A new unified theory of electromagnetic and gravitational interactions

Abstract

In this paper we present a new unified theory of electromagnetic and gravitational interactions. By considering a four-dimensional spacetime as a hypersurface embedded in a five-dimensional bulk spacetime, we derive the complete set of field equations in the four-dimensional spacetime from the fivedimensional Einstein field equation. Besides the Einstein field equation in the four-dimensional spacetime, an electromagnetic field equation is obtained: ∇_aF^ab-ξR^b_aA^a =−4πJ^b with ξ =−2, where F^ab is the antisymmetric electromagnetic field tensor defined by the potential vector A^a, R_ab is the Ricci curvature tensor of the hypersurface, and J^a is the electric current density vector. The electromagnetic field equation differs from the Einstein–Maxwell equation by a curvature-coupled term ξR^b_aA^a, whose presence addresses the problem of incompatibility of the Einstein–Maxwell equation with a universe containing a uniformly distributed net charge, as discussed in a previous paper by the author [L.-X. Li, Gen. Relativ. Gravit. 48, 28 (2016)]. Hence, the new unified theory is physically different from Kaluza–Klein theory and its variants in which the Einstein–Maxwell equation is derived. In the four-dimensional Einstein field equation derived in the new theory, the source term includes the stress-energy tensor of electromagnetic fields as well as the stress-energy tensor of other unidentified matter. Under certain conditions the unidentified matter can be interpreted as a cosmological constant in the four-dimensional spacetime. We argue that, the electromagnetic field equation and hence the unified theory presented in this paper can be tested in an environment with a high mass density, e.g., inside a neutron star or a white dwarf, and in the early epoch of the universe.

Keywords

general relativity / Maxwell’s equations / unified field theory / Kaluza–Klein theory / brane world theory

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