The Triplets Paradox
A trio of triplets – Able, Baker, and Charlie – decide to test time
dilation. After some discussion, they decide Able will stay on
Earth, while and Baker and Charlie head out on spaceships. All
will carry accurate atomic clocks.
Baker will be traveling at 86.6 percent the speed of light relative to
Earth. This number calculates a Lorentz factor of 2, which means
that Baker’s clocks will be ticking at half the rate of Able’s.
Charlie will be traveling at 99.5 percent the speed of light relative
to Earth. This number calculates a Lorentz factor of 10, which
means that Charlie’s clocks will be ticking at one-tenth of the rate of
Able’s.
They depart at the same time. Charlie travels for half a year
(according to his clock), then reverses and travels for another half
year back to Earth. Baker travels for 2.5 years (according to his
clock), then reverses and travels for another 2.5 years back to
Earth. They both arrive back on Earth at the same time.
The triplets then compare their clocks and observe the following:
• Able’s clock has advanced 10 years
• Baker’s clock has advanced 5 years
• Charlie’s clock has advanced 1 year
This appears consistent with the theory of Special Relativity.
The ratio of Able/Baker represents the Lorentz factor of 2, and the
ratio of Able/Charlie represents the Lorentz factor of 10.
But what about the ratio of Baker/Charlie, which is 5? A Lorentz
factor of 5 requires a relative speed of 98.0 percent the speed of
light. But the relative speed between Baker and Charlie is 12.9
percent the speed of light, which calculates a Lorentz factor of only
1.008.
Something we should take into consideration is the relativistic
velocity-addition formula. It is:
Vn = (v+u)/(1+v*u/c2)
Where Vn is the net velocity, and v and u are velocities being added
together, with u being relative to v. In this case Vn is
Charlie’s velocity as observed by Able, v is Baker’s velocity as
observed by Able, and u is Charlie’s velocity as observed by
Baker. Applying this to our situation requires solving the
equation:
0.995c=(0.866c+u)/(1+0.866c*u/c2)
The solution is u=0.932c. This means Baker observes Charlie
moving at 93.2 percent of light speed rather than 12.9 percent.
Now the Lorentz factor for 0.932c is 2.77. Dividing 5 years by
that factor requires Baker to observe Charlie’s clock reading as 1.8
years.
So Baker’s relativity calculations require Charlie’s clock to show 1.8
years. But he must observe it as 1 year because his brothers see
it as that. This appears to be an unsolvable contradiction and a
disproof of Special Relativity. Unlike the conventional Twin
Paradox, there does not appear to be any ‘solutions’ involving
acceleration because the contradiction is still present even if you
argue that Able can be considered ‘at rest’.
To further illustrate the problem, suppose that after Baker and Charlie
obtained a constant cruising speed, Earth (and Able!) was destroyed by
a giant asteroid. We now have two ships moving apart from each
other at 0.932c. This should invoke a time dilation slow-down
factor of 2.77, but they instead slow by a factor of 5. Now it
doesn’t matter which clock you decide is running more slowly or if you
argue that each ‘observes’ the other as running more slowly. The
problem is the factor doesn’t correspond to their relative speed, but
is calculated based on Earth, which no longer exists. Since the
Earth never had a special ‘zero velocity reference frame’, and
certainly has none at this point, this destroys the legitimacy of
Special Relativity.
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