Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

66 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS


To solve this problem, as shown in (2.1.31), we need to add a termΓto the derivative
operator∂k, resulting a new derivative operator∇:


∇jAk=

∂Ak
∂xj

+ΓkijAi,

such that∇={∇j}is a tensor operator. Namely,{∇jAk}is a (1,1) type tensor, and transforms
as


(2.3.21) ∇ ̃jA ̃k=


∂A ̃k
∂ ̃xj

+ ̃ΓkijA ̃i=aklbij

(


∂Al
∂xi

+ΓlriAr

)


=aklbij∇iAl.

By (2.3.20), it follows from (2.3.21) that


(2.3.22) ̃ΓkijA ̃i=aklbijΓlriAr−bij


∂akl
∂xi

Al.

By (2.3.22) we deduce the transformation rule forΓas


(2.3.23) ̃Γkij=aklbribsjΓlrs−bribsj


∂akr
∂xs

.


Fortunately, for a Riemannian space{M,gij}, there exists a set of functions

(2.3.24) Γ={Γkij},


called the Levi-Civita connection, which satisfies the transformation given by (2.3.23), and
are given by


(2.3.25) Γkij=


1


2


gkl

(


∂gil
∂xj

+


∂gjl
∂xi


∂gij
∂xl

)


,


which we have seen in (2.3.7).
Based on the connection (2.3.24)-(2.3.25), we now define the covariant derivatives in the
Riemannian space{M,gij}as follows:


∇ku=

∂u
∂xk

for a scalar fieldu,

∇kuj=
∂uj
∂xk

+Γkljul for a vector field{uj},

∇kuj=

∂uj
∂xk

−Γlk jul for a covector field{uj},

and for a(r,s)type general tensor field{uij^11 ······irjs},


∇kuij^11 ······irjs=

∂uij^11 ······irjs
∂xk

(2.3.26) −Γlk j 1 uil j^12 ······irjs− ··· −Γlk jsuij^11 ······irjs− 1 l


+Γikl^1 ulij 12 ······jisr+···+Γiklruij^11 ······ijss−^1 l.
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