Relativistic minimization with constraints: A smooth trip to Alpha Centauri

Abstract

We consider the relativistic generalization of the problem of the "least uncomfortable" linear trajectory from point A to point B. The traditional problem minimizes the time-integrated squared acceleration (termed the "discomfort"), and there is a universal solution for all distances and durations. This universality fails when the maximum speed of the trajectory becomes relativistic, and we consider the more general case of minimizing the squared proper acceleration over a proper time. The least uncomfortable relativistic trajectory has a rapidity that evolves like the motion of a particle in a hyperbolic sine potential, agreeing with the classical solution at low velocities. We consider the special case of a hypothetical trip to Alpha Centauri, and compare the minimal-discomfort trajectory to one with uniform Earth-like acceleration.