# Understanding of derivation of time dilation

While deriving time dilation using the two postulates of Special theory of relativity we imagine that a light pulse is traveling back and forth between 2 mirrors. In the rest frame (when the observer and this arrangement share the same inertial frame) the figure looks as follows. The velocity of light is taken to be $c$ according to Einstein's second postulate

When this setup is moving towards right with respect to the observer with velocity V, then the figure looks as follows.

Here also we consider the velocity of light as $c$. But we observe that the path of light is not precisely vertical as in the first picture. The path of light is bent towards the horizontal direction. Hence path length has increased and to compensate for this increased path length it comes out that the time is dilated.

What I do not understand is that why the path of light is bent towards horizontal direction? I think that once the light pulse emerges from lower mirror, it is free from the influence of the lower mirror and will move straight up precisely in vertical direction in the space between the mirrors. Why it bends so that it can be reflected precisely form the middle of the upper mirror? According to my assumption the figure would look as follows.

If I consider velocity of light to be c then according to my figure there will be no time dilation.

I know that some where I am committing a mistake. I am following Einstein's 2nd postulate by taking velocity of light as c. So I think somehow I am violating Einstein's 1st postulate. But how? Is it that by keeping the light path precisely vertical I am actually allowing a frame of reference at absolute rest where light moves and such a absolute rest frame does not exist according to Einstein's 1st postulate?

• In the first case (moving mirrors), we view just the mirrors motion across time. In the second case (moving mirrors), we view less mirror motion across time, now that some of the mirror motion is now directed across space. In either case, the mirrors are in motion. – Sean Sep 24 '16 at 4:17

## 1 Answer

In the second case (moving mirrors), the light beam was not vertical from the start. Remember that you are considering a transformation of the reference frame, so everything is moving to the right with constant speed, including the light itself, thus the shift in the initial direction of the light beam. This motion actually reflects that you are moving to the left. According to the principle of relativity, these are the same thing.

This is not specific to Special Relativity though. Special Relativity is taken into account by postulating that the speed (the absolute value of the velocity) of the beam is the same in both reference frames, instead of receiving an additional contribution, as it would have if it were a baseball traveling with neglectable (in comparison to $c$) speed.

• Yes, I could visualize that if I plot the coordinates of position of the light pulse with time as observed from the moving frame of reference, the path will look bent towards horizontal direction. This phenomenon is not specific to relativity. – user103515 Sep 20 '16 at 6:47
• @user103515 yes, that is exactly the point of my answer. Your point being? – Prof. Legolasov Sep 20 '16 at 6:48
• I completely agree with you and my doubt is clarified. Thanks. – user103515 Sep 20 '16 at 6:51