Work in Minkowski spacetime with inertial coordinates

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

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

Problem

Work in Minkowski spacetime with inertial coordinates xμ=(ct,x,y,z)x^\mu=(ct,x,y,z), metric ημν=diag(−1,1,1,1)\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1) and Christoffel symbols that vanish in these coordinates. Let VμV^\mu and WμW^\mu be two arbitrary smooth vector fields. ((a)) Starting from LVWμ=Vν∇νWμ−Wν∇νVμ\mathcal{L}_VW^\mu = V^\nu\nabla_\nu W^\mu - W^\nu\nabla_\nu V^\mu, written in an arbitrary chart, not necessarily inertial, for a general torsion-free, metric-compatible connection Γλμν=Γλνμ\Gamma^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\nu\mu} (not assumed to vanish), show that the connection terms cancel identically, leaving LVWμ=Vν∂νWμ−Wν∂νVμ\mathcal{L}_VW^\mu = V^\nu\partial_\nu W^\mu - W^\nu\partial_\nu V^\mu. ((b)) The vector field ξμ∂μ=x ∂ct+ct ∂x\xi^\mu\partial_\mu = x\,\partial_{ct} + ct\,\partial_x is one of the ten Killing vectors of Minkowski spacetime (a boost generator). A free particle, meaning one on which no four-force acts, moves on a geodesic with constant four-velocity uμ=γ(c,v,0,0)u^\mu=\gamma(c,v,0,0), γ=(1−v2/c2)−1/2\gamma=(1-v^2/c^2)^{-1/2}. Prove, for a general Killing vector ξ\xi and a general geodesic with four-velocity uμu^\mu and affine parameter λ\lambda, that ξμuμ\xi_\mu u^\mu is constant along the worldline. ((c)) State the maximum possible number of independent Killing vectors of an nn-dimensional spacetime and confirm that Minkowski spacetime attains it. Then take the free particle of part (b) to cross x=x0=10 mx=x_0=10\ \mathrm{m} at t=0t=0 with v=0.6cv=0.6c, so its worldline is x(t)=x0+vtx(t)=x_0+vt. Compute ξμuμ\xi_\mu u^\mu as an explicit function of tt using the boost Killing vector above, and confirm that it is the constant part (b) requires.

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