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36.3 The Twin Paradox
Discussion
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Imagine a universe in which space has the shape of a circle (consider one dimension of space for simplicity, as we often do). Then notice that for Gracie to come back to George she does not need to stop, turn around and head back to Earth. Instead, she just keeps on going and because of the circular shape of space she subsequently passes by George (somewhat as traveling on the Earth’s surface in one direction ultimately takes you back to your starting point). That is, neither observer needs to relinquish constant velocity motion; each (apparently) can rightly claim to be at rest. So, in this example, how do you think the Twin Paradox works out?
Update: In this example, I’m imagining that space itself has a circular shape, not that Gracie is traveling along a circular track within space. If you’d like, think of space like a video game screen that “wraps around”: when you reach the right edge of the screen you reappear on the left edge, and so on. The point, then, is that Gracie does NOT accelerate/decelerate to return to George. She maintains constant velocity throughout the journey.
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Both Gracie and George would claim the other sibling is younger. After all, each claims to be stationary while the other is moving. However, they cannot meet up and compare wrinkles. So each can claim to be correct. If one of them accelerates to meet the other, then that acceleration locks in the youth of that traveller. I picture the time-space diagram being like an arch. The stationary perspective (either can argue) shows the moving sibling as an arch, similar to tossing a baseball in the air, except much greater scale. The moving person leaves and magically returns to x=0.
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If space is circular then time must, as well, be circular, and relativity won’t work any more.
We have said that from our perspectives, clocks that are forward in the direction of motion are lagging behind clocks that are backward. When space is circular, there won’t be “forward” and “backward” anymore, and so we will need to modify relativity and Lorentz Transformation to account for that.
For a mathematical example, assume Lambda=13/5, then if Gracie travels 5 years to get back to George, she will find his clock to be t’ = lambda(t – xv/c^2) = lambda(t – 0) = 13, but George will find the same result, and will say that Gracie has passed 13 years. Leading to an actual paradox this time -
My intuition tells me that in this simplified reality the result would be that there would be no paradox. Gracie would continue in constant velocity motion with respect to George for a fixed distance from and on to George. Similarly George would make the same journey from the perspective of Gracie. From each point of view they could both measure the distance, velocity and thus time from each perspective, and each could understand (post process) the alternative view – time dilation and length contraction would be experienced in identical fashion – therefore no paradox 🙂
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so if we consider the earth to be at rest wrt to the circular space we can say that since gracie will be traversing a circular path when viewed from 3D perspective, and sice objects moving in a circular path do experience acceleration, we can say that george will be older (as he is at rest with respect to the circular space).
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I would say that while Gracie is moving, both perspectives are correct (Gracie says George is younger and vice versa). But if Gracie decides to stop to compare her age with George, she will need to deccelerate, so she will no longer be moving with constant velocity. Which means that only George’s perspective can be correct.
However I like Abdulrahman’s idea that Lorentz transformation will not work as before. -
As the late Stephen Hawking said in his book Black Holes and Baby Universes-
The best evidence we have that time travel [into the past] is not possible, and never will be, is that we have not been invaded by hordes of tourists from the future.
Under SRT we see the twin paradox of time altered do the twins are no longer twins of age. Spacelike alterations can be conceived. Being special, the bridge of info gaps really find expression in the ~special~ designation, even if it is only intended to talk of velocities near maximum.
The Einstein- Rosen bridge allows for wormholes.
This may correspond to recent discoveries of turbulent areas of dark matter that also display a whirlpool appearance.
Space as a shape of a circle shows Gracie unwilling to stop after a circumpolar navigation back to George. The point is that is not stopping. She is a candidate for Lorenz transformations under an object in motion with George Burns and his canned salmon diet remaining steadfast for her return.
The trick here seems to be calling this as an at rest state for both G & G (GiGi?). If they are together when apart also as is love, this adds quantum foam.
But how does this apply to the Twins?
The wormholes seem to arrive back where they left off, but faster than the trip around the circumnavigation.
That is the space factor.But the time factor in spacetime is inseparable, so Gracie and George and the twins do not lose time. To the edge of the screen conceptualization.
No aging occurs.
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