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: ∇aFab-ξRbaAa =−4πJb with ξ =−2, where Fab is the antisymmetric electromagnetic field tensor defined by the potential vector Aa, Rab is the Ricci curvature tensor of the hypersurface, and Ja is the electric current density vector. The electromagnetic field equation differs from the Einstein–Maxwell equation by a curvature-coupled term ξRbaAa, 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.
. [J]. Frontiers of Physics, 2016, 11(6): 110402.
Li-Xin Li. A new unified theory of electromagnetic and gravitational interactions. Front. Phys. , 2016, 11(6): 110402.
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In this paper tensor space on a manifold M will generally be denoted by T(M), regardless of the type of tensor (scalar, vector, dual vector, or tensor of any type).
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The Lagrangian in Eq. (65) does not contain any derivatives of N so we do not interpret N as a matter field.
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Note that all ΨabΨab; Ψ2, ΦabΦab, Φ2, ΨabΦab, and ΨΦ are proportional to N﹣2.
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The idea of interpreting the extra geometric terms in a four-dimensional Einstein field equation derived from 5D gravity as representing induced matter in a fourdimensional spacetime has been extensively investigated by Wesson and his collaborators (see [14, 45] and references therein). They proposed that the extra geometric terms are the stress-energy tensors of the induced matter and regarded the fifth dimension as being associated with the rest mass of particles instead of a real space dimension. However, in their theory, they did not derive the field equations of matter and electromagnetic fields.
Strictly, when an electromagnetic field is present the spacetime cannot be exactly Ricci-flat, since the stressenergy tensor of the electromagnetic field will make Rab≠0. However, if the electromagnetic field is weak its effect on the spacetime curvature can be ignored and the spacetime can be approximately Ricci-flat if the mass density of other matter is sufficiently low.
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