Mathematical Principles of Theoretical Physics

50 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

Lorentz Tensors

We now define Lorentz tensors, also called 4-dimensional (4-D) tensors, corresponding
to the Lorentz invariance.

Definition 2.19(Lorentz Tensors).The following quantities are called Lorentz tensors, or
4-dimensional tensors:

1) A function T with 4 kcomponents

T={Tμ 1 ···μk}, μ 1 ,···,μk= 0 , 1 , 2 , 3 ,

is called a k-th order covariant tensor, if under the Lorentztransformation (2.2.13), the

components of T transform as

T ̃μ 1 ···μk=lμν^11 ···lνμkkTν 1 ···νk,

where(lμν) = (L

μ

ν)

− (^1) is the inverse transformation of the Lorentz transformation(Lμ
ν)∈
LG.
2) A tensor
T={Tμ^1 ···μk}, μ 1 ,···,μk= 0 , 1 , 2 , 3 ,
is called a k-th order contra-variant tensor, if under the Lorentz transformation(Lμν)∈
LG, the components of T change as
T ̃μ^1 ···μk=Lμν^1
1 ···L
μk
νkT
ν 1 ···νk.
3) A function
T={Tνμ 11 ······νμsr}, r+s=k,
is called a k-th order(r,s)-type tensor, if under the Lorentz transformation(Lνμ)∈LG,
T ̃νμ 11 ······νμsr=lνα 11 ···lανs
sL
μ 1
β 1 ···L
μr
βrT
β 1 ···βr
α 1 ···αs.
In physics, the most important tensors are first- and second-order tensors. The following
is a list of commonly encountered 4-D tensors, and we always use(gμ ν) = (gμ ν)−^1 with
(gμ ν)being the Minkowski metric given by (2.2.5):

1. Position vectors:

(2.2.15)

xμ= (x^0 ,x^1 ,x^2 ,x^3 ), x^0 =ct,

xμ= (x 0 ,x 1 ,x 2 ,x 3 ) =gμ νxν= (−x^0 ,x^1 ,x^2 ,x^3 ).

2) The 4-D electromagnetic potential:

(2.2.16)

Aμ= (A 0 ,A 1 ,A 2 ,A 3 ),

Aμ= (A^0 ,A^1 ,A^2 ,A^3 ) =gμ νAν,

whereA 0 =−A^0 is the electric potential,(A 1 ,A 2 ,A 3 ) = (A^1 ,A^2 ,A^3 )is the magnetic

vector potential.