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 ; that is, the interval −c < v < c maps onto −∞ < w < ∞. In 1908 Hermann Minkowski explained how the Lorentz transformation could be seen as simply a hyperbolic rotation of the spacetime coordinates, i.e., a rotation through an imaginary angle. This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames. The rapidity parameter replacing velocity was introduced in 1910 by Vladimir Varićak and by Edmund Taylor Whittaker. The parameter was named rapidity by Alfred Robb (1911) and this term was adopted by many subsequent authors, such as Ludwik Silberstein (1914), Frank Morley (1936) and Wolfgang Rindler (2001). The quadrature of the hyperbola xy = 1 by Gregoire de Saint-Vincent established the natural logarithm as the area of a hyperbolic sector or an equivalent area against an asymptote. In spacetime theory, the connection of events by light divides the universe into Past, Future, or Elsewhere based on a Here and Now . On any line in space, a light beam may be directed left or right.