Plane polar coordinates have metric , ,

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

The Geometry of Gravity, Ch Manifolds, vectors, tensor fields and the metric

Problem

Plane polar coordinates have metric

g_{rr}=1

g_{\phi\phi}=r^2

g_{r\phi}=0

, a flat space written in a chart whose components depend on position. Build Riemann normal coordinates centred at the point

r=r_0=4.0

\phi=\phi_0=0

, following the same two-stage construction used on the sphere. ((a)) Let

x^1=r-r_0

and

x^2=r_0(\phi-\phi_0)

. Show this already gives

g_{ij}(P)=\delta_{ij}

, then find the one non-zero first derivative

\partial_1g_{22}

P

and evaluate it numerically. ((b)) Using

A_{\mu\alpha\beta}=-\tfrac12(\partial_\alpha g_{\mu\beta}+\partial_\beta g_{\mu\alpha}-\partial_\mu g_{\alpha\beta})

, find the quadratic change of coordinates

x^1(x'^1,x'^2)

x^2(x'^1,x'^2)

that removes this derivative, and evaluate its numerical coefficients at

r_0=4.0

m. ((c)) Substitute this quadratic change back into

g_{11}

g_{22}

and

g_{12}

and expand to first order in

x'^1,x'^2

, to confirm, as the sphere's own worked example checked, that no linear term in

x'^1

x'^2

survives anywhere.

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