Time Dilation and Length contraction

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). 

The vertical axis shows the distance between the light source and the mirror, measured in light seconds. The time in seconds is given by the distance the light has travelled in light seconds. The mirror is 1 light second from the source. The light will take 1 second to travel to the mirror. It will then take 1 second to return to the light source, so the clock will ’tick’ every two seconds. The x and z axes may be added as required to represent a horizontal displacement, by the x axis or an area by the x and z axes. Clock B, which we will draw in red identical to clock A. Either clock would be considered to be stationary by an observer on that spaceship and the other to be moving. It is when we want to reference measurements of the moving clock, relative to a the stationary observer, that we need to look a bit deeper.

Now, we will take two Light Clocks installed on spaceships, A & B. Remember, these two spaceships are passing in space with a relative velocity of 0.6c. These two clocks are identical. Mirrors, 1 light-second from the light sources, reflect the pulses of light. Each ’tick’ will have a duration of two seconds. Our clocks are synchronised at the moment that A and B pass by one another.

Fig. 1b shows the Frame of Reference of clock A. It shows space and time, as measured by an observer in Spaceship A, with Clock B moving at 0.6c relative to clock A. It is the framework with which we may address what is happening and how it is measured from the viewpoint of the Observer in Spaceship A. The green measurements are those measured by the observer on board Spaceship A, for whom Clock A is stationary. After 1 second, the clocks having been synchronised when they passed, the pulse of light in each clock will be reflected in that clock’s mirror. The two pulses of light, each measured by its local observer, will have travelled 1 light second. Clock B will be 0.6 light seconds displaced from Clock A. But when we add the path taken by the pulse of light in Clock B, as viewed by observer A, we have a difficulty. (Fig. 1c)  Clock B is moving away from Clock A. The progress of the light flash in Clock B, is shown at successive points along its diagonal pathway from Clock A.  The green and red path is that of the light in Clock B as observed from Spaceship A. As measured from Observer A’s Frame of Reference. After 1 second the light in each clock will have reached its mirror.   But when we measure the path of Clock B’s light, as measured from spaceship A, we find, by Pythagoras, that it is √(0.62 + 1.02)= 1.166 light seconds. Which would mean that the light flash must have travelled faster than the speed of light. Yet we know this is impossible.  So how do we resolve this conundrum? Let us examine the diagram again, rotating the Frame of Reference of Clock B so the light in Clock B travels at the Speed of Light, as it has to do. When we add the Frame of Reference of Clock B to the diagram it can no longer be a cartesian diagram, for the time axis of moving Frame B is rotated through angle β (Sin β = v/c), along the red and green line. After 1 second, measured by A, the Clock B will have travelled 0.6 light seconds from Clock A; its light having travelled 1 light second to point 0.6,0.8 (Fig. 1d). So, Measured by observer A, the light flash in Clock B, will have travelled 1.25 light seconds to its mirror, taking 1.25 seconds as measured from A so to do. Yet as we have seen above, as measured by observer B, it will still have taken 1 second to reach its mirror.

Yet this leads to a conundrum; how can the time for the light flash 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. As measured from ship A, the light in B has further to travel and at the speed of light that means taking longer so to do.

Length Contraction

It may seem that we are seeing here the same effect for length as for time, as the distance the light in B travels, as measured from A, is increased, from 1 light second to 1.25 light seconds.

The displacement along the x axis too, is increased from 0.6 light seconds to 7.5 light seconds, so why do we talk about length contraction?

The answer to this lies back with Einstein, who didn't use the terms Time Dilation and Length Contraction when he described these effects.

No, he described how the length of a moving meter rod was measured to be less by a stationary observer.

And that is what we see in Fig. 1d. the displacement of the moving clock, 0.75 light seconds, in the rotated Clock B Frame of Reference, is reduced to 0.6 light seconds in Clock A's Cartesian Frame of reference, and that it is by the Lorentz factor (1.25 at 0.6c) just as the Time Dilation is an increase by the Lorentz factor.

So Time Dilation and Length Contraction are the equivalent effects, on Time and Distance but viewed from opposite sides.  

Time is dilated in a moving clock, Length is contracted in the stationary clock.

But note too that in fact the Duration remains the same and the separation between the clocks remains the same!  It is the unit size and unit quantity that change, and both by the Lorentz factor.