# Representations and Implications of Papers Written by E.T. Whittaker in 1903
and 1904

Mark Titleman

(April 30, 2022)

Abstract. In 1903 and 1904, E.T. Whittaker produced a pair of papers that,
although sometimes considered mere mathematical statements (Barrett 1993),
held important implications if viewed as physical realities. His 1903 paper
united electrostatic and gravitational attraction as both resulting from
longitudinal waves in the medium. His 1904 paper specifically dealt with
electromagnetic waves as resulting from the interference of two such
longitudinal waves as scalar potential functions. These papers, if
representative of physical reality, may have important implications:
gravitational lensing, gravitational waves, the Aharonov-Bohm effect, the
existence of a hyperspace above or behind normal space, and the elimination of
gravitational and point charge singularities.

Keywords: Whittaker, Gravitational Lensing, Gravitational Waves, Aharonov-Bohm
Effect, Singularities

Introduction

In light of the fact that gravitational theories today face several
difficulties – the requirement for dark energy, the requirement for dark
matter, the existence of unphysical gravitational singularities, and
incompatibility with Quantum Mechanics – it may be useful to uncover and
explore older classical theories of gravity that overtly or implicitly offered
several features of General Relativity. One such theory was put forward by
E.T. Whittaker in 1903 and 1904. His 1903 paper on partial differential
equations anticipated General Relativity in many ways by positing both an
undulatory theory of gravity and a static gravitational field that resulted
from effects propagating through the field. His 1904 paper on two scalar
potential functions constructed the electromagnetic wave from two scalar
potential functions, showing that the electromagnetic four-potential of
Relativity may possess a degree of arbitrariness. This paper will serve to
review these two overlooked papers and highlight some of their overt and
implicit consequences.

Representations of the Works of E.T. Whittaker

E.T. Whittaker’s 1903 paper “On the Partial Differential Equations of
Mathematical Physics” found a general harmonic solution to the Wave Equation
and the more specific Laplace Equation in three dimensions in such a way that
both potentials could be analyzed into simple plane waves. An electrostatic or
gravitational field, varying with the inverse square of distance, results from
such propagating effects. The general solution to the Laplace equation was in
the form:

(1)

This is achieved by expanding the function f as a Taylor series with respect
to the first argument and a Fourier series with respect to the second
argument. The v is considered a periodic argument. Using similar techniques,
the general solution to the Wave Equation was found in the form:

(2)

Whittaker considered “gravitation and electrostatic attraction explained as
modes of wave-disturbance” (Whittaker 1903). Once these longitudinal waves
interfere with each other and an action is set up, the disturbance at any
point does not depend on time but only on position; they are standing waves.
Therefore, the force potential can be defined in terms of both standing waves
(global, or non-local solution) and by propagating waves (local solution
changing in time) (Barrett 1993). This produced an undulatory theory of
gravity propagating with a finite velocity and subsumed gravity to the
transmission of electromagnetic radiation. That the aether produced
longitudinal as well as transverse electromagnetic waves was a common belief
of 19th century physicists and was given a mathematically rigorous treatment
by Whittaker in 1903 (Carvalo and Rodrigues 2008).

Hector Munera writes that Whittaker’s claim of generality is unacceptable due
to the assumption of periodicity. Authors subsequent to Whittaker wrote that,
in most practical cases, the solution is expected to vary harmonically in
time. Munera also importantly reminds the reader that equation (1) implies a
rotation in the complex plane $(z,ixcos\theta+iysin\theta)$. Finally, Munera
writes that a time-dependent solution in the form of equation (2) is arrived
at by “projecting z and $xcos\theta+ysin\theta$ onto ray r directed at angle
$\phi$ relative to the Z-axis, thus shifting to spherical coordinates” (Munera
2018).

In 1904, Whittaker’s paper ”On an Expression of the Electromagnetic Field due
to Electrons by Means of Two Scalar Potential Functions” showed that EM fields
specifically could be decomposed into two scalar potential functions that
could be assembled as intersecting beams in which ordinary EM waves appear.
Whittaker achieved this by first defining three scalar fields which are
related by time and space derivatives to the traditional scalar electric
potential and vector magnetic potential. One of the fields is eliminated by
choosing a gauge. The two scalar potentials F and G can derive the magnetic
force h and dielectric displacement d as:

(3)

F and G are represented in terms of magnitude as:

(4)

The summation is taken over all the electrons in the field. In continuous form
they are represented as:

(5)

Whittaker rewrote these two scalar potentials in his 1951 version of A History
of the Theories of Aether and Electricity as:

(6)

Whittaker wrote:

> It will be noted that F and G are defined in terms of the positions of the
> electrons alone, and do not explicitly involve their velocities. Since in
> the above formulae for d and h an interchange of electric and magnetic
> quantities corresponds to a change of G into F and of F into G, it is clear
> that the two functions F and G exhibit the duality which is characteristic
> of electromagnetic theory: thus an electrostatic field can be described by F
> alone, and a magnetostatic field by G alone; again, if the field consists of
> a plane wave of light, then the functions F and G correspond respectively to
> two plane-polarised components into which it can be resolved. Since there
> are an infinite number of ways of resolving a plane wave of light into two
> plane-polarised components, it is natural to expect that, corresponding to
> any given electromagnetic field, there should be an infinite number of pairs
> of functions F and G capable of describing it, their difference from each
> other depending on the choice of the axes of co-ordinates-as is in fact the
> case. Thus there is a physical reason why any particular pair of functions F
> and G should be specially related to one co-ordinate, and cannot be
> described by formulae symmetrically related to the three co-ordinates
> (x,y,z) (Whittaker 1951).

Implications of the Works of E.T. Whittaker

The combination of Whittaker’s 1903 paper on gravitation and electrostatic
attraction and his 1904 paper on electromagnetic wave propagation shed new
light on these phenomena in three dimensions by treating the z direction
(propagation direction) differently than the other two spatial directions.
Gravity may thus manifest itself solely in the y direction (Yurth 2007) due to
the preferred direction of the potentials and the mass-proportionality of the
medium with respect to propagation speed (Laszlo 2003). Gravitational lensing
would result from the two scalar potentials interfering with each other to
create electromagnetic waves by means of their mass-proportionality and
preferred direction.

These papers account not only for gravity and electromagnetism as modes of
disturbance in the same medium, but could provide an explanation for
gravitational waves. Whittaker explicitly claimed that his theory of
gravitational and electromagnetic potentials provided for an “undulatory
theory of gravity” (Whittaker 1903).

These papers directly anticipated the Aharonov-Bohm effect since the
potentials F and G are considered the most basic entities. While fields
require the potentials to exist, the potentials can exist on their own. This
is demonstrated experimentally with the Aharonov-Bohm effect.

Additionally, the scalar potential (normally dimensionless) is given a
discrete and quantized structure as resulting from the interference of two
superluminal longitudinal waves in the medium itself. This medium is implied
as hyperdimensional with its existence defined in terms of frequency
resonance.

Singularities can potentially be eliminated by this theory. A point charge as
one type of singularity would not exist; rather, charge would work
collectively as longitudinal motion in the aether that carries electromagnetic
radiation. The need for a propagation medium for transverse Hertzian waves was
in fact seen by Maxwell and other 19th century physicists since the waves
consist of orthogonal electric (E) and magnetic (B or H) waves. The E waves
are undulating dipolar electric fields and all such dipoles were considered to
require separated and opposite electric charges in the 19th century. The
concept of massless charge as motion in the two scalar potentials would
account for this. Furthermore, gravitational singularities would not need to
exist. Solutions to light propagation around black holes were provided by
Whittaker by first considering Maxwell’s equations in a dielectric medium
(Whittaker 1928). Black holes can be viewed in the same light as the two
scalar potentials, although they are complex and three-dimensional. Thus,
black holes may work collectively as charge does, exist as part of the
hyperspatial structure, and may even generate the Whittaker potentials
(localized around each galaxy’s supermassive black hole) by extracting an
aspect of mass from infalling matter.

Conclusion

The works of E.T. Whittaker have been neglected by the existing community of
physicists, but may hold fruitful avenues for exploration. It is possible that
a model of gravity based on these papers could do away with singularities as
well as dark matter and dark energy. They could be developed and modelled in
order to answer some of the open questions in physics.

References

BARRETT, TERENCE W. “Electromagnetic Phenomena Not Explained by Maxwell’s
Equations.” Essays on the Formal Aspects of Electromagnetic Theory, 1993, pp.
6–86.

Evans, Myron. Modern Nonlinear Optics. Wiley, 2001.

Laszlo, Ervin. The Connectivity Hypothesis Foundations of an Integral Science
of Quantum, Cosmos, Life, and Consciousness. State University of New York
Press, 2003.

MÚNERA, HÉCTOR A. “Neo-Cartesian Unified Fluid Theory: From the Classical Wave
Equation to De Broglie’s Lorentzian Quantized Mechanics and Quantized
Gravity.” Unified Field Mechanics II: Formulations and Empirical Tests, 2018.

TROVON DE CARVALHO, A. L., and W. A. RODRIGUES. “The Non Sequitur Mathematics
and Physics of the ‘New Electrodynamics’ Proposed by the Aias Group.” Random
Operators and Stochastic Equations, vol. 9, no. 2, 2001/

Whittaker, E. T. A History of the Theories of Aether and Electricity / Rev.
and ENL. Ed. Thomas Nelson, 1951.

Whittaker, E. T. “On an Expression of the Electromagnetic Field Due to
Electrons by Means of Two Scalar Potential Functions.” Proceedings of the
London Mathematical Society, vol. s2-1, no. 1, 1904, pp. 367–372.

Whittaker, E. T. “On the Partial Differential Equations of Mathematical
Physics.” Mathematische Annalen, vol. 57, no. 3, 1903, pp. 333–355.

WHITTAKER, E. T. “The Influence of Gravitation on Electromagnetic phenomena1.”
Nature, vol. 121, no. 3061, 1928, pp. 1022–1024.

Yurth, David. ”Y-Bias and Angularity: The Dynamics of Self-Organizing
Criticality From the Zero Point to Infinity.” 2007.