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Mathematics for Physics

109 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.

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

definition [d] (Lorentz Factor) From Emam: the ratio \(\dfrac{v}{c}\) as well as the factor with the square root appears frequently in many of the subsequent equations, so for brevity’s sake it is conventional to define the so-called boost vector (a 3-vector)

  • \(\boldsymbol{\beta} = \dfrac{\mathbf{v}}{c}\)

and the Lorentz factor

  • \(\gamma = \dfrac{1}{\sqrt{1 - \beta^{2}}}\) ,

where

  • \(\beta^{2} = \boldsymbol{\beta}\cdot\boldsymbol{\beta} = \dfrac{v^{2}}{c^{2}}\) .
  • \(\mathbf{v}\) is the relative 3-velocity.
  • \(c\) is the speed of light in vacuum.
  • \(\gamma\) is the Lorentz factor.

definition [d] (Lorentz Factor) From Griffiths: the Lorentz factor is defined by

  • \(\gamma \equiv \dfrac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}\) .

where

  • \(v\) is the relative speed of the frames.
  • \(c\) is the speed of light in vacuum.

definition [d] (Lorentz Factor) From Knight: the Lorentz factor is

  • \(\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}} = \dfrac{1}{\sqrt{1 - \beta^{2}}}\) ,

where

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

109.1 Elementary Example

109.1.1 Simple

For \(v = 0.6c\),

\[ \beta = 0.6,\qquad \gamma = \dfrac{1}{\sqrt{1 - 0.36}} = 1.25 \]

where

  • \(\gamma > 1\) whenever \(v > 0\).

109.1.2 General

As \(v\) approaches \(c\), \(\gamma\) grows without bound.

\[ \lim_{v\to c^{-}}\gamma = +\infty \]

where

  • at \(v = 0\), \(\gamma = 1\) and relativistic corrections vanish.