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).