## Relativity Challenge## Test your thinking skills
## The SituationFig 1.1 Two accurate identical clocks, A and B, move with uniform motion along a shared straight-line path but with different constant velocities such that the distance between them either steadily increases or steadily decreases with time. According to the Theory of Special Relativity, as proposed by Einstein in 1905, these clocks should run at different rates. To quote:
## The QuestionWe wish to know which clock runs more slowly. What information is needed to determine this and how should it be applied?
## RulesYour answer may be expressed in
words, in equations, or a combination
of both. Just so long as it describes which of the clocks
runs more slowly. An answer that does not include this
information will be considered a fail.
## More DetailsTo assist the challenger, more
details are provided.
## Objections and Possible ResponsesNaturally there will be objections
and it is worth anticipating them in
advance.
## Part 2Two identical clocks – A and B – are attached to rocket engines. Affixed to the end of each rocket is a very long pole stretching out behind. The poles are of equal size, measuring several ‘light hours’ in length (relative to their own frames of reference), and at the end of each pole is a flag as shown: Fig 2.1
Clock A will remain ‘at rest’ throughout the experiment. It
faces left in this diagram. Clock B faces right and comes in
from the left at a constant velocity of 40% the speed of light,
relative to A. Fig 2.2
B then continues at constant speed. Many hours later, B draws
level with the flag at the end of A’s tail. At this point, B
presses the Fig 2.3
Meanwhile far away to the left, and presumably at the same time
(although this can be left as ambiguous for supporters of
‘non-simultaneity’), A draws level with the flag at the end of B’s
tail. At this point, A presses its Fig 2.4
A few hours later (allowing time for ‘simultaneity’ to catch up at the
other end, or to compensate for whatever other objections relativists
can think of), B applies the brakes and then reverses all the way back
to A. The two sheets of paper are then compared. Fig 2.5 And B sees this: Fig 2.6
So each clock sees the other pole 80% shorter and each clock records 80
minutes as it passes from one end to the other.
## Part 3
This next part will examine a ‘reverse perspective’. Fig 3.1
As discussed, B will observe 80 minutes passing as he moves from A1 to
A2. Fig 3.2
Note 1: for convenience the invisible periods are shown in grey. Fig 3.3 A2 will see this: Fig 3.4
Due to length contraction, A1 will see B shrunken along its length and
closer. B’s starting position will be level with A1 but its
ending position, where B disappears, will be 80% of the A1-A2 distance.
## Part 4
We’ll now extend the above to a three clock/observer situation. Fig 4.1
Coming in from the right is another observer C (see above
diagram). C has its own stopwatch and is invisible while at
the right of A2. Fig 4.2
At this point B becomes invisible and C visible. B announces
its elapsed time (of 80 minutes), and C sets its stopwatch to start
with B’s reading. Fig 4.3
That is, A1 sees B and C draw level at same location, and at a distance
80% of the way from A1 to A2. The reason for this identical
location is that the length contraction formula (the Lorentz transform)
squares velocity, which makes its direction unimportant. And
since B and C have the same speed relative to A1 (of 0.6c), they get
the same length contraction treatment. Fig 4.4
## Part 5
Now for the final example involving a science-fiction scenario with
space travel. This example describes the ‘Twins Paradox’
scenario, i.e. the passage of a traveller from Earth to a distant point
and back again. Fig 5.1 And here’s a perspective drawing showing Earth’s view of the B/C meeting: Fig 5.2
As can be seen, the total time recorded by Earth between the departure
of B and the arrival of C is 16 years. And the total of times
recorded by B and C add up to 16 years.
## Epilogue
## Thanks for comingThis concludes the
challenge. There are no prizes for ‘solving’ it,
sorry. Its purpose is to be thought-provoking and hopefully
you found it so.
[1]
http://www.fourmilab.ch/etexts/einstein/specrel/www/ |

Copyright © 2014
Bernard Burchell, all rights reserved.