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 functionartanh, 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 range, and so the interval −c < v < c maps onto −∞ < w < ∞.
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[clarification needed]. 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 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 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[clarification needed] 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 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 , where Z 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 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.
As we can see from the Lorentz transformation above, the Lorentz factor identifies with cosh w
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
The relativistic velocity is associated to the rapidity of an object via
where the vector is thought of as Cartesian coordinates on the 3-dimensional subspace of the Lie algebra of the Lorentz group spanned by the boost generators – in complete analogy with the one-dimensional case discussed above – and velocity space is represented by the open ball with radius since . The latter follows from that is a limiting velocity in relativity (with units in which ).
The general formula for composition of rapidities is[nb 1]
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 ,
where the physicist convention for the exponential mapping is employed. This is a consequence of the commutation rule
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 pz 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
^This is to be understood in the sense that given two velocities, the resulting rapidity is the rapidity corresponding to the two velocities relativistically added. Rapidities also have the ordinary addition inherited from , and context decides which operation to use.
Notes and references
^Hermann Minkowski (1908) Fundamental Equations for Electromagnetic Processes in Moving Bodies via Wikisource
Shaw, Ronald (1982) Linear Algebra and Group Representations, v. 1, page 229, Academic PressISBN 0-12-639201-3.
Walter, Scott (1999). "The non-Euclidean style of Minkowskian relativity" (PDF). In J. Gray (ed.). The Symbolic Universe: Geometry and Physics. Oxford University Press. pp. 91–127.(see page 17 of e-link)
Rhodes, J. A.; Semon, M. D. (2004). "Relativistic velocity space, Wigner rotation, and Thomas precession". Am. J. Phys. 72 (7): 93–90. arXiv:gr-qc/0501070. Bibcode:2004AmJPh..72..943R. doi:10.1119/1.1652040. S2CID 14764378.