H2 Twins Again, With Different Rates of Aging
A picture is worth a thousand words:
Epstein’s twins, Peter and Danny, decide they no longer want to be the same age. Since time 'deep' in a gravitational field runs more slowly than 'higher up' in the field they decide while having tea that Peter will spend some time on the first floor while Danny hangs out in the cellar. At point A, they begin to implement this plan. The lines AB and AB' and CD and C'D' are of equal length. But also the arcs BC and B'C' are of equal length, so the two will meet again at the same time on the first floor, i.e., at the place of A, D' and D. Peter is now really a bit older than Danny!
What does it mean, however, ‘to meet again at the same time’? We assume again the position of an observer in OFF. Thus, for us the whole experiment elapses in time AB + BC + CD = AB' + B'C' + C'D'. For the red (Peter) the time elapsed is given by the projection of the arc BC on the time axis (i.e., the red Δt), while for the blue (Danny) even less time has elapsed. For him it is given by the projection of B'C' on the time axis (i.e., the blue Δt').
Did you notice that Peter and Danny were very quick at climbing stairs? How fast were they moving? Well, the whole story would also work if they took it somewhat more relaxed. The drawing would, however, be a bit more complicated.
The Swiss lawyer, philosopher and chansonnier Mani Matter has written a song about a boy running 'as fast as lightning':
es git e bueb mit name fritz, es git e bueb mit name fritz
und dä cha renne wi dr blitz, und dä cha renne wi dr blitz.
är rennt, dä unerhört athlet, är rennt, dä unerhört athlet
so schnäll, das me ne gar nid gseht, so schnäll, das me ne gar nid gseht.
und wil er geng isch grennt bis jitz, und wil er geng isch grennt bis jitz
het ne no niemer gseh, dr fritz, het ne no niemer gseh, dr fritz.
und ig sogar, dr värslischmid, und ig sogar, dr värslischmid,
mues zuegäh: vilich gits ne nid, mues zuegäh: vilich gits ne nid.
there is a boy let's call him Fritz, there is a boy let's call him Fritz
that can run as fast as a blitz, that can run as fast as a blitz.
he runs that outrageous athlete, he runs that outrageous athlete
so fast you can not see his feet, so fast you can not see his feet.
and 'cause he never ends his race, and 'cause he never ends his race
no one has ever seen his face, no one has ever seen his face.
so even I, the bard, agree, so even I, the bard, agree
perhaps the lad can't even be, perhaps the lad can't even be.
Lyrics and tune by Mani Matter [46-41],
translated from the original Schwiizertüütsch (Swiss German) by David Eckstein and Samuel Edelstein
Fritz becomes a victim of Ockham’s razor! (see problem 1)