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{{electromagnetism|cTopic=[[Covariant formulation of classical electromagnetism|Covariant formulation]]}} | |||
In [[special relativity|special]] and [[general relativity]], the '''four-current''' is the 4-dimensional analogue of the [[electric current density]], which is used in the geometric context of ''4d spacetime'', rather than 3d space and time separately. Mathematically it is a [[four-vector]], and is [[Lorentz covariant]]. | |||
Analogously, it's possible to have any form of "current density", meaning the flow of a quantity per unit time per unit area, see [[current density]] for more on this quantity. | |||
This article uses the [[Einstein notation|summation convention]] for indices, see [[covariance and contravariance of vectors]] for background on raised and lowered indices, and [[raising and lowering indices]] on how to switch between them. | |||
==Definition== | |||
;[[Minkowski metric]]: | |||
Using the Minkowski metric <math>\eta_{\mu\nu}</math> of [[metric signature]] (+−−−), its four components are given by: | |||
:<math>J^\alpha = \left(c \rho, j^1 , j^2 , j^3 \right) = \left(c \rho, \mathbf{j} \right)</math> | |||
where ''c'' is the [[speed of light]], ''ρ'' is the [[charge density]] and '''j''' the conventional [[current density]]. The [[Einstein notation|dummy index]] ''α'' labels the [[spacetime]] [[dimension]]s. | |||
;[[Euclidean metric]]: | |||
Some authors{{Who|date=October 2013}} prefer to work under a Euclidean spacetime rather than Minkowski's, in which case the Euclidean metric of signature (++++) is used, so that: | |||
:<math>J^\alpha = J_\alpha </math> | |||
so the four-current is rewritten as | |||
:<math>J^\alpha = \left(ic \rho, \mathbf{j} \right)</math> | |||
where ''i'' is the [[imaginary unit]]. | |||
===Motion of charges in spacetime=== | |||
{{see also|Lorentz transformations}} | |||
This can also be expressed in terms of the [[four-velocity]] by the equation:<ref>Roald K. Wangsness, Electromagnetic Fields, 2nd edition (1986), p. 518, 519</ref><ref>Melvin Schwartz, Principles of Electrodynamics, Dover edition (1987), p. 122, 123</ref> | |||
:<math>J^\alpha = \rho_0 U^\alpha = \rho\sqrt{1-\frac{u^2}{c^2}} U^\alpha </math> | |||
where ''ρ'' is the [[charge density]] measured by an observer at rest observing the [[electric current]], and ''ρ''<sub>0</sub> the charge density for an observer moving at the speed ''u'' (the magnitude of the [[velocity|3-velocity]]) along with the charges. | |||
Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to [[Lorentz contraction]]. | |||
=== Physical interpretation === | |||
Charges (free or as a distribution) at rest will appear to remain at the same spatial position for some interval of time (as long as they're stationary). When they do move, this corresponds to changes in position, therefore the charges have velocity, and the motion of charge constitutes an electric current. This means that charge density is related to time, while current density is related to space. | |||
The four-current unifies charge density (related to electricity) and current density (related to electricity magnetism) in one electromagnetic entity. | |||
== Continuity equation == | |||
{{Main|Continuity equation}} | |||
In special relativity, the statement of [[charge conservation]] is that the [[Lorentz invariant]] divergence of ''J'' is zero:<ref>J. D. Jackson, Classical Electrodynamics, 3rd Edition (1999), p. 554</ref> | |||
:<math>\dfrac{\partial J^\alpha}{\partial x^\alpha} = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0</math> | |||
where <math>\partial/\partial x^\alpha </math> is the [[4-gradient]]. This is the [[continuity equation]]. | |||
In general relativity, the continuity equation is written as: | |||
:<math>J^\alpha{}_{;\alpha}=0\,</math> | |||
where the semi-colon represents a [[covariant derivative]]. | |||
== Maxwell's equations == | |||
{{main|Maxwell's equations}} | |||
The four-current appears in two equivalent formulations of [[Maxwell's equations]], in terms of the [[four-potential]]:<ref>as [ref. 1, p519]</ref> | |||
:<math>\Box A^\alpha = \mu_0 J^\alpha </math> | |||
where <math>\Box </math> is the [[D'Alembert operator]], or the [[electromagnetic field tensor]]: | |||
:<math>\partial_\beta F^{\alpha\beta} = \mu_0 J^\alpha</math> | |||
where ''μ''<sub>0</sub> is the [[permeability of free space]]. | |||
== General Relativity == | |||
{{see also|Maxwell's equations in curved spacetime}} | |||
In [[general relativity]], the four-current is defined as the divergence of the electromagnetic displacement, defined as | |||
:<math>\mathcal{D}^{\mu \nu} \, = \, \frac{1}{\mu_{0}} \, g^{\mu \alpha} \, F_{\alpha \beta} \, g^{\beta \nu} \, \sqrt{-g} \,</math> | |||
then | |||
:<math>J^\mu = \partial_\nu \mathcal{D}^{\mu \nu}</math> | |||
==See also== | |||
* [[Noether's theorem]] | |||
* [[Covariant formulation of classical electromagnetism]] | |||
* [[Ricci calculus]] | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Four-Current}} | |||
[[Category:Electromagnetism]] | |||
[[Category:Minkowski spacetime]] | |||
[[Category:Theory of relativity]] | |||
Revision as of 17:24, 13 January 2014
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In special and general relativity, the four-current is the 4-dimensional analogue of the electric current density, which is used in the geometric context of 4d spacetime, rather than 3d space and time separately. Mathematically it is a four-vector, and is Lorentz covariant.
Analogously, it's possible to have any form of "current density", meaning the flow of a quantity per unit time per unit area, see current density for more on this quantity.
This article uses the summation convention for indices, see covariance and contravariance of vectors for background on raised and lowered indices, and raising and lowering indices on how to switch between them.
Definition
Using the Minkowski metric of metric signature (+−−−), its four components are given by:
where c is the speed of light, ρ is the charge density and j the conventional current density. The dummy index α labels the spacetime dimensions.
Some authorsTemplate:Who prefer to work under a Euclidean spacetime rather than Minkowski's, in which case the Euclidean metric of signature (++++) is used, so that:
so the four-current is rewritten as
where i is the imaginary unit.
Motion of charges in spacetime
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This can also be expressed in terms of the four-velocity by the equation:[1][2]
where ρ is the charge density measured by an observer at rest observing the electric current, and ρ0 the charge density for an observer moving at the speed u (the magnitude of the 3-velocity) along with the charges.
Qualitatively, the change in charge density (charge per unit volume) is due to the contracted volume of charge due to Lorentz contraction.
Physical interpretation
Charges (free or as a distribution) at rest will appear to remain at the same spatial position for some interval of time (as long as they're stationary). When they do move, this corresponds to changes in position, therefore the charges have velocity, and the motion of charge constitutes an electric current. This means that charge density is related to time, while current density is related to space.
The four-current unifies charge density (related to electricity) and current density (related to electricity magnetism) in one electromagnetic entity.
Continuity equation
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In special relativity, the statement of charge conservation is that the Lorentz invariant divergence of J is zero:[3]
where is the 4-gradient. This is the continuity equation.
In general relativity, the continuity equation is written as:
where the semi-colon represents a covariant derivative.
Maxwell's equations
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The four-current appears in two equivalent formulations of Maxwell's equations, in terms of the four-potential:[4]
where is the D'Alembert operator, or the electromagnetic field tensor:
where μ0 is the permeability of free space.
General Relativity
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The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.
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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.
In general relativity, the four-current is defined as the divergence of the electromagnetic displacement, defined as
then