Pseudorapidity

, is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis.

is the component of the momentum along the beam axis (i.e. the longitudinal momentum – using the conventional system of coordinates for hadron collider physics, this is also commonly denoted

In the limit where the particle is travelling close to the speed of light, or equivalently in the approximation that the mass of the particle is negligible, one can make the substitution

(i.e. in this limit, the particle's only energy is its momentum-energy, similar to the case of the photon), and hence the pseudorapidity converges to the definition of rapidity used in experimental particle physics: This differs slightly from the definition of rapidity in special relativity, which uses

One speaks of the "forward" direction in a hadron collider experiment, which refers to regions of the detector that are close to the beam axis, at high

; in contexts where the distinction between "forward" and "backward" is relevant, the former refers to the positive z-direction and the latter to the negative z-direction.

In hadron collider physics, the rapidity (or pseudorapidity) is preferred over the polar angle

because, loosely speaking, particle production is constant as a function of rapidity, and because differences in rapidity are Lorentz invariant under boosts along the longitudinal axis: they transform additively, similar to velocities in Galilean relativity.

if the particles involved are massless) is hence not dependent on the longitudinal boost of the reference frame (such as the laboratory frame).

This is an important feature for hadron collider physics, where the colliding partons carry different longitudinal momentum fractions x, which means that the rest frames of the parton-parton collisions will have different longitudinal boosts.

is the transverse momentum (i.e. the component of the three-momentum perpendicular to the beam axis).

one can approximate rapidity by which makes it easy to see that for relativistic particles with

, pseudorapidity becomes equal to (true) rapidity.

, which is Lorentz invariant under a boost along the longitudinal (beam) direction.

Often, the rapidity term in this expression is replaced by pseudorapidity, yielding a definition with purely angular quantities:

, which is Lorentz invariant if the involved particles are massless.

, is invariant under Lorentz boosts along the beam line (z-axis) because it is measured in a plane (i.e. the "transverse" x-y plane) orthogonal to the beam line.

Here are some representative values: Pseudorapidity is odd about

Hadron colliders measure physical momenta in terms of transverse momentum

, polar angle in the transverse plane

is the longitudinal momentum component, which is denoted

is the standard notation at hadron colliders).

The equivalent relations to get the full four-momentum (in natural units) using "true" rapidity

along the beam-axis of velocity corresponds to an additive change in rapidity of

Under such a Lorentz transformation, the rapidity of a particle will become

and the four-momentum becomes This sort of transformation is common in hadron colliders.

For example, if two hadrons of identical type undergo an inelastic collision along the beam axis with the same speed, then the corresponding rapidity will be where

are the momentum fraction of the colliding partons.

When several particles are produced in the same collision, the difference in rapidity

will be invariant under any such boost along the beam axis, and if both particles are massless (