rapidity

Rapidity is a kinematic variable used in special relativity and high-energy physics to describe the motion of particles along a chosen axis, typically the beam (z) axis in collider experiments. It is defined so that the Lorentz boost along that axis corresponds to a simple addition. For a particle with total energy E and momentum component p_z along the boost axis, the rapidity y is given by y = 1/2 ln[(E + p_z c)/(E − p_z c)]. In units where the speed of light c = 1, this becomes y = artanh(p_z/E). The rapidity is related to the Lorentz boost parameter by cosh y = γ and sinh y = γβ, where β = v/c and γ is the Lorentz factor. Consequently, E = m cosh y and p_z = m sinh y for a particle of mass m.

A key property of rapidity is its behavior under boosts along the same axis: the rapidities of successive boosts add. If a frame is boosted by y_boost, the particle’s rapidity becomes y′ = y − y_boost. This makes differences in rapidity particularly useful, as they are invariant under such boosts, and kinematic separations along the beam direction are often clearer in rapidity space than in velocity space.

Pseudorapidity, η, is a related concept defined from the polar angle θ relative to the beam axis by η = −ln tan(θ/2). For massless particles or highly relativistic ones, rapidity and pseudorapidity coincide (y ≈ η). Pseudorapidity is commonly used in detectors because η depends only on the geometry of the particle’s trajectory, not its energy.

Applications of rapidity and pseudorapidity span event selection, cross-section measurements, and jet and particle correlations in collider experiments.