Normal Coordinates and Local Metric Expansions

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Normal coordinates use the exponential map of the Levi-Civita connection to make geodesics through a chosen point appear as straight radial lines. At that point the connection coefficients and first metric derivatives vanish, but curvature remains in the quadratic metric term. With the site’s curvature convention, $g_{\mu\nu}=\eta_{\mu\nu}-\frac13R_{\mu\alpha\nu\beta}x^\alpha x^\beta+O(\lvert x\rvert^3)$ .

Required background. Tensor Fields and Natural Operations supplies charts, tangent vectors, and Taylor expansion of tensor components; Pseudo-Riemannian Metrics and Signature supplies an orthonormal basis at a point.

Helpful background. Calculus and Linear Algebra for Relativity supplies multivariable Taylor expansions and inverse-function arguments.

Constructing normal coordinates

Fix $p$ in a smooth pseudo-Riemannian manifold and choose an orthonormal basis $e_\mu$ of $T_pM$ . For $v=v^\mu e_\mu$ sufficiently small, let $\exp_p(v)$ be the point reached at affine parameter one by the Levi-Civita geodesic with initial tangent $v$ . Since $d(\exp_p)_0$ is the identity, the inverse-function theorem makes $\exp_p$ a diffeomorphism from a neighborhood of $0\in T_pM$ to a neighborhood of $p$ . Define the coordinates of $\exp_p(v)$ to be $x^\mu=v^\mu$ .

Radial lines $x^\mu(\lambda)=\lambda v^\mu$ are then affinely parametrized geodesics. Substitution into the geodesic equation gives

\Gamma^\rho{}_{\mu\nu}(\lambda v)v^\mu v^\nu=0.

At $\lambda=0$ , this holds for every $v$ . Because the Levi-Civita coefficients are symmetric in $\mu,\nu$ , polarization implies

\Gamma^\rho{}_{\mu\nu}(p)=0.

Metric compatibility then yields

\partial_\alpha g_{\mu\nu}(p)=0, \qquad g_{\mu\nu}(p)=\eta_{\mu\nu}.

Tong constructs these coordinates and states their immediate consequences in Tong 2019, §3.3.2, pp. 119–123, PDF.

Curvature in the quadratic term

Vanishing $\Gamma(p)$ removes no tensorial curvature. Adopt the site convention

R^\rho{}_{\sigma\mu\nu} =\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} +\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}.

Write a comma for a coordinate derivative evaluated at $p$ . Since $\Gamma^\rho{}_{\mu\nu}(p)=0$ , expanding the radial-geodesic identity one order farther gives

0 =\Gamma^\rho{}_{\mu\nu,\alpha}(p) x^\mu x^\nu x^\alpha+O(\lvert x\rvert^4).

Replace $x$ by $tx$ , divide by $t^3$ , and let $t\to0$ . The leading cubic polynomial must vanish for every $x$ , so polarization says explicitly that

\Gamma^\rho{}_{\mu\nu,\alpha}(p) +\Gamma^\rho{}_{\nu\alpha,\mu}(p) +\Gamma^\rho{}_{\alpha\mu,\nu}(p)=0.

At the same point, the quadratic $\Gamma\Gamma$ terms in the curvature vanish, leaving

R^\rho{}_{\sigma\mu\nu}(p) =\Gamma^\rho{}_{\nu\sigma,\mu}(p) -\Gamma^\rho{}_{\mu\sigma,\nu}(p).

Solving these two linear identities, using the symmetry of the Levi-Civita coefficients in their lower indices, yields

\Gamma^\rho{}_{\mu\nu,\alpha}(p) =-\frac13\left( R^\rho{}_{\mu\nu\alpha}(p) +R^\rho{}_{\nu\mu\alpha}(p) \right).

Now lower the output index by writing $\Gamma_{\rho\mu\nu}=g_{\rho\lambda}\Gamma^\lambda{}_{\mu\nu}$ . Differentiating metric compatibility once and evaluating at $p$ gives

g_{\mu\nu,\alpha\beta}(p) =\Gamma_{\nu\alpha\mu,\beta}(p) +\Gamma_{\mu\alpha\nu,\beta}(p) =-\frac13\left( R_{\mu\alpha\nu\beta}(p) +R_{\mu\beta\nu\alpha}(p) \right).

The ordinary Taylor formula therefore makes the factor of two explicit:

\begin{aligned} g_{\mu\nu}(x) &=\eta_{\mu\nu} +\frac12g_{\mu\nu,\alpha\beta}(p)x^\alpha x^\beta +O(\lvert x\rvert^3)\\ &=\eta_{\mu\nu} -\frac13R_{\mu\alpha\nu\beta}(p)x^\alpha x^\beta +O(\lvert x\rvert^3), \end{aligned}

where the last step exchanges the dummy indices $\alpha$ and $\beta$ in one term. This calculation is also a useful diagnostic: the curvature definition, the connection derivative, and the metric expansion must carry compatible signs.

The minus sign is tied to the displayed curvature convention. Importing the formula from a source with the opposite Riemann-tensor sign without translation would reverse it. The local proof needs nondegeneracy rather than positive definiteness. For the normal-coordinate construction with an arbitrary nondegenerate signature, see Wendl 2021–2022, §23.1, Propositions 23.1–23.2, pp. 192–193, PDF.

The notation $O(\lvert x\rvert^3)$ is coordinate-local: it means that the remaining components are bounded by a constant times a Euclidean coordinate norm cubed in a sufficiently small chart. It is not a global distance estimate.

Positive-curvature sign check

On a unit two-sphere, geodesic polar coordinates about the north pole give

ds^2=dr^2+\sin^2r\,d\theta^2.

Normal Cartesian coordinates are $x=r\cos\theta$ , $y=r\sin\theta$ . The radial direction retains unit coefficient, while the tangential direction is scaled by

\left(\frac{\sin r}{r}\right)^2 =1-\frac{r^2}{3}+O(r^4).

For constant sectional curvature $K=+1$ ,

R_{i\alpha j\beta} =\delta_{ij}\delta_{\alpha\beta} -\delta_{i\beta}\delta_{\alpha j}.

The general expansion therefore becomes

g_{ij} =\delta_{ij} -\frac13\left(r^2\delta_{ij}-x_ix_j\right) +O(r^3),

which changes only the tangential coefficient by $-r^2/3$ . Agreement with $(\sin r/r)^2$ independently verifies the quadratic sign.

What can and cannot be transformed away

At $p$ , normal coordinates remove the Levi-Civita coefficients and the first metric derivatives. They do not remove curvature, which depends on derivatives of the coefficients and appears at second order. They also do not make the metric constant on a neighborhood. A freely falling-coordinate interpretation belongs to gravitational physics; geometrically, the statement is simply a local coordinate normalization.

The chart exists only where $\exp_p$ is one-to-one and nonsingular. Beyond a normal neighborhood, distinct radial geodesics can reach the same point or conjugate behavior can spoil the coordinate map. For a torsionful connection, coordinate transformations cannot remove the tensorial antisymmetric coefficient part. For a degenerate metric, there is no Levi-Civita connection from which to construct these coordinates.

Finally, $\Gamma(p)=0$ does not imply $R(p)=0$ . If curvature happens to vanish at $p$ , the quadratic term vanishes in exponential-map normal coordinates, but higher derivatives may still make the neighborhood nonflat. Flatness on an open simply connected region is a much stronger statement.

The volume density supplies another contraction check. Taking the determinant of the quadratic metric expansion gives, in the same normal coordinates and curvature convention,

\sqrt{\lvert g(x)\rvert} =1-\frac16R_{\alpha\beta}(p)x^\alpha x^\beta +O(\lvert x\rvert^3).

This follows from the first determinant variation $\delta\sqrt{\lvert g\rvert} =\tfrac12\sqrt{\lvert g\rvert}\,g^{\mu\nu}\delta g_{\mu\nu}$ and the contraction $g^{\mu\nu}R_{\mu\alpha\nu\beta}=R_{\alpha\beta}$ . It verifies both the factor $1/3$ in the metric and the Ricci contraction sign without calculating every connection derivative separately.

Exponential-map normal coordinates are not unique because the orthonormal basis at $p$ is not unique; changing it applies a constant pseudo-orthogonal transformation (a Lorentz transformation in Lorentzian signature), under which the displayed curvature coefficient transforms as a tensor. This restricted statement must not be extended to every chart that merely satisfies $g(p)=\eta$ and $\partial g(p)=0$ . A coordinate change that starts at cubic order preserves those two pointwise conditions but can alter the quadratic metric coefficients and destroy the radial-geodesic property.

Flat one-dimensional space gives a decisive check. Starting from $ds^2=dx^2$ , set $y=x+a x^3$ . Near the origin, $x=y-a y^3+O(y^5)$ , and hence

g_{yy}=\left(\frac{dx}{dy}\right)^2 =1-6a y^2+O(y^4).

Thus $g_{yy}(0)=1$ and $\partial_y g_{yy}(0)=0$ although the quadratic coefficient is nonzero while curvature vanishes identically. Curvature is the invariant combination of second derivatives; the simple $-R_{\mu\alpha\nu\beta}/3$ coefficient belongs specifically to exponential-map normal coordinates. Cubic and higher coefficients there require derivatives of curvature and further geometric data.

Exercises

For a sphere of radius $a$ , write the normal-coordinate expansion through quadratic order and check it against $ds^2=dr^2+a^2\sin^2(r/a)d\theta^2$ .

Solution

The sectional curvature is $K=a^{-2}$ , so

g_{ij} =\delta_{ij} -\frac{1}{3a^2} \left(r^2\delta_{ij}-x_ix_j\right) +O(r^3).

The exact tangential scale is

\left(\frac{a\sin(r/a)}{r}\right)^2 =1-\frac{r^2}{3a^2}+O(r^4),

while the radial coefficient remains one. This agrees with the tensor expansion and reduces to the flat metric as $a\to\infty$ .

References

Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §3.3.2, pp. 119–123. Official course PDF.
Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022, §23.1, pp. 192–193. Course PDF.