Harmonic coordinates
Encyclopedia
In Riemannian geometry
, a branch of mathematics
, harmonic coordinates are a coordinate system
on a Riemannian manifold
each of whose coordinate functions xi is harmonic
, meaning that it satisfies Laplace's equation
Here Δ is the Laplace–Beltrami operator. Equivalently, regarding a coordinate system as a local diffeomorphism , the coordinate system is harmonic if and only if φ is a harmonic map of Riemannian manifolds, roughly meaning that it minimizes the elastic energy of
"stretching" M into Rn. The elastic energy is expressed via the Dirichlet energy functional
In two dimensions, harmonic coordinates have been well understood for more than a century, and are closely related to isothermal coordinates
, the latter being a special case of the former. Harmonic coordinates in higher dimensions were developed initially in the context of general relativity
by (see harmonic coordinate condition
). They were then introduced into Riemannian geometry by and later were studied by . The essential motivation for introducing harmonic coordinate systems is that the metric tensor
is especially smooth
when written in these coordinate systems.
Harmonic coordinates are characterized in terms of the Christoffel symbols
by means of the relation
and indeed, for any coordinate system at all,
Harmonic coordinates always exist (locally), a result which follows easily from standard results on the existence and regularity of solutions of elliptic partial differential equations. In particular, the equation
has a solution in a ball around any given point p, such that uj(p) and
are all prescribed.
The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space Ck,α when expressed in some coordinate system, then they are in that same Hölder space when expressed in harmonic coordinates.
In general relativity
, harmonic coordinates are solutions of the wave equation
instead of the Laplace . This is known as the harmonic coordinate condition
in physics.
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, harmonic coordinates are a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
on a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
each of whose coordinate functions xi is harmonic
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
, meaning that it satisfies Laplace's equation
Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
Here Δ is the Laplace–Beltrami operator. Equivalently, regarding a coordinate system as a local diffeomorphism , the coordinate system is harmonic if and only if φ is a harmonic map of Riemannian manifolds, roughly meaning that it minimizes the elastic energy of
"stretching" M into Rn. The elastic energy is expressed via the Dirichlet energy functional
In two dimensions, harmonic coordinates have been well understood for more than a century, and are closely related to isothermal coordinates
Isothermal coordinates
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifoldare local coordinates where the metric isconformal to the Euclidean metric...
, the latter being a special case of the former. Harmonic coordinates in higher dimensions were developed initially in the context of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
by (see harmonic coordinate condition
Harmonic coordinate condition
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions xα satisfies d'Alembert's...
). They were then introduced into Riemannian geometry by and later were studied by . The essential motivation for introducing harmonic coordinate systems is that the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
is especially smooth
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
when written in these coordinate systems.
Harmonic coordinates are characterized in terms of the Christoffel symbols
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
by means of the relation
and indeed, for any coordinate system at all,
Harmonic coordinates always exist (locally), a result which follows easily from standard results on the existence and regularity of solutions of elliptic partial differential equations. In particular, the equation
has a solution in a ball around any given point p, such that uj(p) and
are all prescribed.The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space Ck,α when expressed in some coordinate system, then they are in that same Hölder space when expressed in harmonic coordinates.
In general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, harmonic coordinates are solutions of the wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
instead of the Laplace . This is known as the harmonic coordinate condition
Harmonic coordinate condition
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions xα satisfies d'Alembert's...
in physics.