Coordinates on a Manifold| 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 = \(1\over2\)(u2
− v2); 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 p ∈ M,
identify the unique geodesic γp(t)
such that p ∈ γp
and γp(0) ∈ Σ;
- Label p ∈ M 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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send feedback and suggestions to bombelli at olemiss.edu – modified 1 jun 2020