Mathematical Principles of Theoretical Physics

68 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

2.Scalar curvature. Again by contraction of the Ricci tensor with the metric tensor, we
derive an invariant, called scalar curvature:

(2.3.32) R=gμ νRμ ν.

3.Lagrangian action. In the Riemannian space{M,gμ ν}, the scalar curvature (2.3.32)
is a unique invariant which contains up to second-order derivatives ofgμ ν. Hence it is natural
to choose the scalar curvatureRas the main part of the Lagrange density. Namely,Lshould
be in the form

(2.3.33) L=R+S,

whereSis the energy-momentum term of baryonic matter in the Universe. Physically, the
energy-momentum density termSis taken as

(2.3.34) S=

8 πG

c^4

gα βSα β,

whereGis the gravitational constant, andSα βis the energy-momentum stress tensor. There-
fore we obtain the Lagrange action of gravitational fields:

(2.3.35) LEH=

∫

M

[

R+

8 πG

c^4

gα βSα β

]

√

−gdx,

whereg=det(gα β), and

√

−gdxis the volume element.
The functional (2.3.35) is called the Einstein-Hilbert action or Einstein-Hilbert functional.
Historically, the functional was first introduced by David Hilbert in 1915 after he listened to
the lecture on the general theory of relativity by Einstein.In fact, it is not easy to determine
the expression of the variational derivative operatorδLRfor the functional

(2.3.36) LR=

∫

M

R

√

−gdx.

2.3.5 Einstein gravitational field equations

The gravitational field equations based on PLD are the variational equations of the Einstein-
Hilbert actionLEHin (2.3.35):

(2.3.37) δLEH= 0.

By (2.3.35), the equation (2.3.37) can be explicitly expressed as

(2.3.38) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν,

whereRμ νis the Ricci tensor,Rthe scalar curvature, andTμ νis the energy-momentumtensor.
The equations (2.3.38) are the well-known Einstein gravitational field equations.