🚀 go-pugleaf

In article <1992Jul23.050957.6557@nuscc.nus.sg> matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:
>John Baez: What, more reticence? I'm sure I speak for many when I say
>that I would be very interested to hear about those 3 hours of debate on
>the twins. Please post a summary. Thanks rett 

My reticence is never more than momentary.  We started out talking about
relativistic free particles and got into the twin "paradox".  I think
the upshot was what I described in a previous post.  Namely, the only
reason why one might expect the twins to have the same age upon
meeting each other again is some sort of symmetry argument.  When you
realize that one computes a twin's age by calculating the length of his
worldline using the Minkowski metric, one should realize that a symmetry
argument is only applicable when there is an isometry of Minkowski space
mapping one twins worldline to the other.  I.e., when their worldlines
look the same up to translations, rotations, and Lorentz boosts.
There's never any way to map a geodesic (unaccelerated worldline) to a
nongeodesic (accelerated worldline) so if one twin has felt acceleration
and the other not it's clear that no symmetry argument applies.
However, in fancier spacetimes like the torus world there may even be
two timelike geodesics that cannot be mapped to one another via
isometries, so one can have two unaccelerated twins who meet with
different ages.

We also discussed the version of the twin paradox that von Flandern likes so
much.  Here of course the trick is that simultaneity is an
frame-dependent concept.  We also went over the extra things you have to
do to take into account what relativistic observers actually SEE -
namely, take into account the time it takes light to get there.  (All
the things we SEE at t=0 is a different notion than all the
things happening at t=0 in our frame.)  Here the famous case is the fact
that one will SEE a rapidly moving sphere as having a perfectly round
profile, despite the Lorentz length contraction.  

I also showed them how to rapidly draw Lorentz-boosted coordinate
systems in 2-d spacetime diagrams, an invaluable skill.  Hint: call your
coordinates t and x.  A Lorentz boost will map the diagonal lines t = x
and t = -x to themselves.  Since a Lorentz boost is volume preserving,
it simply stretches out the line t = x by some factor k while squishing
in the line t = -x by the factor 1/k.  I.e. in lightcone coordinates
u = x+t, v = t-x, we get something like

u -> u/k
v -> kv

Everyone who hasn't done this twice already is advised to take the
situation von Flandern describes and draw a spacetime diagram of it
first in "our" coordinate system and then the coordinate system in which
the astronaut is at rest, to see his explanation of why he's so much
younger than us stay-at-homes.