Coordinates on a Manifold  

General Coordinates on Rn
* Any dimension: Two general sets are Cartesian coordinates, invented by #R Descartes, and polar or (hyper)spherical coordinates.
* In 2D: Polar coordinates, first used by #J Bernoulli; > s.a. conical sections [elliptical coordinates].
* In 3D: Two common choices are cylindrical and spherical coordinates.
* 3D paraboloidal: The coordinates (u, v, φ) such that x = uv cos φ, y = uv sin φ, z = \(1\over2\)(u2v2); In terms of these coordinates, the euclidean metric is

ds2 = (u2 + v2) du2 + (u2 + v2) du2 + u2 v2 dφ2 .

> 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 ds2 = – dt2 + hij(t, x) dxi dxj.
* Construction:
- Choose one such hypersurface Σ, t = const, and any coordinate system \(x^i\) on it;
- Consider the unit timelike vector na orthogonal to Σ at each point on it;
- Extend each vector to the unique affinely parametrized timelike geodesic it defines;
- Given pM, identify the unique geodesic γp(t) such that pγp and γp(0) ∈ Σ;
- Label pM by the spatial coordinates of γp(0) and the affine parameter value t such that γp(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 TpM to a normal neighborhood of p in M; With them, geodesics through p become straight lines in \(\mathbb R\)n, gab has vanishing first derivatives, and the distance of a point from the origin has the flat-space expression.
* Line element: It has the form ds2 = [ηab + \(1\over3\)Rmanb xmxn + O(x3)] dxadxb, 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
$ Def: Given a 1-parameter subgroup of G generated by γ ∈ TeG and a basis {ea} for TeG, with γ = γa ea, the normal coordinates of an element g(t,γ) = exp(t γaea) of the 1-parameter subgroup are ga(t,γ):= t γa.

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 ∂av = gab dxb/dλ (the choice is not unique), yi two additional coordinates; Then guv = 1, gui = guu = 0, so ds2 = 2 dudv + C dv + 2Ci dyidv + Cij dyi dyj; 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].
@ Quantum coordinate systems: Hardy a1903-proc [for the quantum equivalence principle].
@ 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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