8 Riemann normal coordinates

8.1 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

xˆa = − ea ′(x′)σα′(x,x′), (8.1 ) α
where β′ eaα′ = ηabgα′β′eb 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′σα ′σβ′ = gα′β′σ α′σ β′, or
a b ′ ηabˆx ˆx = 2σ(x,x ). (8.2 )
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 + δˆx = − eα′σ (x + δx, x) = ˆx − eα′σ βδx, so that

dˆxa = − ea σα′dxβ. (8.3 ) α′ β
The coordinate transformation is therefore determined by ∂ ˆxa∕∂xβ = − eaα′σα′β, and at coincidence we have
[ a] [ α ] ∂ˆx-- = ea′, ∂x-- = eα′; (8.4 ) ∂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 Eq. (8.3View Equation), a β J := det (∂ ˆx ∕∂x ), is given by √ --- J = − gΔ (x,x ′), where g is the determinant of the metric in the original coordinates, and Δ (x,x ′) is the Van Vleck determinant of Eq. (7.2View Equation). This result follows simply by writing the coordinate transformation in the form ∂ ˆxa∕∂x β = − ηabeα′σ ′ b α β and computing the product of the determinants. It allows us to deduce that in RNC, the determinant of the metric is given by

∘ ---------- ----1--- − g(RNC ) = Δ (x,x ′). (8.5 )
It is easy to show that the geodesics emanating from x ′ are straight coordinate lines in RNC. The proper volume of a small comoving region is then equal to dV = Δ −1d4ˆx, 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.

8.2 Metric near x′

We now would like to invert Eq. (8.3View Equation) in order to express the line element 2 α β ds = g αβdx dx in terms of the displacements dxˆa. We shall do this approximately, by working in a small neighbourhood of x ′. We recall the expansion of Eq. (6.8View Equation),

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

and in this we substitute the frame decomposition of the Riemann tensor, α′ a α′ c b d R γ′β′δ′ = R cbdea eγ′e β′eδ′, and the tetrad decomposition of the parallel propagator, ′ ′ gββ = eβb ebβ, where ebβ (x ) is the dual tetrad at x obtained by parallel transport of eb′(x ′) β. After some algebra we obtain

α′ α′a 1- a α′b c d 3 σ β = − ea eβ − 6R cbdea eβxˆ ˆx + O(𝜖 ),

where we have used Eq. (8.1View Equation). Substituting this into Eq. (8.3View Equation) yields

[ ] dˆxa = δa + 1-Ra ˆxcxˆd + O (x3) ebdx β, (8.6 ) b 6 cbd β
and this is easily inverted to give
[ 1 ] eaαdx α = δab − -Racbdˆxcˆxd + O (x3) dˆxb. (8.7 ) 6
This is the desired approximate inversion of Eq. (8.3View Equation). It is useful to note that Eq. (8.7View Equation), when specialized from the arbitrary coordinates xα to ˆxa, gives us the components of the dual tetrad at x in RNC. And since α′ α′ ea = δ a in RNC, we immediately obtain the components of the parallel propagator: a′ a 1 a c d 3 g b = δ b − 6R cbdˆx ˆx + O (x ).

We are now in a position to calculate the metric in the new coordinates. We have ds2 = gαβdx αdx β = (ηabeaeb )dx αdxβ = ηab(eadx α)(ebdxβ) α β α β, and in this we substitute Eq. (8.7View Equation). The final result is 2 a b ds = gabdˆx dˆx, with

1 c d 3 gab = ηab − -Racbdˆx ˆx + O (x ). (8.8 ) 3
The quantities Racbd appearing in Eq. (8.8View Equation) are the frame components of the Riemann tensor evaluated at the base point x′,
′ Racbd := R α′γ′β′δ′eαa′eγc′eβb eδd′, (8.9 )
and these are independent of ˆxa. They are also, by virtue of Eq. (8.4View Equation), the components of the (base-point) Riemann tensor in RNC, because Eq. (8.9View Equation) can also be expressed as
[ ] [ ][ ] [ ] ∂xα- ∂x-γ ∂xβ- ∂x-δ Racdb = Rα′γ′β′δ′ ∂ˆxa ∂ ˆxc ∂ˆxb ∂xˆd ,

which is the standard transformation law for tensor components.

It is obvious from Eq. (8.8View Equation) that ′ gab(x ) = ηab and a ′ Γ bc(x) = 0, where a 1 a a d 2 Γ bc = − 3(R bcd + R cbd)ˆx + O (x ) 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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