The surface of a right circular cone is described by the two-dimensional metric ds^2…

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

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

Problem

The surface of a right circular cone is described by the two-dimensional metric ds2=dρ2+b2ρ2 dϕ2,ds^2 = d\rho^2 + b^2\rho^2\,d\phi^2, where ρ≥0\rho \ge 0 is the distance from the apex measured along the cone's surface, ϕ\phi is the azimuthal angle around the cone's axis, 0≤ϕ<2π0 \le \phi < 2\pi, and b=sin⁡αb = \sin\alpha with α\alpha the cone's half-angle (0<b<10 < b < 1). Take b=0.60b = 0.60. ((a)) Use the Christoffel formula to find every non-zero Γλμν\Gamma^\lambda{}_{\mu\nu} for this metric, treating (ρ,ϕ)(\rho,\phi) as the two coordinates. ((b)) Evaluate the non-zero symbols at ρ=2.0 m\rho = 2.0\ \mathrm{m}.

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