- Source: 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 (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
exp
p
:
T
p
M
⊃
V
→
M
{\displaystyle \exp _{p}:T_{p}M\supset V\rightarrow M}
with
V
{\displaystyle V}
an open neighborhood of 0 in
T
p
M
{\displaystyle T_{p}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:
φ
:=
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 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
{\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
γ
V
{\displaystyle \gamma _{V}}
be the geodesic with
γ
V
(
0
)
=
p
{\displaystyle \gamma _{V}(0)=p}
and
γ
V
′
(
0
)
=
V
{\displaystyle \gamma _{V}'(0)=V}
. Then in normal coordinates,
γ
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
δ
i
j
{\displaystyle \delta _{ij}}
, i.e.,
g
i
j
(
p
)
=
δ
i
j
{\displaystyle g_{ij}(p)=\delta _{ij}}
.
The Christoffel symbols vanish at
p
{\displaystyle p}
, i.e.,
Γ
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
=In the neighbourhood of any point
p
=
(
0
,
…
0
)
{\displaystyle p=(0,\ldots 0)}
equipped with a locally orthonormal coordinate system in which
g
μ
ν
(
0
)
=
δ
μ
ν
{\displaystyle g_{\mu \nu }(0)=\delta _{\mu \nu }}
and the Riemann tensor at
p
{\displaystyle p}
takes the value
R
μ
σ
ν
τ
(
0
)
{\displaystyle R_{\mu \sigma \nu \tau }(0)}
we can adjust the coordinates
x
μ
{\displaystyle x^{\mu }}
so that the components of the metric tensor away from
p
{\displaystyle p}
become
g
μ
ν
(
x
)
=
δ
μ
ν
−
1
3
R
μ
σ
ν
τ
(
0
)
x
σ
x
τ
+
O
(
|
x
|
3
)
.
{\displaystyle 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
Γ
λ
μ
ν
(
x
)
=
−
1
3
[
R
λ
ν
μ
τ
(
0
)
+
R
λ
μ
ν
τ
(
0
)
]
x
τ
+
O
(
|
x
|
2
)
.
{\displaystyle {\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
μ
∗
a
(
x
)
=
δ
a
μ
−
1
6
R
a
σ
μ
τ
(
0
)
x
σ
x
τ
+
O
(
x
2
)
,
{\displaystyle e_{\mu }^{*a}(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
ω
a
b
μ
(
x
)
=
−
1
2
R
a
b
μ
τ
(
0
)
x
τ
+
O
(
|
x
|
2
)
.
{\displaystyle {\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−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
∂
/
∂
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&0&\cdots \ 0\\0&&\\\vdots &&g_{\phi \phi }(r,\phi )\\0&&\end{bmatrix}}.}
References
Busemann, 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, 2000
See also
Gauss Lemma
Fermi coordinates
Local reference frame
Synge's world function
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