Mathematical Principles of Theoretical Physics

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.