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, the rapidity of light being infinite.
Using the inverse hyperbolic function , the rapidity corresponding to velocity is where c is the velocity of light. For low speeds, is approximately . Since in relativity any velocity is constrained to the interval the ratio satisfies . The inverse hyperbolic tangent has the unit interval for its domain and the whole real line for its range, and so the interval maps onto .
History.
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 E. T. Whittaker. The parameter was named "rapidity" by Alfred Robb (1911) and this term was adopted by many subsequent authors, such as Silberstein (1914), Morley (1936) and Rindler (2001).
Area of a hyperbolic sector.
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. Take the x-axis as the events passed by the right beam and the y-axis as the events of the left beam. Then a resting frame has time along the diagonal "x" = "y". The rectangular hyperbola "xy" = 1 can be used to gauge velocities (in the first quadrant). Zero velocity corresponds to (1,1). Any point on the hyperbola has coordinates formula_1 where w is the rapidity, and is equal to the area of the hyperbolic sector from (1,1) to these coordinates. Many authors refer instead to the unit hyperbola formula_2 using rapidity for parameter, as in the standard spacetime diagram. There the axes are measured by clock and meter-stick, more familiar benchmarks, and the basis of spacetime theory. So the delineation of rapidity as hyperbolic parameter of beam-space is a reference to the seventeenth century origin of our precious transcendental functions, and a supplement to spacetime diagramming.
In one spatial dimension.
The rapidity arises in the linear representation of a Lorentz boost as a vector-matrix product
The matrix is of the type formula_4 with and satisfying , so that lies on the unit hyperbola. 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. This action may be depicted in a spacetime diagram. In matrix exponential notation, can be expressed as formula_5, where is the negative of the anti-diagonal unit matrix
It is not hard to prove that
This establishes the useful additive property of rapidity: if , and are frames of reference, then
where denotes the rapidity of a frame of reference relative to a frame of reference . The simplicity of this formula contrasts with the complexity of the corresponding velocity-addition formula.
As we can see from the Lorentz transformation above, the Lorentz factor identifies with 
so the rapidity is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using and β. We relate rapidities to the velocity-addition formula
by recognizing
and so
Proper acceleration (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to proper time (time as measured by the object undergoing acceleration itself). 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 and appears frequently, and is from the above arguments
Exponential and logarithmic relations.
From the above expressions we have
and thus
or explicitly
The Doppler-shift factor associated with rapidity is formula_17.
In more than one spatial dimension.
The relativistic velocity formula_18 is associated to the rapidity formula_19 of an object via
where the vector formula_21 is thought of as Cartesian coordinates on the 3-dimensional subspace of the Lie algebra formula_22 of the Lorentz group spanned by the boost generators formula_23 – in complete analogy with the one-dimensional case formula_24 discussed above – and "velocity space" is represented by the open ball formula_25 with radius formula_26 since formula_27. The latter follows from that formula_28 is a limiting velocity in relativity (with units in which formula_29).
The general formula for composition of rapidities is
where formula_31 refers to relativistic velocity addition and formula_32 is a unit vector in the direction of formula_18. This operation is not commutative nor associative. Rapidities formula_34 with directions inclined at an angle formula_35 have a resultant norm formula_36 (ordinary Euclidean length) given by the hyperbolic law of cosines,
The geometry on rapidity space is inherited from the hyperbolic geometry on velocity space via the map stated. This geometry, in turn, can be inferred from the addition law of relativistic velocities. Rapidity in two dimensions can thus be usefully visualized using the Poincaré disk. Geodesics correspond to steady accelerations. Rapidity space in three dimensions can in the same way be put in isometry with the hyperboloid model (isometric to the -dimensional Poincaré disk (or "ball")). This is detailed in geometry of Minkowski space.
The addition of two rapidities results not "only" in a new rapidity; the resultant total transformation is the composition of the transformation corresponding to the rapidity given above and a "rotation" parametrized by the vector formula_38,
where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule
where formula_41 are the generators of rotation. This is related to the phenomenon of Thomas precession. For the computation of the parameter formula_38, the linked article is referred to.
In experimental particle physics.
The energy and scalar momentum of a particle of non-zero (rest) mass are given by:
With the definition of 
and thus with
the energy and scalar momentum can be written as:
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 is the component of momentum along the beam axis. 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. Related to this is the concept of pseudorapidity.
Rapidity relative to a beam axis can also be expressed as