# gEM: Unifying gravity, electricity, and magnetism by analogy(ps, pdf)

Douglas B Sweetser@alum.mit.edu

# The goal

Create one mathematical structure for gravity and electromagnetism that can be quantized.

## The problem

The difference between gravity and electromagnetism is the oldest core problem facing physics, going back to the first studies of electromagnetism in the seventeenth century. The best minds on the planet are trying to merge the Riemannian geometry of general relativity with the quantum field theory of the standard model. Three lines of attack are string theory in ten or eleven dimensions, loop quantum gravity, and non-commutative geometry. These approaches are technically challenging.

## Some history

Gravity was the first inverse square law, discovered by Isaac Newton. After twenty years of effort, he was able to show that inside a hollow massive shell, the gravitational field would be zero. Ben Franklin, in his studies of electricity, demonstrated a similar property for an electrically charged hollow sphere. Joseph Priestly realized this meant that the electrostatic force was governed by an inverse square law just like gravity. Coulomb gets credit for the electrostatic force law modeled on Newton's law of gravity.

Hundreds of years later, Einstein started from the tensor formalism of electromagnetism on the road to general relativity. Instead of an antisymmetric field strength tensor, Einstein used a symmetric tensor. There is a precedence for transforming mathematical structures between gravity and electromagnetism.

## The method

The process of transforming mathematical structures from electromagnetism to gravity will be continued. Specifically, the gravitational analog to the Lorentz force will be written, as well as the gravitational analog to the classic electromagnetic Lagrangian.

## The results

• The Lagrangian contains both terms with a connection and the Fermi Lagrangian of electromagnetism.
• The gravitational field equations are analogues to Gauss' and Ampere's laws, so are not second rank tensor equations like Einstein's field equations.
• The Maxwell source equations result if all gravitational field terms are zero.
• The Maxwell homogeneous equations are unaffected.
• The field equations have been quantized before, but new interpretations will flow from the unification effort.
• A link to the the Lagrangian of the standard model will be detailed.
A weak static gravitational field in a vacuum will be studied using standard methods: normalizing the potential and looking at perturbations. That field in the force equation leads to:

• A geodesic equation where the potential causes the curvature, something which is missing from general relativity.
• Newton's law of gravity if spacetime symmetry is broken.
• A metric equation similar to the Schwarzschild metric if spacetime symmetry is preserved, but the two metrics are different at second order post-Newtonian accuracy.
• A new class of solutions for the gravitational source where velocity is constant, but the distribution of mass varies with distance. This may provide new ways to look at problems with the rotation profiles of disk galaxies and big bang cosmology.

# The Lagrangians

## The structure

Lagrangian = mass - charge * velocity contracted with the potential - field strength tensor contracted with self.

• -q

• -

• ,

## The total Lagrangian

• The kinetic energy term is for one particle experiencing both gravity and electromagnetism, all other terms are added together.
• The Fermi Lagrangian of electromagnetism is a subset.
• Includes the connection squared and the connection contracted with the field strength tensor and velocity.

For locally covariant coordinates,

# The classical field equations

## The classical fields

• row and column of the asymmetric field strength tensor.

• of the off-diagonal field strength tensor terms.

## The Maxwell source equations + analogues for gravity

Source equations: Gauss' law and a dynamic g (or Newton's field equations under certain conditions).

One can recover Newton's field equation, , in isolation if:

Gauss' law, , can also be recover if:

Ampere's law.

• The first rank field equations for gravity are not equivalent to Einstein's second rank field equations.
• Gravitational field equations depend on a symmetric second rank tensor.

• With all derivatives of the g fields equal to zero, the Maxwell source equations result.
• The homogeneous Maxwell equations are unaffected.

# Connonical quantization

## Problems with quantization

• Must constrain modes to Lorenz gauge (Gupta/Bleuler).

• Interpretation of quantization changes.

• only.
• Mass breaks gauge symmetry. Get Maxwell field equations + 4-vector gravitational field equations.
4 modes of transmission:

2 traverse (photons for EM),

1 scalar (gravitons for g),

1 longitudinal (gravitons for g).

Phrase scalar photon" is non-sense because photons transform like vectors.

When gravitational waves are detected, this proposal predicts they will have scalar (timelike) polarity or longitudinal polarity, not transverse polarity as predicted by general relativity.

# Integration with the standard model

The standard model does not obviously deal with curved spacetime. The unitary aspects of the symmetries U(1), SU(2), and SU(3) will be condensed with the 4-vectors and metric. Start with the standard model Lagrangian:

where

with a 4-vector is to use a metric. Since it is complex-valued, use the conjugate like so:

The parity operator flips the sign of the spatial part of a 4-vector.

Normalize the potential.

From this, it can be concluded that the normalized 4-vector is an element of the symmetry group U(1) if the multiplication operator is the metric combined with the parity and conjugate operators. The same logic applies to the 4-vector potentials for the weak and the strong forces which happen to have internal symmetries.

In curved spacetime,

Mass breaks U(1), SU(2), and SU(3) symmetry, but does so in a precise way (meaning you can calculate what should equal). There is no need for the Higgs mechanism to give particles mass while preserving U(1)xSU(2)xSU(3) symmetry, so this proposal predicts no Higgs particle will be found.

# The Forces

## The structure:

Force = charge * velocity * field strength tensor.

• -

• ,

## The analogous force

Note: the gravitational force and the electromagnetic force behave differently under charge inversion, if mass goes to negative m or electric charge goes to negative q.

## The total force

fbox

If
F is zero, this has the form of a Killing's equation.

## Geodesics and their sources

• With no force, spacetime curvature causes acceleration (in general relativity).
• With no force, spacetime curvature is caused by the potential (not in general relativity).
Therefore, general relativity's description of geodesics is incomplete.

Note: A geodesic equation applies to electromagnetism!

# A weak field gravitational force

## Newton's Law

• Solution:

• Singularity: point where

## Solution to gEM field equations

• Want gravity only, no EM, therefore

• Solution: 4D, where .

• Singularity: the lightcone where take over.

## The weak field approximation

Normalize and study perturbations:

Take the derivative with respect to and .

## Gravitational force for a weak field

Plug into the force of gravity proposal, assuming the Christoffel symbol is zero which will be the case for a weak field in Euclidean coordinates. Note the contravariant derivative flips a sign.

fbox

# Newton's law of gravity

Must break spacetime symmetry.

## Three assumptions

• Gravitational source spring constant:

• Static field approximation:

• Break spacetime symmetry:
Lightlike events in Minkowski spacetime.

Lightlike events in Newtonian spacetime.

# A metric equation

The force equation is two second order differential equations (assuming radial symmetry) [sigma squared equals minus tau squared].

Substitute to create first order equations.

Solve.

For flat spacetime, The constraint on relativistic velocities in flat spacetime is:

Solve for the constants, and plug back into the constraint, multiplying through by

## Assumptions

• Gravitational source spring constant:

• Static field approximation:
• Both sigma and tau must have the same magnitude, R. To make the metric extressed in terms of tau real, assume sigma is i R, so tau is R.

Taylor series expansion to second order in

fbox

• Will pass all classical test to first degree of post Newtonian accuracy.
• Will differ in predictions about second degree tests.

# Newtonian" Constant velocity solutions

Apply the chain rule to the force equation.

Assume:

• The weak field approximation for the 1/interval squared potential.

• No change in velocity,

• Break spacetime symmetry:

Both the three-vectors are constants, creating a first order differential equation.

Solve.

• Gravitational source spring constant:

• Static field approximation:
• Both sigma and tau must have the same magnitude, R. To make the mass distribution extressed in terms of tau real, assume sigma is i R. Tau could either be positive or negative R. Choose negative so the mass decays with distance.

For thin disk galaxies

After reaching has a constant velocity while the mass decays exponentially. It might match this equation, but needs to be looked at in detail.

For the early Universe:

The very high density of the early Universe combined with its uniform velocity distribution would be consistent with the constant velocity solution of the gravitational force equation dominating the dynamics. The solution is stable.

# Conclusions

Using a nineteenth century approach, two pillars of twentieth century physics have been fused, general relativity and the standard model. Both required technical modifications. The description of geodesics by general relativity is not complete because it does not explicitly show how the potential source causes curvature. A dynamic metric equation is found but it uses a simpler set of field equations (a rank 1 tensor instead of 2). In the standard model as elsewhere, combining two 4-vectors requires a metric. By normalizing the 4-vectors, the unitary aspect of the standard model can be self-evident. Although not mentioned before, the normalization process is essential to getting the dimensional analysis right.

This theory makes three testable predictions, two subtle, one not. First, the polarity of gravitational waves will be scalar or longitudinal, not transverse as predicted by general relativity. Second, if gravitation effects are measured to secondary post Newtonian accuracy, the coefficients for the metric derived here are different from the Schwarzschild metric in isotropic coordinates. Such an experiment will be quite difficult to do. The third test is to see if the complete relativistic force equation matches all the data for a thin spiral galaxy. It is this test which should be investigated first.