Notes for General Relativity

1 Convention

natural unit system, Einstein's summation convention
Minkowski metric $(- + + +)$

η_{μ ν} = [\begin{matrix} - 1 \\ 1 \\ 1 \\ 1 \end{matrix}]

Curvature

R_{μ ν λ}^{ρ} (t h i s d o c u m e n t) = - Γ_{μ ν, λ}^{ρ} + Γ_{μ λ, ν}^{ρ} + Γ_{μ λ}^{σ} Γ_{σ ν}^{ρ} - Γ_{μ ν}^{σ} Γ_{σ λ}^{ρ} = - R_{μ ν λ}^{ρ} (W e i n b e r g)

R_{μ ν} = R_{μ ν λ}^{λ}

Christoffel symbols

Γ_{μ ν}^{λ} = \frac{1}{2} g^{λ ρ} (g_{μ ρ, ν} + g_{ν ρ, μ} - g_{μ ν, ρ})

g_{α β; μ} = 0

2 $f (R)$ Gravity & Field Eqn.

refer to F(R) theories of gravitation - Scholarpedia

The total action of the known gravitational system is:

S = \int [f (R) + 16 π G \cdot L_{m}] \sqrt{- g} d^{4} x

In $f (R)$ gravity, the field equations are obtained by varying with respect to the metric and not treating the connection independently. The main steps are the same as in the case of the variation of the Hilbert-Einstein action but there are also some important differences.

The variation of the determinant is

δ \sqrt{- g} = - \frac{1}{2} \sqrt{- g} g_{μ ν} δ g^{μ ν}, h e r e g = det (g_{μ ν})

The Ricci scalar is defined as

R = g^{μ ν} R_{μ ν} .

Therefore the variation with respect to the inverse metric $g^{μ ν}$ is given by

\begin{matrix} (1) & δ R = R_{μ ν} δ g^{μ ν} + g^{μ ν} δ R_{μ ν} \overset{P a l a t i n i E q n .}{=} R_{μ ν} δ g^{μ ν} + g^{μ ν} ((δ Γ_{ρ μ}^{ρ})_{; ν} - δ (Γ_{ν μ}^{ρ})_{; ρ}) \end{matrix}

Since $δ Γ_{μ ν}^{λ}$ is the difference of two connections, it should transform as a tensor. Therefore, using $Γ_{μ ν}^{λ} = \frac{1}{2} g^{λ ρ} (g_{μ ρ, ν} + g_{ν ρ, μ} - g_{μ ν, ρ})$ and $δ g^{α β} = - g^{μ α} g^{ν β} δ g_{μ ν}$ , it can be written as

δ Γ_{μ ν}^{λ} = \frac{1}{2} g^{λ α} ((δ g_{α ν})_{; μ} + (δ g_{α μ})_{; ν} - (δ g_{μ ν})_{; α}) .

Substituting into the $E q n . (1)$ yeilds:

δ R = R_{μ ν} δ g^{μ ν} + \underset{g^{μ ν} δ R_{μ ν}}{\underset{⏟}{(δ g^{μ ν})_{; μ; ν} - ◻ (g_{μ ν} δ g^{μ ν})}}

where " $; μ$ " is the covariant derivative and $◻ f = g^{α β} f_{; α; β}$ is the D'Alembert operator acting on the function $f$ . Let us consider now a generic analytic function $f (R)$ obeying the variational principle $δ \int d^{4} x \sqrt{- g} f (R) = 0$ . We have

\begin{aligned} δ \int d^{4} x \sqrt{- g} f (R) & = \int d^{4} x [δ (\sqrt{- g}) f (R) + \sqrt{- g} δ (f (R))] \\ = \int d^{4} x \sqrt{- g} [f^{'} (R) R_{μ ν} - \frac{1}{2} g_{μ ν} f (R)] δ g^{μ ν} + \int d^{4} x \sqrt{- g} f^{'} (R) g^{μ ν} δ R_{μ ν} \end{aligned}

For the second item, performing integration by parts ( $\int d^{4} x \sqrt{- g} A_{; μ}^{μ} B = \underset{d i v e r g e n c e t e r m}{\underset{⏟}{\int d^{4} x \sqrt{- g} (A^{μ} B)_{; μ}}} - \int d^{4} x \sqrt{- g} A^{μ} B_{; μ}$ , $∵ A_{; μ}^{μ} = \frac{1}{\sqrt{- g}} (\sqrt{- g} A^{μ})_{, μ}$ ) and ignoring the divergence term will lead to

\begin{aligned} \int d^{4} x \sqrt{- g} f^{'} (R) g^{μ ν} δ R_{μ ν} & = \int d^{4} x \sqrt{- g} f^{'} (R) ((δ g^{μ ν})_{; μ; ν} - ◻ (g_{μ ν} δ g^{μ ν})) \\ = + \int d^{4} x \sqrt{- g} (f^{'} (R)_{; μ; ν} - g_{μ ν} ◻ f^{'} (R)) δ g^{μ ν} \end{aligned}

Thus

\begin{array}{r} δ \int d^{4} x \sqrt{- g} f (R) = \int d^{4} x \sqrt{- g} [f^{'} (R) R_{μ ν} - \frac{1}{2} g_{μ ν} f (R) + f^{'} (R)_{; μ; ν} - g_{μ ν} ◻ f^{'} (R)] δ g^{μ ν} \end{array}

On the other hand, the variation of the action of the material part is

δ S_{m} = δ \int 16 π G \cdot L_{m} \sqrt{- g} d^{4} x = - 8 π G \int d^{4} x \sqrt{- g} T_{μ ν} δ g^{μ ν}

Yield the $f (R) F i e l d E q n .$ :

- f^{'} R_{μ ν} + \frac{1}{2} g_{μ ν} f - f_{; μ; ν}^{'} + g_{μ ν} ◻ f^{'} = - 8 π G T_{μ ν}

Substituting $f (R) = - R$ into the above eqn., one find

R_{μ ν} - \frac{1}{2} g_{μ ν} R = - 8 π G T_{μ ν}

That's So-called Hilbert-Einstein Field Equation.