Harmonic Coordinate Condition
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The harmonic coordinate condition is one of several coordinate conditions in
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, which make it possible to solve the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
. 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. The parallel notion of a harmonic coordinate system in
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
is a coordinate system whose coordinate functions satisfy
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delt ...
. Since d'Alembert's equation 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 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 In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
instead of the partial derivative in d'Alembert's equation, so we get: :0 = \left(x^\alpha\right)_ g^ = \left(\left(x^\alpha\right)_ - \left(x^\alpha\right)_ \Gamma^_\right) g^ \,. 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 In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
, we get: :0 = \left(\delta^\alpha_ - \delta^\alpha_ \Gamma^_\right) g^ = \left(0 - \Gamma^_\right) g^ = - \Gamma^_ g^ \,. And thus, dropping the minus sign, we get the harmonic coordinate condition (also known as the de Donder gauge after Théophile de Donder ohn Stewart (1991), "Advanced General Relativity", Cambridge University Press, /ref>): :0 = \Gamma^_ g^ \,. This condition is especially useful when working with gravitational waves.


Alternative form

Consider the covariant derivative of the
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
of the reciprocal of the metric tensor: :0 = \left(g^ \sqrt \right)_ = \left(g^ \sqrt \right)_ + g^ \Gamma^_ \sqrt + g^ \Gamma^_ \sqrt - g^ \Gamma^_ \sqrt \,. The last term - g^ \Gamma^_ \sqrt emerges because \sqrt is not an invariant scalar, and so its covariant derivative is not the same as its ordinary derivative. Rather, \sqrt _ = 0 \! because g^_ = 0 \!, while \sqrt _ = \sqrt \Gamma^_ \,. Contracting ν with ρ and applying the harmonic coordinate condition to the second term, we get: :\begin 0 &= \left(g^ \sqrt \right)_ + g^ \Gamma^_ \sqrt + g^ \Gamma^_ \sqrt - g^ \Gamma^_ \sqrt \,\\ &= \left(g^ \sqrt \right)_ + 0 + g^ \Gamma^_ \sqrt - g^ \Gamma^_ \sqrt \,. \end Thus, we get that an alternative way of expressing the harmonic coordinate condition is: :0 = \left(g^ \sqrt \right)_ \,.


More variant forms

If one expresses the Christoffel symbol in terms of the metric tensor, one gets :0 = \Gamma^_ g^ = \frac g^ \left( g_ + g_ - g_ \right) g^ \,. Discarding the factor of g^ \, and rearranging some indices and terms, one gets : g_ \, g^ = \frac g_ \, g^ \,. In the context of linearized gravity, this is indistinguishable from these additional forms: :\begin h_ \, g^ &= \frac12 h_ \, g^ \,; \\ g_ \, \eta^ &= \frac12 g_ \, \eta^ \,; \\ h_ \, \eta^ &= \frac12 h_ \, \eta^ \,. \end However, the last two are a different coordinate condition when you go to the second order in ''h''.


Effect on the wave equation

For example, consider the wave equation applied to the electromagnetic vector potential: :0 = A_ g^ \,. Let us evaluate the right hand side: :A_ g^ = A_ g^ - A_ \Gamma^_ g^ - A_ \Gamma^_ g^ \,. Using the harmonic coordinate condition we can eliminate the right-most term and then continue evaluation as follows: :\begin A_ g^ &= A_ g^ - A_ \Gamma^_ g^ \\ &= A_ g^ - A_ \Gamma^_ g^ - A_ \Gamma^_ g^ - A_ \Gamma^_ g^ - A_ \Gamma^_ \Gamma^_ g^ \,. \end


See also

*
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metri ...
*
Covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
*
Gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
*
General relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
*
General covariance In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the Invariant (physics), invariance of the ''form'' of physical laws under arbitrary Derivative, differentiable coordinate transf ...
*
Holonomic basis In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region In geography, regions, otherwise referred to as areas, zones, land ...
*
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
*
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delt ...
*
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
*
Ricci calculus Ricci () is an Italian surname. Notable Riccis Arts and entertainment * Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin * Christina Ricci (born 1980), American actress * Clara Ross Ricci (1858-1954), British ...
*
Wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...


References

*P.A.M.Dirac (1975), ''General Theory of Relativity'', Princeton University Press, , chapter 22


External links

* https://mathworld.wolfram.com/HarmonicCoordinates.html {{DEFAULTSORT:Harmonic Coordinate Condition Coordinate charts in general relativity