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 are local coordinates on a manifold with an affine connection defined by means of the exponential map
and an isomorphism
given by any basis of the tangent space at the fixed basepoint . 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 neighborhoodU is an open subset of M such that there is a proper neighborhood V of the origin in the tangent spaceTpM, and expp acts as a diffeomorphism between U and V. On a normal neighborhood U of p in M, the chart is given by:
The isomorphism E, and therefore the chart, is in no way unique. A convex normal neighborhoodU is a normal neighborhood of every p in U. The existence of these sort of open neighborhoods (they form a topological base) 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 is a normal neighborhood centered at a point in and are normal coordinates on .
Let be some vector from with components in local coordinates, and be the geodesic with and . Then in normal coordinates, as long as it is in . Thus radial paths in normal coordinates are exactly the geodesics through .
The coordinates of the point are
In Riemannian normal coordinates at a point the components of the Riemannian metric simplify to , i.e., .
The Christoffel symbols vanish at , i.e., . In the Riemannian case, so do the first partial derivatives of , i.e., .
Explicit formulae
In the neighbourhood of any point equipped with a locally orthonormal coordinate system in which and the Riemann tensor at takes the value we can adjust the coordinates so that the components of the metric tensor away from become
The corresponding Levi-Civita connection Christoffel symbols are
Similarly we can construct local coframes in which
and the spin-connection coefficients take the values
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. That is,
for any smooth function ƒ. As a result, the metric in polar coordinates assumes a block diagonal form
References
Busemann, Herbert (1955), "On normal coordinates in Finsler spaces", Mathematische Annalen, 129: 417–423, doi:10.1007/BF01362381, ISSN 0025-5831, MR 0071075.
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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 Contents 1 Geodesic normal coordinates 1 1 Properties 1 2 Explicit formulae 2 Polar coordinates 3 References 4 See alsoGeodesic normal coordinates EditGeodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the exponential map exp p T p M V M displaystyle exp p T p M supset V rightarrow M and an isomorphism E R n T p M displaystyle E mathbb R n rightarrow T p M given by any basis of the tangent space at the fixed basepoint p M displaystyle 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 f E 1 exp p 1 U R n displaystyle 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 base has been established by J H C Whitehead for symmetric affine connections Properties Edit The properties of normal coordinates often simplify computations In the following assume that U displaystyle U is a normal neighborhood centered at a point p displaystyle p in M displaystyle M and x i displaystyle x i are normal coordinates on U displaystyle U Let V displaystyle V be some vector from T p M displaystyle T p M with components V i displaystyle V i in local coordinates and g V displaystyle gamma V be the geodesic with g V 0 p displaystyle gamma V 0 p and g V 0 V displaystyle gamma V 0 V Then in normal coordinates g V t t V 1 t V n displaystyle gamma V t tV 1 tV n as long as it is in U displaystyle U Thus radial paths in normal coordinates are exactly the geodesics through p displaystyle p The coordinates of the point p displaystyle p are 0 0 displaystyle 0 0 In Riemannian normal coordinates at a point p displaystyle p the components of the Riemannian metric g i j displaystyle g ij simplify to d i j displaystyle delta ij i e g i j p d i j displaystyle g ij p delta ij The Christoffel symbols vanish at p displaystyle p i e G i j k p 0 displaystyle Gamma ij k p 0 In the Riemannian case so do the first partial derivatives of g i j displaystyle g ij i e g i j x k p 0 i j k displaystyle frac partial g ij partial x k p 0 forall i j k Explicit formulae Edit In the neighbourhood of any point p 0 0 displaystyle p 0 ldots 0 equipped with a locally orthonormal coordinate system in which g m n 0 d m n displaystyle g mu nu 0 delta mu nu and the Riemann tensor at p displaystyle p takes the value R m s n t 0 displaystyle R mu sigma nu tau 0 we can adjust the coordinates x m displaystyle x mu so that the components of the metric tensor away from p displaystyle p become g m n x d m n 1 3 R m s n t 0 x s x t O x 3 displaystyle g mu nu x delta mu nu frac 1 3 R mu sigma nu tau 0 x sigma x tau O x 3 The corresponding Levi Civita connection Christoffel symbols are G l m n x 1 3 R l n m t 0 R l m n t 0 x t O x 2 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 Similarly we can construct local coframes in which e m a x d a m 1 6 R a s m t 0 x s x t O x 2 displaystyle e mu a x delta a mu frac 1 6 R a sigma mu tau 0 x sigma x tau O x 2 and the spin connection coefficients take the values w a b m x 1 2 R a b m t 0 x t O x 2 displaystyle omega a b mu x frac 1 2 R a b mu tau 0 x tau O x 2 Polar coordinates EditOn 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 f where r 0 is the radial parameter and f f1 fn 1 is a parameterization of the n 1 sphere Composition of r f 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 r displaystyle partial partial r That is d f d r f r displaystyle 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 1 0 0 0 g ϕ ϕ r ϕ 0 displaystyle g begin bmatrix 1 amp 0 amp cdots 0 0 amp amp vdots amp amp g phi phi r phi 0 amp amp end bmatrix References EditBusemann 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 2000See also EditGauss Lemma Fermi coordinates Local reference frame Synge s world function Retrieved from https en wikipedia org w index php title Normal coordinates amp oldid 1101353666, wikipedia, wiki, book, books, library,