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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?

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  • $\begingroup$ 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? $\endgroup$
    – rob
    May 12, 2023 at 23:44
  • $\begingroup$ @rob apologies for the misphrasing, the term "photon" may be more appropriate in this context $\endgroup$
    – JS4137
    May 13, 2023 at 0:40

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$$v_r = c\\v_\vartheta=\vartheta_0\\v_\varphi=\varphi_0$$ Basically by definition, or visual inspection of how the relations are meant to work.

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  • $\begingroup$ Do you mean for the angular velocities to be zero? You seem to have set them equal to the initial angles, but that has the wrong units. $\endgroup$
    – rob
    May 12, 2023 at 23:41
  • $\begingroup$ No, that is correct. For any vector in polar form, the angle parts are, literally, angles. $\endgroup$ May 13, 2023 at 3:06
  • $\begingroup$ 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 $v_\phi=2\pi\rm\,radian/day.$ $\endgroup$
    – rob
    May 13, 2023 at 3:45
  • $\begingroup$ No, you are just wrong, and irrelevant to the purposes of both the OP and my answer. What the OP wanted, is to describe the wavefront propagation vector of what we model as infinite plane waves, and not spherical waves. That is very obvious when you see that there are subscript zeroes on the angles. Those are constants and not changing with time. Also, if you learnt any General Theory of Relativity, you would know that coördinates are not vectors, and the first true contravariant vector is the velocity. Even if the unit vectors change direction, the fix lies in Christoffel symbols. Not itself $\endgroup$ May 13, 2023 at 3:51

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