I was reading Jacob Bronowski’s *The Ascent of Man* this morning and came across a lovely thought experiment that gives intuition into time dilation that probably goes back to Einstein himself. It goes something like this. Imagine there is a clock. You see the second hand move around the face as light continuously streams from the clock to your eyes. Now imagine that you suddenly travel at the speed of light away from the clock. You would be riding with the last photons from the clock face to reach your eyes. Your vision would be filled with a frozen image of the clock because no other light would be able to reach you. Hence the second hand does not move and time stands still.

One can derive the full special relativistic time dilation expression from this kind of argument and a simple spacetime diagram.

This analogy does raise interesting questions. Imagine we were fish and that we put our clock under water. The speed of light in water is about 1.3 times smaller than in vacuum and it would be possible (although difficult) to move at the speed of light in water. So if a fish could travel at the speed of light in water away from our now-submerged clock, they would also see the clock frozen in time.

But would that be the correct interpretation of the relative rates of the ticking of the clock and a watch on the fish’s fin?

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Here’s another observation about this argument: say that you’re moving toward that clock at the speed of light, rather than away from it. (Let’s also assume the clock is very far away.) The same reasoning implies that we would see the clock speed up, but special relativity tells us that the clock stops from our perspective, just as it does when we move away from it at the speed of light. So (as your fish question points out too) there’s a difference between the light we *see* from the clock and what we *infer* about the speed of the clock. And what we infer about the clock based upon what we see depends on the laws of physics we’re using — after all, a clock receding at the speed of light will appear to be stopped even in Newtonian physics. So while this argument certainly gives an intuitive feel for time dilation, I don’t think you can actually use it to get to the relativistic expression for time dilation without assuming that the speed of light is the same for all observers independent of their motion with respect to the source (or some other equivalent axiom). I think an assumption like that is built into the spacetime diagram, but maybe I’m wrong?

Could become a fishy paradox…

I’ve always been confused about the twin paradox. If twin A is moving close to the speed of light away from twin B, isn’t it the same as saying twin B is moving close to the speed of light away from twin A? I.e., if you choose one as the reference point, the other is the one that is moving away.

The implication, in my mind, is that both clocks in fact run at the same speed. I.e. to an observer at twin A (who is considered to be stationary), twin B’s clock is running slower since B is moving away from A at close to the speed of light, according to the above analogy. But the same can be said if the observer switches sides to B: it is A’s clock that is running slower. Therefore, the observer sees time to be slower for whomever travels faster according to the observer’s reference frame. But if there was a complicated arrangement where the observer saw both twins to be travelling away from the observer at the same speed, as well as the twins travelling away from each other at close to the speed of light, both clocks would appear to run at the same speed to the observer.

But we can’t say whether it is A or B that is travelling at all, only that one is travelling relative to the other. It could be that Earth, with twin B on it, is actually travelling at close to the speed of light, but twin A in the space ship is the one that is staying still. Kind of like that Futurama episode..

Could you please point out, where is the fatal flaw in this thinking?

Hi Avc – your logic is almost perfectly right. Their situations are almost exactly symmetric, except that in order to meet again, one of the twins must turn around and return to the other. Special relativity only applies when there is no acceleration involved. It is this acceleration required to tun the ship around in order for the twins to meet again that breaks the symmetry. However this isn’t trivial and historically required a fair amount of work to tease out. You can read the details here: http://en.wikipedia.org/wiki/Twin_paradox

I trust that helps! Of course, if the twins were moving on a doughnut-shaped universe acceleration would not be needed and the paradox deepens…you can read about that resolution here: http://arxiv.org/abs/0910.5847