Relativistic Clocks

A couple of questions about time dilation examples. A common example given is a time clock with light reflecting back and forth between two mirrors. One of these time clocks that is stationary, and one that is moving near the speed of light, lets say half the speed of light, on a moving spaceship. The stationary observer supposedly sees the moving time clocks light taking travelling a further distance from one mirror to the other. Here is where I don’t follow the problem – if the travelling clock is standing vertically, the stationary observer is said to see the light traveling at an angle in order to reach the other mirror. But how does this light ever start ‘angling’. In my mind, when the light leaves the one mirror, it is travelling straight up or down. If the other mirror is moving with the spaceship, the light should miss it, because the mirror would have moved out of the way before the light moving vertically would have hit it. The light would not be angled, because that means it was influenced by the moving spaceship. Another example I’ve seen was a light ray initiated in the same direction as the movement (e.g. horizontally), and it appears to travel longer distance in the time needed. But if the light was initiated in the opposite direction of the movement, wouldn’t it appear to travel shorter distance in the same amount of time?
- Mark Duffy (age 46)
Downingtown

Yes, these are common questions when people first look at relativity. The way you phrase them is misleading, however, even though it's how most people start trying to picture it. Neither clock is "moving" more than the other. Each is, in its own frame, staniding still. In each, its own description simply has the light bouncing back and forth, not angled. Now when each looks at the other clock, they must see all the parts of it, including the light, moving together. Otherwise they'd see the light escaping- which we know it doesn't from the point of view of each clock itself. Some basic things like whether the light escapes are invariant as you switch between different viewpoints. (We'll ignore the GR black-hole information paradox here!) 

So what's strange is not that each observer sees the light in the other clocks angled enough to stay inside the clock. That would be true even in Galileo's old relativity, the one that more closely matches our intuition. What's strange is that each observer sees the light in the other clock as still only traveling at the fixed speed c! That's one of the basic ingredients around which special relativity weaves a self-consistent but anti-intuitive picture.

I didn't exactly follow your second question. I think it's related to the math of the Michelson-Morley experiment. If light and the apparatus (of length L)  move relative to each other at speed v+c one way and v-c the other way, the time for a back-forth bounce isn't 2L/c, which is what you'd get if v=0. It's L/(c+v) +L/(c-v)= 2L/c(1-v²/c²). The effects of v cancel to first order in v, but not to second order.

Mike W.

(published on 05/04/2014)

Thanks Mike. I’m still having trouble following it, because I didn’t that you could add to c (v+c). But I had one idea that is helping me visualize it. I’m not sure if it’s right though – I’m visualizing spacetime itself to be much more fluid than I thought. So that within the spaceship the spacetime itself is traveling along. Like it’s all a soup, and the spaceship is a container that is traveling in the soup, but also filled with the soup. Do you think this is near the mark? If not, could you help me visualize it? Thanks again.
- Mark Duffy (age 46)
Downingtown, PA

Mark- I've gone back to clarify that the (v+c) is the relative velocity of the clock and the light, as seen by the other observer. 

The "soup" idea is really very close to the old "ether" idea that special relativity replaced. Your particular version, in which the soup is dragged along with nearby stuff, was called the "ether drag" theory. It could account for some experimental facts, but contradicted other ones, such as the observations of stellar aberration. To put it very briefly, the effects expected as light went from one part of the soup to another part moving with respect to the first weren't there. So that theory didn't work.

How to picture special relativity? I can give an analogy. Think of looking at an object from different angles. Things like the height, width, and depth all change depending on the angle. If you're careful, you can find some "invariants" like the volume that don't change.  The relativity rules just tell us how things look from different points of view moving with respect to each other. Relativity says that many things we think shouldn't depend on the motion of an observer turn out not to be invariants. Things like the time interval between events turn out to be more like width in our example than like volume. On the other hand, the speed of light, which we would expect not to be an invariant turns out to be one. Our problem is that the usual range of different frames that we use is so small that we develop a strong intuition that works well enough within that little range. Then we mistake our intuition for the real rules of the world.

So I'm afraid that didn't give much of a picture, but at least the analogy had one.

Mike W.

(published on 05/05/2014)