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

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

with $V$ an open neighborhood of 0 in $T_{p}M$ , and an isomorphism

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

given by any basis of the tangent space at the fixed basepoint $p\in M$ . 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 an open 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 a normal neighborhood U of p in M, the chart is given by:

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

The isomorphism E, and therefore the chart, is in no way unique. A convex normal neighborhood U is a normal neighborhood of every p in U. The existence of these sort of open neighborhoods (they form a topological basis) has been established by J.H.C. Whitehead for symmetric affine connections.

Properties

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

Let $V$ be some vector from $T_{p}M$ with components $V^{i}$ in local coordinates, and $\gamma _{V}$ be the geodesic with $\gamma _{V}(0)=p$ and $\gamma _{V}'(0)=V$ . Then in normal coordinates, $\gamma _{V}(t)=(tV^{1},...,tV^{n})$ as long as it is in $U$ . Thus radial paths in normal coordinates are exactly the geodesics through $p$ .
The coordinates of the point $p$ are $(0,...,0)$
In Riemannian normal coordinates at a point $p$ the components of the Riemannian metric $g_{ij}$ simplify to $\delta _{ij}$ , i.e., $g_{ij}(p)=\delta _{ij}$ .
The Christoffel symbols vanish at $p$ , i.e., $\Gamma _{ij}^{k}(p)=0$ . In the Riemannian case, so do the first partial derivatives of $g_{ij}$ , i.e., ${\frac {\partial g_{ij}}{\partial x^{k}}}(p)=0,\,\forall i,j,k$ .

Explicit formulae

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

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

The corresponding Levi-Civita connection Christoffel symbols are

{\Gamma ^{\lambda }}_{\mu \nu }(x)=-{\tfrac {1}{3}}{\bigl [}R_{\lambda \nu \mu \tau }(0)+R_{\lambda \mu \nu \tau }(0){\bigr ]}x^{\tau }+O(|x|^{2}).

Similarly we can construct local coframes in which

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

and the spin-connection coefficients take the values

{\omega ^{a}}_{b\mu }(x)=-{\tfrac {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 $\partial /\partial r$ . That is,

\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

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