Normal coordinates

Normal coordinates, also known as Riemann normal coordinates, are a system of local coordinates on a Riemannian manifold (M, g) centered at a point p \in M, constructed such that the geodesics emanating from p appear as straight radial lines, the metric tensor g_{ij} takes the Euclidean form \delta_{ij} at p, and the Christoffel symbols \Gamma^k_{ij} vanish at p.^[1]^[2] These coordinates are defined using the exponential map \exp_p: T_p M \to M, which sends a tangent vector v \in T_p M to the endpoint \gamma_v(1) of the unique geodesic \gamma_v starting at p with initial velocity v, where \gamma_v(0) = p and \dot{\gamma_v}(0) = v.^[1]^[2] To obtain the coordinates, choose an orthonormal basis \{e_i\} for the tangent space T_p M, identify a neighborhood of the origin in T_p M with \mathbb{R}^n via v = \sum v^i e_i \mapsto (v^1, \dots, v^n), and pull back this identification to a normal neighborhood U \subset M of p via the diffeomorphism \exp_p, yielding coordinates x^i(q) = v^i for q = \exp_p(v) \in U.^[1]^[2] This construction ensures the existence of a normal neighborhood where \exp_p is a diffeomorphism, allowing the manifold's local geometry near p to be analyzed as closely as possible to Euclidean space.^[2] Key properties of normal coordinates include the vanishing of the first partial derivatives of the metric components \partial_k g_{ij}(p) = 0, which follows directly from the zero Christoffel symbols and simplifies the geodesic equation to straight lines through the origin.^[1]^[2] The Gauss lemma further guarantees that radial geodesics are orthogonal to the "geodesic spheres" (level sets of the distance function from p) and that the radial vector field has unit length along these geodesics.^[2] In these coordinates, the metric expands as g_{ij}(x) = \delta_{ij} - \frac{1}{3} R_{ikjl}(p) x^k x^l + O(|x|^3), where R is the Riemann curvature tensor, highlighting how curvature affects the local geometry beyond the first order.^[1] Introduced by Bernhard Riemann in his 1854 habilitation lecture on the foundations of geometry, normal coordinates provide a canonical way to study the intrinsic geometry of manifolds, facilitating computations of curvature, geodesics, and local isometries.^[3] They are essential in theorems such as the local minimization of geodesics and the characterization of spaces of constant curvature, and extend to pseudo-Riemannian settings like spacetime in general relativity.^[2]

Geodesic normal coordinates

Definition

Geodesic normal coordinates, also known as normal coordinates or Riemann normal coordinates, are a system of local coordinates on a Riemannian manifold (M, g) centered at a point p \in M. They are constructed such that the geodesics emanating from p appear as straight radial lines in the coordinates, the metric tensor g_{ij} takes the Euclidean form \delta_{ij} at p, and the Christoffel symbols \Gamma^k_{ij} vanish at p.^[1]^[2] These coordinates are defined using the exponential map \exp_p: T_p M \to M, which maps a tangent vector v \in T_p M to the point \gamma_v(1) on the geodesic \gamma_v starting at p with initial velocity v, where \gamma_v(0) = p and \dot{\gamma_v}(0) = v. By choosing an orthonormal basis for T_p M and identifying a neighborhood in T_p M with \mathbb{R}^n, the coordinates are pulled back via \exp_p to a normal neighborhood of p in M. This ensures a diffeomorphism where the local geometry near p mimics Euclidean space as closely as possible.^[1]^[2]

Construction

The exponential map on a Riemannian manifold (M, g) at a point p \in M is defined as \exp_p: D_p \to M, where D_p \subseteq T_p M is the domain consisting of those tangent vectors v \in T_p M for which the geodesic \gamma_v with \gamma_v(0) = p and \gamma_v'(0) = v is defined on at least the interval [0, 1], and \exp_p(v) = \gamma_v(1).^[4] This map associates each suitable tangent vector at p to the endpoint of the unique geodesic segment of unit parameter length emanating from p in that direction.^[5] To construct geodesic normal coordinates centered at p, first select an orthonormal basis \{e_1, \dots, e_n\} for the tangent space T_p M with respect to the metric g_p. Identify an open neighborhood \tilde{U} \subseteq T_p M of the origin, such as a ball B(0, r) with radius r less than the injectivity radius at p, on which \exp_p is a diffeomorphism onto its image U = \exp_p(\tilde{U}) \subseteq M. Define the coordinate chart (\phi, U) by \phi(q) = (x^1, \dots, x^n) for q \in U, where the coordinates satisfy q = \exp_p\left( \sum_{i=1}^n x^i e_i \right). This identification equates points in U with their corresponding position vectors in \mathbb{R}^n via the basis.^[4]^[5] Such coordinates exist and are unique in a normal neighborhood of p, meaning there is an open set U containing p where \exp_p provides a diffeomorphism from a star-shaped domain in T_p M to U, and the coordinate representation is independent of the choice of orthonormal basis up to orthogonal transformations.^[6] This follows from the local diffeomorphism property of the exponential map, guaranteed by the inverse function theorem applied to its differential at the origin, which is the identity.^[5] In these coordinates, the radial lines from the origin correspond to geodesics: for any v \in T_p M with \|v\| < r, the curve \gamma(t) = \exp_p(t v) for $0 \leq t \leq 1 is the geodesic connecting p to \exp_p(v), and in the coordinate chart, it traces the straight line (t x^1, \dots, t x^n).^[4] This parameterization ensures that geodesics issuing from p appear as linear rays in the tangent space identification.^[6]

Properties

At the origin point

In geodesic normal coordinates centered at a point p on a Riemannian manifold, the origin corresponds to p, and the coordinate basis is chosen to be orthonormal in the tangent space T_p M. At this origin, the metric tensor takes the Euclidean form g_{ij}(p) = \delta_{ij}, where \delta_{ij} is the Kronecker delta. This normalization simplifies local computations by aligning the inner product at p with the standard Euclidean metric.^[7] A key feature is the vanishing of all Christoffel symbols at the origin: \Gamma^k_{ij}(p) = 0 for all indices i, j, k. This property arises because the geodesics through p are radial straight lines in these coordinates, leading to the geodesic equation implying zero connection coefficients at p.^[7]^[8] Consequently, the geodesic equation simplifies dramatically at p. For a curve \gamma(t) with \gamma(0) = p, it reduces to \frac{d^2 x^k}{dt^2} \big|_{t=0} = 0, confirming that initial segments of geodesics from p are straight lines x^k(t) = t v^k for some initial velocity v \in T_p M. This mirrors the behavior in flat space exactly at the origin.^[7]^[1] The vanishing Christoffel symbols also imply that the covariant derivative of the coordinate basis vectors is zero at p: \nabla_{\partial / \partial x^i} \partial / \partial x^j \big|_p = 0. As a result, the coordinate frame is parallel at the origin, facilitating the transport of vectors without torsion or curvature effects precisely at this point.^[7]

In a normal neighborhood

In a Riemannian manifold (M, g), a normal neighborhood U of a point p \in M is an open set containing p such that the exponential map \exp_p: T_p M \to M restricts to a diffeomorphism from a star-shaped open neighborhood \tilde{U} \subset T_p M of the origin onto U. This ensures that every point q \in U is connected to p by a unique minimizing geodesic segment entirely contained in U, parameterized as \gamma(t) = \exp_p(t v) for $0 \leq t \leq 1, where v \in T_p M with \|v\| < \rho for some \rho > 0. In geodesic normal coordinates on U, these radial geodesics appear as straight lines emanating from the origin, facilitating local analysis of the manifold's geometry.^[6]^[1] The size of the normal neighborhood U is bounded by the injectivity radius \operatorname{inj}_p(M, g) at p, defined as the supremum of radii r > 0 such that \exp_p is a diffeomorphism on the ball [B_r(0) \subset T_p M](/page/Ball). This radius is determined by the distance to the first conjugate point along any geodesic from p, where the differential of \exp_p becomes singular, marking the onset of non-uniqueness in geodesic extensions. Within U, there are no conjugate points, preserving the local diffeomorphism property and ensuring the coordinates remain well-behaved without singularities or multiple geodesic representations. For compact manifolds, the injectivity radius is positive and uniformly bounded below across all points.^[9]^[6]^[1] Curvature plays a crucial role in delimiting and distorting the normal neighborhood, as sectional curvatures influence the growth of Jacobi fields along geodesics, which in turn affect the exponential map's injectivity. Positive curvature tends to concentrate geodesics, reducing the injectivity radius (e.g., bounded above by \pi / \sqrt{K} for constant sectional curvature K > 0), while nonpositive curvature allows larger neighborhoods, potentially extending indefinitely in complete manifolds. Away from p, curvature introduces deviations from the Euclidean structure observed at the origin—where Christoffel symbols vanish—manifesting as nonlinear distortions in the metric and geodesic flows that accumulate with distance. These effects highlight the coordinates' utility for probing local curvature invariants without global assumptions.^[9]^[10]^[11]

Explicit formulae

Metric tensor expansion

In geodesic normal coordinates centered at a point p, the metric tensor satisfies g_{ij}(0) = \delta_{ij}, reflecting the Euclidean structure at the origin.^[12] This zeroth-order term arises because the coordinates are chosen such that the tangent space metric at p is the standard flat metric.^[12] The first-order terms in the expansion vanish, with \partial_k g_{ij}(0) = 0 for all indices.^[12] This property follows directly from the vanishing of the Christoffel symbols at p, which implies that the coordinate system aligns geodesics without first-order corrections to the metric.^[12] The second-order Taylor expansion of the metric tensor is given by

g_{ij}(x) = \delta_{ij} - \frac{1}{3} R_{k i l j}(0) x^k x^l + O(|x|^3),

where R_{k i l j}(0) denotes the components of the Riemann curvature tensor evaluated at p.^[12] This formula captures the leading deviation from flatness due to curvature. A sketch of the derivation begins with the geodesic equation in these coordinates, where straight lines represent geodesics, and employs the Koszul formula for the Levi-Civita connection to compute higher derivatives of the metric; the second partial derivatives at the origin relate to the Riemann tensor via the commutator of covariant derivatives along geodesics.^[12] An important consequence of this expansion is that the squared geodesic distance d(p,q)^2 from p to a nearby point q with normal coordinates x approximates the Euclidean norm \sum_i (x^i)^2.^[12] This approximation holds to leading order, with corrections appearing at O(|x|^4) in the quadratic form g_{ij}(x) x^i x^j, underscoring the local flatness encoded by normal coordinates.^[12]

Christoffel symbols

In geodesic normal coordinates centered at a point p, the Christoffel symbols of the Levi-Civita connection vanish at p, satisfying \Gamma^k_{ij}(p) = 0.^[13] This property arises directly from the coordinate construction, where the exponential map ensures that radial geodesics from p follow straight lines in the coordinate chart, eliminating first-order connection terms at the origin.^[13] The Christoffel symbols are defined in terms of the metric tensor by the formula

\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \right),

where g^{kl} is the inverse metric.^[13] In geodesic normal coordinates, the metric satisfies g_{ij}(p) = \delta_{ij} and \partial_m g_{ij}(p) = 0 for all indices, which immediately implies \Gamma^k_{ij}(p) = 0 upon substitution.^[13] This simplification highlights how normal coordinates locally mimic Euclidean space at p, with the connection measuring deviations from flatness. To examine behavior away from p, consider the Taylor expansion of the Christoffel symbols around the origin. The leading non-vanishing term is linear in the coordinates:

\Gamma^k_{ij}(x) = \frac{1}{3} \left( R^k_{i j l}(p) + R^k_{j i l}(p) \right) x^l + O(|x|^2).

This expansion is derived by substituting the second-order Taylor series of the metric tensor into the expression for \Gamma^k_{ij}, yielding first-order contributions from the second derivatives of g_{ij}.^[13] The symmetrization in the indices i and j accounts for the torsion-free symmetry of the Levi-Civita connection, ensuring consistency.^[13] The appearance of the Riemann curvature tensor R^k_{i j l} in this expansion directly connects the Christoffel symbols to the intrinsic geometry of the manifold, as the curvature encodes how parallel transport deviates from flat space.^[13] Specifically, the linear term in \Gamma^k_{ij} reflects the antisymmetry properties of R, such as R^k_{i j l} = -R^k_{i l j}, which influence the rate at which geodesics diverge or converge near p. This relation underscores the role of normal coordinates in revealing curvature effects through higher-order terms in the connection.^[13]

Polar coordinates

Relation to normal coordinates

In geodesic normal coordinates centered at a point p on a Riemannian manifold M, the coordinates x^i parametrize a normal neighborhood via the exponential map \exp_p: T_p M \to M, mapping tangent vectors to points along geodesics emanating from p. These Cartesian-like coordinates can be adapted into a polar form by introducing a radial coordinate r = |x|, representing the geodesic distance from p, and angular coordinates \theta on the unit sphere S^{n-1} in T_p M. This polar representation separates the radial and directional components, providing a natural adaptation for analyzing radial symmetry in the manifold's geometry.^[13] The transformation from normal coordinates to geodesic polar coordinates is given by x^i = r \omega^i(\theta), where \omega^i(\theta) are the components of a unit vector on S^{n-1} determined by \theta, ensuring that points at distance r from p lie along the geodesic in the direction \omega. In this system, radial geodesics correspond to curves with fixed \theta, parameterized as \gamma(t) = t \omega for $0 \leq t \leq r, which are unit-speed and have length exactly r within the injectivity radius. This structure aligns the coordinate lines with the manifold's geodesics, simplifying the description of distances and paths from the origin point.^[13] Geodesic polar coordinates are particularly advantageous in spaces exhibiting radial or spherical symmetry, such as manifolds of constant sectional curvature, where they facilitate explicit computations of the metric as a warped product and reveal isometries to model spaces like spheres or hyperbolic spaces. For instance, in homogeneous spaces, this coordinate choice highlights the invariance under rotations around p, aiding in the study of curvature propagation and volume elements along radial directions without introducing extraneous terms in the geodesic equations.^[13]

Expressions in curved spaces

In Euclidean space, the metric in polar coordinates assumes the simple form

ds^2 = dr^2 + r^2 \, d\Omega^2,

where d\Omega^2 denotes the standard round metric on the unit sphere S^{n-1}. This expression arises from the flat geometry, where geodesics are straight lines radiating from the origin, and the coordinate spheres are concentric spheres with metric scaled by r^2.^[1] On a general Riemannian manifold, geodesic polar coordinates (r, \theta) centered at a point p are constructed via the exponential map, with r representing the geodesic distance from p and \theta parametrizing directions on the unit sphere in the tangent space T_p M. The metric in these coordinates takes the form

ds^2 = dr^2 + r^2 h_{ab}(r, \theta) \, d\theta^a d\theta^b,

where h_{ab}(0, \theta) = \Omega_{ab} is the standard metric on the unit sphere S^{n-1}, and the off-diagonal terms vanish by the Gauss lemma, ensuring radial geodesics are orthogonal to the coordinate spheres. The tensor h_{ab} encodes the geometry of the geodesic spheres \Sigma_r of radius r, with deviations from the Euclidean case driven by the ambient curvature. Near r = 0, expansions of h_{ab} incorporate terms from the Riemann curvature tensor; specifically, to second order,

h_{ab}(r, \theta) = \Omega_{ab} - \frac{1}{3} r^2 K(\theta) \Omega_{ab} + O(r^3),

where K(\theta) is the sectional curvature of the two-plane in T_p M spanned by the radial vector and the tangential direction corresponding to \theta. This correction term reflects how curvature distorts the spheres from their flat counterparts, with positive K contracting them and negative K expanding them.^[14]^[1] Exact expressions for h_{ab} exist in spaces of constant sectional curvature. For the n-dimensional sphere S^n with K = +1, the metric is

ds^2 = dr^2 + \sin^2 r \, d\Omega^2,

where the spherical factor \sin r arises from great-circle geodesics closing up at r = \pi. In n-dimensional hyperbolic space H^n with K = -1, it becomes

ds^2 = dr^2 + \sinh^2 r \, d\Omega^2,

capturing the exponential divergence of geodesic spheres due to negative curvature. These forms highlight the role of sectional curvature in determining the global structure within normal neighborhoods.^[14]

Fermi normal coordinates

Definition

The concept of Fermi normal coordinates originates from Enrico Fermi's 1922 work on local coordinates in special relativity and was extended to general relativity, with a systematic treatment provided by Frank K. Manasse and Charles W. Misner in 1963.^[15] Fermi normal coordinates provide a local coordinate system adapted to an entire geodesic curve in a pseudo-Riemannian manifold, such as spacetime in general relativity.^[15] They are constructed along a timelike or spacelike geodesic \gamma(\tau), where \tau is the affine parameter, typically proper time for timelike paths.^[16] In these coordinates, denoted as (t, x^i) with i = 1, 2, 3 (or up to the manifold's dimension minus one), the coordinate t parametrizes the geodesic such that \gamma(t) = (t, 0, 0, 0), and the x^i represent transverse directions in the orthogonal complement to the tangent vector \gamma'(\tau).^[15] The basis vectors in these transverse directions are Fermi-Walker transported along the geodesic to maintain a non-rotating frame.^[16] A defining property is that the metric tensor along the geodesic takes the standard Minkowski form: g_{\mu\nu}(t, 0) = \eta_{\mu\nu}, where \eta_{\mu\nu} is the diagonal metric (-1, 1, 1, 1) for Lorentzian signature, and the first partial derivatives of the metric components vanish along \gamma: \partial_\rho g_{\mu\nu}(t, 0) = 0.^[15] This ensures that the Christoffel symbols vanish along the curve, mimicking flat spacetime locally to first order.^[16] Unlike geodesic normal coordinates, which achieve similar flatness only at a single point, Fermi normal coordinates extend this property uniformly along the entire geodesic curve, providing an extended local inertial frame suitable for analyzing gravitational effects over finite segments of the path.^[15]

Construction and applications

The construction of Fermi normal coordinates begins with the selection of a timelike geodesic \gamma in a Lorentzian manifold, parameterized by proper time t, with an initial point O at t=0. An orthonormal tetrad \{e_0, e_1, e_2, e_3\} is erected at O, where e_0 is tangent to \gamma. This tetrad is then transported along \gamma using Fermi-Walker transport, which preserves orthonormality and coincides with parallel transport for geodesic motion, ensuring no rotation relative to non-gravitational forces.^[17] To define the coordinates, space-like geodesics are constructed orthogonal to \gamma at each point \gamma(t), emanating in the directions of the spatial basis vectors e_i(t) (for i=1,2,3). A point P near \gamma is reached by following the geodesic from \gamma(t) in the transverse plane, with coordinates (t, x^i) such that the position vector from \gamma(t) to P is x^i e_i(t), and the full coordinate is given by the exponential map \exp_{\gamma(t)}( \sum_i x^i e_i(t) ). This yields a coordinate system where t is the proper time along \gamma, and x^i represent displacements in the Fermi-transported frame.^[17] In these coordinates, the metric takes a canonical form along \gamma (where x^i = 0): g_{tt} = -1 + O(x^2), g_{ti} = O(x^2), and g_{ij} = \delta_{ij} + O(x^2), with the Christoffel symbols vanishing to first order along the geodesic. The quadratic deviations are directly tied to the Riemann curvature tensor components evaluated on \gamma.^[17] Fermi normal coordinates provide local inertial frames for freely falling observers, extending the Minkowski metric along the entire worldline rather than at a single event, which illustrates the equivalence principle by showing how gravity manifests as fictitious forces only away from the geodesic. They are essential for analyzing tidal forces, where the Riemann tensor governs relative accelerations between nearby geodesics, as seen in the geodesic deviation equation. In the Newtonian limit of general relativity, these coordinates recover the weak-field approximation, linking gravitational potentials to metric perturbations like g_{00} \approx - (1 + 2\Phi).^[18]^[17] As an example, in the Schwarzschild metric describing a static black hole, Fermi coordinates can be constructed along a radial timelike geodesic, such as that of an infalling observer. Here, the metric expansion reveals tidal stretching in the radial direction and compression transversely, with explicit forms derived by transforming from Schwarzschild coordinates, highlighting curvature effects near the horizon.^[19]

Normal coordinates

Geodesic normal coordinates

Definition

Construction

Properties

At the origin point

In a normal neighborhood

Explicit formulae

Metric tensor expansion

Christoffel symbols

Polar coordinates

Relation to normal coordinates

Expressions in curved spaces

Fermi normal coordinates

Definition

Construction and applications

References