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 .

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 T p M, 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. 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,\ldots,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,\ldots,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 . 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 {\bold e}^{*a}= e_\mu^{*a}dx^\mu 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 p M. That is, one introduces on T p M the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ 1 ,...,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 f . 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== . . . ==See also== Gauss Lemma Fermi coordinates Local reference frame Synge's world function

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