A two-dimensional surface of constant negative curvature (a model hyperbolic plane)…

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

A two-dimensional surface of constant negative curvature (a model hyperbolic plane) carries coordinates

(\chi,\phi)

and line element

ds^2 = a^2\left(d\chi^2 + \sinh^2\chi\, d\phi^2\right),

with

a

a constant length. Consider a curve

x^\mu(\lambda) = (\chi(\lambda), \phi(\lambda))

and the Lagrangian

L = g_{\mu\nu}\dot x^\mu\dot x^\nu

. ((a)) Write

L

out and derive the two Euler-Lagrange equations for

\chi(\lambda)

and

\phi(\lambda)

. ((b)) By matching each equation to the geodesic equation

\ddot x^\mu + \Gamma^\mu{}_{\alpha\beta}\dot x^\alpha\dot x^\beta = 0

, read off every non-zero Christoffel symbol of this metric (remembering that a mixed pair

\chi\phi

receives contributions from both

\Gamma^\phi{}_{\chi\phi}

and

\Gamma^\phi{}_{\phi\chi}

). ((c)) Evaluate the non-zero symbols numerically at

\chi = 1

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