Classical Electrodynamics · Problem 11.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · relativistic-acceleration-transformation

Problem

A coordinate system

K'

moves with a velocity

\mathbf{v}

relative to another system

K

. In

K'

a particle has a velocity

\mathbf{u}'

and an acceleration

\mathbf{a}'

. Find the Lorentz transformation law for accelerations, and show that in the system

K

the components of acceleration parallel and perpendicular to

\mathbf{v}

are

\mathbf{a}_{\|} = \frac{\left(1 - \dfrac{v^2}{c^2}\right)^{3/2}}{\left(1 + \dfrac{\mathbf{v} \cdot \mathbf{u}'}{c^2}\right)^3} \mathbf{a}'_{\|}

\mathbf{a}_{\perp} = \frac{\left(1 - \dfrac{v^2}{c^2}\right)}{\left(1 + \dfrac{\mathbf{v} \cdot \mathbf{u}'}{c^2}\right)^3} \left(\mathbf{a}'_{\perp} + \frac{\mathbf{v}}{c^2} \times (\mathbf{a}' \times \mathbf{u}')\right)