Imagine this: There is a lighthouse sitting at the North Pole, its beam of light is rotating once per second..

Question

Ignoring attenuation by the Earth's atmosphere, how far out along the beam will the beam be travelling tangentially at the speed of light? Beyond this distance along the beam, will it be travelling tangentially faster than the speed of light? Remember, I am not refering to the speed radially, along the beam, but rather tangentially as it rotates. Also, please ignore the speed of rotation of the Earth, I am only interested in the rotation of the beam.

Answer 1

I think the question you are asking is how does the distance from the light source affect the rotational speed of a beam of light that is rotating at 1 rps at the source.

I think the thing to remember here is that this is not the same thing as a someone putting a rock in a slingshot and the rocking moving at a higher speed than the source due to the conservation of angular momentum. The actual photons are not moving in a circular fashion, they are actually moving straight out from the source at exactly the speed of light.

So I think the basic premise of what you are trying to demonstrate is not actually relevant in this case.

Answer 2

It's as the others said, the light itself is always moving radially, at C. When you rotate the beam a person who thinks in terms of calculus might say you're actually sending out an infinite number of beams of infinitesimal duration. Since the location of the intercept point of the moving beam on a constant radius surface is rotating at 1 rps, if you're shining it from the center to the inside of a north-south aligned cylinder of about 30,000 miles radius (C/2pi) the intercept point is moving at C. You might want to search for "apparent superluminal jet" to see other ways in which the speed of light can seemingly be exceeded. The ref. is one example.

Answer 3

A very interesting question.
It's akin to the old quandry: You are in a spaceship traveling at half the speed of light...if you shine a light beam in the direction of travel, is the beam's light traveling faster than the speed of light? The answer is no, and that can be proved by complex math, which frankly is beyond me.
I think your paradox is similar...perhaps a good mathematician out there can help both of us out. :-))

Later review: Just noticed Chilly Willy's answer, and think he nailed your question...but he assumes light is particulate...the math becomes more onerous if one considers the wave theory of light. Wonder if he can offer a similarly uncomplicated answer to the the paradox that I brought up?

And later yet: Ty, Chilly Willy...That explanation was offered to me by the mathematician involved, but I was never convinced that one should be permitted to arbitrarily 'change one's point of reference' repeatedly in order to prove a theorem. That's when the complexity came into play. I have had math through Differential Equations (while pusuing a chem PhD) , but his equations were so esoteric that I walked away...convinced that I was being stonewalled by math jargon.

Answer 4

You got this all wrong, The "Beam" is not a solid object. There is no lateral motion only forward motion in different directions. What would happen is that the beam will bend just the same that the water flowing from a water hose does not follow the nozzle but lags behind.