Normal coordinates

In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.

A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable (Busemann 1955).

Geodesic normal coordinates

Geodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the exponential map

${\displaystyle \exp _{p}:T_{p}M\supset V\rightarrow M}$

and an isomorphism

${\displaystyle E:\mathbb {R} ^{n}\rightarrow T_{p}M}$

given by any basis of the tangent space at the fixed basepoint p ∈ M. If the additional structure of a Riemannian metric is imposed, then the basis defined by E may be required in addition to be orthonormal, and the resulting coordinate system is then known as a Riemannian normal coordinate system.

Normal coordinates exist on a normal neighborhood of a point p in M. A normal neighborhood U is a subset of M such that there is a proper neighborhood V of the origin in the tangent space T_pM, and exp_p acts as a diffeomorphism between U and V. On the normal neighborhood U of p in M, the chart is given by:

${\displaystyle \varphi$ :=E^{-1}\circ \exp _{p}^{-1}:U\rightarrow \mathbb {R} ^{n}}

The isomorphism E can be any isomorphism between the two vector spaces, so there are as many charts as there are different orthonormal bases in the domain of E.

Properties

The properties of normal coordinates often simplify computations. In the following, assume that ${\displaystyle U}$ is a normal neighborhood centered at a point ${\displaystyle p}$ in ${\displaystyle M}$ and ${\displaystyle x^{i}}$ are normal coordinates on ${\displaystyle U}$.

• Let ${\displaystyle V}$ be some vector from ${\displaystyle T_{p}M}$ with components ${\displaystyle V^{i}}$ in local coordinates, and ${\displaystyle \gamma _{V}}$ be the geodesic at ${\displaystyle t=0}$ pass through the point ${\displaystyle p}$ with velocity vector ${\displaystyle V}$, then ${\displaystyle \gamma _{V}}$ is represented in normal coordinates by ${\displaystyle \gamma _{V}(t)=(tV^{1},...,tV^{n})}$ as long as it is in ${\displaystyle U}$.
• The coordinates of a point ${\displaystyle p}$ are ${\displaystyle (0,...,0)}$
• In Riemannian normal coordinates at a point ${\displaystyle p}$ the components of the Riemannian metric ${\displaystyle g_{ij}}$ simplify to ${\displaystyle \delta _{ij}}$, i.e., ${\displaystyle g_{ij}(p)=\delta _{ij}}$.
• The Christoffel symbols vanish at ${\displaystyle p}$, i.e., ${\displaystyle \Gamma _{ij}^{k}(p)=0}$. In the Riemannian case, so do the first partial derivatives of ${\displaystyle g_{ij}}$, i.e., ${\displaystyle {\frac {\partial g_{ij}}{\partial x^{k}}}(p)=0,\,\forall i,j,k}$.

Explicit formulae

In the neighbourhood of any point ${\displaystyle p=(0,\ldots 0)}$ equipped with a locally orthonormal coordinate system in which ${\displaystyle g_{\mu \nu }(0)=\delta _{\mu \nu }}$ and the Riemann tensor at ${\displaystyle p}$ takes the value ${\displaystyle R_{\mu \sigma \nu \tau }(0)}$ we can adjust the coordinates ${\displaystyle x^{\mu }}$ so that the components of the metric tensor away from ${\displaystyle p}$ become

${\displaystyle g_{\mu \nu }(x)=\delta _{\mu \nu }-{\frac {1}{3}}R_{\mu \sigma \nu \tau }(0)x^{\sigma }x^{\tau }+O(|x|^{3}).}$

The corresponding Levi-Civita connection Christoffel symbols are

${\displaystyle {\Gamma ^{\lambda }}_{\mu \nu }(x)=-{\frac {1}{3}}(R_{\lambda \nu \mu \tau }(0)+R_{\lambda \mu \nu \tau }(0))x^{\tau }+O(|x|^{2}).}$

Similarly we can construct local coframes in which

${\displaystyle e_{\mu }^{*a}(x)=\delta _{a\mu }-{\frac {1}{6}}R_{a\sigma \mu \tau }(0)x^{\sigma }x^{\tau }+O(x^{2}),}$

and the spin-connection coefficients take the values

${\displaystyle {\omega ^{a}}_{b\mu }(x)=-{\frac {1}{2}}{R^{a}}_{b\mu \tau }(0)x^{\tau }+O(|x|^{2}).}$

Polar coordinates

On a Riemannian manifold, a normal coordinate system at p facilitates the introduction of a system of spherical coordinates, known as polar coordinates. These are the coordinates on M obtained by introducing the standard spherical coordinate system on the Euclidean space T_pM. That is, one introduces on T_pM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ₁,...,φ_n−1) is a parameterization of the (n−1)-sphere. Composition of (r,φ) with the inverse of the exponential map at p is a polar coordinate system.

Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to p of nearby points. Gauss's lemma asserts that the gradient of r is simply the partial derivative ${\displaystyle \partial /\partial r}$. That is,

${\displaystyle \langle df,dr\rangle ={\frac {\partial f}{\partial r}}}$

for any smooth function ƒ. As a result, the metric in polar coordinates assumes a block diagonal form

${\displaystyle g={\begin{bmatrix}1&0&\cdots \ 0\\0&&\\\vdots &&g_{\phi \phi }(r,\phi )\\0&&\end{bmatrix}}.}$

References

• Busemann, Herbert (1955), "On normal coordinates in Finsler spaces", Mathematische Annalen, 129: 417–423, doi:10.1007/BF01362381, ISSN 0025-5831, MR 0071075.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3.
• Chern, S. S.; Chen, W. H.; Lam, K. S.; Lectures on Differential Geometry, World Scientific, 2000

See also

• Fermi coordinates
• Local reference frame