Force in Hertzian Electrodynamics

Introduction

Hertz’s field equations, which are first-order invariant under the Galilean transformation,
are formally the same as Maxwell’s covariant equations, except that every appearance of the non-invariant operator

(1)

The “convective” parameter

(2a)

(2b)

(2c)

(2d)

where

(3)

under the Galilean transformation

(4)

where

(5a)

Eq. (2c) allows us to introduce a Hertzian vector potential via

(5b)

Eq. (2b) then yields

(6)

Any quantity having a vanishing curl can be represented as the gradient
of a scalar func-tion, so

(7)

an invariant expression for

(8)

In Maxwell’s theory we know that the definition of electric field is force per unit charge. Thus the force on charge q is

(9a)

where

(9b)

and where *about the* whole physical force on an electric charge may be viewed as a flaw.
The necessity in Maxwell’s theory to introduce an extra “force postulate” is certainly not impressive evidence of logical economy.
It creates a gulf between electromagnetism and electrodynamics that might be judged aesthetically displeasing.
Let us see whether the Hertzian invariant formulation of electromagnetism can do better.

Hertzian Force

Since the electric field is traditionally * defined * as the force on unit electric charge, it seems anomalous in Maxwell’s
theory to say instead that such force involves some-thing additional (*viz*., the magnetic component of the Lorentz force). If in Hertzian theory
the whole of electrodynamics is to be contained in electromagnetism, the simplest realization of this objective requires that the full physical
force on purely-electric charge q be expressible in terms of the purely-electric field solution of the field equations,

(10)

To say that the force on q is something else is inconsistent both with the above verbal definition of electric field and with logical
economy. Since * a priori*. Let us therefore examine how such a “Hertzian” force is related to the Lorentz force. From
(7) we see that (10) implies

(11)

wherein the potentials are source-related and presumably differ from those of Maxwell’s theory at most by a gauge transformation. From Eq. (1) we have

(12)

The vector identity [4]

(13)

Inserting this in (12), we have

(14)

Since we can identify our “field detector” (which has velocity

(15)

To find out more about this, let us subject (14) to a gauge transformation,

(16)

where

(17)

From the fact that

(18)

This evaluates

(19)

and leaves the gauge indeterminate within an additive constant. The consequence of this gauge choice is a physical force of the form

(20a)

or, with (5b),

(20b)

which has the form of the familiar Lorentz force law, Eq. (9), with the Hertzian potential quantities formally appearing for the Maxwellian
ones. Thus by a suitable gauge choice it is possible to relate the Hertzian field quantities to observable force on a charge q by the same formal law as that
traditionally employed to accomplish the same thing in Maxwell’s theory. It is reasonable that this should be possible, since the Hertzian theory is a covering
theory of the Maxwellian, in the sense that in the special case

It remains an open question – which we must leave unresolved here – whether the gauge choice that accomplishes this elimination of the “extra” force
(15) is the physically valid one. That is, Hertzian theory is not fundamentally either gauge-symmetrical or space-time symmetrical, and thus leaves open
the possibility that the observable physical force on a charge might differ from the Lorentz force by a term proportional to

Concerning the possibility of an extra force of the form (15), corresponding to the gauge choice

The important aspects of the Hertzian force law (14) are (a) that it is deduced di-rectly from the Hertzian field equations, without additional postulation, (b)
that it is not distinguishable by ordinary laboratory experiments from the empirically well-confirmed Lorentz force law for the total force on charge q, (c) that
it can be reduced by a gauge transformation to a formal analog, Eq. (20), of the Lorentz law, and (d) that the Hertzian electric field, as given by Eq. (7),
produces the full physical force on the charge, Eq. (10), in agreement with the simple physical interpretation (or definition) that

We note that in Eq. (14) two gradient terms appear. These can be combined to form what we might at first guess to be a “physical potential energy,” denoted by V; i.e.,

(21)

In order to investigate this as a possible physical potential energy, it will be useful to take a brief excursion into the territory of the Lagrangian method.

Lagrangian Formalism

From the well-known theory of the Lagrangian method [5] we shall borrow only two relations, first

(22)

which defines the Lagrangian function L in terms of kinetic energy T and a “generalized” Lagrangian potential energy U, which in the case of velocity-dependent potentials gener-ally differs from the physical potential energy V. Secondly, the Legendre transformation,

(23)

where H is the Hamiltonian, here interpreted as the total physical energy, and the

(24)

where k is an undetermined constant. We may simplify to a one-dimensional equivalent problem by introducing a scalar coordinate r aligned
along the direction of the vector

(25)

where

(26)

whence

(27)

From (23), written as

or

(28)

Notice that the k-terms have canceled; so k can be assigned any value, including zero. Thus our guess embodied in Eq. (21), that the physical
potential energy V (defined as that which enters the Hamiltonian) might contain a term proportional to *U*, the physical potential
energy *V* can only be

(29)

The customary Lagrangian derivation [5] of the Lorentz force law assumes

and the term in

is validated, apparently, by endowing the canonical momentum * U* is nonphysical, one is hardly prepared to learn that the corre-sponding
Lagrangian generalized (“canonical”) momentum

All that can be said, finally, concerning (21) is that it was a bad guess about the physics. In view of (29) and the assumption that H is total physical energy,
it would ap-pear that the extra term in

We are assuming that H in electromagnetic theory represents total observable en-ergy. What can be said about this? With velocity-independent potentials the
Lagrange method is known to be reliable. But there is no question – when one deals with velocity-dependent potentials – that the U-function of the Lagrange method
is non-physical and thus amounts to a contrivance – as does the Lagrangian itself. It seems evident that the canonical momentum is also not the physical momentum,
since it includes a “field mo-mentum” term that does not survive to make a contribution to the total physical energy (28). Why is field momentum absent from
total energy? This may be the case because it is permissible to think of the force-exerting field agent, the photon, either as “virtual” – in which case
it manifests no physical attributes – or as possessed of no degrees of freedom independent of matter – in which case it can affect the momentum of its source
and of its sink, but not of intermediate space. In accord with the spirit of quantum mechanics, there is no way to capture or reify this alleged momentum while
it is acting across space, ex-cept to replace “space” with a detector. And if L is a non-physical contrivance and H does not represent total physical energy, what
has become of the physics in all this carni-val of formalism? It would seem best to hold onto H as total energy and to revise our last-century picture of
electromagnetic force as “propagating” – i.e., as energy flying (lo-cally and causally) through space – such a picture being a metaphysical imposition at
odds with all else we have learned about quantum processes.

It is worth mentioning that in Eq. (6-31) of his book [5] Goldstein expresses the Lorentz force in a form equivalent to

(30)

Thus, had he not been blinded by the science of covariance he might have discovered the science of invariance, simply by recognizing the
closed-circuit unobservability of the last gradient term … for with its elimination Eq. (30) reduces to our presently proposed Eq. (11), the Hertzian
invariant form.

We must leave the subject here with the tentative conclusion that the Hertzian and Lorentz force laws are probably for all observational purposes physically
equivalent. If that is not correct, then experiment must be able to decide whether the Hertzian Eqs. (11) and (14) should be modified by addition of the final
gradient term in (30) … which is the same question as whether Eq. (30) should be modified by omission of its last term. By one way of looking at it, the
physicality of the Lorentz force law depends on the ob-servability of the last term in (30) – for, failing such observability, it would become physically
permissible to omit that term entirely – in which case the Hertzian and Lorentz force laws would become not merely predictively equivalent but formally identical.

Hertzian vs. Maxwellian Fields

To those reared on covariance, the invariance claimed in Eq. (3) may seem counter to known fact. Thus the field “scramblings” (whereby
electric fields can “be-come” magnetic fields or linear combinations of both kinds of fields) asserted by covari-ance are today widely accepted as an inherent
feature of the field; i.e., a fact about the underlying physics. But what substantiates this “fact” is not the changing of one kind of field into another ...
what changes in every case is the state of motion of the field detec-tor. The field itself, conceived as ding an sich – something “out there” that is independent
of the state of motion of its detector – does not exist. (This does not imply that there is “nothing” out there … no ontological profundity is intended.) When we
say that one inertial observer “sees” one kind of field, and another inertial observer sees another kind, what we are in fact saying is that field detectors in
different states of motion (each co-moving with a different inertial observer) detect different mixes, or covariantly-related scramblings, of components of the
field at a common “field point” that is momentarily shared in space and time. The two observers disagree about what the field “is” at that point, because they
are measuring the “same field” with differently-moving instruments. In a sense, each instrument may be said to create its own field. (This is a variant of Bohr’s
famous dictum, “The apparatus as a whole makes the measurement.”) However, if instead we consider a given state of motion, defined by that of a single
field-detection instrument, then all observers must agree on the readings of that chosen instrument; so in this case there is no ambiguity or multi-valuedness.
We can then speak of an observer-independent uniqueness or invariance of the field.

In order to bring out explicitly and quantitatively this invariant aspect of field de-scription it is necessary to abandon covariant (Maxwellian) field equations
and substitute invariant ones. This is what Hertz’s mathematics did at the first order, to which we con-fine attention here. To recapitulate: In Maxwell-Einstein
physics each inertial observer is equipped with his own “private” field detector, permanently at rest at his own co-moving field point. When the field points of
two such relatively-moving observers mo-mentarily coincide – implying collision of their instruments – the measured field compo-nents, as displayed by the two
instruments, are two sets of numbers related covariantly and quantified by Maxwell’s field equations, considered valid in both inertial systems. (It is this dual
validity that underpins the constancy of “c” in all inertial systems – Ein-stein’s second postulate.)

By contrast, in Hertzian physics there is only one “public” field detector under consideration, and there are any number of observers, who need not be inertial
– but may for convenience be considered so. In order to quantify, by Hertz’s field equations, the readings at any moment of that unique “public” instrument
(idealized as a point in space), we may picture a multiple coincidence of field points (co-moving with each of two or more inertial observers) with that
instrument. (Such observer-instrument relative veloci-ties are parameterized by .) Then the various observers involved must at that event of multiple
coincidence read from the display of that particular instrument the same num-bers – for the simple reason that there are no other numbers there to be read.
This is the (trivial) physical meaning of numerical invariance, which is reflected also in form invari-ance [cf. Eqs. (3) and (8), above]. Since the
observers’ field points are notional (mathematical points), their “collisions” with the instrument cause no physical disruption, such as would occur in the
case of covariance (multiple instruments, each a composition of matter, in physical coincidence). Since Maxwell’s equations are not involved, there is no
universal constancy of c. All inertial observers honor the relativity principle by using the same field equations – but these are Hertz’s, not Maxwell’s.
Hertz’s equations lack spacetime symmetry and assert a formal effect upon light speed of detector velocity . (This of course requires the development
of a new kinematics [1], in conjunction with higher-order refinement of the Hertzian equations themselves [2,3].)

It will be apparent that two quite distinct ideas of “invariance,” hence of “relativ-ity” are involved. The Maxwell-Einstein idea is that for different
inertial observers the “laws of nature” describing the field are the same because, when each observer is equipped with his own “private” field-measuring
instrument and performs the same measurement operations, each by replicating the experimental procedures of the others will measure (when field points
coincide) not the same numbers but sets of numbers co-variantly related. Such a conception of what motional “relativity” is about evidently in-corporates
a physical assumption of replicability of experiments, whether in the same frame or in different frames, either simultaneously or sequentially.

The Hertzian idea is entirely different and in a sense more primitive. It is that the job of physics is to describe what is out there in nature
(entirely apart from notional ob-servers), on a one-time, one-place, one-occurrence basis. Such description must be in-variant – that is, independent
of state of motion of any observers. In the case of fields, the essential element that is “out there” is the absorber or field detector – that which
“measures,” or by localizing “creates,” the field. Only a single instrument is involved in any (unique) episode of measurement, although multiple
observers may be present. Rep-licability of experiments is not assumed (nor is it, at the strictly classical level, denied). The mathematics
effectuating the Maxwell-Einstein view interprets “invariance” to mean covariance – the linear “scrambling” of field components – which by definition
discards numerical invariance. The mathematics effectuating the Hertzian view demands true in-variance, whereby all symbols appearing in the “laws of
nature” [e.g., the field equations, Eq. (2)] transform in place without altered relationships, as in Eqs. (3) and (8), so that both formal and numerical
invariance are attained.

At the classical level of physical description there is no obvious basis for prefer-ring one of these rival types of “relativity” and “invariance” over
the other. But when we consider the quantum limit of measurement theory, or address the weak-field (one-quantum) limit of field theory, the superiority
of the more primitive (or less assumption-laden) Hertzian version becomes at once evident. In a word, the Maxwell-Einstein field theory works only where
there are many field quanta present – enough in principle that sufficient numbers are absorbed by each notional detector, co-moving with its own
iner-tial observer, to iron-out statistical fluctuations. In that case the smoothed numbers dis-played on each detector are covariantly related. But,
where there are so few field quanta present that this smoothing ceases to be effective, covariance fails catastrophically. Thus if the field is so weak
that only a single photon is present, and two macroscopic detection instruments compete for it, only one can “win.” That is, only one of these instruments
can successfully “measure” the field. The other must register zero – which cannot be co-variantly related to anything. So, in the weak-field limit the
model of a plurality of “pri-vate” measuring instruments (underlying the covariance-based conception of “relativity”) fails – as does the assumption of
replicability of their measurements.

The Hertzian type of relativity, based on invariance, does not fail in either weak-field or strong-field limit. When a single “public” instrument is
present in a field of arbi-trary strength, it must contend with simple statistical fluctuations (more severe as the field weakens), but in a straightforward
way – without the complication of competition for field quanta with other macro instruments (notionally) present at the same place and time. There is no
assumption of replicability of measurements made simultaneously at a given place by a plurality of macro instruments, such as underlies the
Maxwell-Einstein picture of sets of measured field-component numbers covariantly related. The latter “classical” conception is counter-indicated by all
that the twentieth century has taught us about the physics of the quantum world. To put it in a nutshell, covariance fails prima facie if there are more
observers-cum-detectors present than field quanta – for in that case the competition is too fierce and some observers must fail to detect any field at all.

In sum: At high field intensities either the Maxwell-Einstein picture (covariance with many inertial-observers-cum-detectors) or the Hertz picture
(invariance with many observers of a single detection instrument) will work. But in the low-field limit only the Hertz picture remains conceptually viable and
compatible with quantum physics.

Acknowledgment

The writer is indebted to Dr. M. H. Brill for invaluable criticism bearing on this and related work.

References

[1] T. E. Phipps, Jr., Heretical Verities: Mathematical Themes in Physical Description (Classic Non-fiction Library, Urbana, IL, 1986).

[2] Idem., “Hertzian Invariant Forms of Electromagnetism” in Advanced Electromagnet-ism Foundations, Theory and Applications, ed. by T. W. Barrett and D. M. Grimes (World Scientific, Singapore, 1995).

[3] Idem., “When Four-space Falls Apart” in Has the Last Word Been Said on Classical Electrodynamics? – New Horizons, ed. by A. Chubykalo, A. Espinoza, R. Smirnov-Rueda, V. Onoochin (Rinton, Princeton, NJ, 2004).

[4] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

[5] H. Goldstein, Classical Mechanics (Addison-Wesley, Cambridge, MA, 1950).

Author:

T. E. Phipps, Jr.

908 S. Busey Ave.

Urbana, Illinois 61801

tephipps@sbcglobal.net

7 December, 2004

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