Hyperbolic motion in special relativity is rectilinear motion with constant proper acceleration. A representative positive- world line is , , with proper acceleration . This describes a hyperbola in spacetime, not a hyperbolic Kepler orbit.
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Hyperbolic motion refers to a type of motion that can be described using hyperbolic functions, which are analogous to trigonometric functions but are based on hyperbolas instead of circles. In a physical context, hyperbolic motion is often related to scenarios in special relativity, especially when discussing the relationship between time and space for objects moving at speeds close to the speed of light.