  Relativistic rigid motion suggests a new version for the so-called `twin
paradox', comparing the ages of two astronauts on a very long spaceship.
Although there is always an instantaneous inertial frame in which the whole
spaceship, being rigid, is simultaneously at rest, the twins' ages, measured as
the proper-times along their individual world lines, are different when they
are located at remote parts of the spaceship. The age, or proper-time,
difference depends on the distance at rest between the astronauts and the
rapidity difference between start to end. The relation of the age difference
with the relative Doppler shift of light signals transmitted between the
astronauts, and implications for the possibility to assign common age
(proper-time) to complex, spatially extended, relativistic systems, are also
discussed. The condition for simultaneous arrival of light signals emitted
simultaneously from the opposite ends of a rigidly accelerating spaceship is
resolved.
