This is an introduction to a new simpler way to conceptualize the essential mechanism at the heart of Special Relativity. This can be done using only simple cartesian constructs and a different way of understanding Time as the fourth dimension.
We are all familiar, I am sure, with Cartesian axes; whether we know them as x,y,z; length,width and height; or latitude, longitude and altitude. They are the common way we have of viewing the 3D Universe we live in.
Time, on the other hand, is more difficult as it has no physical direction but is the same anywhere from wherever or whenever we view it. This does not mean that any two people will measure it the same in all circumstances, yet to any one observer the time in their Frame of Reference is the same everywhere .
The best way we have of measuring time is by timing the passage of light in a vacuum, because Einstein’s second Postulate that Relativity is that the speed of light in a vacuum is ‘c’ a fundamental physical constant.
Time has no direction, it expands in all directions. Imagine it as an sphere expanding at the speed of light. The whole of that sphere constitutes a point in time. Its measure is the length of any radius drawn to the surface of that sphere. Thus the time axis can be considered to be any radius of that sphere and it is profitable to use whichever radius best suits one’s purpose.
We see this in Fig. 1. where the time axis labelled ct could lie in any direction. Because the time is the same throughout that sphere it is in effect a point in time that does not move but expands.
Also, as our fourth dimension, the surface of our expanding point in time crosses each of the other three dimensions at right angles!
The Duality of Time
How then do we resolve the spherical expansion of time with the traditional idea of the linear nature of time, with concepts like the arrow of time?
We measure time in two ways: with a clock as the background time that passes everyday, or with a stop watch to measure the duration of any activity or process.
One can see the effect of this disparity in Fig. 2. where diagram A demonstrates this difference, with red lines of the expanding point of local time from the event at (0,0), in contrast to the blue lines which are part of a much wider circle centred on whatever point in the distant past one is measuring from – to give a feel for this I have included diagram B where the blue circle is centred on a point 1 minute in the past; and in diagram B, centred on 1 hour in the past. Yet the background time as we think of it is centred at some vague point in the distant past, which could be anything from the start of the day, or right back to the Big Bang! So one can see just how big that blue circle would be and why in diagram A , the divisions form straight lines.
Frame of Reference
A Frame of Reference is the way that Spacetime is mapped relative to a real or imaginary location. As mapped in a Frame of Reference, spacetime will, a priori, be at rest relative to that Frame of Reference.
In a Space time diagram (plotting time against the x axis) the world-line of any stationary body or particle will be vertical, for as time progresses that body will remain at the same point on the x axis.
A good way of demonstrating this is by looking at light clocks – where each ‘tick’ of the clock is the passage of a pulse of light between two mirrors set 1 light second apart, where the position of the light flash corresponds to the time. (Fig. 3)
Fig. 3 Fig. 4
Clock B is an identical inertial clock, synchronized with Clock A, their light flashes each reaching their mirror after 1 second. This is how it would be measured in the reference frame of either clock. (Fig. 4)
But when Clock B’s Frame of Reference is measured relative to clock A, Clock B’s measurements are transformed; from a stationary observer in Frame B, to a moving observer in Frame A.
The travelling clock
The movement of Clock B away from Clock A reveals a very different – and perhaps surprising difference in the measurements of the light path in Clock B.
As observed from Clock A, when the light in Clock B has travelled 1 light second along the time axis of clock B (in red), Clock B has travelled 0.6 light years from Clock A. The light will then be at point (0.6,0.8) in A’s Frame of Reference. Fig. 5. The time axis of Clock B being rotated through angle β, where sin β = vt/ct (the ratio of the clock’s speed, v to the speed of light, c.
To Summarize; measured in Clock B’s Frame, the light in Clock B travels 1 second to the mirror, Fig.4. But when Clock A measures the movement of the light in clock B, it takes 1.25 seconds.
That is Time dilation. The measurement of the light’s movement in Clock B, 1 light second in one second, is transformed by the Lorentz factor, γ = which at 0.6c = 1.25, to 1.25 light seconds in 1.25 seconds relative to Clock A.
It is important to understand the inceased time measured for the travelling clock is to factor in the time taken for the clock to travel in addition to the time accrued on the clock.
Be aware too, that the use of light clocks is only to make the visualization easier. With the light clocks we can see the passing time measured by the light’s movement. One light second takes one second.
It would be just as valid to have drawn a round clock with a face, or a digital watch, travelling. Only in that case it would only be the time increasing without the light path to measure.