Coordinates on a Manifold |

**General Coordinates on R ^{n}**

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*

*

*

d*s*^{2} =
(*u*^{2} + *v*^{2})
d*u*^{2} + (*u*^{2} + *v*^{2})
d*u*^{2} + *u*^{2}* v*^{2} d*φ*^{2} .

> __Special situations__: see hamiltonian systems, symplectic manifolds [Darboux, or local canonical coordinates].

**Fermi Transport and Normal Coordinates** > s.a. classical
particles;
Fermi-Walker Transport.

* __Idea__: Fermi transport
is parallel transport along a geodesic; It can be used to set up a coordinate
system by choosing a set of basis vectors at
one point of the curve, Fermi transporting them along it, and extending
them away from the curve with geodesics; All connection coefficients vanish
then at all points of the curve.

* __Physical interpretation__:
Represents a freely falling frame, whose spatial orientation is defined by
gyroscopes; Used to transport a test body's angular momentum
along its orbit (> tests of general
relativity with orbits).

@ __General references__: Fermi AANL(22);
Manasse & Misner JMP(63);
in Misner et al 73;
Marzlin GRG(94);
Nesterov CQG(99)gq/00 [tetrads and metric];
Underwood & Marzlin IJMPA(10)-a0706 [Fermi-Frenet
coordinates for spacelike curves].

@ __Generalizations__: Eisenhart 27; Delva & Angonin GRG(12) [extended]; Dai et al JCAP(15)-a1502 [conformal Fermi coordinates].

@ __Specific spacetimes__: Chicone & Mashhoon PRD(06)
[de Sitter and Gödel spacetimes]; Klein & Collas JMP(10)-a0912 [de Sitter,
anti-de Sitter]; Klein & Randles AHP(11)-a1010 [expanding Robertson-Walker spacetimes]; Bini et al GRG(11)-a1408 [in Schwarzschild spacetime].

**Gaussian Normal Coordinates** (Or Synchronous)

* __Idea__: A coordinate system adapted to a foliation of spacetime with
spacelike hypersurfaces, in which d*s*^{2} = – d*t*^{2} + *h*_{ij}(*t*, **x**) d*x*^{i}d*x*^{j}.

* __Construction__:

- Choose one such hypersurface Σ, *t* = const,
and any coordinate system *x*^{i} on it;

- Consider the unit timelike vector *n*^{a} orthogonal to Σ at each point on it;

- Extend each vector to the unique affinely parametrized timelike geodesic it defines;

- Given *p* ∈ *M*,
identify the unique geodesic *γ*_{p}(*t*)
such that *p* ∈ *γ*_{p} and *γ*_{p}(0) ∈ Σ;

- Label *p* ∈ *M* by
the spatial coordinates of *γ*_{p}(0)
and the affine parameter value *t* such
that *γ*_{p}(*t*) = *p*.

@ __References__: Rácz CQG(07)gq [existence of global Gaussian null coordinate systems].

**Riemann Normal Coordinates**

* __Idea__: Coordinates obtained
using a given point *p* on a manifold *M* and
the exponential map from T_{p}*M* to
a normal neighborhood of *p* in *M*; With them, geodesics
through *p* become straight lines in \(\mathbb R\)^{n},
*g*_{ab} has vanishing first derivatives,
and the distance of a point from the origin has the flat-space expression.

* __Line element__: It has the form
d*s*^{2} =
[*η*_{ab}
+ \(1\over3\)*R*_{manb}* x*^{m}*x*^{n}
+ *O*(*x*^{3})] d*x*^{a}d*x*^{b}, and \(\sqrt{-g\vphantom!} = 1 - {1\over6}\,x^kx^l\,R_{kl}(0) + O(x^3)\).

@ __General references__: in Eisenhart 26;
Robinson GRG(90);
Mueller et al GRG(99)gq/97 [closed formula];
Hatzinikitas ht/00;
Iliev 06-m.DG [handbook of normal frames and coordinates];
Nester JPA(07) [complete accounting];
Brewin CQG(09)-a0903 [expansion to sixth order in the curvature tensor using Cadabra].

@ __Related topics__: Hartley CQG(95)gq [for
non-metric connection]; Nesterov CQG(99)gq/00 [tetrad
and metric].

**Other Coordinates on a Manifold** > s.a. reference frames;
gauge choices; harmonic
coordinates; Isotropic Coordinates.

* __Connection normal coordinates__: Coordinates
in which the geodesics of a (possibly non-metric) connection are straight lines.

@ __Toroidal coordinates__: Krisch & Glass JMP(03) [spacetime with fluid and cosmological constant].

**Normal Coordinates on a Lie Group G**

$

**Spacetime Coordinates** > s.a. non-commutative
geometry and quantum spacetime [as operators]; Positioning Systems.

* __Null coordinates__: Given
any spacetime and a null geodesic in it, one can choose coordinates in a
neighborhood of that geodesic and adapted to it, *u* = value of affine
parameter *λ* along geodesic,
*v* = function such that ∂_{a}*v* = *g*_{ab} d*x*^{b}/d*λ* (the
choice is not unique), *y*^{i} two
additional coordinates; Then *g*_{uv} =
1, *g*_{ui} = *g*_{uu} =
0, so d*s*^{2} = 2 d*u*d*v* + *C* d*v* +
2*C*_{i} d*y*^{i}d*v* + *C*_{ij} d*y*^{i}d*y*^{j};
This form is used to define Penrose limits.

* __In quantum theory__:
Spacetime coordinates can exhibit very few types of short-distance structures,
if described by linear operators; They can be continuous, discrete,
or "unsharp" in one of only two ways.

@ __General references__: Westman & Sonego AP(09)-a0711 [and
symmetries, observables]; Grant & Vickers CQG(09)-a0809 [block-diagonal
form]; Gralla & Wald CQG(11)-a1104 [coordinate freedom in treating the motion of small particles];
Pooley a1506 [diffeomorphism invariance, background
independence and the meaning of coordinates].

@ __Coordinate transformations__: Pelster & Kleinert qp/96 [non-holonomic];
Erlacher & Grosser a1003-conf
[discontinuous coordinate transformations, inversion];
Garofalo & Meier MNRAS(10)-a1004 [misconceptions in black-hole astrophysics literature];
> s.a. gauge choices.

@ __Unsharp coordinates__: Kempf PRL(00) [propagating fields].

@ __Application to celestial mechanics__:
Soffel & Langhans 13;
in Kopeikin ed-14 [post-Newtonian celestial mechanics].

@ __GPS coordinates__: Rovelli PRD(02)gq/01;
Lachièze-Rey CQG(06)gq [covariance].

@ __Geodesic lightcone coordinates__:
Preston & Poisson PRD(06)gq;
Nugier a1509-conf [and cosmology];
Fleury et al JCAP(16)-a1602 [and the Bianchi I spacetime].

> __Specific types of spacetimes__:
see Gordon Ansatz; Kerr-Schild Metric;
schwarzschild spacetime [Eddington-Finkelstein coordinates];
spherical spacetimes.

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