# Q & A: clocks in relativity

Q:
In the light beam that bounces between two mirrors experiment, why is the 'clock' considered to tell the correct time? why shouldn't the light beam just bounce normally between the mirrors at angle but gradually be left behind by the moving mirrors? why isn't the case where the clock is just flawed and reports the wrong time taken into account?
- The Burst (age 18)
A:
Let's start with the simplest point. Of course there could be a light beam bouncing back and forth in our frame so that as the mirrors move the light would be left behind. There can also be light bouncing back and forth at an angle in our frame so that it keeps up with the mirrors. That's the light that would be used in the mirror frame to make this very simple clock.

Now we get to the interesting part. You could imagine a universe where the laws describing electromagnetism, including the propagation of light, only worked in one reference frame. Then the simple bouncing-light clock in the mirror frame would be "wrong". A symptom would be that various different types of clocks in that frame would not agree with each other.

In our universe, however, all the clocks of all types in the mirror frame agree with each other, just as do all the clocks in our frame. Furthermore in the mirror frame, all the electromagnetic laws, including light propagation, work exactly the same way as in ours. It turns out that all the physical laws for all the forces work the same way in either frame.

Thus in our universe there's no objective way at all to pick out one frame and say that its clocks are more right than those in other frames.

Mike W.

(published on 08/21/2012)

## Follow-Up #1: what are clocks?

Q:
What does clock mean in the above discussion?
- Razin (age 13)
Navsari, Gujarat, India
A:
That's an important question. Here a clock is any device which keeps putting out some signal which can be counted to keep track of time: tick-tick-tick.... A good clock should take the same time period for each tick. How do we check if a clock is good, if we're using that clock to measure the passage of time, since it will always agree with itself? It's done by making a big collection of different types of clocks to make sure that they agree with each other.

The particular type of clock The Burst mentioned consists of two parallel mirrors with light bouncing back and forth between them. Something (maybe a photodetector) ticks each time the light blip bounces off one of the mirrors. It's not really a practical type of clock, but Einstein used it to reason about the behavior of time because it's so extremely simple.

Mike W.

(published on 08/22/2012)

## Follow-Up #2: time in special relativity

Q:
I want to understand time dilation correctly, but I have a few issues with the "light clock" thought experiment which is mentioned in so many text books. If I take the set-up of the experiment literally then there appears to be some conflicting points: a) Let's say the clock is on the train. When the train is stationary the light path is normal to the train tracks. When the train starts moving, as shown in the diagrams, the light path is no longer normal to the tracks (as seen by the stationary observer), but has a directional component in the axis parallel to the tracks. Since this modified light path is used to justify the derivation of the time dilation formula by Pythagoras theorem, the light path is considered real and constrained by the speed of light. But how is the train able to impart forward motion to the light beam? Since Newtonian mechanics cannot be applied to light beams, the train shouldn't be able to impart velocity to it. b) The justification of the derivation is that the stationary observer "sees" the moving clock ticking more slowly. I considered how the observer sees the clock ticking. Let's say the clock emits an omni-directional flash of light for each tick, conveying this information to the observer. Since this information is itself bound by the speed of light, and the clock is moving relative to the observer, he would receive these pulses of light at different times from different distances, at a rate defined by Doppler shift. Of course, this is not factored in the formula for time dilation. c) Since the time dilation should occur for any clock, why not take the light clock in the thought experiment and rotate it 90 deg so that the light beam travels parallel to the train tracks, instead of normal to them. If so, it is no longer possible to use Pythagoras theorem in deriving the formula since no triangle is formed for the light path. Yet if the formula is correct it should still be possible to derive the same formula from my slightly modified arrangement. I will be impressed if you can answer these questions (in a way I understand!) Thanks
- debiddo
UK
A:
These are all standard questions from introductions to special relativity, so don't be too impressed that we can answer them  Understandability is another matter. If the answers aren't yet clear, please follow up.

a. Yeah, if you kept using the same light blip and then accelerated the clock, it'd get lost. This argument isn't about accelerated clocks yet. If the source of the blip is a diode or something mounted with the clock, in the clock frame its light won't go the wrong direction just because somebody else says the clock is moving.

b. Yes, the time needed to transmit the information varies as the clock approaches and recedes from you. One can go through the calculation to see that if you make a simple allowance for the transmission time, as anyone trying to calculate the clock-rate should, you get that the clock beats regularly at the slower rate. And that other guy, looking back at you sees your clock beat slow. You can agree to disagree because you assign different information transmission times because you each say the light moves at c with respect to yourself, not the other guy.

As the other clock goes by, it will initially get closer and then farther away. Assuming that it passes at some distance, there will be a period in between in which its distance is changing little, so the transmission time isn't changing much.  The spacing of the flashes you see then doesn't need to be corrected, and thus directly gives the clock period. But wait- if no correction is needed then, how can you both say the other one is slow? The key is in the word "then". You don't agree on when this time is.

c. Yes, the result must also work out for the rotated clock. The analysis is slightly more complicated here, because there's no simple argument to prove that the distance between the mirrors along the direction of relative motion is the same in both  frames. It was just such an argument that simplified the case for the usual clock orientation, although some discussions skip that part. Here, in fact, the distance does depend on the frame. Since here you have two factors, the length contraction and the time dilation, considering this case by itself isn't enough to separate them. That's why we usually start with the other clock, for which only the time dilation can exist. Once you have the time dilation, the agreement of the clocks tells you that there must be the (1-v2/c2)1/2 length contraction. You might look at discussions of the Michelson-Morley experiment, in which this issue came up even before special relativity.

The bottom line is that people run into all sorts of problems when they apply relativistic transforms to some but not all the variables. Everything is self-consistent, no paradoxes, when you use the complete transforms.

Mike W.

p.s. You wouldn't happen to be someone I know in Islington?

(published on 01/29/2013)