u′μ=γ(c,v,0,0)u'^\mu = \gamma(c, v, 0, 0). If CμC_\mu really did keep the same numbers (mc,0,0,0)(mc,0,0,0) in S′S', as claimed, compute Cμu′μC_\mu u'^\mu in S′S' and show it does not equal mc2mc^2, refuting the claim.

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