Rapidity

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. It was used by Varićak 1910 by Whittaker 1910 It was named "rapidity" by Alfred Robb 1911 and this term was adopted by many subsequent authors, such as Varićak 1912, Silberstein (1914), Morley (1936) and Rindler (2001). The development of the theory of rapidity is mainly due to Varićak in writings from 1910 to 1924.

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 ( e w ,   e − w ) 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 x 2 − y 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 w arises in the linear representation of a Lorentz boost as a vector-matrix product

The matrix Λ(w) is of the type ( p q q p ) with p and q satisfying p² − q² = 1, so that (p, q) 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, Λ(w) can be expressed as Λ ( w ) = e Z w , where Z is the anti-diagonal unit matrix

It is not hard to prove that

This establishes the useful additive property of rapidity: if A, B and C are frames of reference, then

where w_PQ 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.

As we can see from the Lorentz transformation above, the Lorentz factor identifies with cosh w

so the rapidity w is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using γ 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 β and γ 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 w is k = e w .

In more than one spatial dimension

The relativistic velocity β is associated to the rapidity w of an object via

where the vector w is thought of as Cartesian coordinates on the 3-dimensional subspace of the Lie algebra o ( 3 , 1 ) ≈ s o ( 3 , 1 ) of the Lorentz group spanned by the boost generators K 1 , K 2 , K 3 – in complete analogy with the one-dimensional case o ( 1 , 1 ) discussed above – and velocity space is represented by the open ball B 3 with radius 1 since | β | < 1 . The latter follows from that c is a limiting velocity in relativity (with units in which c = 1 ).

The general formula for composition of rapidities is

where β 1 ⊕ β 2 refers to relativistic velocity addition and β ^ is a unit vector in the direction of β . This operation is not commutative nor associative. Rapidities w 1 , w 2 with directions inclined at an angle θ have a resultant norm w ≡ | w | (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 3-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 θ ,

where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule

where J k , k = 1 , 2 , 3 , are the generators of rotation. This is related to the phenomenon of Thomas precession. For the computation of the parameter θ , the linked article is referred to.

In experimental particle physics

The energy E and scalar momentum |p| of a particle of non-zero (rest) mass m are given by:

With the definition of w

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 p_z 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.