Flat space in cylindrical coordinates has metric

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

The Geometry of Gravity, Ch The covariant derivative, parallel transport and geodesics

Problem

Flat space in cylindrical coordinates (ρ,ϕ,z)(\rho,\phi,z) has metric ds2=dρ2+ρ2dϕ2+dz2ds^2 = d\rho^2 + \rho^2d\phi^2 + dz^2, so g=ρ\sqrt{g} = \rho. A radially expanding velocity field has orthonormal (physical) component V^ρ=kρ2\hat V^\rho = k\rho^2 and V^ϕ=V^z=0\hat V^\phi = \hat V^z = 0, with k=0.50 m−1 s−1k = 0.50\ \mathrm{m^{-1}\,s^{-1}}. ((a)) Write the coordinate component VρV^\rho (noting gρρ=1g_{\rho\rho}=1, so it equals V^ρ\hat V^\rho) and use ∇μVμ=(1/g)∂μ(g Vμ)\nabla_\mu V^\mu = (1/\sqrt{g})\partial_\mu(\sqrt{g}\,V^\mu) to find ∇μVμ\nabla_\mu V^\mu as a function of ρ\rho. ((b)) Evaluate it at ρ=2.0 m\rho = 2.0\ \mathrm{m}.

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