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Mathematics for Physics

176 Lorentz Factor

A scalar factor that is used to convert classical expressions for time, length, and momentum into their relativistic forms at high relative speed.

The relativistic correction factor. The Lorentz factor is the dimensionless number that scales space and time measurements between inertial frames. A dimensionless number is a pure numerical coefficient with no units. This principle is used to convert a time interval or a length from one inertial frame to another.

The Lorentz factor is

\[ \gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}} = \dfrac{1}{\sqrt{1 - \beta^{2}}} \]

where

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

Frame speed versus particle speed. The Lorentz factor may be computed from frame speed or from particle speed. Frame speed is the constant speed of one inertial coordinate system relative to another. Particle speed is the speed of a body in a chosen frame. This principle is used to keep boosts distinct from the motion of the particle.

The cosmic speed limit. For a body with rest mass, \(\gamma\) grows without bound as the speed approaches the speed of light. Rest mass is the mass measured in the body’s rest frame. This principle is used to show that reaching the speed of light would require unbounded kinetic energy.

Newtonian correspondence. When the speed is much smaller than the speed of light, \(\gamma\) is nearly equal to \(1\). This principle is used to recover Newtonian time, length, and momentum at everyday speeds.

Dynamic scaling of energy and momentum. Relativistic energy and momentum are the rest energy and the Newtonian momentum scaled by \(\gamma\). Relativistic energy is the total energy \(E=\gamma mc^{2}\) of a free particle. Relativistic momentum is the conserved spatial momentum \(p=\gamma mv\). This principle is used to compute energy and momentum at speeds comparable to \(c\).

Note: Also called gamma. Also called the relativistic factor.

176.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — boost vector \(\boldsymbol{\beta}=\dfrac{\mathbf{v}}{c}\) and Lorentz factor \(\gamma=\dfrac{1}{\sqrt{1-\beta^{2}}}\); \(\gamma\to\infty\) as \(v\to c\); \(\gamma\approx 1\) at low speed.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(\gamma\equiv \dfrac{1}{\sqrt{1-\dfrac{v^{2}}{c^{2}}}}\); \(E=\gamma mc^{2}\); \(p=\gamma mv\).
  3. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(\gamma=\dfrac{1}{\sqrt{1-\dfrac{v^{2}}{c^{2}}}}=\dfrac{1}{\sqrt{1-\beta^{2}}}\); frame speed versus particle speed.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — relativistic energy and momentum scaled by \(\gamma\).