If you sweep a laser pointer across the moon, will the spot move faster than the speed of light? Every physics major encounters this question at some point, and the answer is yes, it will. If you sweep the laser pointer at it in an arc, the velocity of the spot increases with the distance to the surface you point at. On Earth, you only have to rotate the laser in a full arc within a few seconds, then it will move faster than the speed of light on the lunar surface!

This simplified explanation would be all there is to say were the moon a disk, but the moon isn’t a disk and this makes the situation more interesting. The speed of the spot also increases the more parallel the surface you aim at is relative to the beam’s direction. And so the spot’s speed increases without bound as it reaches the edge of the visible part of the moon.

If you sweep a laser pointer across the moon, the velocity of the image on the moon’s surface will get arbitrarily large towards the boundary.

That’s the theory. In practice, of course, your average laser pointer isn’t strong enough to still be visible on the moon.

This faster-than-light motion is not in conflict with special relativity because the continuous movement of the spot is an illusion. What actually moves are the photons in the laser beam, and they move at always the same speed: the speed of light. But different photons illuminate different parts of the surface in a pattern synchronized by the photons’ collective origin, which appears like a continuous movement that can happen at arbitrary speed. It isn’t possible in this way to exchange information faster than the speed of light because information can only be sent from the source to the surface, not between the illuminated parts on the surface.

That is, at least, how it works concerning the movement of the spot on the surface. Trick question: If you sweep a laser pointer across the moon, what will you see? Note the subtle difference — now you have to take into account the travel time of the signal.

Let us assume for the sake of simplicity that you and the moon are not moving relative to each other, and you sweep from left to right. Let us also assume that the moon reflects diffusely into all directions, so you will see the spot regardless of where you are. This isn’t quite right but good enough for our purposes.

Now, if you were to measure the speed of the spot on the surface of the moon it would appear on the left moving faster than the speed of light initially, then slowing down as it approaches the place on the moon’s surface that is orthogonal to the beam, then speed up again. But that’s not what you would see on Earth. That’s because the very left and very right edges are also farther away and so the light takes longer to reach us. You would instead see a pair of spots appear close by the left edge and then separate, one of them disappearing at the left edge, the other moving across the moon to disappear on the other edge. The point where the spot pair seems to appear is the position where the velocity of the spot on the surface drops from above the speed of light to below.

Space-time diagram to illustrate the spot-pair creation. The sweep is in the direction perpendicular to the x-axis (into the page) and NOT shown. Shown is the time-dependence of the distance to the reflection point. Since reflection from farther away points takes longer to reach the observer, the time sequence of the sweep appears mixed up. The first point to become visible (3) is the point where the velocity of the spot on the surface drops from above to below the speed of light.

This pair creation of spots happens for the same reason you hear a sonic boom when a plane passes by faster than the speed of sound. That’s because the signal (the sound or the light) is slower than what is causing the signal (the plane or the laser hitting the surface of the moon). The spot pair creation is thus signal of a “photonic boom,” a catchy phrase coined by Robert Nemiroff, Professor for astrophysics at Michigan Technological University, one of the two people behind the Astronomy Picture Of the Day that clogs our Facebook feeds every morning.

The most surprising thing about this spot pair creation is that nobody ever thought through this until December 2014, when Nemiroff put out a paper in which he laid out the math of the photonic booms. The above considerations for a perfectly spherical surface can be put in more general terms, taking into account also relative motion between the source and the reflecting surface. The upshot is that the spot pair creation events carry information about the structure of the surface that they are reflected on.

But why, you might wonder, who cares about spots on the Moon?

To begin with, if you were to measure the structure of any object, say an asteroid, by aiming at it with laser beams and recording the reflections, then you would have to take into account this effect. Maybe more interesting, these spot pair creations probably occur in astrophysical situations. Nemiroff in his paper for example mentions the binary pulsar 3U 0900–40, whose x-rays may be scattering off the surface of its companion, a signal that one will misinterpret without knowing about photonic booms.

Image credit: X-ray: NASA/U. Hamburg/J.Ness et al; Optical: NASA/STScI, of X-rays reflecting off of Saturn (L); these X-rays match the spectrum from the Sun, indicating that there is a reflection at play, and that the X-rays are not generated by Saturn itself.

The above considerations don’t only apply to illuminated spots but also to shadows. Shadows can be cast for example by opaque clouds on reflecting nebula, resulting in changes of brightness that may appear to move faster than the speed of light. There are many nebula that show changes in brightness thought to be due to such effects, like for example the Hubble Nebula (HVN: NGC 2260) or the Helix Nebula (image below). Again, one cannot properly analyze these situations without taking into account the spot pair creation effect.

Image credit: ESO, via http://www.eso.org/public/images/eso0907a/. The Helix Nebula, one of the objects that might show variations in brightness due to pair creation of superluminal shadow pairs.

In his January paper, Nemiroff hints at an upcoming paper “in preparation” with a colleague, so I think we will hear more about the photonic booms in the near future.

While Special Relativity turns 110 years old this year, it still holds a number of surprises for us, including observable signals that we’re only now coming to understand how they appear!