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\title{DRAFT: Deducing a Unified Field Theory from Electromagnetism}


\author{Douglas B Sweetser}


\email{sweetser@alum.mit.edu}


\address{39 Drummer Road, Acton, MA 01720}

\begin{abstract}
Logic will dictate the equations for gravity based on those for electromagnetism.
The classical action of electromagnetism is broadened to include a
symmetric tensor and a charge where like charges attract. Since the
action breaks U(1) symmetry for massive particles, the Higgs mechanism
is unnecessary. The 4D wave equation contains the classical static
laws of Gauss and Newton's law of gravity within the context of a
tensor equation consistent with special relativity. It is shown how
the Rosen metric is a solution to the field equations, and thus is
passes weak field tests of gravity to first-order Paraterized Post-Newtonian
(PPN) accuracy, and is distinguisable from general relativity at second-order
PPN accuracy. A spin 1 and spin 2 field are required to have like
electric charges repel and like mass charges attract. The linear field
equations should be quantizable using standard techniques.
\end{abstract}
\maketitle

\section{Introduction}

Classical electromagnetic theory has been remarkably robust to modern
innovations. It was the inspiration for special relativity and did
not need alteration. As quantum mechanics evolved, the field equations
remained the same. One technical change was in the action, a gauge
had to be choosen. The advent of general relativity did not alter
electromagnetism. Instead, general relativity provided the background
metric needed. The U(1) symmetry of electromagnetism served as the
starting point for the formation of the standard model, which unifies
electromagnetism, the weak, and the strong forces. A problem with
this picture is the separation of gravity from the forces of the standard
model. In this paper, I use electromagnetic theory to deduce the laws
that govern gravity.


\section{The Action}

The classical action for electromagnetism has two terms, a contraction
of the irreducible antisymmetric field strength tensor and a coupling
of electric charge with the potential. To be logically complete, Nature
must also exploit a contraction of an irreducible symmetric field
strength tensor. The symmetric field will be associated with a different
charge current that couples with the potential because the tensor
is irreducible and thus an independent fundamental force. This new
force must be long range like electromagnetism. The only such force
is gravity.

Like charges repel in the electromagnetic action for three reasons.
First, the signs of the charge coupling and the field strength tensor
contraction are the same. The field equations that arise from varying
the action with respect to the potential will have like charges repel.\cite{landau1975}
Second, the signs of the charge coupling term and the inertia term
are the same. Varying the action with respect to velocity keeping
the potential fixed generates the Lorentz force where like charges
repel. Third, the field strength tensor is antisymmetric, so it must
be represented by an odd spin particle, the spin 1 photon. For gravity,
like charges attract. The coupling term must have the sign opposite
to the field strength contraction and inertia terms so both the field
and force equations will have like charges attract. The symmetric
rank 2 tensor will be represented by a spin 2 particle that travels
at the speed of light like the photon, so the particle matches the
description of a graviton. Based on these deductions, the action for
gravity and electromagnetism (GEM) can be written:

\begin{equation}
S_{GEM}=\int d^{4}x\sqrt{-g}(-\frac{\rho_{m}}{\gamma}-\frac{1}{c}(J_{q}^{\mu}-J_{m}^{\mu})A_{\mu}-\frac{1}{2c^{2}}\nabla^{\mu}A^{\nu}\nabla_{\mu}A_{\nu}).\label{GEM_action}\end{equation}


There are two charge currents, one for electromagnetism where like
charges repel, and another for gravity where like charges attract.
To get the units correct, the mass current density must be multiplied
by the square root of Newton's gravitational constant $G$. The reducible
asymmetric tensor can be split into an antisymmetric tensor for the
spin 1 photon and a symmetric tensor for the spin 2 graviton.

The deep insight into gravity provided by general relativity is that
the geometry of spacetime is not presumed. In special relativity,
the metric must be constant, the same for all inertial observers.
In general relativity, the metric can vary depending on where the
observer is in spacetime: closer to a mass source will have more spacetime
curvature. The Riemann curvature tensor is a rank 4 tensor that contains
second order derivatives of the metric. The Hilbert action has the
Ricci scalar, a contraction of the Riemann curvature tensor. By varying
the Hilbert action with respect to the metric tensor, the geometry
of spacetime can change.

Electromagnetism works no matter what the metric happens to be. The
antisymetric tensor cannot characterize how the symmetric metric tensor
changes, so the possibly dynamic metric must be supplied as part of
the background mathematical structure. By including a symmetric field
strength tensor in the GEM action, it becomes possible for the action
to account for the changes in metric tensor, and thus remove the metric
from the background structure. The idea is not to treat the metric
as an active field, but instead to use a symmetry of the action to
provide a constraint on a metric that varies in spacetime.

The way gravity in the GEM action works mathematically is different
from general relativity. The Hilbert action requires only one field,
the metric field, to be varied in the action to generate the Einstein
field equations. The GEM action does not treat the metric as a field.
Instead there is a symmetry of the action that allows one to chose
how much of a covariant derivative is due to a dynamic potential or
a dynamic metric. Where there is a symmetry, there is a conserved
charge. Since the symmetry involves the metric, the charge must be
mass. Energy and momentum also arise from a symmetry of the action.
Since mass is the square of the energy minus the square of the momentum,
it is logically consistent that energy, momentum, and mass all can
be viewed as different symmetries of the same action.

The action has U(1) symmetry for massless particles. Mass charge breaks
U(1) symmetry. This is an attractive feature because it means the
Higgs mechanism for introducing mass into the standard model is unnecessary.
The GEM proposal predicts the Higgs boson will not be found in particle
field experiments. Currently, electric charge is measured to ten significant
digits.\cite{eidelman2004} The mass of an electron is sixteen orders
of magnitude smaller than electric charge when measured in the same
units, so the broken symmetry is beyond our ability to measure directly.


\section{Field Equations and Solutions}

The field equations can be determined in the usual way by fixing the
metric tensor up to a diffeomorphism and then varying the action with
respect to the potential:\begin{equation}
J_{q}^{\mu}-J_{m}^{\mu}=\nabla_{\nu}\nabla^{\nu}A^{\mu}.\label{4D_wave_eq}\end{equation}


This 4D wave equation has the ability to describe 4 modes of emission
of particles that travel at the speed of light. The two transverse
modes are electromagnetism. That leaves the scalar and longitudinal
modes for the graviton. A longitudinal wave is one where the particle's
acceleration is in the direction of its velocity. Considered a tunnel
drilled though the center of the Earth. Assuming no energy losses,
it would take approximately an hour an a half to journey to the far
side of the Earth and return. The acceleration is in the direction
of the velocity, so the wave is longitudinal, with a period of an
hour and a half and a wavelenth double the diameter of the Earth.

It is critical to note in eq. \ref{4D_wave_eq} the scalar D'Alembertian
operator is not acting alone on the potential. Instead, two covariant
derivatives are applied to the potential. A covariant derivative contains
both the normal derviative and the connection. The complete wave equation
is composed of the D'Alembertian operator acting on the potential,
the divergence of the connection, the connection of the first derivative
of the potential, and the connection of the connection. Spacetime
is nearly flat if one excludes regions near black holes, so the connection
of the connection term can be ignored. The divergence of the connection
contains second derivatives of the metric, so in principle could provide
an equation that must be solved to determine a smoothly changing metric.

Because eq. \ref{4D_wave_eq} is valid up to a diffeomorphism of the
metric, the equation cannot be solved until that degree of freedom
is eliminated. In this paper, I will work with metric compatible,
torsion-free connections as is done in general relativity. If one
chooses to work with a flat Minkowski metric in Euclidean spacetime,
the connection will be zero everywhere. For a static point charge,
the first field equation is a sum of the static form of Gauss' law
of electric charge and Newton's law of gravity:

\begin{equation}
\rho_{q}-\rho_{m}=-c\nabla^{2}\phi\label{Gen_Gauss_law_static}\end{equation}


The solution to this equation was known to Poisson: the potential
is a charge over a distance. The repulsive force of like electric
charges is decreased by mass attraction. This field equation can account
for all classical effects of Gauss' and Newton's law because it is
identical to their superposition. There are two articles in the literature
that assert without proof that one cannot form vector field equations
where like charges attract.\cite{gupta1957}\cite{thirring1961} Perhaps
there was an adversion to putting a negative sign in front of the
mass current. What matters are the relative signs, that the charge
density has the same sign as the Laplacian operator in eq. \ref{Gen_Gauss_law_static}.

A flaw of Newton's law of gravity is that it requires changes in the
charge density to propegate instanteously, which is not consistent
with special relativity. One road to Einstein's field equations is
to make Newton's law respect special relativity.\cite{weinberg1964}
Here is the dynamic form of the equation (I will refer to it as {}``General
Gauss's law'' out of respect for both the Swiss master and Chinese
food):

\begin{equation}
\rho_{q}-\rho_{m}=\frac{1}{c}\frac{\partial^{2}\phi}{\partial t^{2}}-c\nabla^{2}\phi\label{Gen_Gauss_law}\end{equation}


The waves for gravity and light both propagate at the speed of light.
This equation transforms as the first term of a 4-vector equation
and so is manifestly covariant under a Lorentz transformation.

The gauge choice - how one measures things - for the GEM action concerns
how to measure a covariant derivative: how much should be due to the
standard derivative versus how much is due to the connection, the
changes in the metric? In the preceding paragraph, a choice was made
that everything was due to the standard derivative of the potential.
This time I choose to allow all of the change be due to the connection,
nothing due to the standard derivative of the potential. The first
field equation under this gauge choice for a static charge is:

\begin{equation}
\rho_{q}-\rho_{m}=-\partial_{\mu}\Gamma_{\nu}^{\:0\mu}A^{\nu}.\label{ChristoffelDivergence}\end{equation}
 A metric must be found whose connection solves this first-order partial
differential equation. This might appear like a difficult task since
the connection has three derivatives of the metric. Yet one of the
derivatives is zero because the charges are static. If the metric
were diagonal, another of the derivatives would be zero. The metric
must reduce to the Minkowski metric as the charges go to zero. The
exponential metric below provides a solution with these properties
to eq. \ref{ChristoffelDivergence}:

\begin{equation}
g_{\mu\nu}=\left(\begin{array}{cccc}
exp(2(\frac{\sqrt{G}q-GM}{c^{2}R})) & 0 & 0 & 0\\
0 & -exp(-2(\frac{\sqrt{G}q-GM}{c^{2}R})) & 0 & 0\\
0 & 0 & -R^{2} & 0\\
0 & 0 & 0 & -R^{2}sin^{2}\theta\end{array}\right).\label{exponential_metric}\end{equation}


For an electrically neutral source, the metric becomes the Rosen metric.\cite{rosen1973}
That metric is not a solution to the Einstein field equations, but
has been studied in the literature because it is consistent with experimental
tests of gravity to first order parameterized post-Newtonian (PPN)
accuracy. At second order PPN accuracy, eq. \ref{exponential_metric}
predicts $0.7\mu$microarcseconds more bending of star light by the
Sun.\cite{epstein1980} An experiment such as LATOR, could confirm
or reject this proposal on purely experimental grounds.\cite{turyshev-2006-}

The bimetric theory of Rosen is not consistent with strong field tests
of gravity. The fixed background metric can store momentum, allowing
for bipolar modes of emission of gravity waves which is ruled out
by observations of binary pulsars. The GEM action does not have an
additional field to store momentum. For an isolated source, the lowest
mode of emission that conserves momentum will be the quadrapole mode,
consistent with observation.


\section{Quantization}

The 4D wave equation, eq. \ref{4D_wave_eq}, has already been quantized
by Gupta and Blueler.\cite{gupta1950}\cite{bleuler1950} They were
quantizing the field only to do the work of electromagnetism in a
manifestly covariant way. The transverse modes of emission do the
work of electromagnetism. The scalar mode of a spin 1 field caused
a serious technical problem: it suggested negative probablitity states.
To make sure those modes were always virtual, a supplimentary condition
was imposed.

With the GEM action, there are two spin fields, a spin 1 field for
electromagnetism where like charges repel, and a spin 2 field for
gravity where like charges attract. The scalar and longitudinal modes
of emission will do the work of gravity. General relativity predicts
transverse modes of emission for gravity waves. Should a gravitational
wave detection experiment succeed, the polarization of the gravity
wave will serve as an experimental test of this proposal.

There is a strong belief that field equations for gravity must be
nonlinear. To be more precise, this means that gravity fields gravitate.
The same is not true for electromagnetism: the electromagnetic field
cannot be a source of charge. There are thought experiments to support
the notion that gravity must be nonlinear. An example from a review
paper on general relativity had two boxes with six neutral particles
in each.\cite{silberstein1912} Imagine that the rest mass of one
particle were to be converted entirely into the kinetic energy of
the other five. Would the first box be able to tell any difference
in energy density and spacetime curvature between the two? If the
answer is no, gravity must be governed by a nonlinear field equation.

Repeat the thought experiment, but this time make all the particles
positively charged. Now the thought experiment cannot be done because
it would involve destroying an electric charge. There are solid theoretical
and experimental tests demonstrating that electric charge cannot be
destroyed. Because the thought experiment cannot be done, the conclusion
is not supported.


\section{A New Implementation}

The experimental tests of the equivalence principle indicate that
gravity must be a metric theory. This paper questions how to implement
a metric theory. General relativity takes a direct approach: start
with the Riemann curvature tensor that characterizes how a metric
changes, and put its contraction the Ricci scalar into the action.
This paper uses the symmetry found in the definition of a covariant
derivative to allow spacetime geometry to change based on what mass
is in that spacetime. It is the flexibility of the spacetime geometry
that indicates the approach can be characterized as a metric theory.

There is a well-known tought experiment of a man in a closed box who
would be unable to tell if he was in a smoothly accelerating rocket
ship or the box was sitting on the surface of a planet (ignoring tidal
effects). For the box on the planet, general relativity would be able
to generate a metric equation that in a limit process, would become
a potential theory. For the GEM approach, one could choose a dynamic
metric which would be equivalent to first order PPN accuracy to the
metric from general relativity, or with the practical power of a diffeomorphism,
choose to work in a flat spacetime and have a 4-potential completely
characterize the gravitational field. A scalar potential like Newton's
law of gravity is inadequate because there is only one parameter,
so it gets the answer to light bending around the Sun half right.
With four parameters, a 4-potential can describe the way both time
and space are bent by mass charge.

The GEM action is to a surprising degree exactly what Einstein searched
for over the last half of his life.\cite{pais1982} He tried a truly
impressive range of ideas, from five dimentional Kaluza-Klien variations,
to asymmetric Riemann curvature tensors, all to no avail. The radical
approach used here is to abandon the Riemann curvature tensor, and
work only with the building blocks of the curvature tensor, the connection
as found in a covariant derivative. The 4D wave equation in this paper
was called beautiful by Feynman in his lecture series.\cite[Volume II, 18-11]{feynman1964}
Elegance governs the heavens.

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