2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 67
Consequently, physical laws obeying Principle2.26of Einstein general relativity must be
in a form of covariant partial differential equations:
L(u,∇u,···,∇mu) = 0.Now, what remains is to establish the field equations governing the Riemann metric{gij}
(the gravitational potential), which will be introduced inthe next two subsections.
2.3.4 Einstein-Hilbert action
In view of Principle2.3(PLD), to derive the gravitational field equations it suffices to derive
the Lagrange action for the gravitational potential{gμ ν}.
To this end, we first introduce the Ricci curvature tensorRμ νand the scalar curvatureR
in a Riemannian manifold.
- Ricci tensor. It is natural to conjecture that the Lagrange densityLfor the field
equations depends on the terms∂gμ ν, i.e.
(2.3.27) L=L(gμ ν,···,∂mgμ ν).
However, it is known that all covariant derivatives of the Riemann metric are zero:
(2.3.28) ∇gμ ν= 0 , ∇gμ ν= 0.
Hence, we are not able to directly use the terms∇gμ ν,···,∇mgto construct the density
(2.3.27), and have to look for invariants depending on∂gμ ν,···,∂mgμ ν(m≥ 1 )in a different
fashion.
The Ricci curvature tensor provides a natural way for us to find the Lagrange action. For
a covector fieldA={Ak}we have
∇μ∇νAα=∂
∂xμ(∇νAα)−Γβμ ν(∇βAα)−Γβμ α(∇νAβ).Note that
∇νAα=∂Aα
∂xν−Γ
γ
ν αAγ.Then we can deduce that
(2.3.29) [∇μ∇ν−∇ν∇μ]Aα=Rβα μ νAβ,
where
(2.3.30) Rβα μ ν=
∂Γβα μ
∂xν−
∂Γβα ν
∂xμ+Γγα μΓβγ ν−Γγα γΓβγ μ.The tensor in (2.3.30) is called the curvature tensor, and its self-contraction given by(2.3.31) Rμ ν=Rαμ α ν
is the Ricci tensor.