In the flat plane with plane-polar coordinates

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

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

Problem

In the flat plane with plane-polar coordinates, a straight line at perpendicular distance d=3.0 md = 3.0\ \mathrm{m} from the origin is parametrised by arc length ss as r(s)=d2+s2,ϕ(s)=arctan⁡(s/d).r(s) = \sqrt{d^2+s^2}, \qquad \phi(s) = \arctan(s/d). The geodesic equations of the plane in polar coordinates are r¨−rϕ˙2=0\ddot r - r\dot\phi^2 = 0 and ϕ¨+(2/r)r˙ϕ˙=0\ddot\phi + (2/r)\dot r\dot\phi = 0, where a dot is d/dsd/ds. ((a)) Verify that this curve satisfies both geodesic equations identically (for any ss). ((b)) Evaluate rr, r˙\dot r, r¨\ddot r, ϕ˙\dot\phi and ϕ¨\ddot\phi at s=4.0 ms = 4.0\ \mathrm{m} and confirm both equations hold numerically at that point.

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