Rapidity

In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.

For one-dimensional motion, rapidities are additive whereas velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.

Using the inverse hyperbolic function $artanh$ , the rapidity $w$ corresponding to velocity $v$ is $w = artanh(v / c)$ where c is the velocity of light. For low speeds, $w$ is approximately $v / c$ . Since in relativity any velocity $v$ is constrained to the interval $- c < v < c$ the ratio $v / c$ satisfies $-1 < v / c < 1$ . The inverse hyperbolic tangent has the unit interval $(-1, 1)$ for its domain and the whole real line for its image; that is, the interval $- c < v < c$ maps onto $-\infty < w < \infty$ .

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