The upper half-plane with line element is another surface of constant negative curvature

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 upper half-plane

y>0

with line element

ds^2 = (dx^2 + dy^2)/y^2

is another surface of constant negative curvature, now in a chart where both metric components vary. Form the Lagrangian

L = (\dot x^2 + \dot y^2)/y^2

(a dot is

d/d\lambda

for an affine parameter

\lambda

), write its two Euler-Lagrange equations, and match them to the geodesic equation

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

to read off every non-zero Christoffel symbol. ((a)) State the two Euler-Lagrange equations. ((b)) Give every non-zero

\Gamma^\lambda{}_{\mu\nu}

as a function of

y

. ((c)) Evaluate

\Gamma^x{}_{xy}

\Gamma^y{}_{xx}

and

\Gamma^y{}_{yy}

y = 2.0

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