Special Relativity the Headlight Paradox

The Headlight Paradox

I received a list of funny paradoxes that was meant to be a joke.  While the others in the list had no serious answer, this one does.  With the help of Dr. Albert Einstein and his Theory of Special Relativity, I decided to write an explanation as I see it.  The paradox follows:

What happens when a space ship traveling away from you at the speed of light turns on the headlights?

Because nothing can travel faster than the speed of light, this sounds funny.  It does, though, point out some of the consequences of special relativity.

First of all, you can’t travel at the speed of light; you can only travel at NEARLY the speed of light.  If a ship travels away from you at, say, 99% of the speed of light, you would say “Gee, that ship is traveling at nearly the speed of light”.  If the crew of the ship would turn on the headlights (probably to illuminate the star toward which they are rushing, and which they will reach in 50 or 60 years by our estimate), they might say, “Gee, our headlights are traveling away from us at the speed of light”.  As we observe the ship from our vantage point here on earth, we might say “Gee, that headlight beam is traveling away from us at the speed of light! 

How is it that both we and the ship’s crew see the beam of light traveling away at the same speed?

The answer is that the ship’s crew is experiencing TIME DILATION.  Time is not a constant.  How time is experienced depends upon relative motion (as well as gravitational effects).  I said that the crew will get to the star in 50 or 60 years by our estimate.  The crew, on the other hand, will arrive at the star in perhaps only a few months by their clocks.  How can this be?

I like to envision time as a fog, with each particle in the fog being not moisture, but an incredibly tiny bit of time.  As the ship rushes through the time-fog faster and faster, the fog in front seems to compress, rather like a shock wave at the leading edge of a wing traveling at supersonic speed.   As it compresses, it becomes denser and it requires more power to accelerate through it.  If the ship could accelerate through the time-fog to the speed of light, it would require an infinite amount of power but the ship would be through it instantaneously.   (Certain particles, such as photons, can travel at the speed of light because they have no mass.)   At the speed of light, if that were possible, the universe in front of the ship would appear to the crew as only an incredibly small point, while they would see nothing in the rear view mirror.

Please remember that the time-fog is only an aid to visualization.  I do not mean it to be an explanation of the nature of time.

If the crew had launched a scout ship at 99% of the speed of light, it would be 99% BY THEIR PERCEPTION OF TIME (i.e. their frame of reference).  Because we see time differently from earth, (we are in a different frame of reference), we would see the scout ship traveling away from us at 99.4% of the speed of light.  If the scout launched a scout, we would see that ship traveling at perhaps 99.5% of the speed of light.  And so on ad infinitum.  It’s kind of like taking a percentage of the remaining percentage; you never quite reach 100%.

Now, what about that stupid headlight?

To a headlight photon moving through the vacuum of space at the speed of light (from either our frame of reference or that of the ship’s crew), everything seems stationary.   The universe, after all, is only a single point and there is nowhere to move (from the photon’s point of view).  The photon, as long as it travels in a vacuum – and is not slowed by passing through some medium such as air, water, or glass – does not experience time at all.  To use the time-fog analogy, the time-fog has compressed down to a single point.  To the crew of the ship, the photon traveled away at the speed of light, because the photon is not experiencing time while the crew is still experiencing time.  To us here on earth, the photon is traveling away from us at the speed of light for the same reason; the photon is not experiencing time, while we are experiencing time (even though it is not the same time as that experienced by the ship’s crew).   If any object does not experience the effect of time, it will appear to any observer in any other frame of reference that the object is traveling at the speed of light.