 Relativity Science Calculator - Harmonic Coordinates System in General Relativity .  Relativity Science Calculator Web Relativity Science Calculator site 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. 3. 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;

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

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

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

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

8. 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;

2. 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 ). Your ip address is: 44.197.101.251 This document was last modified on: (none) Your browser is: CCBot/2.0 (https://commoncrawl.org/faq/) Domains: relativitycalculator.com, relativityphysics.com, relativityscience.com, relativitysciencecalculator.com, einsteinrelativityphysicstheory.com Urls: https://www.relativitycalculator.com, http://www.relativityphysics.com, http://www.relativityscience.com, http://www.relativitysciencecalculator.com, http://www.einsteinrelativityphysicstheory.com ss note: for a secure encrypted connection type 'https' in the url address bar - i.e., https://www.relativitycalculator.com/

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