A free particle of rest mass kg travels through a straight drift tube

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Symmetry: Lie derivatives, Killing vectors and conserved quantities

Problem

A free particle of rest mass m=1.20m = 1.20 kg travels through a straight drift tube, described by Minkowski coordinates xμ=(ct,x,y,z)x^\mu = (ct,x,y,z) with metric ημν=diag(−1,+1,+1,+1)\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1) and c=3.00×108 m s−1c = 3.00\times10^8\ \mathrm{m\,s^{-1}}. Consider the vector field ξ=∂x\xi = \partial_x. ((a)) Compute Lξημν\mathcal{L}_\xi \eta_{\mu\nu} and decide whether ξ\xi satisfies Killing's equation. ((b)) State which one of Minkowski spacetime's ten independent Killing vectors ξ\xi is. ((c)) The particle moves with vx=0.600cv_x = 0.600c and vy=vz=0v_y = v_z = 0. Compute the numerical value of the conserved quantity that ξ\xi produces along its worldline.

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