# Hyperbolic motion (relativity)

Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram.

The proper acceleration of a particle is defined as the acceleration that a particle "feels" as it accelerates from one inertial reference frame to another. This can be derived mathematically as

$\alpha=\frac{1}{\left(1-u^2/c^2\right)^{3/2}}\frac{du}{dt}$,

where $u$ is the instantaneous speed of the particle. Solving for the equation of motion results in

$x^2-c^2t^2=c^4/\alpha^2$,

which is a hyperbola.

Hyperbolic motion is easily visualized on a Minkowski diagram, where the motion of the accelerating particle is along the $x$-axis. Each hyperbola is defined by

$X=c^2/\alpha$.