The 'Twin Paradox' of relativity is around a century old, but it still gets a lot of attention on many forums. I have also written on it a few times in this Relativity and Cosmology Blog, e.g. here and here. An interesting variant of the classical twinparadox is to consider the situation at the halfway stage, which may be labeled the 'HalfTwin Puzzle' of relativity.
On the right is a graphic of what I call the 'RGB scenario', an allinertial variant of the classical twin paradox. It differs only in that
the 'awaytwin' (Red) does not make a quick turnaround to come home again, but a
third inertial observer (Blue) does a 'R/B flyby' in the opposite
direction and performs the return leg. Both speeds are 0.866c relative to Green, giving a Lorentz factor γ = 1/√(10.866^{2}) = 2 in both directions. This means that the 'moving' clock appears to tick at half the rate of the 'stationary' clock.
At the R/B flyby, Blue sets her
clock to read the same as Red's clock (2 units) and since they are both inertial, presumably their clocks will tick at the same rate relative to Green. When Blue later passes Green,
they compare clocks firsthand and conclude that Green has aged 8 units,
double the sum of Red's and Blue's aging during the two halves of the
test (a sum of 4 units). This is standard, verified relativity and not disputable.
The 'half twin puzzle' is where we simply ask the question: at the halfway (R/B) flyby event, which twin (Green or Red)
has aged less since the original (G/R) flyby event? Here are some issues, lurking in very subtle, perhaps confusing logic. To understand them, we must look at the full GRB scenario, after which Green, Red and Blue can hold a
teleconference and compare results. They will conclude that the proper
time that has elapsed for Red until the R/B flyby (from 0 to 2) equals the proper time that has elapsed for Blue from the R/B to the B/G
flybys (from 2 to 4). This is inevitable, since Red and Blue flew the same distance at the same speed relative to Green.
Green, Red and Blue will further agree that Green's elapsed proper
time between the G/R and G/B flybys was 8 units, as measured by Green's own clock. It seems logical that between the G/R and R/B flybys, Green "really aged" 4 units while Red "really
aged" only 2 units (Green's dotted line of simultaneity). After all, Red and Blue each recorded half of the 4 units, so how can Green not record half of the 8 units at the halfway point?
The first apparent paradox is this: since twins Green and Red were both stationary in their own inertial frames for the duration of the test, they should be equivalent and the one cannot age more (or less) rapidly than the other over the duration of this test. To argue that they age differently will mean giving preference to one inertial frame over another. Since the 2 units for Red is a given, it means that Green should also have aged only 2 units.
Secondly, according to special relativity, Red has the 'right' to view Green as moving at v = 0.866c, with a Lorentz factor of γ = 2. So per Red, Green should have aged only 2/γ = 1 unit between the G/R and R/B flybys (Red's dotted line of simultaneity). Thirdly, to make matters even worse, according to Blue's dotted line of simultaneity, Green's clock should have read 7 units when the R/B flyby occurred. After all, Green flew at 0.866c relative to Blue and while Blue aged 2 units, Green should have aged only 2/γ = 1 unit.
How do we reconcile these apparently paradoxical conclusions, i.e., did Green age 1, 2, 4 or 7 units during this 'HalfTwin test'? We will let interested readers puzzle a little, before offering attempted resolutions of the paradoxes.
J

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