Working in units where

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

Working in units where c=1c=1, so that times and distances share the same unit (here taken as seconds, one second of distance meaning 3.00×1083.00\times10^8 m), consider Minkowski spacetime with coordinates (t,x)(t,x) (suppress y,zy,z) and metric ds2=−dt2+dx2ds^2=-dt^2+dx^2. Let ξ=t ∂x+x ∂t\xi = t\,\partial_x + x\,\partial_t. ((a)) Verify that ξ\xi satisfies Killing's equation for this metric. ((b)) State which family of Minkowski spacetime's ten Killing vectors ξ\xi belongs to. ((c)) A cosmic-ray track crosses the event t=3.00t=3.00 s, x=1.00x=1.00 s, with four-velocity components ut=1.25u^t=1.25, ux=0.75u^x=0.75 there. Compute the numerical value of the conserved quantity ξμuμ\xi_\mu u^\mu at that event.

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