Time dilation for the beginner.

Time Dilation:

What is it? We have all heard of it, although to most, I suspect it is little more than a sound bite that we associate with Relativity. 

Let us diagram it and see how it works. We will use Einstein's Light Clock that has a light source that emits a flash of light that is reflected back to the source by a mirror a set distance away. On its return the flash of light is reflected over the same path by a mirror at the light source. One round trip is counted as one 'tick' of the clock. 

(Fig. 1). 

Now, we will take two Light Clocks installed on spaceships, A & B. These spaceships are passing in space at a relative velocity of 0.6c. These two clocks are identical with mirrors 1 light-second apart, therefore each 'tick' will have a duration of two seconds. Our two clocks are synchronised at the moment that A and B pass by one another.

Fig. 1 is a diagram of a light clock. The vertical scale measures light seconds.  The mirror is 1 light second from the source. Therefore light will take 1 second to travel to the mirror and the position of the light flash may also be used to measure the time since emission.  Fig. 1 shows a light clock after 1 second, when the light flash (Shown in green) has travelled 1 light second to the mirror. This diagram could represent either clock A or Clock B, taken in isolation; for either clock may be considered to be stationary and the other to be moving.

Fig. 2a shows clock B moving at 0.6c to the right of clock A. The green path and green measurements are those made by the observer on board Spaceship A for whom Clock A is stationary. The red path and measurements are those made by the observer on board the Spaceship B for whom Clock B is stationary. The blue path is what is observed from Spaceship A. When the light has travelled 1 light second, Clock B will have travelled 0.6 light seconds on board ship B and 0.8 of the distance to its mirror.

(Our old friend Pythagoras is in evidence here in the shape of a 3,4,5 right angled triangle,  where 12 = 0.62 + 0.82 or 1 = 0.36 + 0.64) Yet this leads to a conundrum; how can the time for the light in clock B take 1 second to reach the mirror as measured by an observer, at rest with the clock, yet 1.25 seconds when measured by an observer relative to whom the clock is moving? The answer of course is that they are measured under different conditions. As measured by the observers on ship A, clock B is moving at 0.6c, a very high speed. Einstein pointed out that the elapsed time between two ticks of a moving clock was greater than that of a stationary clock; which we see here as 1.25 seconds for the moving clock which is greater than the 1 second for the stationary clock. It is reasonable to conclude that this is because the moving clock also needs the time to move 0.6 light seconds displacement from the observer and so the light path is longer when measured in a moving clock, and as light is travelling at c, it must take longer to travel further. Two different elapsed times then, but between the same two events - the light being emitted and the light hitting the mirror! So Einstein realised that time is not absolute, but varies depending on how it is measured. That more time is measured between successive ticks on a moving clock or as he described it: As a consequence of its motion the clock goes more slowly than when at rest. And there we have it - time slows as speed increases.