This thought experiment
is a twist on the Twin "Paradox"
, a thought experiment that some have attempted to use to show the inconsistency of special relativity
but instead demonstrates its consistency quite nicely.
Several explanations for the apparent paradox
give a role to acceleration
which plays absolutely no part in it (it wouldn't be special relativity otherwise).
I thought this up to demonstrate to myself how acceleration played no role.
I wouldn't be surprised if this has been presented elsewhere but AFAICT
I thought it up on my own.
s on Planet A make a remarkable discovery: two star
s (call them B and C) approaching from opposite directions (call them B and C), both at half the speed of light
! (star C is slightly farther away from A than B is).
B-> A <-C
. The stars are not on a collision course. They'll pass within several light-day
s of each other, providing a little gravitational
shove but nothing to worry about).
Not only that, after adjusting for blue shift
, they notice radio
transmissions coming from each star, indicating that there is a planet
with a techhnological civilization
circling each star.
Miraculously, they set up radio communication with both civilizations.
Scientists on all three planets, interested in special relativity
, set up some experiments to demonstrate its effects.
Eventually B closes on and passes A.
This is a moment of great significance to both of them. At the ceremonies for the moment of closest approach, a switch is thrown and both planets start using the same time units, and all of the clocks on both planets are synchronized to the same moment.
The next significant event is that five years later (on B), B passes C (at 8/10 the speed of light).
C synchronizes its time units and clocks to those of B.
When C finally passes A (after 11.54 years),
18.76 years have passed on A!
Where did the other two years go?
Since B is approaching C at a faster rate than A is, it is closer to C than A is!
This is due to the distance contraction
inherent in being a relativistic frame of reference, and despite the fact that A and B are almost at the same place..
Since B and C are both approaching A at half the speed of light, their gamma factor
s, relative to A are both about 1.1547.
C's gamma factor, relative to B, is about 1.6667.
At the time that B passes A, people on B think C is 4 light-years away, but people on A think C is
(1.6667/1.1547)*4 = 5.77 light years away.
At the time that B passes C, people on B observe the trip from A (at 8/10 the speed of light) as taking 5 years, but people on A observe it as taking 7.22 years.
This is where the other 2.22 years on A come from.
Now, C also thinks A is 5.77 light years away, so the trip to A at half the speed of light appears to take 11.54 years, to people on both C and A.
A resident of planet A who got in a spaceship and travelled to B when it passed A, then travelled to C when that planet passed B, and then returned to A at the end
would have experienced 16.54 years of time passing. A resident remaining on A would have experienced 18.76 years of time passing.
The way this thought experiment is set up, no-one has to travel between the planets, and no-one has to accelerate to a relativistic speed.
s on planets B and C both measure the entire exchange, from B's passing A in one direction to C's passing A in the other direction, as taking 16.54 years, but
Clocks on planet A, on the other hand, measure the entire exchange as taking 18.76 years.
A is not in a preferred frame of reference (which doesn't exist), however.
A is different only because it was in the middle.