The Lorenz Gauge

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If we represent the electric and magnetic fields pervading some volume of space in terms of their potentials,

(Eq'n 1)

and

(Eq'n 2)

then we can rewrite Maxwell's Equations as

(Eq'n 3)

and

(Eq'n 4)

Suppose that a scalar field pervades space in such a way that it alters the potentials in accordance with

(Eq'n 5)

and

(Eq'n 6)

That change, which physicists call a gauge transformation, does not change the electric and magnetic fields, as you can see by substituting Equations 5 and 6 into Equations 1 and 2. That change also does not change Maxwell's Equations when we apply them to the transformed potentials.

If we try to work backward, though, and calculate the potentials from a description of the forcefields, then we find that we cannot devise a unique solution. In essence we integrate our descriptions of the forcefields, so we must include constants of integration; those constants comprise the gradient and time derivative of any scalar field that we can differentiate with respect to the four cardinal dimensions. But the potentials represent energy and linear momentum, which come under the dominion of conservation laws. We know that we can redefine the zero point of energy by adding an arbitrary constant to the potential energies of all of the bodies in the space under consideration without compromising the conservation law, so we can generalize that fact by ensuring that any α-field that we use in our gauge transformations uphold conservation of energy.

Let's contemplate an element of electric charge which generates a potential δφ. In some part of space away from that elemental charge we establish a test distribution of electric charge that has charge density ρ. That test distribution gains a potential energy whose density equals ρδφ due to its immersion in the elemental charge's potential field. If the elemental charge moves with velocity v, then the potential field flows through the test distribution in a current of energy whose density equals ρvδφ. Because energy comes under the dominion of a conservation law, we can put those quantities into an equation of continuity to express that fact mathematically;

(Eq'n 7)

In devising that equation we have treated the potential field of the elemental charge as if it corresponded to an incompressible fluid. But energy represents a relationship between the elemental charge and our test distribution: Can we treat a relationship as a material thing, a liquid that flows with the motion of its generating charge?

In this case we can say yes. The relationship between an electric charge and its forcefield does not change: it may vanish under the fields of other charges in accordance with the superposition principle, but it continues to exist nonetheless. Because the field has the permanence of matter, we may treat it as matter, as something that moves and flows and obeys the continuity equation.

Our test distribution has an arbitrary density, so we choose to make the density constant, so that all of its derivatives equal zero. Then Equation 7 becomes

(Eq'n 8)

But we know that the elemental charge, due to its motion, also generates a magnetic vector potential,

(Eq'n 9)

so we can exploit that fact and divide Equation 8 by ρ and c2 to obtain

(Eq'n 10)

Take all of the versions of that equation that apply to all of the elemental charges in space and add them up. The principle of superposition allows us to commute the operation of addition with the differentiations, so we get at last

(Eq'n 11)

That equation expresses the Lorenz condition, which tells us that the potentials generated by any arbitrary distribution of electric charges and currents must conform to the Lorenz gauge, the gauge in which the constants of integration in Equations 5 and 6 must obey the equation

(Eq'n 12)

which we obtain by substituting Equations 5 and 6 into Equation 11. We recognize that equation as D'Alembert's version of Laplace's Equation or, more to the point, as D'Alembert's version of Poisson's Equation with zero sources. We also recognize it as the propagation equation, which describes waves propagating in space.

We can see what that means if we use Equation 11 to rewrite Equations 3 and 4 as

(Eq'n 13)

and

(Eq'n 14)

Those versions of Maxwell's Equations tell us that the electrostatic and magnetic vector potentials must always originate in sources that we regard as properties of matter. Equation 12, on the other hand, tells us that the α-field has no sources, does not emanate from any property of matter.

We should expect that result. We add the gradient and the time derivative of the α-field as constants of integration to the potentials, but we must do so in a way that upholds conservation of energy. If the α-field has no source in a property of matter, then it cannot represent a relationship among bodies mediated by a force or anything like a force. So long as the potentials conform to the Lorentz gauge, expressed in Equation 12, then so long will Existence uphold conservation of energy.

In light of that fact, many physicists regard Equation 11 as the fifth Maxwell Equation, the true completion of classical electromagnetic theory. Given that level of importance of the Lorenz condition, I now want to present the integral version of Equation 11, just as I presented the integral versions of the standard Maxwell Equations:

(Eq'n 15)

in which S represents the closed surface bounding the volume V.

Finally I need to note that physicists often attribute the Lorenz condition to Hendrik Antoon Lorentz (with a tee), the Dutch physicist who devised the Lorentz Transformation of Relativity. In fact the Lorenz condition was discerned in 1867 as an addendum to Maxwell’s Equations by the Danish physicist Ludwig Valentin Lorenz (without the tee) (1829 Jan 18 – 1891 Jun 09), who then used it to calculate a value of the speed of light more accurate than the value Maxwell had calculated.

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