Rapidity
In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light.
Velocities must be combined by Einstein's velocity-addition formula.
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.
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.
[1] This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames.
[2] The rapidity parameter replacing velocity was introduced in 1910 by Vladimir Varićak[3] and by E. T.
[4] The parameter was named rapidity by Alfred Robb (1911)[5] and this term was adopted by many subsequent authors, such as Ludwik Silberstein (1914), Frank Morley (1936) and Wolfgang Rindler (2001).
Rapidity is the parameter expressing variability of an event on the hyperbola which represents the future events one time unit away from the origin O.
Note that as speed and w increase, the axes tilt toward the diagonal.
In fact, they remain in a relation of hyperbolic orthogonality whatever the value of w. The appropriate x-axis is the hyperplane of simultaneity corresponding to rapidity w at the origin.
When the unit hyperbola is interpreted as a one-parameter group that acts on the future, and correspondingly on the past and elsewhere, then the Minkowski configuration expresses the relativity of simultaneity and other features of relativity.
The transformations relating reference frames are associated with Hendrik Lorentz.
To make a moving frame with rapidity w into the rest frame with perpendicular axes of time and space, one applies a hyperbolic rotation of parameter −w.
Since cosh (–w) = cosh w and sinh –w = – sinh w, the following matrix representation of the hyperbolic rotation will bring the moving frame into perpendicularity (though all frames keep hyperbolic orthogonality since that relation is invariant under hyperbolic rotation).
Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra.
In matrix exponential notation, Λ(w) can be expressed as
This establishes the useful additive property of rapidity: if A, B and C are frames of reference, then
where wPQ denotes the rapidity of a frame of reference Q relative to a frame of reference P. The simplicity of this formula contrasts with the complexity of the corresponding velocity-addition formula.
so the rapidity w is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using γ and β.
Therefore, the rapidity of an object in a given frame can be viewed simply as the velocity of that object as would be calculated non-relativistically by an inertial guidance system on board the object itself if it accelerated from rest in that frame to its given speed.
The product of β and γ appears frequently in the equations of special relativity.
As a result, some authors define an explicit parameter
α = β γ = tanh w cosh w = sinh w
The Doppler-shift factor, for the longitudinal case with source and receiver moving directly towards or away from each other, that is associated with rapidity w is
The energy E and scalar momentum |p| of a particle of non-zero (rest) mass m are given by:
So, rapidity can be calculated from measured energy and momentum by
However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis
where pz is the component of momentum along the beam axis.
[6] This is the rapidity of the boost along the beam axis which takes an observer from the lab frame to a frame in which the particle moves only perpendicular to the beam.
Rapidity relative to a beam axis can also be expressed as
