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Imagine there are two cars travelling "straight" at the speed of light*, $A$, and $B$. $B$ is following directly behind $A$.

Suddenly, $B$ switches on its headlights. Will $A$ be able to see this light?


My answer is no, since $A_v = B_v = c$ (the light will always stay stationary relative to $B$. This will probably lead to it gathering up, and intensifying.


*I realize this is impossible, but it's a question my Grade 9 [Honours] teacher asked, so we don't need to get into Relativity, $m = \frac{m_0}{\sqrt{1 – (v / c)^2}}$, cough cough. (I think.)

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I really don't think omitting relativity when considering objects moving at speed of light is the right way to go. –  SF. Apr 29 '11 at 11:49
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up vote 4 down vote accepted

I can think of three ways to answer this:

  1. It can't happen.

  2. It really can't happen.

  3. See #1.

Okay, that's probably enough ;-) Since you say we don't need to consider special relativity, suppose that the universe actually obeys Galilean relativity. That's the technical term for the intuitive way to think about motion, where velocities are measured with respect to some absolute rest frame, and there's nothing special about the speed of light or any other speed. If that were the case, then yes, the light beam would never catch up to car A. The energy contained in the light would presumably pile up in the headlight where it was emitted at first, but afterwards perhaps it would spread out sideways, or would be reabsorbed by the headlight as heat. We don't really have a good answer, because that's not the way the universe works - in fact, there's a lot of physics, both experimental and theoretical, that has been done to prove that it can't work that way. No matter how you try to resolve the problem, at some point you will run into a contradiction.

The best thing you could probably do would be to draw a parallel to some sort of wave that travels with respect to some fixed reference frame, at a speed much less than that of light. Sound, for instance. Sound waves travel with a certain speed with respect to the air, which defines a single absolute reference frame, and their speed is much less than that of light, so there are no special relativistic effects to worry about. Your headlight scenario would then be roughly equivalent to an airplane traveling at the speed of sound. What happens in that case is that the airplane creates a sonic boom, a shock wave which results from the energy in the emitted sound waves piling up at the airplane and eventually being forced to spread out sideways. So one might guess that in your hypothetical situation, the headlights of car B would create a light shock wave that would spread out perpendicular to the direction of motion.

This actually can happen in certain physical situations, namely when something is traveling through a transparent material that slows down the speed of light. This means that light itself travels at a slower speed, but not that the "universal speed limit" is any different. The effect is called Cherenkov radiation and it does indeed work out much like a sonic boom would.

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dammit, David. Second time today you've beaten me to the buzzer by a minute or so! Nice answer, btw. –  Mark Eichenlaub Apr 28 '11 at 5:37
    
lol ;-) just random chance. It'll probably be the other way around tomorrow. –  David Z Apr 28 '11 at 5:39
    
Just to make David's discussion of the faster-than-sound example more explicit: a supersonic airplane in front would not be able to hear music coming from the speakers from a supersonic airplane behind it. (Assuming a big distance between the planes, so we don't have to worry about entrained air between them). –  Anonymous Coward Apr 28 '11 at 20:29
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The question is, as you stated, impossible. However, something roughly like it occurs when the cars are driving at sub-light speeds, but going faster than the group velocity of light in a medium with a refractive index. It's called Cerenkov radiation.

If the cars are moving at the speed of light, they must be massless, and also from our perspective they don't experience any proper time. They aren't cars and can't turn on their lights - the problem simply doesn't make sense.

If the cars are moving very close to the speed of light in our reference frame, (perhaps $(1 - 10^{-10})c$), then by the principle of relativity in their own reference frame everything is business as usual and the lead car will see the headlights turn on.

Another interpretation might be "what if one photon split into two photons?" This could conserve energy and momentum if and only if the two photons continued traveling next to each other in the same direction. If you have photon A in front and photon B in back, and photon B split into two lower-energy photons, then no, neither of them would catch up to photon A.

(note: this answer is pure special relativity. I'm not actually sure whether one photon can simply turn into two photons. Maybe someone could fill me in on this.)

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I'm tempted to say no, a photon cannot turn into two photons (i.e. the process $\gamma\to\gamma\gamma$ does not occur), but I can't seem to identify a reason so maybe I'm wrong. That could make a decent standalone question. –  David Z Apr 28 '11 at 5:45
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Scratch that: angular momentum conservation. Silly me ;-) Although apparently it can happen when there is another object involved, e.g. an atom or a crystal. –  David Z Apr 28 '11 at 5:49
    
@David Cool, thanks –  Mark Eichenlaub Apr 28 '11 at 5:53
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I think Asimov's answer to the age-old question "What happens if an unstoppable force hits an unmovable object" (in his book Please Explain) is applicable here: If you are to play the scientific game you have to agree terms that make sense. So no good science is going to come out of saying "going at the speed of light is impossible, but what if ..."

Note that this is different than the line of thought "What if the sum of angles in a triangle add to more than 180 degrees?" This is because mathematics is axiomatic (i.e. with different axioms you can arrive at different, but equally valid results) whereas science (e.g. physics) is (mostly) not: We know that objects with nonzero mass cannot reach the speed of light. Note that this is different than knowing why this is true.

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I'd put it a little differently: physics ultimately has to take its axioms from nature. Some people do invent physical theories based on arbitrary axioms, the way mathematicians do, but when either the axioms or the consequences disagree with experimental results, you have to ditch the theory. The thing about angles of a triangle is allowed by experiment, but reaching the speed of light is not (and not just because we've tried and failed; experiments support the Poincare symmetry of spacetime which makes it logically impossible to reach the speed of light). –  David Z Apr 29 '11 at 2:58
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