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Mathematics for Physics

194 Rapidity

A hyperbolic parameter for a boost that is used to rewrite Lorentz transformations so successive boosts add by summing parameters.

Additivity of collinear rapidities. Collinear boosts add by summing rapidities, while the corresponding velocities combine non-linearly. Rapidity is the hyperbolic angle of a boost. A boost is a Lorentz transformation for a constant relative velocity. This principle is used to replace velocity addition with ordinary addition of parameters.

The rapidity \(\psi\) satisfies

\[ \tanh\psi = \beta \]

\[ \cosh\psi = \gamma \]

\[ \sinh\psi = \beta\gamma \]

where

  • \(\psi\) is the rapidity.
  • \(\gamma\) is the Lorentz factor.
  • \(\beta = \dfrac{v}{c}\) is the relative speed in units of \(c\).
  • \(v\) is the boost speed.
  • \(c\) is the speed of light.

Boost as hyperbolic rotation. A Lorentz boost is a hyperbolic rotation in spacetime. A hyperbolic rotation is a coordinate map that slides points along hyperbolas of constant interval. This principle is used to write the boost as a rotation through the rapidity angle.

The boost in hyperbolic form is

\[ x^{\prime 0} = x^{0}\cosh\psi - x^{1}\sinh\psi \]

\[ x^{\prime 1} = -x^{0}\sinh\psi + x^{1}\cosh\psi \]

where

  • \(x^{0}\) and \(x^{1}\) are the time and longitudinal space coordinates in the first frame.
  • \(x^{\prime 0}\) and \(x^{\prime 1}\) are the corresponding coordinates in the boosted frame.

The speed-of-light bound on \(\tanh\psi\). Rapidity may be arbitrarily large, while \(\tanh\psi\) stays strictly below \(1\). This principle is used to keep every massive particle slower than the speed of light.

Hyperbolic parameterization of energy and momentum. Energy and momentum of a massive particle are hyperbolic functions of its rapidity. This principle is used to write four-momentum conservation in additive hyperbolic variables.

The hyperbolic energy and momentum are

\[ E = mc^{2}\cosh\psi \]

\[ p = mc\sinh\psi \]

where

  • \(E\) is the total energy.
  • \(p\) is the longitudinal momentum.
  • \(m\) is the rest mass.

Note: Also denoted \(\psi\). Also denoted \(\theta\). Also denoted \(\phi\).

194.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(\cosh\psi=\gamma\), \(\sinh\psi=\beta\gamma\), \(\tanh\psi=\beta\); boost as hyperbolic rotation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(\theta\equiv\tanh^{-1}(v/c)\); collinear additivity.
  3. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — boost written with \(\cosh\phi\), \(\sinh\phi\), and \(v=\tanh\phi\).
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — collinear rapidity addition; \(E=mc^{2}\cosh\psi\); \(p=mc\sinh\psi\).