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
- [math]\displaystyle{ \exp_p : T_{p}M \supset V \rightarrow M }[/math]
and an isomorphism
- [math]\displaystyle{ E: \mathbb{R}^n \rightarrow T_{p}M }[/math]
given by any basis of the tangent space at the fixed basepoint [math]\displaystyle{ p\in M }[/math]. 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 TpM, and expp acts as a diffeomorphism between U and V. On a normal neighborhood U of p in M, the chart is given by:
- [math]\displaystyle{ \varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n }[/math]
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 [math]\displaystyle{ U }[/math] is a normal neighborhood centered at a point [math]\displaystyle{ p }[/math] in [math]\displaystyle{ M }[/math] and [math]\displaystyle{ x^i }[/math] are normal coordinates on [math]\displaystyle{ U }[/math].
- Let [math]\displaystyle{ V }[/math] be some vector from [math]\displaystyle{ T_p M }[/math] with components [math]\displaystyle{ V^i }[/math] in local coordinates, and [math]\displaystyle{ \gamma_V }[/math] be the geodesic with [math]\displaystyle{ \gamma_V(0) = p }[/math] and [math]\displaystyle{ \gamma_V'(0) = V }[/math]. Then in normal coordinates, [math]\displaystyle{ \gamma_V(t) = (tV^1, ... , tV^n) }[/math] as long as it is in [math]\displaystyle{ U }[/math]. Thus radial paths in normal coordinates are exactly the geodesics through [math]\displaystyle{ p }[/math].
- The coordinates of the point [math]\displaystyle{ p }[/math] are [math]\displaystyle{ (0, ..., 0) }[/math]
- In Riemannian normal coordinates at a point [math]\displaystyle{ p }[/math] the components of the Riemannian metric [math]\displaystyle{ g_{ij} }[/math] simplify to [math]\displaystyle{ \delta_{ij} }[/math], i.e., [math]\displaystyle{ g_{ij}(p)=\delta_{ij} }[/math].
- The Christoffel symbols vanish at [math]\displaystyle{ p }[/math], i.e., [math]\displaystyle{ \Gamma_{ij}^k(p)=0 }[/math]. In the Riemannian case, so do the first partial derivatives of [math]\displaystyle{ g_{ij} }[/math], i.e., [math]\displaystyle{ \frac{\partial g_{ij}}{\partial x^k}(p) = 0,\,\forall i,j,k }[/math].
Explicit formulae
In the neighbourhood of any point [math]\displaystyle{ p=(0,\ldots 0) }[/math] equipped with a locally orthonormal coordinate system in which [math]\displaystyle{ g_{\mu\nu}(0)= \delta_{\mu\nu} }[/math] and the Riemann tensor at [math]\displaystyle{ p }[/math] takes the value [math]\displaystyle{ R_{\mu\sigma \nu\tau}(0) }[/math] we can adjust the coordinates [math]\displaystyle{ x^\mu }[/math] so that the components of the metric tensor away from [math]\displaystyle{ p }[/math] become
- [math]\displaystyle{ g_{\mu\nu}(x)= \delta_{\mu\nu} - \frac{1}{3} R_{\mu\sigma \nu\tau}(0) x^\sigma x^\tau + O(|x|^3). }[/math]
The corresponding Levi-Civita connection Christoffel symbols are
- [math]\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). }[/math]
Similarly we can construct local coframes in which
- [math]\displaystyle{ e^{*a}_\mu(x)= \delta_{a \mu} - \frac{1}{6} R_{a \sigma \mu\tau}(0) x^\sigma x^\tau +O(x^2), }[/math]
and the spin-connection coefficients take the values
- [math]\displaystyle{ {\omega^a}_{b\mu}(x)= - \frac{1}{2} {R^a}_{b\mu\tau}(0)x^\tau+O(|x|^2). }[/math]
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 TpM. That is, one introduces on TpM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ1,...,φn−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 [math]\displaystyle{ \partial/\partial r }[/math]. That is,
- [math]\displaystyle{ \langle df, dr\rangle = \frac{\partial f}{\partial r} }[/math]
for any smooth function ƒ. As a result, the metric in polar coordinates assumes a block diagonal form
- [math]\displaystyle{ g = \begin{bmatrix} 1&0&\cdots\ 0\\ 0&&\\ \vdots &&g_{\phi\phi}(r,\phi)\\ 0&& \end{bmatrix}. }[/math]
References
- Busemann, Herbert (1955), "On normal coordinates in Finsler spaces", Mathematische Annalen 129: 417–423, doi:10.1007/BF01362381, ISSN 0025-5831.
- Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, 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
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