The chapter justifies varying the quadratic Lagrangian in place of by comparing the…

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

The chapter justifies varying the quadratic Lagrangian L=gμνx˙μx˙νL=g_{\mu\nu}\dot x^\mu\dot x^\nu in place of ∫dτ=∫−gμνx˙μx˙ν dλ\int d\tau=\int\sqrt{-g_{\mu\nu}\dot x^\mu\dot x^\nu}\,d\lambda by comparing the two Euler-Lagrange expressions in one line. Carry out the square-root variation in full, in geometric units, for a timelike curve. ((a)) Take L=−gμνx˙μx˙νL=\sqrt{-g_{\mu\nu}\dot x^\mu\dot x^\nu} literally and compute ∂L/∂x˙σ\partial L/\partial\dot x^\sigma and ∂L/∂xσ\partial L/\partial x^\sigma by the chain rule. ((b)) Form the Euler-Lagrange equation for xσx^\sigma, and show that when λ\lambda is affine, so that LL itself is a constant along the whole curve, the equation reduces to gσνx¨ν+∂ρgσνx˙ρx˙ν−12∂σgμνx˙μx˙ν=0,g_{\sigma\nu}\ddot x^\nu+\partial_\rho g_{\sigma\nu}\dot x^\rho\dot x^\nu-\tfrac12\partial_\sigma g_{\mu\nu}\dot x^\mu\dot x^\nu=0, independent of the value of that constant, and hence, after the same symmetrisation and index-raising already used for the quadratic Lagrangian, to x¨λ+Γλμνx˙μx˙ν=0\ddot x^\lambda+\Gamma^\lambda{}_{\mu\nu}\dot x^\mu\dot x^\nu=0. ((c)) Apply the result of part (b) to the two-sphere of radius aa, ds2=a2(dθ2+sin⁡2θ dϕ2)ds^2=a^2(d\theta^2+\sin^2\theta\,d\phi^2), with λ\lambda taken to be arc length (the sphere's metric is positive definite, so use L=+gμνx˙μx˙νL=\sqrt{+g_{\mu\nu}\dot x^\mu\dot x^\nu}; the equation of part (b) is unchanged, since it does not depend on the sign under the root), and evaluate the resulting Γθϕϕ\Gamma^\theta{}_{\phi\phi} and Γϕθϕ\Gamma^\phi{}_{\theta\phi} at θ=50∘\theta=50^\circ.

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