Lorentz Transformation of Fields

a) Write down the Lorentz transformation equations relating the space time coordinates of frames K and K', where K' moves with velocity v relative to K. (Take v to point along a coordinate axis for simplicity.) Explicitly define your 4-vector conventions.

b) Use the fact that the electromagnetic field components E and B form an antisymmetric tensor to show that

where  and the subscripts label directions parallel and perpendicular to v.

c) Consider the particular case of a point charge q and recover an appropriate form of the law of Biot and Savart for small v.

SOLUTION

a) Using a 4-vector of the form

    (contravariant form)

and recalling that for v pointing along the x-axis, we have for the Lorentz transformation of a 4-vector

(1)

(2)

(3)

For space–time coordinates,

and we have from (1)–(4)

(5)

(6)

(7)

(8)

b) Writing the explicitly antisymmetric field-strength tensor,

(9)

where F⁰¹ and F²³ do not change under the Lorentz transformation and F⁰², F⁰² and F¹², F¹³ transform as x⁰ and x¹ respectively.

(10)

Substituting (9) into (10), we obtain

(11)

We can rewrite (11) is terms of the parallel and perpendicular components of the fields:

(12)

c) In the case of a point charge q, we have to transform from K' to K which is equivalent to changing the sign of the velocity in (12). For a small velocity v, we may write

where we have changed the signs in (12) and taken γ ≈ 1. For a point charge in K', B' = 0 and

which is the magnetic field for a charge moving with velocity v.