In the flat plane with Cartesian coordinates and metric

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

The Geometry of Gravity, Ch Symmetry: Lie derivatives, Killing vectors and conserved quantities

Problem

In the flat plane with Cartesian coordinates (x,y)(x,y) and metric ds2=dx2+dy2ds^2=dx^2+dy^2, a puck of mass mm slides with no force acting on it, so its trajectory is a geodesic, a straight line traversed at constant speed, parametrised by time tt. ((a)) Write Killing's equation ∇(iξj)=0\nabla_{(i}\xi_{j)}=0 out in these coordinates as three equations for ξx(x,y)\xi_x(x,y) and ξy(x,y)\xi_y(x,y), and solve them: show that every Killing vector of the plane is ξ=(b1−ky) ∂x+(b2+kx) ∂y\xi=(b_1-ky)\,\partial_x+(b_2+kx)\,\partial_y for constants b1b_1, b2b_2 and kk. ((b)) Hence state how many independent Killing vectors the plane has, compare with the maximum for a two-dimensional space, and identify them. ((c)) At t=0t=0 the puck is at (x,y)=(1.0 m,2.0 m)(x,y)=(1.0\ \mathrm{m},2.0\ \mathrm{m}) moving with velocity (vx,vy)=(3.0 m s−1,4.0 m s−1)(v_x,v_y)=(3.0\ \mathrm{m\,s^{-1}},4.0\ \mathrm{m\,s^{-1}}). Using the rotation x∂y−y∂xx\partial_y-y\partial_x and the chapter's result that ξix˙i\xi_i\dot x^i is constant along a geodesic, evaluate L≡xvy−yvxL\equiv xv_y-yv_x at t=0t=0 and again at t=5.0 st=5.0\ \mathrm{s}, and confirm it is unchanged.

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