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3.1 Riemann normal coordinates

3.1.1 Definition and coordinate transformation

Given a fixed base point x′ and a tetrad ′ eαa (x′), we assign to a neighbouring point x the four coordinates

ˆxa = − ea′(x′)σα′(x,x′), (103 ) α
where β′ eaα′ = ηabgα′β′e b is the dual tetrad attached to x′. The new coordinates ˆxa are called Riemann normal coordinates (RNC), and they are such that ηabˆxaˆxb = ηabeaα′ebβ′σ α′σ β′ = gα′β′σ α′σ β′, or
a b ′ ηabˆx xˆ = 2σ(x,x ). (104 )
Thus, ηabˆxaxˆb is the squared geodesic distance between x and the base point x′. It is obvious that x ′ is at the origin of the RNC, where xˆa = 0.

If we move the point x to x + δx, the new coordinates change to a a a α′ ′ a a α′ β ˆx + δ ˆx = − e α′σ (x + δx,x ) = ˆx − eα′σ β δx, so that

dˆxa = − ea σα′dx β. (105 ) α′ β
The coordinate transformation is therefore determined by ∂ ˆxa∕∂xβ = − eaα′σα′β, and at coincidence we have
[ a] [ α ] ∂xˆ- = ea′, ∂x-- = eα′; (106 ) ∂x α α ∂ˆxa a
the second result follows from the identities a α′ a eα′eb = δ b and α′ a α′ ea eβ′ = δ β′.

It is interesting to note that the Jacobian of the transformation of Equation (105View Equation), a β J ≡ det(∂ˆx ∕∂x ), is given by √ --- J = − g Δ (x, x′), where g is the determinant of the metric in the original coordinates, and Δ (x,x ′) is the van Vleck determinant of Equation (95View Equation). This result follows simply by writing the coordinate transformation in the form ∂ ˆxa∕∂xβ = − ηabeα′σ ′ b α β and computing the product of the determinants. It allows us to deduce that in the RNC, the determinant of the metric is given by

∘ ---------- ----1--- − g(RNC ) = Δ (x,x ′). (107 )
It is easy to show that the geodesics emanating from x′ are straight lines in the RNC. The proper volume of a small comoving region is then equal to dV = Δ −1d4 ˆx, and this is smaller than the flat-spacetime value of d4ˆx if Δ > 1, that is, if the geodesics are focused by the spacetime curvature.

3.1.2 Metric near ′ x

We now would like to invert Equation (105View Equation) in order to express the line element ds2 = gαβ dx αdxβ in terms of the displacements dˆxa. We shall do this approximately, by working in a small neighbourhood of x ′. We recall the expansion of Equation (89View Equation),

( ) α′ β′ α′ 1 α′ γ′ δ′ 3 σ β = − gβ δβ′ + --R γ′β′δ′σ σ + 𝒪 (ε ), 6

and in this we substitute the frame decomposition of the Riemann tensor, ′ ′ R αγ′β′δ′ = Racbdeαa ecγ′ebβ′edδ′, and the tetrad decomposition of the parallel propagator, gβ′= eβ′eb β b β, where eb (x ) β is the dual tetrad at x obtained by parallel transport of b ′ eβ′(x ). After some algebra we obtain

′ ′ 1 ′ σ αβ = − eαa eaβ −--Racbdeαa ebβˆxcˆxd + 𝒪 (ε3), 6

where we have used Equation (103View Equation). Substituting this into Equation (105View Equation) yields

[ ] dˆxa = δa + 1Ra ˆxcˆxd + 𝒪 (x3) eb dxβ, (108 ) b 6 cbd β
and this is easily inverted to give
[ 1 ] eaαdx α = δab − -Racbdˆxcˆxd + 𝒪 (x3) d ˆxb. (109 ) 6
This is the desired approximate inversion of Equation (105View Equation). It is useful to note that Equation (109View Equation), when specialized from the arbitrary coordinates xα to ˆxa, gives us the components of the dual tetrad at x in the RNC.

We are now in a position to calculate the metric in the new coordinates. We have ds2 = gαβ dxαdx β = (ηabeaαebβ)dxαdx β = ηab(eaαdx α)(ebβ dxβ ), and in this we substitute Equation (109View Equation). The final result is ds2 = g dˆxadˆxb ab, with

1- c d 3 gab = ηab − 3Racbdˆx xˆ + 𝒪 (x ). (110 )
The quantities Racbd appearing in Equation (110View Equation) are the frame components of the Riemann tensor evaluated at the base point ′ x,
′ ′ β′ ′ Racbd = R α′γ′β′δ′ eαa eγc eb eδd , (111 )
and these are independent of a ˆx. They are also, by virtue of Equation (106View Equation), the components of the (base-point) Riemann tensor in the RNC, because Equation (111View Equation) can also be expressed as
[ ] [ ][ ][ ] ∂x-α ∂x-γ ∂xβ- ∂xδ- Racdb = Rα′γ′β′δ′ ∂xˆa ∂xˆc ∂ˆxb ∂ˆxd ,

which is the standard transformation law for tensor components.

It is obvious from Equation (110View Equation) that gab(x′) = ηab and Γ abc(x ′) = 0, where Γ abc = − 13(Rabcd + Racbd)ˆxd + 𝒪 (x2) is the connection compatible with the metric gab. The Riemann normal coordinates therefore provide a constructive proof of the local flatness theorem.

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