Harmonic Functions |
In General
$ Def: A function f is
harmonic if it satisfies ∇2f
= 0, the Laplace equation with respect to some Riemannian metric.
* Conjugate harmonic functions:
The harmonic conjugate of a function u(x, y)
is another function v(x, y) such that f(x,
y) = u(x, y) + i v(x, y)
satisfies the Cauchy-Riemann conditions; Given by v(z)
= ∫ (ux
dy − uy
dx).
@ On manifolds:
Colding & Minicozzi AM(97).
Results and Applications > s.a. Earnshaw's Theorem.
* In general: A harmonic
function can have no maxima or minima.
* On a non-compact manifold:
A bounded harmonic function on a non-compact manifold is constant; A positive
harmonic function on a non-compact manifold with non-negative R is
constant (& Yau).
* In Euclidean space: The
average of f over a sphere is equal to its value at the center.
Harmonic Coordinates > s.a. coordinates
/ D'Alembertian; gauge choice.
* History: Introduced by Lanczos,
DeDonder and Georges Darmois for the Einstein equation, which in vacuum then looks
like a quasidiagonal, quasilinear system of second-order partial differential equations
hyperbolic for a Lorentzian metric; Initially called "isotherm".
* Idea: A choice of gauge
(a.k.a. Lorenz gauge) for generally covariant theories.
* In general: Defined by
∂a(|g|1/2
gab) = 0, or equivalently
∇2xa
= 0; Thus, they always give Γk
:= gij
Γkij = 0.
* For 2D manifolds: It
is particularly useful to use harmonic functions as coordinates; Given a harmonic
function α there exists always (locally) a conjugate harmonic
function β, such that gab
is explicitly written in the conformally flat form [@ Wald 84,
problem 3.2, p53].
* For flat space: They are actually
harmonic functions of the Cartesian coordinates.
* Applications: Harmonic coordinates
are used in applied relativity as a practical tool for calculating motion of celestial
bodies (planets, Moon, satellites); The International Astronomical Union has adopted
these coordinates as the basis for doing numerical ephemerides and time metrology in
the solar system.
@ General references: in Weinberg 72;
Bičák & Katz CzJP(05)gq [stationary asymptotically flat, with matter].
@ In relativistic celestial mechanics:
Kopeikin CelM(88),
Brumberg & Kopeikin NCB(89) [in curved spacetime];
Soffel et al AJ(03)ap [adoption by the IAU];
Kopeikin et al 11;
Soffel & Langhans 13.
Harmonic Analysis
@ References: Stein ed-84;
Deitmar 05 [II, primer].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 dec 2015