General Relativity: An Introduction for Physicists · Problem 6.5

Electromagnetism — Hobson, Efstathiou & Lasenby

Physics · lorentz-transformation-fields

Problem

Two Cartesian inertial frames

S

and

S'

are in standard configuration. Show that the components of electric and magnetic fields in the two frames are related as follows:

E'^1 = E^1, \qquad B'^1 = B^1,

E'^2 = \gamma(E^2 - vB^3), \qquad B'^2 = \gamma\left(B^2 + \frac{v}{c^2}E^3\right),

E'^3 = \gamma(E^3 + vB^2), \qquad B'^3 = \gamma\left(B^3 - \frac{v}{c^2}E^2\right).

Show further that

c^2\vec{B}^2 - \vec{E}^2

is Lorentz invariant.