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Context

I am studying special relativity using [1]. In [1], Gray writes that,

The Minkowski diagram can help us see what is going on in a given special relativity problem: if we plot the relevant events on the diagram, we can see their relationship more clearly. However, it can also support some arguments directly. For example (this arguement is taken from Rindler), imagine a flashing lighthouse beam being swept across a distant shore. If the shore is far enough away and the beam is turned quickly enough, the illuminated points can be made to travel arbitrarily fast---faster than the speed of light.

I have searched for Rindler's lighthouse online. I found citations to Rinder---including links to non-inertial frames or reference (of which this turning lighthouse would be one). On this cite, I have found [2]. The question in [2] may be related to mine, but I do not know.

Question:

I do not understand what is written here.

Can you explain the example in your own words? In particular how can the illuminated points can be made to travel arbitrarily fast---faster than the speed of light?

Bibliography

[1] N. Gray, A Student's Guide to Special Relativity, Cambridge University Press, 2022, p. 55.

[2] Rindler Coordinates Derivation

Michael Levy
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2 Answers2

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There is no connection between this thought-experiment and Rindler coordinates (except that Rindler coordinates are named after Wolfgang Rindler and the thought-experiment is also his).

It doesn't matter to the argument that the lighthouse rotates. You can replace the lighthouse with a stationary movie projector that projects an image of a circle of light moving horizontally across the frame. The farther the screen is from the projector, the larger the projected image, and so the larger its apparent speed. If the screen is distant enough, the apparent speed will exceed $c$.

It's just an optical illusion. Your visual system interprets the input from your retinas as caused by a rapidly moving object that emits or scatters light, but no such object actually exists. The only moving object in the experiment is the light, and it moves from the projector to the screen to your eye; it doesn't move horizontally across the screen.

See also this answer.

benrg
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Consider a laser you hold in your hand. You can flick it around at the moon, say, and the incident point of the laser beam will appear to move superluminally on the moon's surface if you flicked it fast enough. Just a very straightforward result from rotational mechanics.

Of course, no particles are actually moving faster than light, and one is unable to transmit information faster than light using this. If this were a solid mechanical beam, instead, you might find more quantitative insight on here: https://en.wikipedia.org/wiki/Rindler_coordinates#A_%22paradoxical%22_property. You would have to deal with length contraction, etc, but this is not relevant for disjointed particles.

I can't find a PDF of [1], but is there more context for your question? I hope that answered it, otherwise.

jazzloaf
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