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Electromagnetic Force on a Brane

A fundamental assumption in the theory of brane world is that all matter and radiation are confined on the four-dimensional brane and only gravitons can propagate in the five-dimensional bulk spacetime. The brane world theory did not provide an explanation for the existence of electromagnetic fields and the origin of the electromagnetic field equation. In this paper, we propose a model for explaining the existence of electromagnetic fields on a brane and deriving the electromagnetic field equation. Similar to the case in Kaluza-Klein theory, we find that electromagnetic fields and the electromagnetic field equation can be derived from the five-dimensional Einstein field equation. However, the derived electromagnetic field equation differs from the Maxwell equation by containing a term with the electromagnetic potential vector coupled to the spacetime curvature tensor. So it can be considered as generalization of the Maxwell equation in a curved spacetime. The gravitational field equation on the brane is also derived with the stress-energy tensor for electromagnetic fields explicitly included and the Weyl tensor term explicitly expressed with matter fields and their derivatives in the direction of the extra-dimension. The model proposed in the paper can be regarded as unification of electromagnetic and gravitational interactions in the framework of brane world theory.