**EFFECTS DUE TO THE FINITE SPEED OF LIGHT**

*updated 28-May-1992 by SIC*

There are two well known phenomena which are due to the finite speed of electromagnetic radiation, but are essentially classical in nature, requiring no other facts of special relativity for their understanding.

- Apparent Superluminal Velocity of Galaxies
- A distant object can appear to travel faster than the speed of
light relative to us, provided that it has some component of motion towards
us as well as perpendicular to our line of sight. Say that on Jan. 1 you
make a position measurement of galaxy X. One month later, you measure it
again. Assuming you know it's distance from us by some independent
measurement, you derive its linear speed, and conclude that it is moving
faster than the speed of light.
What have you forgotten? Let's say that on Jan. 1, the object is D km from us, and that between Jan. 1 and Feb. 1, the object has moved d km closer to us. You have assumed that the light you measured on Jan. 1 and Feb. 1 were emitted exactly one month apart. Not so. The first light beam had further to travel, and was actually emitted (1 + d/c) months before the second measurement, if we measure c in km/month. The object has traveled the given angular distance in more time than you thought. Similarly, if the object is moving away from us, the apparent angular velocity will be too slow, if you do not correct for this effect, which becomes significant when the object is moving along a line close to our line of sight.

Note that most extragalactic objects are moving away from us due to the Hubble expansion. So for most objects, you don't get superluminal apparent velocities. But the effect is still there, and you need to take it into account if you want to measure velocities by this technique.

- Terrell Rotation
- Consider a cube moving across your field of view with speed near
the speed of light. The trailing face of the cube is edge on to your line
of sight as it passes you. However, the light from the back edge of that
face (the edge of the square farthest from you) takes longer to get to your
eye than the light from the front edge. At any given instant you are
seeing light from the front edge at time t and the back edge at time
t-(L/c), where L is the length of an edge. This means you see the back
edge where it was some time earlier. This has the effect of
*rotating*the*image*of the cube on your retina.This does not mean that the cube itself rotates. The

*image*is rotated. And this depends only on the finite speed of light, not any other postulate or special relativity. You can calculate the rotation angle by noting that the side face of the cube is Lorentz contracted to L' = L/gamma. This will correspond to a rotation angle of arccos(1/gamma).It turns out, if you do the math for a sphere, that the amount of apparent rotation exactly cancels the Lorentz contraction. The object itself is flattened, but then you see

*behind*it as it flies by just enough to restore it to its original size. So the image of a sphere is unaffected by the Lorentz flattening that it experiences.Another implication of this is that if the object is moving at nearly the speed of light, although it is contracted into an infinitesimally thin pancake, you see it rotated by almost a full 90 degrees, so you see the complete backside of the object, and it doesn't disappear from view. In the case of the sphere, you see the transverse cross-section (which suffers no contraction), so that it still appears to be exactly a sphere.

That it took so long historically to realize this is undoubtedly due to the fact that although we were regularly accelerating particle beams in 1959 to relativistic speeds, we still do not have the technology to accelerate any macroscopic objects to speeds necessary to reveal the effect.

References:

- Considerations about the Apparent 'Superluminal Expansions' in Astrophysics, E. Recami, A. Castellino, G.D. Maccarrone, M. Rodono, Nuovo Cimento 93B, 119 (1986).
- Apparent Superluminal Sources, Comparative Cosmology and the Cosmic Distance Scale, Mon. Not. R. Astr. Soc. 242, 423-427 (1990).
- J. Terrell, Phys Rev.
*116*, 1041 (1959). For a textbook discussion, see Marion's*Classical Dynamics*, Section 10.5.