A free particle moves in the plane of Minkowski spacetime

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 moves in the z=0z=0 plane of Minkowski spacetime. At a certain instant its spatial position is (x,y)=(3.0 m,4.0 m)(x,y) = (3.0\ \mathrm{m}, 4.0\ \mathrm{m}) and its momentum components are px=2.0 kg m s−1p_x = 2.0\ \mathrm{kg\,m\,s^{-1}}, py=1.0 kg m s−1p_y = 1.0\ \mathrm{kg\,m\,s^{-1}}. Consider ξ=x ∂y−y ∂x\xi = x\,\partial_y - y\,\partial_x, the generator of rotation about the zz-axis. ((a)) Explain why ξ\xi satisfies Killing's equation for the Minkowski metric. ((b)) State which family of Minkowski spacetime's ten Killing vectors ξ\xi belongs to. ((c)) Compute the numerical value of the conserved quantity that ξ\xi produces along the particle's geodesic.

Browse the problem library