Mathematical Principles of Theoretical Physics

54 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

whereP= (P^1 ,P^2 ,P^3 )is as in (2.2.28). Thus, it follows that the relativistic motion law is
given by

(2.2.32)

d

dt

Pk=

√

1 −v^2 /c^2 Fk, Pk=

mvk

√

1 −v^2 /c^2

fork= 1 , 2 , 3.

2.2.5 Lorentz invariance of electromagnetism

2.6.3 PHD for Maxwell electromagnetic fields

curlE=−

1

c

∂H

∂t

(2.2.33) ,

(2.2.34) divH= 0 ,

curlH=

1

c

∂E

∂t

+

4 π

c

(2.2.35) J,

(2.2.36) divE= 4 π ρ,

whereE,Hare the electric and magnetic fields, andJis the current density, andρthe charge
density.
To show the Lorentz invariance of the Maxwell equations (2.2.33)-(2.2.36), we need to
express them in the form of the 4-D electromagnetic potential and current density.
The electromagnetic potential and current density are briefly introduced in (2.2.16) and
(2.2.17):

(2.2.37)

Aμ= (A 0 ,A 1 ,A 2 ,A 3 ),

Jμ= (J 0 ,J 1 ,J 2 ,J 3 ), J 0 =−cρ.

Using the Lorentz tensor operator

∂μ=

∂

∂xμ

=

(

∂

∂x^0

,

∂

∂x^1

,

∂

∂x^2

,

∂

∂x^3

)

,

we can construct two second-order Lorentz tensors:

(2.2.38)

Fμ ν=

∂Aν

∂xμ

−

∂Aμ

∂xν

,

Gμ ν=

1

2

εμ ν α βgα κgβ λFκ λ,

wheregα βis the Minkowski metric, and

εμ ν α β=







1 (μ,ν,α,β)is an even permutation of( 0123 ),

− 1 (μ,ν,α,β)is an odd permutation of( 0123 ),

0 otherwise,

is a 4-th order Lorentz tensor. Thanks to the relations:

H=curl~A, E=∇A 0 −

1

c

∂~A

∂t

, ~A= (A 1 ,A 2 ,A 3 ),