In this final section we give a brief account of Minkowski's energy tensor, and show how the electromagnetic field itself possesses energy, momentum, and stress—just like a material continuum. In this way the well-validated conservation laws of mechanics can be extended to interactions between charged matter and electromagnetic fields. For example, if we simultaneously release two oppositely charged particles from rest and they accelerate towards each other, where does the kinetic energy come from? Or, relative to another inertial frame where one of these charges begins to move before the other, where does the momentum come from? One can use 'potential' energy and even potential momentum as bookkeeping devices, but there are good reasons (both formal and physical) for considering the field itself able to exchange energy and momentum with matter. According to Einstein's general relativity, energy (that is, mass), momentum, and stress all curve spacetime (measurably, in principle) at their location, and so that location is no longer a matter of convention.
Consider, then, a charged 'fluid' in the presence of an electromagnetic field but subject to no other external forces. We define a (Lorentz-)4-force density vector K1 by a procedure analogous to that which we used in the definition of J1 [cf. after (7.38)], namely dividing all moving charges into classes with unique p0 and U1 (and corresponding u). For each class, K1 is the Lorentz force (7.29) on unit proper volume,4 which thus contains a charge p0; and the effective K1 exerted on the whole fluid at any event by the field is defined as the sum of these partial K^s, and hence is itself a 4-vector:
c c the last equation follows from (7.38) and subsequent text. On the other hand, if we write k for the partial 3-forces per unit proper volume, we have, from (6.44),
K1 = J2 Y(u)(k, c-1k ■ u) = (k, c-1 dW/dt), (7.71)
where k is the total 3-force per unit lab volume, and dW/dt is the rate of work done by the field on the fluid in a unit lab volume; for the effect of each y-factor in the summation in (7.71) is to convert from the various unit proper volumes to the unit volume fixed in the lab.
[From here to the end of (7.77) we now are in for some serious calculating!] By use of Maxwell's equation (7.40) we can next eliminate all reference to the sources from the RHS of eqn (7.70), and write that equation in the form
Kl = 4nElVE°V,a = 4n [(ElVE°V),a - EiV,aE°V]- (7.72)
The second term in the bracket can also be transformed into a derivative thus:
F Fav — I (F —E )F°V — 1F EaV — 1 (F FaV) (7 73)
ElV,aE — 2 \FlV,a Fia,V)F — 2 fov,If — 4 \fOVf ) ^, where in the first step we used the antisymmetry of Eav, in the second step the Maxwell equation (7.41), and finally the see-saw rule. So, combining (7.71), (7.72), and (7.73), we have it1 = (k, c-1 dW/dt) = -MlV,V, (7.74)
Eqn (7.74) is the Lorentz force law adapted to a charged continuum instead of a particle, with all reference to the charge or velocity of the continuum eliminated. When J1 or ElV vanish, so must all terms in (7.74), by (7.70).
4 What is meant by a 'quantity per unit volume' is, of course, the limit of that quantity for a finite volume V divided by V, as V ^ 0. On the other hand, heuristically, we can regard it as the actual quantity over an actual unit volume, provided we choose our units sufficiently small; and we are always at liberty to do so. A corresponding remark applies to a 'quantity per unit time'.
The tensor Mxv which has surfaced here, and which is easily shown to be symmetric and trace-free,
is the fundamentally important (Minkowski-)energy tensor of the electromagnetic field. For its components we find, directly from the definition and from (7.32):
4i c cM = — (e x b); =: s; = energy-current density (Poynting vector) 4n
Mij = —■1[eiej + bibj + 2gij(e2 + b2)] =: pij = Maxwell stress tensor,
where, however, the last entries in each line (the physical interpretations) have yet to be justified.
To that end, let us look at the separate components of eqn (7.74). First, setting X = 4 yields (after a sign-change)
dt dt which leads us to identify a as the energy density and s as the energy-current density of the field. For if dW/dt is the rate of work done by the field on the fluid, —dW/dt can be regarded as the rate of work done by the fluid on the field. And this should equal the increase of field energy, da/dt, in the unit volume, plus the outflux of field energy, div s, from that volume, in unit time. Next, let us set x = i in eqn (7.74). This yields (again after a sign-change)
1 dt dxj
Since k is the force of the field on the fluid, —k can be regarded as the force of the fluid on the field, and this should equal the rate at which field momentum is generated inside a unit volume. The first term on the RHS of eqn (7.79) should therefore represent the increase of field momentum inside a unit volume, and the divergence-like second term the outflux of field momentum from that volume. Accordingly we recognize c—2s; as the momentum density of the field and pij as the momentum-current density; that is, the flux of i-momentum through a unit area normal to the j-direction per unit time.
We note how c—2 times the energy current s; of eqn (7.78) serves as momentum density in eqn (7.79). This is a striking manifestation of Einstein's mass-energy equivalence. For an ordinary fluid we would have: energy current = energy density x velocity = c2 mass density x velocity = c2 x momentum density. Even though in general there is no way to associate a unique velocity with a given electromagnetic field (cf. Exercise 7.15), or, equivalently, to assign a rest-frame to it, it still has momentum and energy current and the two are related 'as usual'.5
Recall that force equals rate of change of momentum. If a machine-gun fires bullets into a wooden block, that block experiences a force equal to the momentum absorbed in unit time; that is, equal to the momentum current. Maxwell accordingly regarded Pij as the ¿-component of the total force which the field (!) on the negative side of a unit area normal to the j -direction exerts on the field on the positive side. Thus pij is called the Maxwell stress tensor of the field.
One can take this idea quite seriously. For example, consider a pure e-field parallel to the x-axis at some point of interest. Then from (7.77), p11 = -(1/8)ne2. But p11 is pressure in the x-direction. This leads to the idea that there is tension (negative pressure) along the electric field lines. Such tension 'explains' the attraction between unlike charges. Similarly, p22 = +(1/8)ne2. So there is pressure at right angles to the field lines, tending to separate them. This 'explains' the repulsion between like charges. (The reader will recall the well-known field-line patterns—materialized by iron filings—between both equal or opposite 'magnetic charges'; and the pattern is the same for electric charges.)
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