Einstein’s dream of a unified field theory accomplished?

During the latter part of the 20th century, string theory was put forward as a unifying theory of physics foundations. String theory has not, however, fulfilled expectations. That is why we are of the view that the scientific community needs to reconsider what comprises elementary forces and particles.

Since the early days of general relativity, leading physicists, like Albert Einstein and Erwin Schrödinger, have tried to unify the theory of gravitation and electromagnetism. Many attempts were made during the 20th century, including by Hermann Weyl.

Finally, it seems that we have found a unified framework to accommodate the theory of electricity and magnetism within a purely geometric theory. This means that electromagnetic and gravitational forces are both manifestations of ripples and curvatures in spacetime geometry.

Dreams of a unified field theory

Einstein’s aim was to explain electromagnetism as a geometric property of four-dimensional spacetime. He continued this work until his death in 1955. The work was not completed. Arthur Eddington, Theodor Kaluza and others have also put forward their theories on how to unify gravity and electromagnetism, but none of these theories have been universally accepted.

Schrödinger, the father of quantum mechanics, put forward his unified field theory in the 1940s, but without complete success. Many different approaches have been proposed, including five-dimensional theories and theories based on asymmetric metrics.

A new perspective, new nonlinear Maxwell’s equations

In our approach, electric charge and electric currents, as well as electromagnetic forces, are seen as purely geometrical and immanent properties of spacetime itself, and not as some external objects. This approach was supported by the late physicist John Wheeler, in his vision of geometrodynamics. It turns out that the four-dimensional electromagnetic potential is really a building block of the metric tensor of spacetime.

Utilizing an approach from calculus of variations, we have put forward an aesthetically appealing geometric formulation of electromagnetism. When the variation of the metric tensor is optimized using functional derivatives, the necessary optimality conditions yield a new, nonlinear generalization of Maxwell’s equations. Our work is published in the Journal of Physics: Conference Series.

In the classical theory of electromagnetism, Maxwell’s equations governing electric and magnetic fields are linear partial differential equations. In our approach, optimal metrics are required to be harmonic, which gives nonlinear field equations for the electromagnetic potentials and Maxwell’s equations as a special linear case. The field equations then give the correct dynamics for the electromagnetic field.

A generalization of geometry solves the riddle

When Albert Einstein formulated his theory of gravity, he used mathematics known as pseudo-Riemannian differential geometry. We found in our research that pseudo-Riemannian geometry is not sufficiently general for a purely geometric theory of electromagnetism. A more general differential geometry was needed.

A purely local geometry was invented in 1918 by the famous German mathematician Weyl. We took Weyl’s ideas and combined them with our earlier research on this subject, and the puzzle seemed to open to us. In a Weyl geometry, lengths are local properties of spacetime, so it is in line with the principles of relativity theory.

We found out that Weyl geometry allowed us to inspect the local compression of spacetime. The same results are produced in our research using the so-called geometric algebra. Geometric algebra and Weyl geometry therefore seem to be equally usable in formulating a geometric theory of electromagnetism.

Electric charge as a local compression of spacetime

We discovered that on top of the new nonlinear field equations, electric charge is related to the local divergence or compression of spacetime. Charge is therefore a field, which has its own laws of motion.

The familiar Lorentz force law governing forces on charged particles is shown to be a condition for the test particle to travel on geodesics, just as in general relativity. This feature completes the geometric description of electromagnetism.

Conclusions

Our results indicate that light and all of electromagnetic radiation are really oscillations of spacetime itself. In terms of the older theories of “aether,” it seems that Einstein was correct when he concluded that the “aether” is the spacetime. Electric charge is a local compression of spacetime and the forces on electric charges correspond to motion on the shortest paths, that is, on geodesics.

We believe that a sufficiently complete geometric theory of electromagnetism is now available for further research. Furthermore, assuming spacetime fluctuations in the metric tensor at Planck scales leads to randomly fluctuating electromagnetic field in the vacuum.

The model predicts random fluctuations of the electromagnetic field at Planck scales and thus random creation and annihilation of charge at Planck scale due to the random covariant divergence of the electromagnetic four-potential. Finally, our theory predicts “forces” acting on charges even without an electromagnetic field, that is, it explains and predicts the Aharonov-Bohm effect.

This story is part of Science X Dialog, where researchers can report findings from their published research articles. Visit this page for information about Science X Dialog and how to participate.

Jussi Lindgren works at the Ministry of Finance Finland, and he holds a D.Sc. degree from Aalto University in applied mathematics.

Andras Kovacs works at the ExaFuse start-up company, in applied physics based energy research role. He studied physics at Columbia University.

Jukka Liukkonen holds a PhD in applied physics, and he works full-time at Nuclear and Radiation Safety Authority, STUK, Vantaa, Finland.