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According to relativity, Light does not experience any time. So it must travel any distance in no time.

But, we know that light has finite speed $c$. So it should take finite time.

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Speed is measured with respect to some reference frame. Relativity does not allow for a reference frame that is comoving with a photon, so there's no sense within the theory talking about the time or speed "experienced" by the photon in "its own frame". From any valid frame, the speed of light is $c$.

The statement that light does not experience time is not really physically precise. It captures something about limiting behavior of certain formulas, but it is a limit that isn't realizable in the way that the simplified statement suggests.

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The spatial distance between 2 points depends on the frame of reference. So, while it is true that light has always the same speed c, the time it takes to travel the distance between that points is not a universal constant.

The notion of "zero time for the photon" must be understood like the definition of the limit of a function in a point in maths.

For example, it takes 4 years for the light from Alpha Centaury to come to us. But for a spaceship close to the speed of light towards the star, and that happens to be nearby the Earth, and for any arbitrarily small imagined time of travel for the light, there is a ship speed that corresponds to an even smaller time of travel for the light in the ship frame.

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You are saying that photons do not experience time. In reality, what we can say, is that photons do not have a reference frame. We cannot ride along the photon, and check what the universe would look like from that view, and we cannot check our clock how much time elapses, because there is no such frame.

What we can do, is check here on Earth, how much time it takes for a photon to reach us from the Sun. It takes 8 minutes. But this is because we view it from our frame here on Earth, divide the known distance by the time elapsed on our clocks between emission (in reality between when the photon exits the Sun's outer layers, it takes the photon millions of years to get our from the core where it was originally emitted) and absorption here on Earth. We get 8 minutes.

Without having seen the series, I think what is envisioned is that a photon created in the innermost parts of the Sun (where the fusion happens) could take millions of years to reach the surface due to (really heavy) scattering in warm plasma that is the suns interior.

Very old photons from the Sun

Now let's try to do a calculation from another frame. For example a frame of a neutrino, riding along the same path from the Sun to the Earth. On the neutrino's clock, the time elapsed is much less (relative to the 8 minutes). The neutrino's frame is practically as close you can get to the speed of light. So you get the idea. The closer you get to the speed of light, the less time elapses on your clock.

There is no such thing as an observer traveling with a photon.

How does a photon experience space and time?

Now the photon does not have a frame, but what you can say, is that the spacetime distance for the photon is 0 from the Sun to the Earth. But it is not correct to say that the photon does not experience time. What we can say, is that the closer you get to the speed of light, the less time will elapse on your clock (relatively), or that your clock ticks relatively slower.

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