Verdier duality - Revision history

en>K9re11 at 08:33, 25 October 2014

2014-10-25T08:33:15Z

← Older revision		Revision as of 10:33, 25 October 2014
Line 1:		Line 1:
	~~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 ''x''<sup>α</sup> (regarded as scalar fields) satisfies [[Wave equation\|d'Alembert's equation]]~~. ~~The parallel notion~~ of ~~a [[harmonic coordinate system]] in [[Riemannian geometry]] is a coordinate system whose coordinate functions satisfy [[Laplace's equation]]. Since [[Wave equation\|d'Alembert's equation]] is~~ the ~~generalization of Laplace's equation to space-time, its solutions are also called "harmonic".~~		Hi there. Let me start by introducing the author, her title is Myrtle Cleary. One of the very best things in the world for him is to gather badges but he is struggling to find time for it. For many years he's been living in North Dakota and his family members loves it. For years he's been working as a receptionist.<br><br>Here is my weblog [http://streaming.iwarrior.net/user/WMcneil std home test]

	~~==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 conditions\|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]] instead of the partial derivative in d'Alembert~~'s ~~equation, so we get:~~

	~~:<math>0 = (x^\alpha)_{; \beta ; \gamma} g^{\beta \gamma} = ((x^\alpha)_{, \beta , \gamma} - (x^\alpha)_{, \sigma} \Gamma^{\sigma}_{\beta \gamma}) g^{\beta \gamma} \,.</math>~~

	Since the coordinate ''x''<sup>α</sup> 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:

	:<math>0 = (\delta^\alpha_{\beta , \gamma} - \delta^\alpha_{\sigma} \Gamma^{\sigma}_{\beta \gamma}) g^{\beta \gamma} = (0 - \Gamma^{\alpha}_{\beta \gamma}) g^{\beta \gamma} = - \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} \,.</math>

	And thus, dropping the minus sign, we get the '''harmonic coordinate condition''' (also known as the [[Théophile de Donder\|de Donder]] gauge <ref> [John Stewart (1991), "Advanced General Relativity", Cambridge University Press, ISBN 0-521-44946-4 ]</ref>):

	~~:<math>0 = \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} \,.</math>~~

	~~This condition is especially useful when~~ working ~~with gravitational waves.~~

	~~==Alternative form==~~
	~~Consider the covariant derivative of the [[tensor density\|density]] of the reciprocal of the metric tensor:~~

	:<math>0 = (g^{\mu \nu} \sqrt {-g})_{; \rho} = (g^{\mu \nu} \sqrt {-g})_{, \rho} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \rho} \sqrt {-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \rho} \sqrt {-g} - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt {-g} \,.</math>

	The last term <math> - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt {-g} \!</math> emerges because <math> \sqrt {-g} \!</math> is not an invariant scalar, and so its covariant derivative is not the same as its ordinary derivative. Rather, <math> \sqrt {-g}_{; \rho} = 0 \!</math> because <math> g^{\mu \nu}_{; \rho} =0 \!</math>, while <math> \sqrt {-g}_{, \rho} = \sqrt {-g} \Gamma^{\sigma}_{\sigma \rho} \,.<~~/math~~>

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

	:<~~math~~>0 = (g^{\mu \nu} \sqrt {-g})_{, \nu} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \nu} \sqrt {-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \nu} \sqrt {-g} - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \nu} \sqrt {-g} \,</math>
	~~:<math>= (g^{\mu \nu} \sqrt {-g})_{, \nu} + 0 + g^{\mu \alpha} \Gamma^{\beta}_{\alpha \beta} \sqrt {-g} - g^{\mu \alpha} \Gamma^{\beta}_{\beta \alpha} \sqrt {-g} \,.</math>~~

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

	~~:<math>0 = (g^{\mu \nu} \sqrt {-g})_{, \nu} \,.</math>~~

	~~==More variant forms==~~
	~~If one expresses the Christoffel symbol in terms of the metric tensor, one gets~~
	:<math>0 = \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} = \tfrac12 g^{\alpha \delta} ( g_{\gamma \delta , \beta} + g_{\beta \delta , \gamma} - g_{\beta \gamma , \delta} ) g^{\beta \gamma} \,.</math>
	~~Discarding the factor of <math>g^{\alpha \delta} \,</math> and rearranging some indices and terms, one gets~~
	~~:<math> g_{\alpha \beta , \gamma} \, g^{\beta \gamma} = \tfrac12 g_{\beta \gamma , \alpha} \, g^{\beta \gamma} \,.</math>~~
	~~In the context of~~ [~~[linearized gravity]], this is indistinguishable from these additional forms:~~
	~~:<math> h_{\alpha \beta , \gamma} \, g^{\beta \gamma} = \tfrac12 h_{\beta \gamma , \alpha} \, g^{\beta \gamma} \,;</math>~~
	~~:<math> g_{\alpha \beta , \gamma} \, \eta^{\beta \gamma} = \tfrac12 g_{\beta \gamma , \alpha} \, \eta^{\beta \gamma} \,;</math>~~
	~~:<math> h_{\alpha \beta , \gamma} \, \eta^{\beta \gamma} = \tfrac12 h_{\beta \gamma , \alpha} \, \eta^{\beta \gamma} \,.</math>~~
	~~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:~~

	~~:<math>0 = A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} \,.</math>~~

	~~Let us evaluate the right hand side:~~

	:<math>A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} = A_{\alpha ; \beta , \gamma} g^{\beta \gamma} - A_{\sigma ; \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma} - A_{\alpha ; \sigma} \Gamma^{\sigma}_{\beta \gamma} g^{\beta \gamma} \,.</math>

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

	~~:<math>A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} = A_{\alpha ; \beta , \gamma} g^{\beta \gamma} - A_{\sigma ; \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma}</math>~~

	:~~<math> = A_{\alpha , \beta , \gamma} g^{\beta \gamma} - A_{\rho , \gamma} \Gamma^{\rho}_{\alpha \beta} g^{\beta \gamma} - A_{\rho} \Gamma^{\rho}_{\alpha \beta , \gamma} g^{\beta \gamma}~~
	~~- A_{\sigma , \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma}~~
	~~- A_{\rho} \Gamma^{\rho}_{\sigma \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma} \,.<~~/~~math>~~

	~~==See also==~~
	*[[Christoffel symbols]]
	*[[Covariant derivative]]
	*[[Gauge theory]]
	*[[General relativity]]
	*[[General covariance]]
	*[[Kronecker delta]]
	*[[Laplace's equation]]
	*[[Laplace operator]]
	*[[Ricci calculus]]
	*[[Wave equation]]

	~~==References==~~
	~~<references~~/>
	*P.A.~~M.Dirac (1975), ''General Theory of Relativity'', Princeton University Press, ISBN 0-691-01146-X, chapter 22~~

	~~==External links==~~
	* http://~~mathworld.wolfram.com/HarmonicCoordinates.html~~

	~~{{DEFAULTSORT:Harmonic Coordinate Condition}}~~
	~~[[Category:Coordinate charts in general relativity]~~]

en>Frietjes at 00:15, 16 January 2014

2014-01-16T00:15:46Z

← Older revision		Revision as of 02:15, 16 January 2014
Line 1:		Line 1:
	~~Let me first begin by introducing myself~~. ~~My title~~ is ~~Boyd Butts even though it~~ is not [~~http~~://~~www~~.zs-~~imports~~.~~com~~/~~blogs~~/~~post~~/~~70017 over~~ the ~~counter std test~~] ~~name on my birth~~ ~~[http:~~//~~mtomtech~~.~~co.kr~~/~~bbs~~/~~?document_srl~~=~~92816 std testing at home] certification~~. ~~He used~~ to ~~be unemployed~~ ~~[http~~://~~www~~.~~fuguporn~~.~~com~~/~~user/RMitchell std testing at home~~] ~~home test but now he~~ is ~~a pc operator but his [http~~://~~women~~.~~webmd~~.~~com~~/~~features~~/~~could~~-~~you~~-~~have~~-~~std~~-~~not~~-~~know marketing~~] ~~by no means arrives~~. ~~California is exactly where I~~'~~ve always been residing and I love each day living right here~~. ~~What I adore doing is performing ceramics but I haven't made a dime with over the counter std test it~~.		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 ''x''<sup>α</sup> (regarded as scalar fields) satisfies [[Wave equation\|d'Alembert's equation]]. The parallel notion of a [[harmonic coordinate system]] in [[Riemannian geometry]] is a coordinate system whose coordinate functions satisfy [[Laplace's equation]]. Since [[Wave equation\|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 conditions\|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]] instead of the partial derivative in d'Alembert's equation, so we get:

			:<math>0 = (x^\alpha)_{; \beta ; \gamma} g^{\beta \gamma} = ((x^\alpha)_{, \beta , \gamma} - (x^\alpha)_{, \sigma} \Gamma^{\sigma}_{\beta \gamma}) g^{\beta \gamma} \,.</math>

			Since the coordinate ''x''<sup>α</sup> 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:

			:<math>0 = (\delta^\alpha_{\beta , \gamma} - \delta^\alpha_{\sigma} \Gamma^{\sigma}_{\beta \gamma}) g^{\beta \gamma} = (0 - \Gamma^{\alpha}_{\beta \gamma}) g^{\beta \gamma} = - \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} \,.</math>

			And thus, dropping the minus sign, we get the '''harmonic coordinate condition''' (also known as the [[Théophile de Donder\|de Donder]] gauge <ref> [John Stewart (1991), "Advanced General Relativity", Cambridge University Press, ISBN 0-521-44946-4 ]</ref>):

			:<math>0 = \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} \,.</math>

			This condition is especially useful when working with gravitational waves.

			==Alternative form==
			Consider the covariant derivative of the [[tensor density\|density]] of the reciprocal of the metric tensor:

			:<math>0 = (g^{\mu \nu} \sqrt {-g})_{; \rho} = (g^{\mu \nu} \sqrt {-g})_{, \rho} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \rho} \sqrt {-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \rho} \sqrt {-g} - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt {-g} \,.</math>

			The last term <math> - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt {-g} \!</math> emerges because <math> \sqrt {-g} \!</math> is not an invariant scalar, and so its covariant derivative is not the same as its ordinary derivative. Rather, <math> \sqrt {-g}_{; \rho} = 0 \!</math> because <math> g^{\mu \nu}_{; \rho} =0 \!</math>, while <math> \sqrt {-g}_{, \rho} = \sqrt {-g} \Gamma^{\sigma}_{\sigma \rho} \,.</math>

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

			:<math>0 = (g^{\mu \nu} \sqrt {-g})_{, \nu} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \nu} \sqrt {-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \nu} \sqrt {-g} - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \nu} \sqrt {-g} \,</math>
			:<math>= (g^{\mu \nu} \sqrt {-g})_{, \nu} + 0 + g^{\mu \alpha} \Gamma^{\beta}_{\alpha \beta} \sqrt {-g} - g^{\mu \alpha} \Gamma^{\beta}_{\beta \alpha} \sqrt {-g} \,.</math>

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

			:<math>0 = (g^{\mu \nu} \sqrt {-g})_{, \nu} \,.</math>

			==More variant forms==
			If one expresses the Christoffel symbol in terms of the metric tensor, one gets
			:<math>0 = \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} = \tfrac12 g^{\alpha \delta} ( g_{\gamma \delta , \beta} + g_{\beta \delta , \gamma} - g_{\beta \gamma , \delta} ) g^{\beta \gamma} \,.</math>
			Discarding the factor of <math>g^{\alpha \delta} \,</math> and rearranging some indices and terms, one gets
			:<math> g_{\alpha \beta , \gamma} \, g^{\beta \gamma} = \tfrac12 g_{\beta \gamma , \alpha} \, g^{\beta \gamma} \,.</math>
			In the context of [[linearized gravity]], this is indistinguishable from these additional forms:
			:<math> h_{\alpha \beta , \gamma} \, g^{\beta \gamma} = \tfrac12 h_{\beta \gamma , \alpha} \, g^{\beta \gamma} \,;</math>
			:<math> g_{\alpha \beta , \gamma} \, \eta^{\beta \gamma} = \tfrac12 g_{\beta \gamma , \alpha} \, \eta^{\beta \gamma} \,;</math>
			:<math> h_{\alpha \beta , \gamma} \, \eta^{\beta \gamma} = \tfrac12 h_{\beta \gamma , \alpha} \, \eta^{\beta \gamma} \,.</math>
			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:

			:<math>0 = A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} \,.</math>

			Let us evaluate the right hand side:

			:<math>A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} = A_{\alpha ; \beta , \gamma} g^{\beta \gamma} - A_{\sigma ; \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma} - A_{\alpha ; \sigma} \Gamma^{\sigma}_{\beta \gamma} g^{\beta \gamma} \,.</math>

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

			:<math>A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} = A_{\alpha ; \beta , \gamma} g^{\beta \gamma} - A_{\sigma ; \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma}</math>

			:<math> = A_{\alpha , \beta , \gamma} g^{\beta \gamma} - A_{\rho , \gamma} \Gamma^{\rho}_{\alpha \beta} g^{\beta \gamma} - A_{\rho} \Gamma^{\rho}_{\alpha \beta , \gamma} g^{\beta \gamma}
			- A_{\sigma , \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma}
			- A_{\rho} \Gamma^{\rho}_{\sigma \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma} \,.</math>

			==See also==
			*[[Christoffel symbols]]
			*[[Covariant derivative]]
			*[[Gauge theory]]
			*[[General relativity]]
			*[[General covariance]]
			*[[Kronecker delta]]
			*[[Laplace's equation]]
			*[[Laplace operator]]
			*[[Ricci calculus]]
			*[[Wave equation]]

			==References==
			<references/>
			*P.A.M.Dirac (1975), ''General Theory of Relativity'', Princeton University Press, ISBN 0-691-01146-X, chapter 22

			==External links==
			* http://mathworld.wolfram.com/HarmonicCoordinates.html

			{{DEFAULTSORT:Harmonic Coordinate Condition}}
			[[Category:Coordinate charts in general relativity]]

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2012-05-05T22:17:09Z

ISBNs (Build KC)

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