Lecture Eleven: Theory of Relativity

11.1 The Gravitational Field Equations

We are now in a position to tackle the problem of deriving a set of field equations for gravity. Let’s recall a few results from previous lectures:

• We showed that in a weak gravitational field

g_{00} = (1 + \frac{2ϕ}{c^{2}}) ,

(Henceforth in these lecture notes I will (usually) set c = 1.)

• We have constructed an energy-momentum tensor

T^{μν}, T_{μν},

with a T_{00} component which, for ‘dust’, measures the mass density

T_{00} = ρ.

• The gravitational field equation of Newtonian gravity is

∇^{2}ϕ = 4πGρ,

i.e., for a weak gravitational field

∇^{2}g_{00} = 8πGT_{00}.  (1)

• The last equation suggests that the field equations are tensorial equations of rank 2,

of the form

G_{μν} = 8πGT_{μν},  (2)

where the tensor G_{μν} must be consistent with the Newtonian limit (1).

But what is the form of G_{μν}? We have a few clues:

• G_{μν} must involve second derivatives of the metric tensor so that in the weak field limit G_{00} → ∇^{2}g_{00}.

• The vanishing of a gravitational field is equivalent to the vanishing of the curvature tensor, which as we know involves the second derivatives of g_{μν}. This suggests that G_{μν} is related to rank two tensors constructed from the curvature tensor.

• T_{μν} is symmetric and so G_{μν} must be symmetric.

• T_{μν} is conserved, T_{μν;ν} = 0, hence G_{μν} must also be conserved G_{μν;ν} = 0.

As a guess, let’s write down a rank 2 tensor constructed from the curvature tensor:

Ricci  Curvature
tensor  Scalar
↓  ↓
G_{μν} = α R_{μν} + β g_{μν} R
↖  ↗

constants

This guess automatically satisfies the first three conditions. The tensor G_{μν} involves just the metric tensor and its first two derivatives. The tensor G_{μν} is symmetric and it vanishes if all components of the curvature tensor vanish.

Now let’s make use of the condition that G_{μν} satisfies a conservation equation. Raise an index

G_{ν}^{κ} = g^{κμ}G_{μν} = αg^{κμ}R_{μν} + βg^{κμ}g_{μν}R
= αR_{ν}^{κ} + βδ_{ν}^{κ}R,

and take the covariant derivative,

G_{ν;κ}^{κ} = αR_{ν;κ}^{κ} + βδ_{ν}^{κ}R_{;κ}.

Now recall the Bianchi identity

(R_{η}^{μ} − \frac{1}{2}δ_{η}^{μ}R)_{;μ} = 0,

this gives

G_{ν;κ}^{κ} = (\frac{1}{2}α + β) R_{;ν},

and so if G_{μν} satisfies a conservation equation

G_{ν;κ}^{κ} = 0, either  \{ \begin{matrix}β = −\frac{1}{2}α, \\ R_{;ν} = 0. \end{matrix}

We can easily rule out the idea that R_{;ν} is always equal to zero starting from the field equations (2).

G_{κ}^{κ} = 8πGT_{κ}^{κ}
= αR_{κ}^{κ} + βδ_{κ}^{κ}R
= αR + βR
= α′R,  where α′ = α + β.

So, taking the covariant derivative of this last relation

α′R_{;ν} = 8πGT_{κ;ν}^{κ} = 0,  (3)

But (3) requires that \frac{∂T_{κ}^{κ}}{∂x^{ν}} = 0 always, which cannot be satisfied in general since it requires ∂ρ_{0}/∂x = 0.

We must therefore select the solution

β = −\frac{1}{2}α

and so the field equations must look like

α (R_{μν} − \frac{1}{2}g_{μν}R) = 8πGT_{μν}.  (4)

To fix the constant α, we can look at the weak field limit of equations (4). For non-relativistic matter,

T_{ij} ≪ T_{00} ( \begin{matrix}T_{00} & ∼ & ρ_{0}c^{2} \\ T_{ij} & ∼ & ρ_{0}v^{2} \end{matrix} )

and for weak gravitational fields

g_{μν} ≈ η_{μν},

i.e. the metric tensor is nearly that of Minkowski space-time. The field equations look like

α (R_{μν} − \frac{1}{2}g_{μν}R) = 8πGT_{μν},

and so if T_{ij} ≈ 0, the spatial components on the left hand side must be approximately zero, i.e.

R_{ij} ≈ \frac{1}{2}g_{ij}R,
≈ \frac{1}{2}η_{ij}R.

The definition of the curvature scalar is

R = g^{μκ}R_{μκ}
≈ R_{00} − R_{ii}
= R_{00} + \frac{3}{2}R,  using R_{ij} ≈ \frac{1}{2}η_{ij}R.

Hence

R ≈ −2R_{00},

and since

G_{μν} = α (R_{μν} − \frac{1}{2}g_{μν}R) ,
G_{00} = α (R_{00} − \frac{1}{2}g_{00}R) ,
≈ α (R_{00} + \frac{1}{2}2R_{00})
= 2αR_{00}.

The curvature tensor is, by definition,

R_{λμνκ} = \frac{1}{2} [\frac{∂^{2}g_{λν}}{∂x^{κ}∂x^{μ}} − \frac{∂^{2}g_{μν}}{∂x^{κ}∂x^{λ}} − \frac{∂^{2}g_{λκ}}{∂x^{ν}∂x^{μ}} + \frac{∂^{2}g_{μκ}}{∂x^{ν}∂x^{κ}} ] + g_{ησ} [Γ_{νλ}^{η}Γ_{μκ}^{σ} − Γ_{κλ}^{η}Γ_{μν}^{σ}] .

In a weak static field, this expression simplifies significantly because

• all time derivatives vanish,

• products of Γ’s can be ignored (they are second order in g_{μν}).

• we know that in the weak field limit,

g_{00} ≈ (1 + 2ϕ).

so, since

R_{μν} = g^{λκ}R_{λμκν},

the 00 component is

R_{00} = g^{λκ}R_{λ0κ0} ≈ R_{0000} − R_{i0i0}.

But

R_{0000} ≈ 0, R_{i0j0} = \frac{1}{2} \frac{∂^{2}g_{00}}{∂x^{i}∂x^{j}} ,

and so for a static weak gravitational field,

G_{00} = 2αR_{00} = −α∇^{2}g_{00},

i.e.

−α∇^{2}g_{00} = 8πGT_{00}.

But, in a weak gravitational field the field equations must take the form

∇^{2}g_{00} = 8πGT_{00},

hence

α = −1.

Having fixed the constant α, this means that the field equations must have the form

(R_{μν} − \frac{1}{2}g_{μν}R) = −8πGT_{μν}.

These are Einstein’s field equations of General Relativity.

11.2 Sign Conventions

There is no accepted system of sign conventions in General Relativity. Different books use different sign conventions for the metric tensor, curvature tensors and for the field equations. We can summarize these sign conventions in terms of three sign factors S1, S2 and S3. These are defined as follows:

\begin{aligned}η^{μν} & = & [S1] (−1 + 1, +1, +1) \\ R_{αβγ}^{μ} & = & [S2] (\frac{∂Γ_{αγ}^{μ}}{∂x^{β}} − \frac{∂Γ_{αβ}^{μ}}{∂x^{γ}} + Γ_{σβ}^{μ}Γ_{γα}^{σ} − −Γ_{σγ}^{μ}Γ_{βα}^{σ}) \\ G_{μν} & = & [S3] 8πGT_{μν} \\ R_{μν} & = & [S2] [S3] R_{μαν}^{α} \end{aligned}

I have chosen to use a convention in which all three sign factors are negative. This because of the way that I learned General Relativity, but it is exactly the opposite of the convention used by, for example, Misner, Thorne and Wheeler. My convention is similar to d’Inverno and Rindler, except that I define the curvature tensor with the opposite sign. Here is a summary of the sign conventions used in various books:

Us d’Inverno/Rindler Misner, Thorne, Wheeler

Weinberg

[S1]  -  -  +  +
[S2]  -  +  +  -
[S3]  -  -  +  -