Normal coordinates explained

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 → M

with

V an open neighborhood of 0 in T _{p} M , and an isomorphismE: R ^{n} → 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_pM, 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 ^{-1} _{p} : U → 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 is a normal neighborhood centered at a point p in M and x ^{i} are normal coordinates on U .

• 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,} ... , 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 , ..., 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 ^{k(p) =0} _{ij} . In the Riemannian case, so do the first partial derivatives of g _{ij} , i.e., (\partial g _{ij})/(\partial x ^{k}) (p) = 0 ,\forall i,j,k .

Explicit formulae

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} ( 0 ) we can adjust the coordinates x ^{\mu} so that the components of the metric tensor away from p becomeg _{\mu\nu} (x) = \delta _{\mu\nu} - \tfrac{ 1 }{ 3 } R _{\mu\sigma} ( 0 ) x ^{\sigma} x ^{\tau} + O(|x| ^{3).}

The corresponding Levi-Civita connection Christoffel symbols are

{\Gamma ^{λ} }_(x) = -\tfrac \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 ^{*a} _{\mu(x) =} \delta _{a} - \tfrac{ 1 }{ 6 } R _{a} ( 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_pM. That is, one introduces on T_pM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ₁,...,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.

\partial/\partial r . That is,\langle df, dr\rangle = (\partial f)/(\partial r) for any smooth function ƒ. As a result, the metric in polar coordinates assumes a block diagonal formg = \begin{bmatrix} 1 & 0 & … 0 \\ 0 &&\\ \vdots &&g _{\phi\phi} (r,\phi)\\ 0 && \end{bmatrix}.

References

• .
• .
• 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