# Does the Einsteins theory of a clock tower observed from a tram really explain time dilation?

We were shown a video in our class about time dilation which explained a Einstein's theory - the one where he was travelling in a tram and realized that if he travels at light speed, time will flow differently for him. He hypothesized this as the movement of the hands of the clock seemed to get slower with increase in the trams speed. He thought that if he is able to travel at light speed, the hands of the clock will freeze for him.

But, on this, I thought that as Einstein would be travelling at the speed of light, the light reflected from the clock will never reach his eyes. The light reflected from the clock before he achieves light speed, will be the last he sees of the clock's original state, hence, to him, the hands of the clock will appear still, while actually they still will be moving. Time still will be flowing, irrespective of the hands' apparent positions, and at the same rate; only Einstein would not be able to view the indicator(i.e. clock hands) showing this!

Could someone please explain why the theories above are right/wrong?

## 2 Answers

First, the time of flight of light is irrelevant to time dilation (due to uniform relative motion) so I understand why this is confusing to you.

The essential nature of time dilation due to uniform relative motion is that, relative to Einstein, the clock is moving (with speed $$v \lt c$$) and so, Einstein must use two, spatially separated and synchronized clocks, at rest with respect to him, to observe the elapsed time according to the tower clock.

Initially, the tower clock is (instantaneously) co-located with the first of Einstein's clocks and it records the current time as well as the time given by the tower clock.

Some time later, the tower clock is (instantaneously) co-located with the second of Einstein's clocks and it records the current time as well as the time given by the tower clock.

Now, Einstein computes the elapsed time, as found by his two clocks, by subtracting the second clock's recorded times from the first clock's recorded times.

He finds that the elapsed time calculated for the tower clock is less than the elapsed time as calculated on his two spatially separated, synchronized clocks.

This is time dilation - moving clocks run slow. Note that time of flight of light is not involved here since the readings on the tower clock are made with (instantaneously) co-located clocks.

What is crucial to understand is that, according to the tower clock, Einstein's clocks are not synchronized (relativity of simultaneity) and this is why someone in the tower clock observes Einstein's clocks to run slow.

Something to keep in mind here is that time dilation is not because it takes longer for light to reach your eyes while moving faster (if this was the case then you could have time contraction too, depending on what direction you were traveling relative to what you are looking at).$$^*$$

It does not make sense to try to move to a frame moving at the speed of light. First, we cannot get anything with mass to move at the speed of light relative to us. This would be impossible. Even then, trying to move to a frame of reference moving the speed of light relative to something else does not work. The equations for SR break down at light speed.

The best you can do in this situation is just replace people saying "moving at light speed" with "approaching light speed" or "moving near the speed of light". Then you can think about what happens as your relative velocity approaches the speed of light. In this case you are on the right track. Due to time dilation, the closer you get to light speed the slower time for things moving relative to you will appear. However, you will still see light moving at the same speed regardless due to the effects of this time dilation as well as length contraction.

$$^*$$

The light reflected from the clock before he achieves light speed, will be the last he sees of the clock's original state, hence, to him, the hands of the clock will appear still, while actually they still will be moving.

This statement is somewhat misguided even if we use the language of moving "near" the speed of light as I mention above. It seems to suggest that there is a "true" frame of reference where the clock hands are actually moving "normal", but someone on the train is moving relative to this "true" frame of reference, so they do not see what is actually going on.

This is not the case. The whole point of SR is that, in an inertial frame of reference, we cannot perform any type of experiment to know if we are moving. All inertial reference frames are on equal footing. So there is no "actually the clock is ticking away at some rate". The slower ticking clock as seen by someone on the train is just as legitimate as someone standing next to the clock watching it tick "normally".