Mathematical Principles of Theoretical Physics

52 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

Theorem 2.20(Lorentz Invariants).

1) The contractions

(2.2.22) Aμ 1 ···μkBμ^1 ···μk,

is a Lorentz invariant, and so are the following contractions:

(2.2.23) gα βAαBβ, gα βAαBβ.

2) Let Aμνbe a (1,1) Lorentz tensor, then the contraction

(2.2.24) Aμμ=A^00 +A^11 +A^22 +A^33

is a Lorentz invariant.

3) The following differential operators

(2.2.25)

Aμ∂μ=

(

A 0

∂

∂x 0

,A 1

∂

∂x 1

,A 2

∂

∂x 2

,A 3

∂

∂x 3

)

,

Aμ∂μ=

(

A^0

∂

∂x^0

,A^1

∂

∂x^1

,A^2

∂

∂x^2

,A^3

∂

∂x^3

)

,

∂μ∂μ=−

1

c^2

∂^2

∂t^2

+

∂^2

∂x^21

+

∂^2

∂x^22

+

∂^2

∂x^23

are Lorentz invariant operators.

Theorem2.20provides a number of typical Lorentz invariants through contractions, and
in fact, all Lagrange actions in relativistic physics are combinations of the Lorentz invariants
in (2.2.22)-(2.2.25).

2.2.4 Relativistic mechanics

First, the 4-D velocity is defined by

uμ= (u^0 ,u^1 ,u^2 ,u^3 ) =

(

dx^0

ds

,

dx^1

ds

,

dx^2

ds

,

dx^3

ds

)

,

wheredsis the arc-length element in (2.2.6), and is given by

ds=c

√

1 −v^2 /c^2 dt, v= (v^1 ,v^2 ,v^3 ), vk=

dxk

dt

.

It is clear that the 4-D velocityuμcan be expressed as

(2.2.26)

uμ= (u^0 ,u^1 ,u^2 ,u^3 ),

u^0 =

1

√

1 −v^2 /c^2

, uk=

vk/c

√

1 −v^2 /c^2

for 1≤k≤ 3.