Lesson 18

Lesson 18 Electromagnetic Fields in Matter

Lesson 17 assembled the electromagnetic field in vacuum, with the field strength as a 2-form, the potential as a 1-form, the Lorentz force, the Maxwell equations, the stress-energy tensor, and the variational principle that ties them together. All of that structure lives on Minkowski spacetime and is indifferent to the presence of ordinary matter. Real materials, however, are composed of charges that can be displaced, of currents that can be induced, and of dipoles that can be aligned. The field that acts on a test charge inside such a material is not the microscopic field of the vacuum theory; it is an average field whose sources have been reorganized into polarization and magnetization. The present lesson examines how that reorganization is performed, how the resulting macroscopic fields transform, and how the conservation laws of Lesson 15 survive the averaging.

We begin by recalling the action of the Lorentz group on vectors and tensors. Because every material response must be consistent with special relativity, the transformation properties of the polarization, the magnetization, and the susceptibility tensors are fixed once the vacuum field strength is known. Noether’s theorem, already applied to time translation and to spatial translation, is now applied to Lorentz transformations themselves; the associated conserved charges are the components of the angular-momentum tensor. That tensor is not symmetric. The electromagnetic stress-energy tensor constructed in Lesson 17 is likewise not symmetric. Both defects are removed by the same improvement term, and the improved tensors become the sources of angular momentum carried by the field and by polarized matter.

With the transformation law in hand we turn to the average fields inside matter. The microscopic current is expanded in multipoles about each molecule. The monopole term reproduces the free charge and free current already present in vacuum electrodynamics. The dipole term produces the polarization and the magnetization . Higher multipoles are neglected in the long-wavelength approximation that defines macroscopic electrodynamics. The average field equations then take the familiar form in which and appear as the fields whose divergences and curls are sourced only by the free charges and free currents. The linear response of an isotropic medium is summarized by the electric and magnetic susceptibility matrices; their transformation properties under boosts follow at once from the tensor character of .

Throughout, the geometric language remains that of Lesson 15: configuration space, phase space, Noether charges, and the symplectic structure that guarantees conservation. The new element is the systematic replacement of microscopic sources by their macroscopic averages, performed so that every equation stays covariant and every conserved quantity retains its interpretation as a flux through a three-surface. The result is a complete spacetime account of electromagnetic fields in matter.

Lorentz Transformations

A finite Lorentz transformation acts on the coordinates by a constant matrix Λ,

(18.1)

that preserves the Minkowski metric,

(18.2)

In matrix language the same condition reads . Every such matrix that can be reached continuously from the identity may be written as the exponential of a generator A,

(18.3)

Substitution into the defining relation then shows that the generator itself must be antisymmetric with respect to the metric,

(18.4)

(The six independent components of an antisymmetric 4 × 4 matrix are precisely the three rotations and the three boosts.)

Because the exponential is the limit of repeated infinitesimal steps,

(18.5)

it is enough to study the generators. A convenient basis is furnished by the six matrices whose only non-zero entries lie in the αβ and βα positions and satisfy

(18.6)

Any infinitesimal Lorentz transformation is then a linear combination of these six matrices, and any finite transformation in the connected component of the Lorentz group is the exponential of such a combination.

Figure 18.1: The six independent generators of the Lorentz algebra. The three boost generators occupy the time-space slots; the three rotation generators occupy the space-space slots.

The same generators reappear when the action of a Lorentz transformation on tensors is computed. Because every index is transformed by Λ or by , the infinitesimal change of a tensor is obtained by letting each generator act on one index at a time. The resulting formula will be needed both for the transformation law of the electromagnetic field strength and for the Noether charge associated with Lorentz symmetry. Once the generators are in hand, the passage from infinitesimal to finite transformations is purely algebraic, we exponentiate the appropriate linear combination of the . That completes the kinematic preparation required for the dynamical application of Noether’s theorem to Lorentz transformations.

Transforming Vectors and Tensors

A finite Lorentz transformation belonging to the connected component of the Lorentz group is obtained by exponentiating a linear combination of the six generators introduced in the previous subsection,

(18.7)

where the coefficients are antisymmetric, . Specifying the six independent numbers completely determines the transformation. Three of them parametrize boosts; the other three parametrize rotations.

A contravariant vector transforms by the matrix Λ itself,

(18.8)

Because each index of a tensor transforms independently, the action on a rank-2 tensor is the tensor product of two copies of Λ. It is convenient to package that action into a single generator acting on the pair of indices. Define

(18.9)

Then

(18.10)

The same construction extends immediately to tensors of any rank, where each index contributes one factor of the fundamental generator , and the exponential of the sum generates the finite transformation.

Figure 18.2: Action of a Lorentz transformation. A contravariant vector is multiplied by Λ; a rank-2 tensor is transformed by the tensor product of two copies of Λ. Boost generators appear in blue, rotation generators in green.

The infinitesimal form is often the most useful for calculations. Expanding the exponential to first order shows that the change in a vector is

(18.11)

while the change in a tensor is the sum of the generator acting on each index separately. These variations are exactly the ones that enter the Noether procedure when the symmetry under consideration is a Lorentz transformation. Once the finite transformation law and its infinitesimal generator are both in hand, the construction of the associated conserved charges can proceed exactly as it did for translations.

Lorentz Invariance

A Lorentz transformation acts simultaneously on the coordinates and on the components of any field that lives on spacetime. If the coordinates transform by

(18.12)

then a field with several components transforms by the corresponding representation of the same generators,

(18.13)

The matrices are block-diagonal. Consequently the only non-vanishing entries of connect components that belong to one and the same irreducible multiplet—scalar, vector, or tensor. Components belonging to different fields do not mix under a Lorentz transformation.

This block-diagonal structure is the precise statement of Lorentz invariance for a multi-component field, where each irreducible piece transforms into itself, and the Lagrangian density, being a scalar, is unchanged when every field is replaced by its transformed counterpart and the coordinates are simultaneously transformed. The same property guarantees that the Euler–Lagrange equations retain their form in every inertial frame.

Figure 18.3: Block-diagonal action of a Lorentz transformation. A scalar, a vector, and a tensor each transform among their own components; components belonging to different fields do not mix.

Because the generators never mix distinct irreducible representations, the classification of fields by their Lorentz transformation properties is unambiguous. A scalar field has a single component and M'=0; a vector field transforms with the fundamental generators ; a rank-2 tensor transforms with the tensor-product generators constructed in the previous subsection. These transformation laws will be used both to verify the covariance of the Maxwell equations and to construct the Noether charges associated with Lorentz symmetry.

Applying Noether’s Theorem to Lorentz Transformations

When the Lagrangian density is a Lorentz scalar, Noether’s theorem supplies one conserved current for each independent parameter of a Lorentz transformation. Because those parameters are the six independent components of the antisymmetric matrix , there exist six conserved currents , themselves antisymmetric in the pair αβ,

(18.14)

Their divergences vanish on shell, and the associated charges are the components of the angular-momentum tensor.

The general Noether formula for a current associated with an infinitesimal variation generated by a parameter ε reads

(18.15)

For a pure Lorentz transformation we set (and all other parameters to zero). The coordinate variation and the field variation are then given by the corresponding generators,

(18.16)

and

(18.17)

Substitution yields the explicit current

(18.18)

The terms that multiply the coordinates and are precisely the components of the canonical stress-energy tensor already obtained from translational invariance. The remaining term is the spin contribution arising from the transformation of the field components themselves. Consequently the conserved current may be written compactly as

(18.19)

Figure 18.4: The Lorentz Noether current. The orbital contribution (blue) is built from the canonical stress-energy tensor; the spin contribution (green) arises from the action of the generators on the field components. Their sum is divergenceless on shell.

The six currents are the relativistic generalization of angular-momentum density. Their spatial integrals yield the total angular momentum of the system, including both the orbital part associated with the motion of the energy-momentum distribution and the intrinsic spin carried by the fields. Because the canonical stress-energy tensor is not symmetric, the orbital piece alone is not conserved; the spin piece is required to cancel the antisymmetric part of and to restore conservation. The same cancellation will reappear when the electromagnetic stress-energy tensor is improved by the Belinfante procedure, confirming that the Noether charge obtained here is identical to the angular momentum constructed from the symmetrized tensor.

Scalar Fields

For a theory containing only scalar fields the spin generators vanish identically. The Lorentz Noether current therefore reduces to its orbital part,

(18.20)

where is the canonical stress-energy tensor already obtained from translational invariance.

Consider first the spatial component . The corresponding finite transformation is a rotation about the -axis,

(18.21)

Invariance of the action under this rotation implies conservation of . Its charge density is

(18.22)

Recognizing as the momentum density, one sees that the spatial integral of is precisely the third component of . Defining the three-vector

(18.23)

then yields the familiar expression

(18.24)

Conservation of the projection follows from invariance under rotations about the axis .

The mixed components arise from boosts. For a boost in the -direction one has

(18.25)

The associated charge is

(18.26)

The integral of is the total momentum and the integral of is the total energy E. Writing for the center-of-energy coordinate then gives

(18.27)

Differentiating with respect to time produces

(18.28)

The velocity of the center of energy is the constant , so the derivative vanishes and is conserved.

Finally, the divergence of the current itself implies a symmetry property of the stress-energy tensor. On shell one has

(18.29)

[Explain on shell]

Figure 18.5: Conserved charges for scalar fields. Rotations (blue) produce the orbital angular momentum ; boosts (green) produce the center-of-energy relation -t P+E X.

Thus, for a theory of scalar fields, conservation of angular momentum forces the stress-energy tensor to be symmetric. For fields that carry intrinsic spin the same argument no longer holds; an additional spin term appears in the current and the canonical tensor need not be symmetric. That case is examined next.

Vector and Tensor Fields

For theories that contain vector or tensor fields the Lorentz generators no longer vanish. The Noether current associated with a Lorentz transformation therefore acquires an additional piece,

(18.30)

where the spin current is

(18.31)

The first two terms continue to represent the orbital angular momentum carried by the momentum density of the fields. The new term is the intrinsic angular momentum, or spin, that resides in the tensorial character of the fields themselves.

Because the generators act on the field indices, the spin current is nonzero whenever the fields transform nontrivially under the Lorentz group. For a vector field the generators are the fundamental matrices ; for a rank-2 tensor they are the tensor-product generators constructed earlier. In either case the spin contribution is local—it depends on the field values and their first derivatives at a single point—and does not involve the coordinate factors that appear in the orbital piece.

Conservation of the total current still holds on shell. The orbital piece alone, however, is no longer divergenceless, because the canonical stress-energy tensor of a spinning field is not symmetric. The divergence of the orbital contribution is precisely cancelled by the divergence of the spin current, so that only their sum is conserved. This cancellation is the relativistic statement that the total angular momentum—orbital plus spin—is the quantity protected by Lorentz invariance.

Figure 18.6: Angular momentum for fields with spin. The orbital contribution (blue) is built from the canonical stress-energy tensor; the spin contribution (green) arises from the Lorentz generators acting on the field indices. Only their sum is divergenceless.

The same decomposition will reappear when the electromagnetic field is examined. The photon carries spin 1, and the spin current constructed from the field strength supplies the term needed to symmetrize the Maxwell stress-energy tensor. The pattern established here—orbital piece from the coordinates, spin piece from the generators—therefore governs every relativistic field theory.

Electrodynamics

The general decomposition of angular momentum into orbital and spin pieces becomes concrete once it is applied to the electromagnetic field. The question is whether the field itself carries intrinsic angular momentum. To answer it one begins with the Lagrangian that couples the electromagnetic potential to charged matter,

(18.32)

The matter fields are scalars under Lorentz transformations, while the potential transforms as a vector. Consequently the only non-vanishing generators in the spin current are those that act on the vector index of .

The finite transformation that mixes the components may be written

(18.33)

where the generators are block-diagonal,

(18.34)

Substitution into the general expression for the spin current then yields

(18.35)

The spatial components of this current define a spin density

(18.36)

In terms of the electric field and the vector potential one finds the compact relation

(18.37)

To see that the spin density is nonzero for a realistic field configuration, consider a circularly polarized plane wave propagating in the -direction. A suitable potential is

(18.38)

The upper (lower) sign corresponds to right (left) circular polarization. The associated electric field is simply

(18.39)

and the spin density reduces to

(18.40)

Thus a circularly polarized electromagnetic wave carries a nonzero spin angular momentum density directed along its direction of propagation. The magnitude is proportional to the energy density of the wave, and the sign is fixed by the sense of polarization. This is the concrete realization, within classical electrodynamics, of the spin term that appears in the general Noether current for Lorentz transformations.

Figure 18.7: Spin density of a circularly polarized wave. The rotating electric field (blue) produces a spin density (green) aligned with the direction of propagation; the sign reverses with the sense of polarization.

The same spin current that yields is precisely the term required to cancel the antisymmetric part of the canonical electromagnetic stress-energy tensor. Once that cancellation is performed, the improved tensor becomes symmetric and the total angular momentum—orbital plus spin—is conserved. In this way the abstract decomposition introduced for general vector fields acquires a direct physical interpretation inside Maxwell theory.

The Electric Field in Matter

Charged matter produces an electromagnetic field and, when accelerated, radiates. When the same matter is placed in an external field it experiences forces; if the material has internal structure, those forces act between its constituent parts as well. In the present treatment irreversible currents inside the matter are set aside; only the reversible response associated with bound charge is retained.

In the local rest frame of a material element the charge density is the product of the charge per atom and the number density of atoms. The four-current of charge is therefore

(18.41)

where is the particle-number current already introduced for the material. Because the charge is bound to the atoms, the spatial components of vanish in the local rest frame. The relation is covariant, so it holds in every frame. Conservation of particle number together with the fact that the material coordinates are constant along the flow then imply

(18.42)

In the absence of interaction the total Lagrangian would be the sum of a pure-matter piece and a pure-field piece,

(18.43)

with

(18.44)

and

(18.45)

with the coupling introduced by the interaction term

(18.46)

Variation with respect to the potential immediately yields the inhomogeneous Maxwell equation

(18.47)

and this is the form required once the charge current of the material is taken as the source.

The stress-energy tensor likewise splits into three contributions,

(18.48)

The matter piece is the one already obtained for a continuous medium,

(18.49)

The electromagnetic piece is the canonical tensor constructed from the field strength,

(18.50)

while the interaction piece is

(18.51)

Four-momentum balance for the material then reads

(18.52)

The right-hand side is precisely the Lorentz force density exerted by the field on the charges bound in the matter. Thus the same variational principle that produces Maxwell’s equations also produces the force law that governs the motion of the charged continuum.

Figure 18.8: Charged continuum in an external field. The bound four-current (blue) sources the field; the field exerts the Lorentz force density (red) that appears on the right-hand side of the material momentum balance.

The construction just given still treats the microscopic charges as structureless. When the charges inside each molecule are allowed to separate under the influence of the local field, polarization appears and the macroscopic Maxwell equations acquire the familiar form involving and . That refinement is the subject of the following subsections.

Symmetrizing the Stress-Energy Tensor

Whenever a field theory possesses a nonzero spin current , the canonical stress-energy tensor fails to be symmetric. The asymmetry is not an obstacle to the conservation of four-momentum, but it is inconvenient for the electromagnetic canonical tensor, for example, is not gauge-invariant. A symmetric improvement can be constructed that yields the same total four-momentum, is divergenceless on shell, and, in the electromagnetic case, is gauge-invariant.

The spin current is antisymmetric in its first pair of indices. From it one builds an auxiliary tensor that is antisymmetric in its last pair,

(18.53)

The improved stress-energy tensor is then defined by

(18.54)

Because is antisymmetric in νλ, the divergence of the added term vanishes identically,

(18.55)

Thus is conserved whenever is conserved.

The two tensors also give the same total four-momentum. Integrating the difference over a spacelike slice and using the antisymmetry of G produces only surface terms that vanish at infinity,

(18.56)

Hence the integrated charges coincide.

It remains only to verify symmetry. A short calculation that again employs the conservation of the angular-momentum current yields

(18.57)

The right-hand side is precisely the divergence of , and that vanishes on the shell. What does, :on the shell” mean? It means that the system is governed by a classical equation of motion. Therefore

(18.58)

The improved tensor is symmetric, conserved, and equivalent to the canonical tensor for all global charges. In electrodynamics it is, in addition, gauge-invariant, removing the last practical defect of the canonical expression.

Figure 18.9: Belinfante symmetrization. The canonical tensor (blue) is improved by the divergence of constructed from the spin current (green). The result is symmetric, conserved, and yields the same total four-momentum.

The same improvement will be applied to the electromagnetic field in a later subsection, where the explicit form of reduces to the familiar Maxwell stress-energy tensor written entirely in terms of .

More About Angular Momentum

Once a symmetric, conserved stress-energy tensor is available, an angular-momentum current can be written that contains only an orbital piece,

(18.59)

No explicit spin term appears. Because is both symmetric and divergenceless, the current is itself conserved. Its spatial integral therefore supplies a conserved angular-momentum charge built solely from the improved tensor.

To see that this charge coincides with the one obtained from the original Noether current, substitute the definition of Θ,

(18.60)

The combination of G terms that is free of derivatives is precisely the spin current,

(18.61)

Hence

(18.62)

When the difference is integrated over a spacelike slice, the divergence term becomes a pure surface integral. Antisymmetry of G in its last pair of indices forces that surface integral to vanish, so

(18.63)

The two currents therefore generate identical total angular momenta. One may use in place of the original Noether current whenever only the integrated charges are required. The advantage is that every contribution is now expressed through the single symmetric tensor

Figure 18.10: Equivalence of angular-momentum currents. The original Noether current (blue) contains both orbital and spin contributions; the improved current (green) is built solely from the symmetric tensor . Their spatial integrals coincide.

The same construction will be applied to the electromagnetic field, where reduces to the familiar Maxwell tensor written entirely in terms of . The resulting angular-momentum density then automatically includes the spin carried by the field.

The Symmetrized Electromagnetic Stress-Energy Tensor

The general Belinfante procedure is now applied to the coupled system of charged matter and the electromagnetic field. The Lagrangian is the one already introduced,

The associated spin current is

(18.64)

From it one constructs the improvement tensor

(18.65)

Adding the divergence of G to the canonical stress-energy tensor produces the improved tensor

(18.66)

The field equations set the last parenthesis to zero, so the interaction term drops out. The improved tensor therefore splits cleanly into a matter piece and a pure-field piece,

(18.67)

where

(18.68)

and

(18.69)

The electromagnetic contribution is symmetric by inspection and is built solely from the gauge-invariant field strength; it is therefore itself gauge-invariant.

In an inertial frame the components of recover the familiar expressions. The energy density is

(18.70)

The mixed components are the Poynting flux,

(18.71)

while the spatial block is the Maxwell stress tensor,

(18.72)

Four-momentum balance between the two subsystems now reads

(18.73)

The Lorentz force density appears as the rate at which four-momentum is transferred from the field to the matter; no residual interaction term remains in the stress-energy tensor itself.

Figure 18.11: The symmetrized electromagnetic stress-energy tensor. All components are written in terms of the gauge-invariant field strength : energy density (blue), Poynting flux (green), and Maxwell stresses (red).

Because is both symmetric and conserved (when the matter current vanishes), the angular-momentum current constructed from it in the previous subsection automatically includes the spin carried by the field. The canonical tensor, the spin current, and the Belinfante improvement have thereby been reduced to a single, gauge-invariant object that governs the energy, momentum, and angular momentum of the electromagnetic field.

Angular Momentum in Electromagnetic Fields

With the symmetrized electromagnetic stress-energy tensor in hand, the angular-momentum current is given entirely by the orbital expression constructed in the preceding subsection,

(18.74)

No separate spin term need be written; the spin carried by the field is already incorporated through the improvement that produced .

Consider a left-circularly polarized plane wave propagating in the -direction. The spin density obtained earlier from the canonical current is

(18.75)

[Explain this a bit better]

so the wave carries a nonzero intrinsic angular momentum density parallel to its direction of propagation. When the same configuration is examined with the improved current one finds

(18.76)

The vanishing is immediate once the momentum density is inspected where is proportional to and therefore points strictly along . The vector product with the transverse coordinates and is identically zero. The orbital contribution measured by therefore vanishes, while the total angular momentum of the wave remains equal to the spin obtained from the original Noether current. The two descriptions differ only by a spatial divergence; their integrated charges coincide.

The same pattern holds for any free electromagnetic field. The improved tensor automatically accounts for both the orbital motion of the energy-momentum distribution and the intrinsic spin associated with the polarization. Consequently the single current furnishes a complete, gauge-invariant account of the angular momentum carried by the field.

Figure 18.12: Angular momentum of a circularly polarized wave. The Poynting vector (green) lies along the propagation direction, so the orbital contribution built from vanishes; the total angular momentum is pure spin.

The construction just completed shows that every conserved quantity associated with the Poincaré group—energy, momentum, and angular momentum—can be expressed through the single symmetric, gauge-invariant tensor . The same tensor will serve as the source of the gravitational field once the Einstein equations are introduced.

Polarized Materials

Many of the distinctive electromagnetic properties of bulk matter arise from the partial alignment of the electric and magnetic dipole moments of its constituent volume elements. That alignment is called polarization. The present section develops the continuum description of polarized media; the same framework will later be applied to plasmas.

The starting point is a dipole-moment tensor constructed directly from the microscopic currents that circulate inside each material element. Once that tensor is in hand, the microscopic Maxwell equations can be averaged to produce a closed set of macroscopic field equations. The averaging replaces the rapidly varying microscopic sources by a polarization and a magnetization; the macroscopic fields then satisfy equations whose sources are only the free charges and free currents. The same averaging procedure yields a macroscopic Lagrangian whose Euler–Lagrange equations are precisely the macroscopic Maxwell system.

Figure 18.13: Polarization of a material. Microscopic dipoles that are randomly oriented (blue) produce no net moment; partial alignment (green) yields a macroscopic polarization .

The construction that follows therefore accomplishes three things at once: it supplies the dipole-moment tensor in terms of the underlying currents, it produces the macroscopic Maxwell equations that govern the averaged fields, and it furnishes the Lagrangian from which those equations are derived. With those elements in place the response of polarized matter—and later of plasmas—can be examined within the same geometric framework already used for the vacuum field.

The Average Electromagnetic Field in Matter

Until now the electromagnetic field inside a material has been treated as though it varied smoothly on every scale. A classical atomic model shows that this cannot be literally true, where the microscopic field can fluctuate violently on atomic distances. The macroscopic field that appears in the continuum theory is therefore defined as a local average of the microscopic field about an observation event ,

(18.77)

The weighting function Δ is appreciable only inside a spacetime region whose spatial radius Λ is large compared with atomic dimensions yet small compared with any macroscopic length of interest, and whose temporal width Λ’ is long compared with atomic periods yet short compared with macroscopic times. Typical values are m and sec. An explicit normalized choice is the Gaussian

(18.78)

which satisfies

(18.79)

So long as every macroscopic length greatly exceeds atomic lengths, the precise values of Λ, Λ’ and the detailed shape of Δ drop out of the averaged field. To see why, consider the Fourier transform of a microscopic electric field,

(18.80)

The macroscopic field is the convolution

(18.81)

whose transform is simply

(18.82)

The Fourier transform of the averaging kernel is itself a Gaussian that equals unity for all wave-numbers small compared with 1/Λ and falls to zero for wave-numbers large compared with 1/Λ. Consequently every Fourier component that varies on macroscopic scales is transmitted unchanged, while the atomic-scale fluctuations are suppressed. The macroscopic field defined by the average is therefore independent of the precise choice of averaging parameters once those parameters lie in the intermediate range described above.

Figure 18.14: Spatial averaging. The microscopic field (red) fluctuates on atomic scales; convolution with the kernel Δ of width Λ produces the smooth macroscopic field (blue)..

The same averaging may be applied to every component of the field strength and to the microscopic current. The resulting macroscopic quantities satisfy a closed set of equations whose sources are only the free charges and the polarization and magnetization of the medium. Those equations, and the Lagrangian from which they follow, are constructed in the following subsections.

The Multipole Expansion of the Average Current

The macroscopic field is the average of the microscopic field; the same averaging applied to the microscopic current produces the macroscopic source. Because the microscopic Maxwell equation identifies the source with the divergence of the field strength, the averaged current

(18.83)

is precisely the source that appears on the right-hand side of the macroscopic inhomogeneous equation.

The material is regarded as a collection of N small elements. The microscopic current is the sum of the contributions of each element,

(18.84)

A representative world-line is assigned to the nth element, with the time component chosen so that . Each element is spatially localized where vanishes outside a region of diameter m about . Since d≪Λ, the weighting function may be expanded about the representative point,

(18.85)

Each successive term is smaller than the preceding one by a factor of order d/Λ.

Charge conservation supplies two integral identities that allow the expansion to be rearranged. After those identities are used, the macroscopic current takes the form

(18.86)

The first term is the free current carried by the motion of the elements themselves,

(18.87)

The second term is the divergence of the dipole-moment tensor

(18.88)

These are the first two terms in the multipole expansion of the averaged current. The monopole piece describes the free charges and currents; the dipole piece describes the polarization and magnetization of the material. Higher multipoles are suppressed by additional powers of d/Λ and may be neglected for ordinary macroscopic applications.

Figure 18.15: Multipole expansion. Each material element (size d≪Λ contributes a free current along its world-line (blue) and a dipole moment arising from internal currents (red). The macroscopic source is the free current plus the divergence of the dipole tensor .

The dipole tensor will be decomposed into its electric and magnetic parts in the following subsections; those parts become the polarization and magnetization that enter the macroscopic Maxwell equations.

Monopole Terms

The free part of the macroscopic current admits a simple interpretation once the spatial integral of the charge density of each element is performed. That integral is the total charge carried by the nth element, so

(18.89)

The combination , is precisely the infinitesimal displacement along the world-line of the element. The expression therefore collapses to a sum of line integrals,

(18.90)

Each term is the contribution of a point charge that travels along the representative world-line . In other words, the monopole piece of the averaged current is exactly the current that would be produced by a collection of structureless charged particles located at the centers of the material elements.

Figure 18.16: Monopole current. Each material element contributes as a point charge moving along its representative world-line; the macroscopic free current is the sum of these contributions, smoothed by the averaging kernel.

Higher multipoles arise only from the internal structure of the elements and are deferred to the next subsection. For many purposes the monopole term alone already supplies the free charge and free current that appear as sources in the macroscopic Maxwell equations.

The Dipole Term

The dipole contribution to the macroscopic current is completely determined by the antisymmetric tensor

(18.91)

where the dipole moment of the nth element is

(18.92)

In the instantaneous rest frame of the element, where , the components simplify. The mixed components become the electric dipole moment,

(18.93)

while the purely spatial components become the magnetic moment,

(18.94)

Thus the single relativistic tensor unifies the electric polarization and the magnetization of each material element. When these microscopic moments are smoothed by the averaging kernel they produce the macroscopic polarization and magnetization that source the macroscopic Maxwell equations.

Figure 18.17: Dipole moments of a material element. In the rest frame the mixed components are the electric dipole (blue) and the spatial components are the magnetic moment (green).

The same tensor will be decomposed into its irreducible pieces under the rotation group in the next subsection, making the identification with the conventional polarization and magnetization completely explicit.

Frame Dependence

The dipole tensor written in the preceding subsection refers to a preferred frame in which the time axis is fixed by a one-form . The question is whether that choice affects the tensor itself.

A world-line may be parametrized by any monotonic parameter . The map is independent of . When the four-velocity is written with respect to this parameter, plays the role of a proper-time coordinate that is constant on the planes of constant . The level-set condition

(18.95)

then determines the parameter, and the chain rule supplies the identity

(18.96)

Substitution of this identity into the definition of produces an expression that still appears to depend on .

An infinitesimal change induces a variation of the parameter. After the resulting variation of is integrated by parts, every term that survives is proportional to a derivative of the averaging kernel. Because that kernel varies only on the scale Λ, while the support of each microscopic current is of size d, the variation satisfies

(18.97)

The dipole tensor therefore changes only by a quantity of relative order d/Λ when the preferred time axis is altered. To the accuracy already adopted in the multipole expansion, is independent of the choice of frame.

Figure 18.18: Frame independence. Changing the preferred time axis (blue to green) alters the dipole tensor only by a relative amount of order d/Λ, which is already neglected in the multipole expansion.

The same conclusion applies to the macroscopic polarization and magnetization constructed from . They may be computed in any convenient frame; the result, to the order retained here, is unique.

The Field-Dipole Moment Interaction

Once the averaged current has been decomposed into free and dipole pieces, Maxwell’s equations take the form

(18.98)

The dipole tensor therefore acts as an additional source. The corresponding interaction energy density is the scalar contraction

(18.98)

Adding this term to the free-field and free-current contributions produces the macroscopic Lagrangian

(18.100)

In the rest frame of the material the contraction reduces to the familiar expression

(18.101)

where and are the electric polarization and the magnetization constructed from the mixed and spatial components of . The same scalar appears as the interaction energy of a dipole with an external field, confirming that the relativistic coupling introduced above is the correct continuum generalization of the elementary dipole interaction.

Figure 18.19: Field-dipole interaction. The contraction reduces in the rest frame to , the energy density of polarized matter in an external field.

The Matter Dipole-Moment Interaction

Polarizing a material requires energy. The simplest continuum description of that energy is a quadratic form in the electric and magnetic dipole densities,

(18.102)

The coefficients κ and χ function as spring constants; they are material-dependent and may vary from point to point.

To obtain a Lorentz-covariant expression one replaces the rest-frame squares by the appropriate contractions with the four-velocity of the material,

(18.103)

The complete Lagrangian of matter, field, and polarization is then

(18.104)

Variation with respect to the potential recovers Maxwell’s equations. Variation with respect to the dipole tensor itself yields the linear constitutive relations. In the local rest frame they reduce to

(18.105)

and

(18.106)

The covariant statement of the same stationarity condition is

(18.107)

The dimensionless coefficient κ is the electric susceptibility; for ordinary materials it ranges from to . The magnetic susceptibility χ is typically far smaller: negative for diamagnetic substances and positive for paramagnetic ones.

Figure 18.20: Linear response. In the rest frame the polarization is proportional to the electric field and the magnetization is proportional to the magnetic field; the constants of proportionality are the electric and magnetic susceptibilities.

Other Choices of Polarization Fields

The components of the dipole tensor have so far been treated as the fundamental fields that describe polarization. Polarization, however, is a property of the matter itself. It is therefore natural to introduce fields that live on the material manifold and are dragged along with the flow.

Define the Lagrangian-frame polarization and magnetization by

(18.108)

In the rest frame the inverse relations are simply

(18.109)

The same relations, written covariantly, reconstruct the spacetime dipole tensor from the material fields and the particle current,

(18.110)

When these expressions are substituted into the earlier Lagrangian one obtains

(18.111)

The polarization fields and are now the independent variables that characterize the material response; the spacetime tensor is a derived object. Variation with respect to and again yields the linear constitutive relations, but the underlying variational principle is expressed entirely in terms of quantities that transform as scalars under changes of the material coordinates.

Figure 18.21: Material-frame polarization. The fields and are defined on the body manifold; the spacetime dipole tensor is obtained from them by the flow of the material.

The Electric Susceptibility Matrix

The electric contribution to the polarization energy may be generalized by replacing the deformation tensor with an arbitrary symmetric matrix . With n the number of particles per Lagrangian volume element the term becomes

(18.112)

Variation with respect to the material polarization then yields the linear relation

(18.113)

where the Lagrangian-frame electric field is the pull-back

(18.114)

In the special case of an undeformed, stationary element this reduces to the familiar laboratory expression

(18.115)

For a deformed element observed in its local rest frame the same relation acquires the Jacobian factors of the deformation,

(18.116)

The field equations remain unchanged if the susceptibility matrix is allowed to depend on the material coordinates; such dependence simply describes an inhomogeneous medium.

When the strain is small the susceptibility may be expanded to first order,

(18.117)

The unstrained matrix is symmetric and therefore possesses six independent components. The tensor is symmetric in each pair i j and k l, giving thirty-six independent components in the most general anisotropic solid.

Isotropy imposes a strong restriction. For every rotation matrix A one must have

(18.118)

The only tensors compatible with this requirement are

(18.119)

and

(18.120)

The two scalars and measure the change in susceptibility under pure compression and under pure shear, respectively. For a dielectric fluid the susceptibility is isotropic in the current configuration and takes the still simpler form

(18.121)

Figure 18.22: Strain dependence of the susceptibility. In an isotropic solid the unstrained response is a pure multiple of the identity (blue); small strain adds independent compression and shear corrections (green).

The Magnetic Susceptibility Matrix

The magnetic contribution to the polarization energy is treated in exact analogy with the electric term. The simplest covariant expression is

(18.122)

Replacing the deformation tensor by a general symmetric matrix and introducing the particle density n yields the more flexible form

(18.123)

Variation with respect to the material magnetization then produces

(18.124)

where the Lagrangian-frame magnetic field is the pull-back of the dual field strength,

(18.125)

In the local rest frame the same relation becomes

(18.126)

As in the electric case, the susceptibility matrix may depend on the material coordinates, thereby describing a magnetically inhomogeneous medium, and may be expanded in powers of the strain when the deformation is small. Isotropy again reduces the independent coefficients to a pair of scalars that govern the response to compression and to shear.

Figure 18.23: Magnetic response. The magnetization is obtained from the magnetic field by contraction with the susceptibility matrix ; deformation of the material mixes the laboratory components through the deformation gradients.

Another Look at the Stress-Energy Tensor

The complete Lagrangian of the coupled system is the sum of seven contributions,

(18.127)

The pure-field and free-current pieces are the familiar expressions

(18.128)

and

(18.129)

while the field-dipole couplings are

(18.130)

The remaining terms are the quadratic polarization energies and the matter internal energy already introduced.

The canonical stress-energy tensor is constructed in the usual way,

(18.131)

Its associated spin current is

(18.132)

After the Belinfante improvement that employs Maxwell’s equations, the symmetric tensor splits into the same seven pieces,

(18.133)

The pure electromagnetic contribution recovers the vacuum Maxwell tensor,

(18.134)

the free-current piece vanishes identically, and the remaining pieces are the interaction and material terms listed in the original calculation.

In the material rest frame the energy density, momentum density, and stresses reduce to

(18.135)

and

(18.136)

together with the corresponding spatial stresses. Introducing the auxiliary fields

(18.137)

and

(18.138)

rewrites the energy density and the Poynting flux in the compact forms

(18.139)

and

(18.140)

When the susceptibilities themselves depend on strain, additional stress contributions appear. To first order in the strain one has

(18.141)

and

(18.142)

The polarization-matter and magnetization-matter stresses then become

(18.143)

and

(18.144)

where the barred coefficients absorb the deformation gradients. The complete electromagnetic stress in a solid is therefore

(18.145)

The electric and magnetic sectors enter symmetrically.

For a polarizable fluid the susceptibilities are isotropic in the current configuration,

(18.146)

and

(18.147)

The interaction stresses simplify to

(18.148)

and

(18.149)

Collecting terms yields the electromagnetic stress of a fluid,

(18.150)

All contributions are now expressed through the macroscopic fields , and the derivatives of the susceptibilities with respect to specific volume.

Figure 18.24: Electromagnetic contributions to the stress-energy tensor in matter. Energy density (blue), energy flux (green), and the stresses built from and (red), together with the strain corrections that arise when the susceptibilities depend on deformation.