## Barebones mathematics of unification of electromagnetism and gravity

April 22, 2014 by zulfahmed

Electromagnetism and gravity can be represented by geometric structures so although the scientific question of unification is profound, the technical mathematical question may be considered fairly simply. As I am still a student of attempts of Nordstrom, Einstein, Weyl, Kaluza for this unification, I have a tentative picture of my own abstracted from these attempts. We have a compact four-manifold which will be our universe and we must produce a metric on it. A three dimensional manifold will be the physical universe where the metric will represent gravity. Electromagnetism will be represented by a potential. In the Nordrom case, the metric itself contains the components of the potential.

If is a four-manifold and is a hypersurface representing the physical universe, a metric on can represent gravity and a 1-form on could represent electromagnetism. Rather that take the approach where components of a metric on contain data for electromagnetism, I think it is better to consider the Ricci curvature equation for to be the gravitational field equations and then seen the relation between gravity and electromagnetism equating second fundamental form terms with the field strength components of .

In principle, the geometry of could be arbitrary for the issue of coherent unification of electromagnetism and gravity, i.e. perhaps for many different possible geometries of , locally one could produce a metric and a 1-form that are related by the Ricci curvature equation for submanifolds on . The evidence that has a specific geometry — that of a scaled 4-sphere — could be thought to be independent of this unification scheme. Then we can say that falls in a class of manifolds where the unification scheme works.

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