A straight ray from the origin in the flat plane

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Vector calculus and field geometry

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

Problem

A straight ray from the origin in the flat plane, at the fixed angle

\phi=\phi_0=35^\circ

, is described in plane polar coordinates by

r(\sigma)=R_0\sigma^2

\phi(\sigma)=\phi_0

, with

R_0=3.0\ \mathrm{m}

and

\sigma>0

a dimensionless parameter along the curve. A student checks whether this curve is a geodesic using the plane's geodesic equations in polar coordinates,

\ddot r-r\dot\phi^2=0

and

\ddot\phi+(2/r)\dot r\dot\phi=0

(a dot meaning

d/d\sigma

), finds

\dot\phi=0

and

\ddot r=2R_0=6.0\ \mathrm{m}

, so the first equation gives

6.0\ \mathrm{m}-0\neq0

, and concludes: "A ray from the origin is not a geodesic of the flat plane." ((a)) This arithmetic is correct. Identify the error in the reasoning that produces the final conclusion. ((b)) Find an affine parameter for this same curve, and show explicitly that the geodesic equations are satisfied exactly once it is used. ((c)) Confirm that the residual just found is exactly the extra term

(f''/f')\,dr/d\sigma

that the chapter's reparametrisation formula predicts for a non-affine parameter, for the specific relation between

\sigma

and the affine parameter implied by

r(\sigma)=R_0\sigma^2

, and say what this shows about checking a curve this way in general.

Browse the problem library