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.

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 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:


 * $$\varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n$$

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


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

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 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&minus;1) is a parameterization of the (n&minus;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 &fnof;. 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}.$$