Local existence of a classical solution to the Einstein–Maxwell–Klein–Gordon system in higher dimensions

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In this paper, we study the local existence of a classical solution to the coupled Einstein and Maxwell–Klein–Gordon system in higher dimensions. This system provides a toy model to study the dynamics of the complex scalar fields under the influence of the interaction of the gravitational and electromagnetic fields. In the starting point, we introduce the metric of the space-time in the Bondi coordinate. Then, we construct the evolution equation in the form of a single first-order nonlinear integro-differential equation. Furthermore, we show that there exists a unique fixed point that is the solution of the main problem based on the contraction mapping arguments. Finally, for a given initial data, we prove the existence of a local classical solution.