The Twin Paradox: No Acceleration

Thursday, November 01, 2018

The Twin Paradox: No Acceleration

The Twin Paradox Part 2 - No Acceleration

Check out the first version of the twin paradox if you haven't seen it already!

The twins hate special relativity and so want to find a paradox in the theory which shows it to be wrong. They contact an alien friend and tell it to fly to Earth. Rob then gets in his rocket again and flies away from Earth. On the way he passes the alien who takes a reading on Rob's clock. The alien then continues its journey and arrives at Earth to compare its clock so Sam's clock. This way we have avoided acceleration and so we can't use the excuse of saying Rob was not in an inertial frame of reference.

Sam's View

Rob moves away from him and so his clocks run slower. The Alien then takes this reading of the slow clock and then travels towards him. The Alien moves at the same speed as Rob except in the opposite direction and so the Alien's clock also runs slowly. When the Alien gets to Earth, Sam's clock shows more elapsed time than the Alien's and so he concludes he is older than Rob.

Rob's View

Sam is the one moving and so clocks run slower for Sam. When Rob reaches the Alien, the time elapsed for Sam is less than the time elapsed for him and he gives this data to the Alien.
The Alien adds the time the Rob claimed elapsed for Sam and then continues to measure the time elapsed for Sam.
The Alien sees Sam moving towards him and so again Sam's clocks run slowly. When the Alien gets to Earth; it concludes that Rob is older than Sam because Sam's clocks were dilated the whole time.

This is clearly a paradox and so there must be something that we have missed. There is no way the Alien and Sam can stand next to each other and disagree on which twin is older and so we have to look carefully at what kind of assumptions we may have made that weren't warranted.

The Correct Answer

The assumption we have made was about the relativity of simultaneity. As Rocket Rob moves away from Earth, he claims to be at rest while Sam moves away from him. Remember clocks towards the back of motion are ahead of clocks towards the front? Well imagine an infinite line of clocks in Sam's frame of reference. The clock's will all be dilated but the clocks at the point where Rob and the Alien meet will be ahead of the clock where Sam is.

When we change frames of reference so that it appears that Sam is coming towards the Alien, the line of clocks are still dilated but the clocks at the point where Rob and the Alien meet are now behind Sam's clock. To account for this change of frame of reference the Alien should add on some time to account for the relativity of simultaneity.

In reality Sam is correct in claiming Rob is younger because Rob and the Alien did the wrong calculation in trying to measure Sam's age. If they accounted for the relativity of simultaneity, then they would have agreed with Sam's calculations. Once again the twin's can't find a paradox.

We can show this using the space-time diagram below.

The space-time diagram shows the time that Rob says elapsed for Sam. When the Alien comes and then continues measuring the time elapsed for Sam, he also claims Sam is moving and so experiences slower time. The diagram shows the time that the Alien measures as well. The diagram shows the mistake Rob and the Alien made however. The part of the time axis between the red and blue line shows the amount of time that the Alien should have added to its clock to account for the relativity of simultaneity.

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