A carousel platform turns at constant angular velocity while a lift raises it along…

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 carousel platform turns at constant angular velocity Ω\Omega while a lift raises it along its axis at constant speed ww. Cylindrical coordinates (t,ρ,ϕ,z)(t,\rho,\phi,z) that turn and rise with the platform give flat spacetime the form ds2=−(c2−Ω2ρ2−w2) dt2+2Ωρ2 dt dϕ+2w dt dz+dρ2+ρ2dϕ2+dz2.\begin{aligned}ds^2 &= -(c^2-\Omega^2\rho^2-w^2)\,dt^2 + 2\Omega\rho^2\,dt\,d\phi\\ &\quad + 2w\,dt\,dz + d\rho^2 + \rho^2 d\phi^2 + dz^2.\end{aligned} An engineer argues: "This metric has two cross terms, dt dϕdt\,d\phi and dt dzdt\,dz, and ∂t\partial_t is the only candidate timelike Killing vector here, so removing them is impossible while keeping tt as a time coordinate. The platform spacetime is therefore stationary but not static." ((a)) Say whether the engineer's conclusion is correct. ((b)) If it is not, exhibit explicitly a Killing vector of this spacetime, other than ∂t\partial_t, and show that a coordinate change built from it removes both cross terms.

Browse the problem library