Time Dilatation

Perhaps the best known and most surprising result of special relativity is time dilatation. Time Dilatation is simply the name of a phenomenon: moving clocks run slowlier.This phenomenon seems to have no logic and to be just a false supposition made by a crazy old man because it goes against our 'absolute' intuition, however we will demonstrate it is true.

Obviously what most of you are thinking right now is what are the clocks moving relative to? and slowlier relative to what? Those are two good questions, so once again we will need the help of our trailer laboratory and our group of observers, only that this time we will change our trailer for a super-spaceship that can reach velocities up to 80% the speed of light. We will put half of our observers in space and the other half inside the ship; the ship is completely transparent, so that the observers outside of the ship can see what's going on in there. Since our experiment is related to time we will also need the help of a clock or another time-measuring unit, in this case we will use a light clock. A light clock works like this: two mirrors are set horizontally with some distance between them and facing one towards the other, a beam of light is sent from the first mirror (which has incorporated some sort of lantern) and it is reflected in the second one and the time it takes the light to complete the run between the two mirrors will be our time unit.

We will start our experiment by setting the altitude of our light clock (the distance between the mirrors) to be 0.3[m]. Next we will measure the time it takes light to complete the run when the clock is at rest. In this case our time unit will be equivalent to 2 nanoseconds (1 nanosecond is equal to 1×10^-9[s]). As you probably figured out a nanosecond is a really small time unit, if not imagine that in a second there are 1000000000 nanoseconds, so using this unit could get to be hard, however our lab is well equipped and we were sufficiently funded to buy portatil equipement for those who won't be inside our spaceship-lab, we will suppose that with this equipement we can measure nanoseconds rather accurately. Now we put our light clock inside the ship  in a place where everybody can see it and everybody can measure the time it takes the light to complete the run. Now we ask the pilot of the ship to accelerate to a velocity of 240000[km/s]. The part of the experiment we will consider and develop is when we pass by the observers who aren't inside the ship. At all times the light clock will be running.

Here comes the interesting part, if we were moving at a speed of 240000 [km/s] the observers outside instead of seeing the beam of light going in a vertical straight line at all time (what the ones inside the ship see), they see that the path of the light is a diagonal path which starts at the point the beam was originated and ends at the position the light clock was when the light beam completed the run. To put this in the right terms we will say that relative to the frame of reference of the ship (the ship being the frame of reference) the light was completing runs that described vertical lines, this is what all the observers inside the ship saw; relative to the other observers' frame of reference (space if you want), the light described a path consisting of two diagonal lines that came as a result of the motion of the light clock when this was inside the ship. Each of the diagonal lines the light beam travels through is 0.5 [m] long, so relative to the observers outside the ship the light traveled a distance of 1[m] in each run, while relative to the observers inside the ship it traveled 0.3 [m] in each run.

Here it is important to remember that the speed of light remains a constant for both the observers outside the ship and the observers inside the ship and that the velocity of the ship does not have to do anything to the speed of light relative to any of the observers. The speed of light will remain being c=3×10⁸ [m/s]

Doing the math for the experiment we arrive that with the equipement used by the observers outside the ship the beams of light we say:

The clock inside the ship measured 2 [ns] while relative to the observers outside the ship it measured 3.333 [ns]. Which of the two is correct? Let's go back to our lesson about Classical Relativity and frames of reference, there we established that one observer observed a ball's trajectory as a straight line while another observed the ball's trajectory as a curve; we also concluded that both of them were correct and that there wasn't a 'true' frame of reference. This leads to say that both of the time measurements in our experiment are correct.What we can conclude from this is that there is such a thing as absolute time as much as there is a 'true' frame of reference: there isn't!

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