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 a common age (proper-time) to complex, spatially
extended, relativistic systems are also discussed.