Hyperbolic motion (relativity)

Hyperbolic motion can be visualized on a Minkowski diagram, where the motion of the accelerating particle is along the $X$ -axis. Each hyperbola is defined by $x=\pm c^{2}/\alpha$ and $\eta =\alpha \tau /c$ (with $c=1,\alpha =1$ ) in equation (2).

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 whose coordinates represent a suitable inertial (non-accelerated) frame. This motion has several interesting features, among them that it is possible to outrun a photon if given a sufficient head start, as may be concluded from the diagram.^[1]

^ Misner, Thorne & Wheeler 1973, Chapter 6.

[1] Misner, Thorne & Wheeler 1973, Chapter 6.

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