Mathematical Principles of Theoretical Physics

3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 149

By the Riemannian Geometry, for each pointx 0 ∈Mthere exists a coordinate system
under which

(3.4.23) Γkij(x 0 ) = 0 ∀ 1 ≤i,j,k≤n.

It is known that the covariant derivatives ofgijandgijare zero, i.e.Dgij=0 andDgij=0.
Hence we infer from (3.4.33) that

∂gij(x 0 )

∂xk

= 0 ,

∂gij(x 0 )

∂xk

= 0 ∀ 1 ≤i,j,k≤n.

By (3.4.22), atx 0 , we have

gijδRij=gij

(

∂

∂xj

δΓkik−

∂

∂xk

δΓkij

)

=

∂

∂xk

(

gikδΓlil−gijδΓkij

)

(3.4.24).

AlthoughΓkijare not tensor fields, the variations

δΓkij(x) =Γkij(x+δx)−Γkij(x)

are (1,2)-tensor fields. Therefore atx 0 , (3.4.24) can be rewritten as

(3.4.25) gijδRij=
∂uk
∂xk

=divu atx 0 ∈M.

where

uk=gikδΓlil−gijδΓkij

is a vector field. Since (3.4.25) is independent of the coordinate systems, in a general coordi-
nate system the relation (3.4.25) becomes

(3.4.26) gijδRij=divu=

1

√

−g

∂

∂xk

(

√

−guk) atx 0 ∈M.

Asx 0 ∈Mis arbitrary, the formula (3.4.26) holds true onM. Hence we have

∫

M

gijδRij

√

−gdx=

∫

M

divu

√

−gdx.

SinceMis closed, i.e.∂M=/0, it follows from (3.2.37) that

∫

M

gijδRij

√

−gdx= 0.

Thus we derive (3.4.21), and the derivative operator of the Einstein-Hilbert functional (3.4.12)
is as given by (3.4.15).