On Fri, May 20, 2016 at 06:03:52PM -0700,  (via tml list) wrote:
> But only the "it took two weeks" observers would say that the ship
> had a tau of 0.5   :-)

I did make an error in my previous post, but not that great of one.

With tau 0.5, the relative velocity is sqrt(3/4) c.  If the velocity
is parallel with the jump direction, the Lorentz transformation gives

 x' = (x - v t) / sqrt(1 - (v/c)^2)

so that the distance is 1.99 parsecs.  However, the time interval they
observe between the ship going into jump and coming out is

 t' = (t - vx/c^2) / sqrt(1 - (v/c)^2),

which is -293 weeks.  If the relative velocity is in the opposite
direction, the distance observed is 2.01 parsecs and time +297 weeks.

A very simple sanity check can be made without involving the Lorentz
transformations: A relative velocity corresponds to a rotation of
spacetime axes, preserving the x^2 - (c t)^2 invariant interval.
This is similar to Euclidean rotations preserving Euclidean distance
x^2 + y^2, except with a conversion factor of c between space and
time, and a different sign.

So if in one frame, it emerges at a point 1 parsec away and 1 week
later, that's an invariant interval of extremely close to 1 parsec
(it's about 0.99998 parsecs).  If some other observer sees the
distance as being 2 parsecs, then they must see the time as being
+/-295 weeks to preserve the invariant interval.  2 parsecs and 2
weeks is not possible.

- Tim