Riemannian normal coordinates

Riemann normal coordinates (after Bernhard Riemann ; also normal coordinates or exponential coordinates ) form a special coordinate system , which is considered in differential geometry . Here the tangent space an is used as a local map of the manifold in a neighborhood of . Such coordinates are easy to handle and are therefore also used in general relativity . ${\displaystyle p}$ ${\displaystyle p}$

definition

Let be a differentiable manifold with an affine connection and let be an arbitrary curve which satisfies the geodesic equation. With will denote the tangent space at the point and for will with ${\displaystyle M}$ ${\displaystyle \nabla }$ ${\displaystyle \gamma }$ ${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}$ ${\displaystyle T_{p}M}$ ${\displaystyle p\in M}$ ${\displaystyle {\mathcal {E}}_{p}\subset T_{p}M}$

${\displaystyle \exp _{p}\colon {\mathcal {E}}_{p}\to M,\quad V\mapsto \exp _{p}(V):=\gamma _{V}(1)}$

denotes the exponential map. An isomorphism is obtained by choosing an orthonormal basis of ${\displaystyle (E_{i})_{i}}$ ${\displaystyle T_{p}M}$

${\displaystyle E\colon \mathbb {R} ^{n}\to T_{p}M,}$

which is defined by . Let further be an open neighborhood of , on which the exponential map is a diffeomorphism , and for which holds. Then you get a picture ${\displaystyle \textstyle E(x^{1},\ldots ,x^{n})=\sum _{i}x^{i}E_{i}}$ ${\displaystyle U\subset M}$ ${\displaystyle p}$ ${\displaystyle 0\in \exp _{p}^{-1}(U)\subset T_{p}M}$

${\displaystyle \phi \colon U\to \mathbb {R} ^{n},\quad U\ni q\mapsto E^{-1}\circ \exp _{p}^{-1}(q).}$

Since or defined in the corresponding definition areas an isomorphism or diffeomorphism, is also diffeomorphic, and thus can be used as map image to be viewed. The local coordinates obtained from these maps are called Riemannian normal coordinates. ${\displaystyle E}$ ${\displaystyle \exp _{p}}$ ${\displaystyle \phi }$

properties

Let be a Riemannian or pseudo-Riemannian manifold and define centered Riemannian normal coordinates in . The following applies: ${\displaystyle (M,g)}$ ${\displaystyle (U,(x^{i})^{i})}$ ${\displaystyle p\in M}$

• For all of them , the geodesic , which begins with the velocity vector, is represented in Riemannian normal coordinates ${\displaystyle \textstyle V:=\sum _{i}V^{i}\partial _{i}\in T_{p}M}$ ${\displaystyle \gamma _{V}}$ ${\displaystyle p}$ ${\displaystyle V}$

${\displaystyle \gamma _{V}(t)=(tV^{1},\ldots ,tV^{n})}$

as long in stays. ${\displaystyle \gamma _{V}}$ ${\displaystyle U}$

• The coordinates of are . ${\displaystyle p}$ ${\displaystyle (0,\ldots ,0)}$
• The components of the Riemannian metric are in . ${\displaystyle p}$ ${\displaystyle g_{ij}=\delta _{ij}}$
• The Christmas symbols in are zero. ${\displaystyle p}$
• If the Levi-Civita relationship (or any other metric relationship ) then applies ${\displaystyle \nabla }$ ${\displaystyle \partial _{x_{k}}(g_{ij})|_{p}=0.}$

Physical view

From a physical point of view, normal coordinates in space-time describe the rest system of a freely falling observer at the point . This point is set as the origin of the coordinate system. Normal coordinates are suitable for describing the equivalence principle of general relativity. In normal coordinates, all geodesics through the origin are straight lines in four-dimensional space-time. This makes it understandable what the equivalence of freely falling observers with observers in inertial systems means. Since only the geodesics through a single space-time point are straight lines, the equivalence principle is only exactly valid in a single space-time point. The crooked geodesics, which do not run through the origin, are explained by the observer by tidal forces . ${\displaystyle p}$ ${\displaystyle p}$

In normal coordinates , the metric tensor at a point of the pseudo-Riemannian manifold can be given as a series expansion in . Up to the 5th order you have: ${\displaystyle x^{\alpha }}$ ${\displaystyle P}$ ${\displaystyle x^{\alpha }}$

${\displaystyle g_{\mu \nu }(x)=\eta _{\mu \nu }+{\tfrac {1}{3}}R_{\mu \alpha _{1}\alpha _{2}\nu }x^{\alpha _{1}}x^{\alpha _{2}}+{\tfrac {1}{6}}R_{\mu (\alpha _{1}\alpha _{2}|\nu ;|\alpha _{3})}x^{\alpha _{1}}x^{\alpha _{2}}x^{\alpha _{3}}+{\tfrac {1}{20}}{\bigl [}R_{\mu (\alpha _{1}\alpha _{2}|\nu ;|\alpha _{3}\alpha _{4})}+{\tfrac {8}{9}}R_{\mu (\alpha _{1}\alpha _{2}|\delta }R^{\delta }{}_{|\alpha _{3}\alpha _{4})\nu }{\bigr ]}x^{\alpha _{1}}x^{\alpha _{2}}x^{\alpha _{3}}x^{\alpha _{4}}+{\mathcal {O}}(x^{5}),}$

here are the components of the Minkowski metric and the components of the Riemann curvature tensor , using Einstein's summation convention . With increasing distance of the point from the origin of coordinates at , the metric tensor deviates more and more from the flat Minkowski metric, whereby the first order coefficient (given by a Christoffel symbol ) just vanishes in these coordinates, and the first non-zero correction to the flat Minkowski metric The metric thus only occurs in a quadratic order and is given by the Riemann tensor. The coefficients in the higher orders are given by non-commutative tensor polynomials in the Riemann tensor and its covariant derivatives , which are represented here in compact form using semicolon notation, ie . Indices in round brackets are used to symmetrize and indices enclosed in vertical lines are excluded from symmetrization. ${\displaystyle \eta _{\mu \nu }}$ ${\displaystyle R_{\mu \lambda \nu \rho }}$ ${\displaystyle x}$ ${\displaystyle x=0}$ ${\displaystyle R_{\mu \alpha _{1}\alpha _{2}\nu ;\alpha _{3}}:=\nabla _{\alpha _{3}}R_{\mu \alpha _{1}\alpha _{2}\nu }}$

literature

• John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228 .