Headlight Effect

We live in a universe in which very distant stars have enormous cosmological redshifts of their light. This fact is interpreted as a cosmological recession velocity at nearly the speed of light. Unusual relativistic effects can be observed when looking at such fast-moving light sources. We consider a more local version here.

Suppose you are standing next to a straight test track that carries a vehicle with a light that shines in a cone with an apex angle 45 degrees about the forward direction. In the past, you have always seen the light as the vehicle approached. One day the vehicle for the first time is able to reach its highest speed ever: v = 0.9999 times the speed of light. But this time you do not see the light as it approaches. Why not? Do you see the vehicle as it passes and then recedes into the distance? Suppose a distant star or galaxy is approaching you at this speed. What would you see? And if receding?

Answer

In the special theory of relativity (STR) a result called the headlight effect occurs. One considers the Lorentz- Fitzgerald contraction of distances in the direction parallel to the constant velocity and no change in the perpendicular direction. If the primed frame is the vehicle frame, then the angle in the two frames are related by cos φ = (cos φ′ + v/c)/(1 + v/c cos φ′ ). Substituting the appropriate values tells us that cos φ ~ 1, or φ ~ 0°! Therefore, all the light is in a very small solid angle in the forward direction, and only an observer directly along the line of motion will see the light. You will not see the light from the relativistic vehicle passing nearby unless your eye is within the very narrow light cone.

In the rest frame of the source, the star emits light in all directions, yet the calculation reveals that for an observer of a very fast-approaching star, practically all its light will be shining along the direction of motion! A fast approaching star or galaxy will possess a very narrow bright headlight beam that could miss Earth. Meanwhile, a fast-receding star or galaxy may not be seen at all because its light is redshifted out of the visible range and practically all its light shines away from us! So in observing stars, there is this STR headlight effect to consider. There also are the different clock rates for the two frames of reference, so the number of photons emitted per second on the star and received at Earth will differ. Moreover, the spectrum of light will be different as well.

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