Normal coordinates

{{Short description|Special coordinate system in Differential Geometry}}

{{about|Differential geometry|use in classical mechanics|Normal mode}}

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 {{harv|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 sorts 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

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

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

{{Citation | last1=Busemann | first1=Herbert | title=On normal coordinates in Finsler spaces |mr=0071075 | year=1955 | journal=Mathematische Annalen | issn=0025-5831 | volume=129 | pages=417–423 | doi=10.1007/BF01362381}}.
{{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry|volume=1| publisher=Wiley Interscience | year=1996|edition=New|isbn=0-471-15733-3}}.
{{citation | last1=Chern|first1=S. S.|last2=Chen|first2=W. H.|last3=Lam|first3=K. S.| title =Lectures on Differential Geometry| publisher=World Scientific |year=2000|edition=hardcover|isbn=978-981-02-3494-2}}.