Mathematical Principles of Theoretical Physics

148 CHAPTER 3. MATHEMATICAL FOUNDATIONS

It follows that

(3.4.19) g ̃ij=−gikgjlg ̃kl.

Thus, (3.4.18) is rewritten as

(3.4.20)

d

dλ

∣

∣

∣

λ= 0

det(gij+λ ̃gij) =−ggijg ̃ij.

For the Einstein-Hilbert functional (3.4.12), we have

d

dλ

∣

∣

∣

λ= 0

F(gij+λg ̃ij)

=

∫

M

[

Rij ̃gij

√

−g−

1

2

√

−g

R

d

dλ

det(gij+λ ̃gij)+gij

d

dλ

Rij(gij+tg ̃ij)

√

−g

]

dx

∣

∣

∣

λ= 0

=(by( 3. 4. 20 ))

=

∫

M

(Rij−

1

2

gijR) ̃gij

√

−gdx+

∫

M

gij

d

dλ

∣

∣

∣

λ= 0

Rij(gij+λ ̃gij)

√

−gdx.

In view of (3.4.5) and

〈δF,g ̃ij〉=

∫

M

δF ̃gij

√

−gdx,

d

dλ

∣

∣

∣

λ= 0

Rij(gij+λ ̃gij) =

∂Rij

∂gkl

̃gkl,

we arrive at

∫

M

[

δF−(Rij−

1

2

gijR)

]

̃gij

√

−gdx=

∫

M

gij

∂Rij

∂gkl

̃gkl

√

−gdx.

To verify (3.4.15), it suffices to prove that

(3.4.21)

∫

M

gijδRij

√

−gdx= 0 ,

whereδRijis the variational element

δRij=Rij(gkl+δgkl)−Rij(gkl),

which are equivalent to the following directional derivative:

d

dλ

∣

∣

∣

λ= 0

Rij(gij+λ ̃gij).

To get (3.4.21), we takeRijin the form

(3.4.22) Rij=

∂Γkki

∂xj

−

∂Γkij

∂xk

+ΓlikΓkl j−ΓlijΓklk.