# Components of velocity 3-vector of light in spherical coordinates

The components of the velocity 3-vector of a light ray in cartesian coordinates can be found given a polar angle and azimuthal angle as follows:

$$v_x = ||c|| \sin (\theta_0) \cos (\phi_0) \\ v_y = ||c|| \sin (\theta_0) \sin (\phi_0) \\ v_z = ||c|| \cos (\theta_0)$$

How would the components of the velocity 3-vector of light be found in spherical coordinates?

• When people say "light ray," they usually mean a plane wave of light, where all of the light goes parallel. Do you mean that, or are you asking about light emitted spherically from a pointlike source?
– rob
May 12, 2023 at 23:44
• @rob apologies for the misphrasing, the term "photon" may be more appropriate in this context May 13, 2023 at 0:40

$$v_r = c\\v_\vartheta=\vartheta_0\\v_\varphi=\varphi_0$$ Basically by definition, or visual inspection of how the relations are meant to work.
• That’s incorrect. In spherical coordinates, the directions of the unit vectors vary with the coordinates $(\theta,\phi)$. The derivatives of non-radial vectors are complicated because you must also differentiate the unit vectors. Radial vectors are nice because the entire magnitude points in the $\hat r$ direction; a spherical light wave has zero velocity in the $\hat\theta$ or $\hat\phi$ directions. But consider your motion as you follow the rotation of the Earth. Your $\theta,\phi$ are related to your latitude and longitude, but your angular speed is $v_\phi=2\pi\rm\,radian/day.$