# Normal coordinates

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{{Short description|Special coordinate system in differential geometry}}
{{about|differential geometry|use in classical mechanics|Normal mode}}
In [differential geometry](/source/differential_geometry), '''normal coordinates''' at a point ''p'' in a [differentiable manifold](/source/differentiable_manifold) equipped with a [symmetric](/source/torsion_tensor) [affine connection](/source/affine_connection) are a [local coordinate system](/source/local_coordinate_system) in a [neighborhood](/source/neighborhood_(mathematics)) of ''p'' obtained by applying the [exponential map](/source/exponential_map_(Riemannian_geometry)) to the [tangent space](/source/tangent_space) at ''p''.  In a normal coordinate system, the [Christoffel symbols](/source/Christoffel_symbols) of the connection vanish at the point ''p'', thus often simplifying local calculations.  In normal coordinates associated to the [Levi-Civita connection](/source/Levi-Civita_connection) of a [Riemannian manifold](/source/Riemannian_manifold), one can additionally arrange that the [metric tensor](/source/metric_tensor) is the [Kronecker delta](/source/Kronecker_delta) at the point ''p'', and that the first [partial derivative](/source/partial_derivative)s 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](/source/Albert_Einstein) in the [general theory of relativity](/source/general_theory_of_relativity): the  [equivalence principle](/source/equivalence_principle) uses normal coordinates via [inertial frame](/source/inertial_frame)s.  Normal coordinates always exist for the Levi-Civita connection of a Riemannian or [Pseudo-Riemannian](/source/Pseudo-Riemannian) manifold. By contrast, in general there is no way to define normal coordinates for [Finsler manifold](/source/Finsler_manifold)s 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](/source/exponential_map_(Riemannian_geometry))

: <math>\exp_p : T_{p}M \supset V \rightarrow M</math>

with <math> V </math> an open neighborhood of '''0''' in <math> T_{p}M </math>, and an isomorphism

: <math>E: \mathbb{R}^n \rightarrow T_{p}M</math>

given by any [basis](/source/basis_of_a_vector_space) of the tangent space at the fixed basepoint <math>p\in M</math>.  If the additional structure of a Riemannian metric is imposed, then the basis defined by ''E'' may be required in addition to be [orthonormal](/source/orthonormal_basis), 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](/source/tangent_space) ''T<sub>p</sub>M'', and exp<sub>''p''</sub> acts as a [diffeomorphism](/source/diffeomorphism) between ''U'' and ''V''. On a normal neighborhood ''U'' of ''p'' in ''M'', the chart is given by:

: <math>\varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n</math>

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](/source/topological_basis))   has been established by [J.H.C. Whitehead](/source/J.H.C._Whitehead) for symmetric affine connections.

=== Properties ===

The properties of normal coordinates often simplify computations. In the following, assume that <math>U</math> is a normal neighborhood centered at a point <math>p</math> in <math>M</math> and <math>x^i</math> are normal coordinates on <math>U</math>.

* Let <math>V</math> be some vector from <math>T_p M</math> with components <math>V^i</math> in local coordinates, and <math>\gamma_V</math> be the [geodesic](/source/geodesic) with <math>\gamma_V(0) = p</math> and <math>\gamma_V'(0) = V</math>. Then in normal coordinates, <math>\gamma_V(t) = (tV^1,\ldots,tV^n)</math> as long as it is in <math>U</math>. Thus radial paths in normal coordinates are exactly the geodesics through <math>p</math>.
* The coordinates of the point <math>p</math> are <math>(0,\ldots,0)</math>.
* In Riemannian normal coordinates at a point <math>p</math> the components of the [Riemannian metric](/source/Metric_tensor) <math>g_{ij}</math> simplify to <math>\delta_{ij}</math>, i.e., <math>g_{ij}(p)=\delta_{ij}</math>.
* The [Christoffel symbols](/source/Christoffel_symbols) vanish at <math>p</math>, i.e., <math> \Gamma_{ij}^k(p)=0 </math>.  In the Riemannian case, so do the first partial derivatives of <math>g_{ij}</math>, i.e., <math>\frac{\partial g_{ij}}{\partial x^k}(p) = 0,\,\forall i,j,k</math>.

=== Explicit formulae ===

In the neighbourhood of any  point <math>p=(0,\ldots 0)</math>  equipped with   a locally  orthonormal coordinate system   in which  <math>g_{\mu\nu}(0)= \delta_{\mu\nu}</math> and  the    Riemann tensor at <math>p</math> takes the value <math> R_{\mu\sigma \nu\tau}(0) </math>  we  can adjust the  coordinates <math>x^\mu </math> so that   the components of the metric tensor away from 
<math>p</math> become

: <math>g_{\mu\nu}(x)= \delta_{\mu\nu} - \tfrac{1}{3} R_{\mu\sigma \nu\tau}(0) x^\sigma x^\tau + O(|x|^3).</math>

The corresponding Levi-Civita connection Christoffel symbols are

: <math>{\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).</math>

Similarly we can construct local coframes in which

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

and the spin-connection coefficients take the values

: <math>{\omega^a}_{b\mu}(x)= - \tfrac{1}{2} {R^a}_{b\mu\tau}(0)x^\tau+O(|x|^2).</math>

==Polar coordinates==
On a Riemannian manifold, a normal coordinate system at ''p'' facilitates the introduction of a system of [spherical coordinates](/source/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''<sub>''p''</sub>''M''.  That is, one introduces on ''T''<sub>''p''</sub>''M'' the standard spherical coordinate system (''r'',φ) where ''r''&nbsp;≥&nbsp;0 is the radial parameter and φ&nbsp;=&nbsp;(φ<sub>1</sub>,...,φ<sub>''n''&minus;1</sub>) is a parameterization of the [(''n''&minus;1)-sphere](/source/N_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](/source/Gauss's_lemma_(Riemannian_geometry)) asserts that the [gradient](/source/gradient) of ''r'' is simply the [partial derivative](/source/partial_derivative) <math>\partial/\partial r</math>.  That is,
:<math>\langle df, dr\rangle = \frac{\partial f}{\partial r}</math>
for any smooth function <math>f</math>.  As a result, the metric in polar coordinates assumes a [block diagonal](/source/block_diagonal) form
:<math>g = \begin{bmatrix}
1&0&\cdots\ 0\\
0&&\\
\vdots &&g_{\phi\phi}(r,\phi)\\
0&&
\end{bmatrix}.</math>

==References==
* {{Citation | last1=Busemann | first1=Herbert | title=On normal coordinates in Finsler spaces |mr=0071075 | year=1955 | journal=[Mathematische Annalen](/source/Mathematische_Annalen) | issn=0025-5831 | volume=129 | pages=417–423 | doi=10.1007/BF01362381}}.
* {{citation | last1=Kobayashi|first1=Shoshichi|author-link=Shoshichi Kobayashi|last2=Nomizu|first2=Katsumi |author2-link=Katsumi Nomizu | title = [Foundations of Differential Geometry](/source/Foundations_of_Differential_Geometry)|volume=1| publisher=[Wiley Interscience](/source/Wiley_Interscience) | year=1996|edition=New|isbn=0-471-15733-3}}.
* {{citation | last1=Chern|first1=S. S.|author-link=Shiing-Shen Chern|last2=Chen|first2=W. H.|last3=Lam|first3=K. S.| title =Lectures on Differential Geometry| publisher=[World Scientific](/source/World_Scientific) |year=2000|edition=hardcover|isbn=978-981-02-3494-2}}.

==See also==
*[Gauss Lemma](/source/Gauss's_lemma_(Riemannian_geometry))
*[Fermi coordinates](/source/Fermi_coordinates)
*[Local reference frame](/source/Local_reference_frame)
*[Synge's world function](/source/Synge's_world_function)

Category:Riemannian geometry
Category:Coordinate systems in differential geometry

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Adapted from the Wikipedia article [Normal coordinates](https://en.wikipedia.org/wiki/Normal_coordinates) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normal_coordinates?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.