Theories that Relate Gravity, Electricity, and Magnetism

Question

This is not mainstream physics. There are some people who (without having a stated theory that I know of) insist that Gravity, Electricity, and Magnetism are related. Some point to symmetry in Maxwell's Equations as a potential indication of this (intuitive) connection. Is it possible that there are "Maxwell-like" equations that relate Electricity and Magnetism directly to Gravity? All forms of electromagnetic energy do interact with gravity. Are there any physicists who are working on this sort of theory?

Answer 1

I know two (non-mainstream) distinct ways.
Because this simple Question atracted downvotes I will have to say something in defense:

A particular physics viewpoint that 'is not mainstream' do not cast in itself a ban, at least it should not be so. To a physicist minded person there are no taboos. To be mainstream means that, in the present, the main focus of the mindforce is concentrated in the exploration of certain labyrinths that once were hidden behind a closed door. Other doors were closed without further exploration and a sign was placed at the entrance: Dead End. Some solitary minds pursue the difficult job of digging other labyrints. If they succeed they will be our heros. The only quality that is required is honesty, IMO, as is posted at the entrance of this site.

Polarizable-Vacuum approach:
Polarizable-Vacuum (PV) representation of general relativity by Puthoff, 1999

Having shown by specific calculation that the PV approach to the three classical tests of GR reproduces the traditional GR results

from the Discussion section:

alternatively, one can define space as Euclidean and time as the same everywhere, and discover (from exactly the same measurements) how the velocity of light, and natural clocks, rods, and particle inertias 'really' behave in the neighborhood of large masses. There is just as much (or as little) content for the word 'really' in the one approach as in the other; provided that each is self-consistent, the ultimate appeal is only to convenience and fruitfulness, and even 'convenience' may be largely a matter of personal taste..."

Linked with this paper Polarizable vacuum analysis of electric and magnetic fields by Xing-Hao Ye, 2009

The electric and magnetic fields are investigated on the basis of quantum vacuum. The analysis of the electromagnetic energy and force indicates that an electric field is a polarized distribution of the vacuum virtual dipoles, and that a magnetic field in vacuum is a rearrangement of the vacuum polarization. It means that an electromagnetic wave is a successional changing of the vacuum polarization in space. Also, it is found that the average half length of the virtual dipoles around an elementary charge is $a = 2.8 × > 10^{−15}m$

The other approach that I'm aware is explored in Douglas Pinnow book 'Our Resonant Universe' it follows after the steps of Kaluza-Klein, Goedecke and Haus (and Maxwell ;-) and ...
It is a monography of a model of particles, based only on Electromagnetism (EM), that has only one parameter (electron mass) and derives the particle properties to within 1% of their values (barion masses bellow 0.1%) and does not suffer of the barionic spin crisis. The model uses only three building blocks: Electron, Pion and Muon.
The concepts of Mass, Charge and Gravitational Force became crystal clear, IMO.

You can have a glimpse of the model and explore further starting from here.

I'm used to see two arguments against the possibility of unification of EM and gravity:
The different scale of the forces and the fact that gravity is only attractive.
IMO both of them can be easily dismissed.