Can anyone help me with what goes wrong with the following Moon paradox thought experiment ?

You all know the faster than light (FTL) Moon "paradox": send a beam to the moon through a "hand laser," and swipe the laser sufficiently fast from some point $$A$$ to some point $$B$$ on the lunar surface.

The beam can then easily made to move faster than light on this trajectory.

But it is not a paradox, since no actual photon travels that way, and no information is passed from $$A$$ to $$B$$.

But what if we change the experiment a little: call the person with the laser "Angie" and an observer on the moon "Billy."

Suppose Angie swipes the laser very fast from $$A$$ to $$B$$, or suppose she draws a little letter, or symbol. (The swipe from $$A$$ to $$B$$ could be a symbol for some letter.)

She can do this such that the symbol is drawn faster than the speed of light.

If Billy observes this symbol, it seems at first sight that he gets the information FTL.

What goes wrong ?

• it would take about 3 seconds for light to go to moon and back – Adrian Howard Oct 1 '19 at 11:27
• @AdrianHoward : One could start with a beam which already intersects the moon prior to the experiment, as part of the setting, no ? – THC Oct 1 '19 at 11:30
• One way of seeing that there is no problem here is rather than sweeping a laser just project an image of a letter: when the light from the projector reaches the Moon the whole image appears simultaneously in the rest frame of someone on the surface! – tfb Oct 1 '19 at 11:44
• @THC: yes one could see something that had already been sent to the moon (3 seconds ago), but to send new information the image would have to be changed, then you would see it 3 seconds later – Adrian Howard Oct 1 '19 at 11:52
• @tfb: the person on the moon would still see the closer part of the image before he saw the farther part, as the image on the moon would still travel to him at light speed – Adrian Howard Oct 1 '19 at 11:56

Suppose that Angie flips a coin and wants to communicate to Billy the result of that flip. This is what is meant by “information”. She may communicate the result of the flip by drawing different symbols on the moon. The time between the flip and the symbol is not limited by the time it takes for the light to trace the symbol, but by the time it takes for the light to go from earth to the moon. This is not FTL.

The only thing that appears FTL here is the tracing of the symbol from A to B. However, this is not information transfer from A to B. Someone at A who is flipping a coin cannot use Angie's symbol drawing to inform a person at B about the outcome of the flip FTL. The information in the symbol is traveling from Earth to the moon, not from A to B. As such, the information transfer is not FTL.

• One could start with a beam which already intersects the moon prior to the experiment, as part of the setting, no ? – THC Oct 1 '19 at 11:32
• You could, but then how could anyone use a beam already intersecting the moon to communicate the result of a coin flip anywhere? You need to focus less on the beam and more on the information. What is prohibited is FTL information transfer. – Dale Oct 1 '19 at 12:21

The beam you are talking about, will not hit the position on the lunar immediately. Laser light, as most other light, has the same speed as ordinary photons. It means, that when you change the angle of the laser, it will take some time for the light to travel from you hand to the moon. Therefore, you will not get any special effect.

The FTL-moon paradox forgets that the laser beam has to travel from earth to the moon again and again every time it changes position.

Paradoxes appear when one is mixing up frames of reference.

Suppose Angie swipes the laser very fast from A to B,

You have already said that this is no paradox as it is not the same photons that travel in the image on the moon

or suppose she draws a little letter, or symbol. (The swipe from A to B could be a symbol for some letter.)

If she traces a letter with her laser (mind you the drawing mechanism needs some thinking) it will be completed as a recognized letter by Angie, or by a picture of the moon, calculations giving faster than light the motion of the writing beam on the moon. For an observer on the moon to see the information, i.e. that it is a letter, will need detecting systems and local time taken. Maybe individual detectors in the kilometers of surface, and the information that the whole plots a letter, will arrive to Bob much slower than the speed of light, as all detectors are limited by the speed of light.

So it is mixing up the observer frame of Angie with the observer frame of Bob on the moon that makes the paradox.

She can do this such that the symbol is drawn faster than the speed of light.

There is no paradox. You are confusing two separate quantities, 1) being the time taken for information to travel to the moon from Angie, and 2) being the elapsed time for parts of the message to arrive at different points on the moon.

1) is subject to the rule that nothing can travel faster than light.

2) is not subject to any limits whatsoever.

Angie could use a laser to write 'The answer is 42' on the surface of the moon, as you suggest, which might take a certain time to complete. She could just as easily project it in huge letters that arrive on the surface simultaneously with each other. There is nothing in SR which prevents spatially separated events in one reference frame from happening simultaneously.

What counts is that in either case the time taken for the message to go from the Earth to the Moon is determined by the speed of light.