A particle at

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A particle at r=3.0 mr = 3.0\,\mathrm{m}, θ=60∘\theta = 60^\circ in spherical coordinates has coordinate-basis velocity components Vr=r˙=2.0 m s−1V^r = \dot r = 2.0\,\mathrm{m\,s^{-1}}, Vθ=θ˙=0.50 rad s−1V^\theta = \dot\theta = 0.50\,\mathrm{rad\,s^{-1}} and Vϕ=ϕ˙=0.30 rad s−1V^\phi = \dot\phi = 0.30\,\mathrm{rad\,s^{-1}}, using the metric grr=1g_{rr}=1, gθθ=r2g_{\theta\theta}=r^2, gϕϕ=r2sin⁡2θg_{\phi\phi}=r^2\sin^2\theta. ((a)) Find the covariant components VrV_r, VθV_\theta, VϕV_\phi (state the unit of each). ((b)) Find the physical (orthonormal) velocity components vrv_r, vθv_\theta, vϕv_\phi, each with the unit m s−1\mathrm{m\,s^{-1}}, using vi=gii Viv_i = \sqrt{g_{ii}}\,V^i (no sum). ((c)) Find the particle's speed.

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