Question

Find killing equations for spherical polar coordinates

Answer 1

Killing vector field is a vector field on a Riemannian manifold that preserves the metric.

Killing equation:

$\nabla_{\mu}X_{v}+\nabla_{v}X_{\mu}=0$

Killing fields on a 2-sphere:

The conventional metric on the sphere is

$g(\Omega)=d\theta^2+sin^2\theta d\phi^2$

Rotation about the pole should be an isometry. The infinitesimal generator of a rotation can then be identified as a generator of the Killing field. This can be

$U=\partial_ \phi$

The surface of the sphere is two-dimensional, and there is another generator of isometries; it can be taken as

$V=\partial_ \theta$

conventional 3-space coordinate system is given by

$x=sin\theta,y=sin\theta sin \phi,z=cos\theta$

first generator, rotation about the  z-axis:

$R=x\partial _y-y\partial_x=sin^2\theta \partial \phi$

second generator, rotation about the  x-axis:

$S=z\partial _y-y\partial_z$

third generator, rotation about the  x-axis:

$T=z\partial _x-x\partial_z$

Normalizing this, and expressing these in spherical coordinates:

$R'=\frac{R}{sin^2\theta}=\partial_ \phi$

$S'=\frac{S}{sin^2\theta}=sin\phi \partial_ \theta +cot\theta cos\phi \partial _\phi$

$T'=\frac{T}{sin^2\theta}=cos\phi \partial_ \theta +cot\theta sin\phi \partial _\phi$