How The Twin Paradox Of Relativity Changes In An Expanding Universe.

I'm sure most of you have heard of the twin paradox "in which a twin makes a journey into space in a high-speed rocket and returns home to find he has aged less than his identical twin who stayed on Earth."  This paradox has been worked out for special relativity in Minkowski spacetime.  Recently, Boblest et al. worked out the details using general relativity for an expanding universe. (de Sitter spacetime.)

First a review of the standard Minkowski version:  In this case the whole universe is flat Minkowski spacetime and can therefore be handled with special relativity.

The twins in the paper have names: Eric and Tina. Eric stays on Earth while Tina accelerates away from Earth with constant acceleration α = 9.8 m/s²  until her clock shows 5 years have past.  Then she decelerates by the same magnitude coming to a complete stop after ten years then begins her journey back to earth accelerating then decelerating in the same 5 year intervals.  Finally, after 20 years has transpired on her clock she has returned to earth being now 20 years old.  The plot above might help make this more clear.  It shows how far she she is compered to Earth versus the time recorded on her clock.

This plot above shows the time on Eric's clock versus the time on Tina's clock.  As you see, Eric is nearly 350 years old when Tina returns.  Furthermore, he ages quickest relative to Tina when she is traveling at peak velocities.

Now consider an expanding universe:  For an expanding de Sitter spacetime, the universe is no longer flat and so the authors have to appeal to general relativity.  The Hubble expansion parameter, quantifying how fast the expansion is happening, is denoted by H.  For our universe, H =  H₀ = 71 km/s/Mpc.  Larger H means faster expansion.

The plot above shows how the results change in an expanding universe.  Interestingly H must be greater than 10⁷ H₀ , before we see significant differences compared with the flat case.  One very important thing to notice is that, since the universe is expanding quickly, after 20 years Tina is not able to return home.  This should make sense since the distance she has to travel in an expanding universe is further then in the flat case.

This next plot above shows how Eric's clock changes compared to Tina's in the expanding universe case.  As you can see, Eric does not age compared to Tina nearly as much.  This should also make sense because the expansion of the universe inhibits Tina's relative velocity to become to significantly different from Eric's.

Now For A Proper Round Trip. Now let us demand Tina accelerates and decelerates in such a way that Tina is able to return home in 20 years.  Now Tina must turn around before the 10 year mark.  Here the authors compare three "trips" where the constant acceleration/deceleration is greater for each trip.  IE, Tina accelerates faster for trip 3 than trip 2 which is faster than trip 1.  In each case, H =  10⁹ H_0.

The plot above shows how Tina's position compared to earth changes with time in an expanding universe.

And lastly, the plot above shows how Eric's clock changes compared with Tina's for the three trips.

Conclusion:  This paper shows how the twin paradox is altered for an expanding universe.  Interestingly, the expansion of the universe lessons the distance Tina can travel in 20 years and makes it so that Tina's change in age changes much more closely to how Eric's age changes.  However, it should also be noticed that to see significant differences compared to the special relativity case the universe must be expanding significantly faster than our own universe is.

Sebastian Boblest, Thomas Müller, & Günter Wunner (2010). Twin Paradox in de Sitter Spacetime E-Print arXiv: 1009.3427v1