Coordinates on a Manifold

For Rⁿ
* Cartesian: Invented by #R Descartes.
* 2D polar: First used by #J Bernoulli.
* 3D cylindrical and spherical:

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).
@ References: Fermi AANL(22); Eisenhart 27 [generalization]; Manasse & Misner JMP(63); in Misner et al 73; Marzlin GRG(94); Nesterov CQG(99)gq/00 [tetrads and metric]; Chicone & Mashhoon PRD(06) [for de Sitter and Gödel spacetimes]; Underwood & Marzlin a0706 [Fermi-Frenet coordinates for spacelike curves].

Gaussian Normal Coordinates (Or Synchronous)
* Idea: A coordinate system adapted to a foliation of spacetime with spacelike hypersurfaces.
* Construction:
- Choose one such hypersurface , and any coordinate system on it;
- Consider the unit timelike vector orthogonal to at each point on it;
- Extend each vector to the unique affinely parametrized timelike geodesic it defines;
- Given p M, identify the unique geodesic _p(t) such that p _p and _p(0) ;
- Label p M by the spatial coordinates of _p(0) and the affine parameter value t such that _p(t) = p.
@ References: Rácz gq/07 [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_pM to a normal neighborhood of p in M; With them, geodesics through p become straight lines in Rⁿ, and g_ab has vanishing first derivatives.
* Line element: It has the form ds² = [_ab + R_manb x^mxⁿ + O(x³)] dx^adx^b.
@ 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].
@ 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. Frames; gauge choices; harmonic coordinates; Isotropic Coordinates.
@ 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 T_eG and a basis {e_a} for T_eG, with = ^a e_a, the normal coordinates of an element g(t,) = exp(t ^ae_a) of the 1-parameter subgroup are g^a(t,):= t ^a.

Spacetime Coordinates > s.a. non-commutative geometry and quantum spacetime [as operators].
* 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 along geodesic, v = function such that _av = g_ab dx^b/d (the choice is not unique), yⁱ two additional coordinates; Then g_uv = 1, g_ui = g_uu = 0, so ds² = 2 dudv + C dv + 2C_i dyⁱdv + C_ij dyⁱdy^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 a0711 [and symmetries, observables].
@ Coordinate transformations: Pelster & Kleinert qp/96 [non-holonomic]; > s.a. gauge choices.
@ Unsharp coordinates: Kempf PRL(00) [propagating fields].
@ GPS coordinates: Rovelli PRD(02)gq/01; Lachièze-Rey CQG(06)gq [covariance].
@ Null coordinates: Preston & Poisson PRD(06)gq [based on a geodesic world-line].
> Specific types of spacetimes: see schwarzschild [Eddington-Finkelstein]; spherical spacetimes.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 11 jun 2008