The harmonic coordinate condition is one of several coordinate conditions

Coordinate conditions

In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions...

 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...

, which make it possible to solve the Einstein field equations

Einstein field equations

The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions x^α (regarded as scalar fields) satisfies d'Alembert's 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...

. The parallel notion of a harmonic coordinate system in Riemannian geometry

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...

 is a coordinate system whose coordinate functions satisfy 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...

. Since d'Alembert's 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...

 is the generalization of Laplace's equation to space-time, its solutions are also called "harmonic".

Motivation

The laws of physics can be expressed in a generally invariant form. In other words, the real world does not care about our coordinate systems. However, for us to be able to solve the equations, we must fix upon a particular coordinate system. A coordinate condition

Coordinate conditions

In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions...

 selects one (or a smaller set of) such coordinate system(s). The Cartesian coordinates used in special relativity satisfy d'Alembert's equation, so a harmonic coordinate system is the closest approximation available in general relativity to an inertial frame of reference in special relativity.

Derivation

In general relativity, we have to use the covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

 instead of the partial derivative in d'Alembert's equation, so we get:


Since the coordinate x^α is not actually a scalar, this is not a tensor equation. That is, it is not generally invariant. But coordinate conditions must not be generally invariant because they are supposed to pick out (only work for) certain coordinate systems and not others. Since the partial derivative of a coordinate is the Kronecker delta, we get:


And thus, dropping the minus sign, we get the harmonic coordinate condition (also known as the de Donder

Théophile de Donder

Théophile Ernest de Donder was a Belgian mathematician and physicist famous for his 1923 work in developing correlations between the Newtonian concept of chemical affinity and the Gibbsian concept of free energy.-Education:...

 gauge ):


This condition is especially useful when working with gravitational waves.

Alternative form

Consider the covariant derivative of the density of the reciprocal of the metric tensor:


The last term emerges because is not an invariant scalar, and so its covariant derivative is not the same as its ordinary derivative. Rather, because , while .

Contracting ν with ρ and applying the harmonic coordinate condition to the second term, we get:


Thus, we get that an alternative way of expressing the harmonic coordinate condition is:

Effect on the wave equation

For example, consider the wave equation applied to the electromagnetic vector potential:


Let us evaluate the right hand side:


Using the harmonic coordinate condition we can eliminate the right-most term and then continue evaluation as follows:

See also

• 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...
  
  
• Covariant derivative
  Covariant derivative
  In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
  
  
• Gauge theory
  Gauge theory
  In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
  
  
• 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...
  
  
• General covariance
  General covariance
  In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
  
  
• Kronecker delta
• 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...
  
  
• Laplace operator
  Laplace operator
  In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
  
  
• 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...
  
  

External links

• http://mathworld.wolfram.com/HarmonicCoordinates.html