# Relativistic minimization with constraints: A smooth trip to Alpha Centauri

## TL;DR

This paper extends the classical least uncomfortable trajectory problem into the relativistic regime, deriving solutions for minimizing proper acceleration during interstellar travel, exemplified by a trip to Alpha Centauri.

## Contribution

It introduces a relativistic formulation of the least uncomfortable trajectory problem, providing analytical solutions and comparing them to uniform acceleration paths.

## Key findings

- Relativistic least uncomfortable trajectory follows a hyperbolic sine potential.
- Classical solutions are recovered at low velocities.
- Relativistic solutions differ significantly from uniform acceleration paths.

## 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.