Special Relativity – Examining the Four-Velocity Vector of Light and Its Implications

photonsspecial-relativityspeed-of-lightvelocityvisible-light

Please note that my question is not a duplicate, it is not about the speed of light, my question is only technical about the four velocity vector for light, its definedeness, value and constantness.

I have read these questions:

What is the time component of velocity of a light ray?

Where Izhov says:

Four-velocity actually isn't well-defined for light.

And where ClassicStyle says in a comment:

The four velocity of light is perfectly well defined. You just can't use proper time to parameterize the world line. Four velocity is just the tangent vector to a world line

Are components of the velocity of light equal to $c$?

https://en.wikipedia.org/wiki/Four-vector

The four-velocity defined here using the proper time of an object does not exist for world lines for objects such as photons travelling at the speed of light

Why is light affected by time dilations in space-time curvatures

Where Safesphere says in a comment:

The magnitude of the 4-velocity of light is always zero (see my comment above).

The (always) non-zero time component of the 4-velocity of light does NOT mean that light moves in time. To calculate the 4-velocity of light, we have to use a different affine parameter instead of proper time, because the proper time of light is always zero.

Now this is confusing. Light must have a four velocity vector, but it seems to be either well defined or not, and it seems to have a magnitude of 0 or c and it seems to be always constant or not.

Questions:

  1. Which one is right, is the four velocity of light well defined or not?

  2. Is the magnitude of the four velocity vector for light always constant?

  3. Is the magnitude 0 or c?

Best Answer

From "A First Course in General Relativity":

2.3 The four velocity

A particularly important vector is the four-velocity of a world line. ... In our four-geometry we define the four-velocity $\vec U$ to be a vector tangent to the world line of the particle, and of such a length that it stretches one unit of time in that particle's frame.

The immediate problem for the case of a photon is that it does not have a frame. Schutz makes this explicit here:

2.7 Photons

No four-velocity. Photons move on null lines, so, for a photon path,

$$\mbox{d}\vec x \cdot \mbox{d}\vec x = 0$$

Therefore $\mbox{d}\tau$ is zero and Eq. (2.31) $[\vec U = \mbox{d}\vec x / \mbox{d}\tau]$ shows that the four-velocity cannot be defined. Another way of saying the same thing is to note that there is no frame in which light is at rest (the second postulate of SR), so there is no MCRF for a photon. Thus, no $\vec e_0$ in any frame will be tangent to a photon's world line.

Note carefully that it is still possible to find vectors tangent to a photon's path (which, being a straight line, has the same tangent everywhere): $\mbox{d}\vec x$ is one. The problem is finding a tangent of unit magnitude, since they all have vanishing magnitude.

So, by the above, the answer to your first question is: the four-velocity is not defined for photons.

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