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Mathematics for Physics

66 Lorentz Transformations

A change of inertial frame that is used to convert electromagnetic fields from one observer to another moving at constant relative velocity, where an inertial frame is a frame in which free particles move at constant velocity.

Invariance of the parallel fields. Under a boost, the components of \(\mathbf{E}\) and \(\mathbf{B}\) parallel to the relative velocity are unchanged. A boost is a Lorentz transformation between frames that differ by a constant velocity. This principle is used to keep the longitudinal fields the same for both observers.

The parallel field transformations are

\[ \bar{E}_{\parallel} = E_{\parallel},\qquad \bar{B}_{\parallel} = B_{\parallel} \]

where

  • \(E_{\parallel}\) and \(B_{\parallel}\) are the field components along the boost.
  • bars mark the fields in the moving frame.

Mixing of the transverse fields. The transverse fields mix. A field that is purely electric in one frame has a magnetic part in a frame in relative motion. This principle is used to transform the fields perpendicular to the boost.

The field transformation for a boost along \(x\) is

\[ \bar{E}_{y} = \gamma(E_{y} - v B_{z}),\qquad \bar{E}_{z} = \gamma(E_{z} + v B_{y}) \]

\[ \bar{B}_{y} = \gamma\Bigl(B_{y} + \dfrac{v}{c^{2}}E_{z}\Bigr),\qquad \bar{B}_{z} = \gamma\Bigl(B_{z} - \dfrac{v}{c^{2}}E_{y}\Bigr) \]

where

  • \(\mathbf{E}\) and \(\mathbf{B}\) are the fields in the original frame.
  • \(\bar{\mathbf{E}}\) and \(\bar{\mathbf{B}}\) are the fields in the boosted frame.
  • \(v\) is the relative speed along \(x\).
  • \(\gamma\) is the Lorentz factor.
  • \(c\) is the speed of light.

The tensor transformation of \(F^{\mu\nu}\). The same mixing is the tensor transformation of \(F^{\mu\nu}\). This principle is used to obtain the field transformations from the Lorentz transformation of a second-rank tensor.

The Lorentz transformation of the field tensor is

\[ \bar{F}^{\mu\nu} = \Lambda^{\mu}{}_{\alpha}\Lambda^{\nu}{}_{\beta}F^{\alpha\beta} \]

where

  • \(F^{\mu\nu}\) is the electromagnetic field tensor.
  • \(\Lambda^{\mu}{}_{\alpha}\) is the Lorentz transformation matrix.

Note: Also called a Lorentz boost of the electromagnetic field.