Riemann normal coordinates

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

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

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

The coordinate transformation is therefore determined by $∂ ˆxa∕∂xβ = − eaα′σα′β$ , and at coincidence we have

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 (105 ), $a β J ≡ det(∂ˆx ∕∂x )$ , 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 (95 ). 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

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 (105 ) 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 (89 ),

( ) 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 (103 ). Substituting this into Equation (105 ) yields

and this is easily inverted to give

This is the desired approximate inversion of Equation (105

). It is useful to note that Equation (109

), 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 (109 ). The final result is $ds2 = g dˆxadˆxb ab$ , with

The quantities $Racbd$ appearing in Equation (110

) are the frame components of the Riemann tensor evaluated at the base point $′ x$ ,

and these are independent of $a ˆx$ . They are also, by virtue of Equation (106

), the components of the (base-point) Riemann tensor in the RNC, because Equation (111

) 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 (110 ) 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.