Harmonic Coordinates System in General Relativity

"What really interests me is whether God had any choice in the creation of the world " - Albert Einstein ( 1879 - 1955 )

[ source: Einstein to an assistant as quoted by G. Holton ( 1971 ), pg. 20, as referenced in "Gravitation", by Misner, et al. ]

Brief Thumbnail Definition of the Harmonic Function and its Coordinates

1. A system of local Minkowski coordinates on a Riemannian manifold surface is harmonic if and only if the affine ( connection ) coefficients satisfy the condition



2. Harmonic coordinates is a set of coordinates ( out of sundry others! where all other coordinate frames are relatively - speaking 'equal' ) which are "gauge chosen" or "gauge fixed" so that there's a straightforward, mathematically operational technique for calculating spacetime curvature;


3. In other words, harmonic coordinates are by definition pseudo - Euclidean Minkowskian suitable for study under conditions of the weak gravitational field;


4. every harmonic function is infinitely differentiable for both the real and imaginary components of the function;


5. the Mean - Value Theorem for harmonic functions states that harmonic functions are always equal to the average of their nearby values;


6. if the function is harmonic and has a local maximum or minimum, then the function is constant;


7. harmonic functions can't have local extrema - i.e., they're smooth.

A Brief Few Examples of the Harmonic Function and its Coordinates

1. and are all vacuum equations;

The Covariant D'Alembertian Operator Applied to a Scalar Wave Function is Harmonic

 A Critical Harmonic Function Derivation in General Relativity

The Linear Scalar Wave Equation

§ References:

1. "General Relativity", by Dr. Valeria Ferrari, Leonardo Gualtieri ( Research Associate ), Dipartimento di Fisica, Sapienza Universita' di Roma.


2. "Lecture Notes on General Relativity", by Sean M. Carroll, Institute for Theoretical Physics, University of California, 1997. Specifically see: pgs. 130 - 131 of the pdf for equations ( 4.84 ) to ( 4.91 ).

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