Archanum bifocal  




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Imagine a lift (or elevator) which is, say, one metre across. Its walls are made of glass so that an observer can see into it. On one side of the lift is attached a small mirror, and immediately opposite is a lamp and a light sensitive cell. When the lift is stationary, a pulse of light radiating from the lamp will travel out to the mirror, then back – in other words it will travel 2-metres to reach the cell beside the lamp. When the lift is falling, the pulse will again radiate out and hit the mirror in the same place - but now at a lower point (by the amount the lift has descended while the light pulse travelled across the lift) - and the reflected pulse will arrive at the cell even lower (by the same distance again).

Thus the light travels further when the lift is falling. Anyone in the lift would not notice any change from when the lift was stationary. Either way, for them, the pulse would merely appear to travel out and reflect back horizontally across the lift. They might have a clock to measure the duration of the pulse going out and back – but whatever they do, however fast the lift falls, the clock will always read the same (this has been verified by experiment). Only someone who remains stationary outside the lift would say that the light pulse was travelling further than merely across the lift and back.

Since the speed of light has been found to be constant, regardless of anything except the medium it propagates in, then because for the outside observer the light travels further, it will also, clearly, take longer to do so than for the person in the lift.


This means that the person in the lift will have aged less.


This is easy to prove mathematically:


L is the width of the lift. c is the speed of light.

If the distance the lift falls - in the time the light-pulse goes across the lift and back - is d, then you can draw a triangle so d is a vertical line between the lamp when it emits a pulse and the photocell when it receives the pulse, with L as a one-metre horizontal line from the centre of d across to the mirror (at the point when it reflects the pulse). Joining the ends of d up to the mirror gives the path of the beam (as seen by an outside observer) as the lift falls – which makes two right-angled triangles.

Time the pulse takes to cross the lift and back for person in lift:


t = 2 L/c............or............ L= ct’/2................. [equation 1]


When the lift falls, the distance the light travels from the lamp to the mirror (as observed by someone outside) will be increased. The new distance is h.


So new Time, t = 2h/c   or ..... h = ct/2 …...........…[equation 2]


Note also that for the outside observer, the distance the lift falls d is the lift’s velocity multiplied by the time it takes.


tha......t.....So.................. ... d = v t …..........…. [equation 3]


By Pythagoras’s theorem – ie, the square of the long side of a right-angled triangle equals the square of the other sides, added together

..................Hence............. h= (d/2)2+ L2........[equation 4]


Substituting from equation 2…. Where h = c t/2
........... And from equation 3.....Where d = v t
.......... .And from equation 1.....Where L = c t’/2


equasion 4 ..becomes:.. (ct/2)2= (vt/2)2+ (ct'/2)2................


Rearranging this to make t the subject (see detail ) results in:


.........t = t'/(1 - v2/ c2)


So if you travel at 0.8 times the speed of light for 30 years, say, then the time that will have elapsed for those you left behind will be:..


.......t = 30 /√(1- (0.8c)2/ c2) = 50 years


or if 0.999 times c for 1 year then..... t = 22....years
or    0.9999 c  .    for 1 year ............t = 70.. .-years
or    0.99999 c    . for 1 year ............t = 223..years


This is one of those profound laws of nature that people are rarely told about (much too interesting and thought provoking for dull compulsory day-prison - ie, school). It’s just the kind of simple logic that Einstein considered when working on relativity and related theories. And in this instance means, ultimately, that there is no limit to how far or how fast we can actually travel - so long as we develop the technology. The consequences, though, of travelling at speed are also profound – as can be seen – and would temporally separate us from the world we left by an amount depending on that relative speed.

THE TRICK is to realise that while for the observer the traveller never exceeds c, for the traveller there is no limit - speed just increases and increases with no barrier at all. Which means the speed of light is actually NOT a limit (as everyone mistakenly believes), only the observation or measurement of it!

So 0.9999 c as measured by a 'static' observer for 70-years, which takes the traveller 1-year to cover, means the average speed for the traveller is 69.993 c