After much help on the ATM forum I think I am seeing the basics fitting together. So I wondered if I might presume to show how it looks to me?
For me it all starts with what I agree is counter intuitive:
The Speed of Light
The speed of light. An innocuous phrase. Stating a seemingly obvious fact that light moves at a particular speed, in the same way that anything that moves has a speed.
Yet what does it mean?
The speed of light in a vacuum is constant; but relative to what?
The speed of light is constant and the same relative to any observer.
If we measure the speed of light between two points in space, without regard to whether those points are moving, then we are measuring how long it takes to travel a fixed distance without regard to any movement of the source nor of the recipient.
Take, for example the light from a star (that lies in the solar plane). As it orbits the sun the Earth is travelling at around 70,000 mph, sometimes towards that start and sometimes away from it. Yet, contrary to all common sense and expectations, the speed of the light from that star is always 'c', the speed of light.
How can that be?
Einstein gave us the answer when he said that time is not absolute. This new way of regarding time supported Einstein's Special Relativity, which showed that all could be resolved once one accepted that the measurement of time could depend on where it was measured from.
Einstein's Light Clock, which he defines as part of his 'thought experiments' serves to show how this works.
The light clock comprises no more than pulses of light sent to a distant mirror and thence reflected back to the light source. The mirror is a fixed distance so that the light returns in a set time, 1 second is commonly used.
But if the clock, in its Frame of Reference, is moving at velocity v, relative to an observer; that observer will see the clock travel the distance vt in the time t that the light takes to reach the mirror.
So, during the time, t, that the light travels to reach the mirror, the clock has travelled an additional distance,vt.
The Greek letter τ (tau) is commonly used to denote Proper Time, that is time measured by an observer on a standard clock that he is travelling with. A clock that he is stationary to.
But is the time τ the same as time, t, for the moving clock?
That is, is the time in the clock's Frame of Reference the same when measured by an observer, relative to whom, the clock is moving?
Fig 1 above, gives us the answer in no uncertain terms, for by the application of Pythagoras it tells us
τ² = (ct)² - (vt)²
τ = ct√ 1 - v²/ c² or
t = γτ
where c = 1
and γ = 1/√ 1 - v²/ c²
γ is also known as the Lorentz factor which is much used in Special (and indeed in General) Relativity
But that is just formulae, which are difficult to picture, so let us put some figures into the scene:
Part 1 shows the traditional (Galileian) view of how the light pulse would appear for a clock moving at 0.6c.
In the second that the light would travel to the mirror, the clock will have moved 0.6 light seconds along the x axis, combining to create a diagonal path, that the Pythagoras theorem would give a length of: √1 + (0.6)² = √1.36 = 1.166 light seconds. Which is further than the light could travel in 1 second.
However, in Part 2 we can see the Relativistic view and how far the moving light will have travelled, along the diagonal path in that second; and we see it will have reached the point 0.8,0.6 in the coordinates, or Frame of Reference, of the observer for whom the clock is moving.
Yet an observer travelling with that clock would see the light reach the mirror, 1 light second away, in that time, so we know that in that second the light has reached the mirror.
This seeming paradox is what took the genius of Einstein to resolve.
Indeed we can see this in Part 2 of Fig 2. For the y axis which is measuring the distance travelled by light in light seconds can also represent the time in seconds of the stationary observer.
And we see that, while the light in the moving clock has travelled for 1 second, only only 0.8 seconds have passed for the stationary observer.
This means that, when the Moving clock has travelled for one second, it will have travelled 0.6 light seconds relative to the stationary observer; one light second along the diagonal path, in the moving frame of reference of the clock; yet only 0.8 light seconds in 0.8 seconds in the stationary observer's frame of reference.
And this leads us to the inevitable conclusion that the time it takes for the pulse of light to reach the mirror, is measured differently in the two frames of reference. An observer moving with the clock would measure 1 second, while an observer for whom the clock was moving would measure only 0.8 seconds.
So how well does this agree with the formula we deduced earlier?
We said that: t = γτ
and when v = 0.6c, γ = √1 – (0.6)² = √1 – 0.36 = √ 0.64 = 0.8
Therefore: t = 0.8τ
τ = 1.25 t
or 1 second in the clock's Frame of Reference, measured from a moving (relative to the clock) observer, is the same duration as 0.8 seconds, measured by an observer in that Frame of reference.