Faster than Light

A recent New York Times article describes the fascinating possibility that some non-things, such as shadows, can travel faster than the speed of light.

NY Times Article

At least one physicist I talked to was uninterested and unimpressed by this fact, but I found it fun to think about. Since the Times article lacked sufficient, clear explanation of how shadows might travel faster than light, I’ll explain it here.

First you must convince yourself that a point on the edge of a spinning record, CD, or other rotating disk, is moving faster than a point near the center of the rotating disk. This makes sense because a point, A, near the center of the disk and a point, B, near the edge make the same number of rotations per minute, but point B, on the edge, is traveling a greater distance. In other words, point B is tracing a circle with a larger circumference.

With that fact in mind, imagine a powerful spotlight with a wide beam such that a cone of light is produced (see picture). Assume the beam is unimpeded and can be seen far, far out in space. At some far vantage the beam would be very wide.

Now imagine putting a cut-out, such as the “bat signal” over the spotlight. The shadow of the “bat signal” would grow very large as the beam extends out into space. Finally, rotate the spotlight. Now at some significant distance we have a large rotating shadow in the shape of a bat. At some very significant distance, that shadow will be rotating faster than the speed of light.

Here is how:

Google tells me the speed of light is 299,792,458 m/s.

Assume our spotlight has a relatively meager circumference of one meter and we are rotating the spotlight one full revolution each second. Therefore, the initial rotational speed of a point on the edge of the spotlight, such as where the tip of the “bat signal” touches the edge of the spotlight, is one meter/second. As the cone of light expands the shadow continues to rotate once every second, but the circumference increases as the diameter of the cone increases! When the diameter of the cone of light expands larger than 95,427 kilometers (speed of light / pi) then the shadow will be rotating faster than the speed of light!

Note:
Every photon coming out of the light source (spotlight) is traveling in a straight line. The shadow, however, is tracing a corkscrew pattern as it simultaneously flies away from the light source and rotates.

The photons travel at the speed of light, but never faster and they are never actually rotating in our thought experiment. Sadly this means information still cannot be transmitted faster than the speed of light.

Shadows aren’t inherently special. Any expanding, rotating, non-physical pattern can travel faster than the speed of light. If, for example, a square color filter were placed over the light source then the square of a different color would eventually rotate faster than the speed of light.

J0n0’s illuminating thoughts: That's pretty cool. For the concept to work, you have to define the distance r from the source where you want to consider the shadow observed. The shadow then, at any point in time, is the region of the spherical surface of radius r centered on the source that light has been blocked from hitting. Then that shadow (or the segment in it that lies in the equator of the rotation) moves at a speed rw, where w is the angular speed of the klieg's rotation. At a certain point, of course, rw will exceed the speed of light, you just have to find a large enough r.

The thing that makes this kind of mind-warping is that if you don't define the shadow as dependent on a distance, then weird things happen. For instance, if you consider the shadow at a given point in time to instead be the whole volume of space where the light of the klieg has been blocked, so that it's a sort of expanding spiral horn, then you can't give it a uniform speed, of course, which brings you back to having to define a distance.

On the other hand, you could also think of the shadow as defined by the time that the light around it emerges from the source, and take it to be the surface region that is bounded by those same light particles as they move out into space - a cohort-like definition. Then its area expands, but it is only ever going at the speed of light.

In conclusion: Dang!
-j0n0

Megan adds: Yes, I agree with Jono's explanation (I'm not sure why she [the author] never explicitly stated v=r*omega where omega=angular frequency, because that would clarify the concept, but I guess that equations frighten some math-phobic readers). I am visualizing the shadow as a (time-dependent) 2D patch of "nothing" directly adjacent to a line segment of photons on the edge of the bat design. The photons have to move at v=c, but the shadow could move faster at a very large distance. Weird
-Megan

Here is another simple (simpler) way to make some fast shadows, potentially faster than the speed of light.

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