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Mathematics for Physics

110 Lorentz Transformations

The coordinate transformation that is used to convert spacetime coordinates between inertial frames moving at constant relative velocity.

Note: Also called a Lorentz boost when the frames differ by a constant velocity along one axis.

definition [d] (Lorentz Transformations) From Knight: the Lorentz transformations relating the coordinates \((x,y,z,t)\) of an event in one inertial frame to the coordinates \((x',y',z',t')\) in a second inertial frame moving at constant velocity \(v\) along the shared \(x\)-axis are

  • \(x' = \gamma(x - vt)\) ,
  • \(y' = y\) ,
  • \(z' = z\) ,
  • \(t' = \gamma\left(t - \dfrac{vx}{c^{2}}\right)\) ,

with inverse

  • \(x = \gamma(x' + vt')\) ,
  • \(y = y'\) ,
  • \(z = z'\) ,
  • \(t = \gamma\left(t' + \dfrac{vx'}{c^{2}}\right)\) ,

where

  • \((x,y,z,t)\) are spacetime coordinates in the first inertial frame.
  • \((x',y',z',t')\) are spacetime coordinates in the second inertial frame.
  • \(v\) is the constant relative velocity of the primed frame along \(x\).
  • \(c\) is the speed of light in vacuum.
  • \(\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}} = \dfrac{1}{\sqrt{1 - \beta^{2}}}\) is the Lorentz factor.
  • \(\beta = \dfrac{v}{c}\) is the relative speed in units of \(c\).

definition [d] (Lorentz Transformations) From Griffiths: with the Lorentz factor

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

the boost takes the form

  • \(\bar{x} = \gamma(x - vt)\) ,
  • \(\bar{y} = y\) ,
  • \(\bar{z} = z\) ,
  • \(\bar{t} = \gamma\left(t - \dfrac{v}{c^{2}}x\right)\) .

where

  • \((\bar{x},\bar{y},\bar{z},\bar{t})\) are the coordinates in the moving frame.
  • \(v\) is the relative velocity along \(x\).
  • \(c\) is the speed of light.

110.1 Elementary Example

110.1.1 Simple

For a relative speed \(v = 0.6c\), the Lorentz factor is

\[ \gamma = \dfrac{1}{\sqrt{1 - 0.36}} = 1.25 \]

\[ x' = 1.25(x - 0.6ct) \]

where

  • \(\gamma\) stretches both space and time mixing terms.

110.1.2 General

An event at the origin of the moving frame, \(x' = 0\), satisfies \(x = vt\) in the lab frame, and the lab time and moving time are related by

\[ t' = \gamma\left(t - \dfrac{vx}{c^{2}}\right) = \dfrac{t}{\gamma} \]

when \(x = vt\).

where

  • this is the time-dilation relation for a clock at rest in the primed frame.