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In physics a conserved current is a current, , that satisfies the continuity equation =.The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume , large enough to have no net currents through its surface, leads to the conservation law =, where = is the conserved quantity.
The axial current resulting from the motion of an axially charged moving particle is formally defined as = ¯, where is the particle field represented by Dirac spinor (since the particle is typically a spin-1/2 fermion) and and are the Dirac gamma matrices.
In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form: + = where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity, and a(y) is called the current coefficient, usually corresponding to the partial derivative in the ...
In modern terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current. The Noether current is defined up to a solenoidal (divergenceless) vector field. In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants: [7]
Current events; Conserved current, a concept in physics and mathematics that satisfies the continuity equation; Current density, a mathematical concept unifying electric current, fluid current, and others; Current River (Ozarks), Missouri, US; Currant (disambiguation) Kurrent, German handwriting style
Equations in conservation form take the form + = for any conserved quantity , with a suitable function . An equation of this form can be transformed into an integral equation d d t ∫ V ξ d V = − ∮ ∂ V f ( ξ ) ⋅ ν d S {\displaystyle {\frac {d}{dt}}\int _{V}\xi ~dV=-\oint _{\partial V}\mathbf {f} (\xi )\cdot {\boldsymbol {\nu }}~dS ...
The term on the right is the divergence of the current density J at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a current of charge to flow into or out of the point. This statement is equivalent to a conservation of four-current.
In particular it allows the total of matter plus the gravitating energy–momentum to form a conserved current within the framework of general relativity, so that the total energy–momentum crossing the hypersurface (3-dimensional boundary) of any compact space–time hypervolume (4-dimensional submanifold) vanishes.