[anonytrary]

I think those answers are actually wrong. When the object casting the shadow moves, the shadow remains in the same place for an observer in it until the light from the source reaches the observer inside the shadow.

I know that's a weird explanation, so consider:

  t0: S~~~>O     U (shadow exists)

  t1: S~~~~~~>   U (shadow exists)
      
  t2: S~~~~~~~~~>U (shadow !exist)

Where "S" is a light source, "O" is an opaque object, "~" are photons traveling to the right, "U" is the observer, and "t0", "t1", and "t2" are times (increasing).

At time "t0", "U" thinks he's in the shadow.

At time "t1", "U" thinks he's in the shadow.

At time "t0", "U" thinks he isn't in the shadow, since the photons are now hitting him.

A similar calculation/thought-experiment can be done for shadows with "angular momentum", in case you think the tangential velocity of the shadow will exceed the speed of light.

[lenkite]

Thanks for that nice diagram! What happens when the light source moves and the shadow is far bigger than the object casting it ? Wouldn't the speed of the shadow on the surface be faster than the speed of light ?

[anonytrary]

For example, when a light source is super close to an object and the shadow gets super big really far away, and the light source moves?

  t0: SO          U1 (U1 in shadow)
                  
  t1: S~~~~~~~~~~>U1 (shadow leaves top first)
       O~~~~~~>   .
         ~>      .
                .
              .
             '
            U2

Dotted line is the path of a fixed point on the shadow during the time S moves.

I didn't do the precise math, but I'm pretty sure the tangential velocity of the shadow along the dotted line won't be greater than the speed of light. The curvature of the "wave front" formed by the tips of the arrows ">" above will be lesser than the dotted line curvature, so the photons near the top of the diagram hit the dotted edge before the ones towards the bottom. This is because the source, S, takes time to move away from O.

Note that the wavefront formed by the photons moves radially outward from S, but ascii art is limiting.

[foobarrio]

No information can be conveyed by the wavefront and so nothing is actually moving than the speed of light. What you diagrammed is called the Lighthouse Paradox:

* https://en.wikipedia.org/wiki/Lighthouse_paradox

There are similar things that appear to exceed the speed of light:

* https://en.wikipedia.org/wiki/Faster-than-light

See "group" and "phase" velocities for similar things to the lighthouse.