By using Bianchi I type of homogenous and anisotropic background metric having cylindrical symmetry in x direction of a local cartesian coordinates system, we solve metric field equations for a non-minimally coupled Einstein–Maxwell gravity. To do so we choose long wavelength EM waves where spatial dependence of the waves are negligible at the expansion duration. Motivation of this work is study directional effects of the EM vector potential on anisotropy trajectory of the space time expansion scale factor. This problem is checked just for de Sitter inflationary epoch because of its importance in the expansion of the universe. By applying the dynamical system approach we investigate stability nature of the cosmic system in two different cases: (a) direction of the EM vector potential is parallel to x direction of the spacetime and (b) it is perpendicular to x direction. We obtained at stable critical points for some of obtained inflationary solutions the EM waves behave as dark energy in cases a and b where the anisotropy is negligible by expanding of the universe while there are some stable critical points with inflationary stable solutions in case b for which the EM waves behave as baryonic visible matter with non-vanishing anisotropy.

中文翻译：

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具有非最小耦合爱因斯坦-麦克斯韦引力理论的 Bianchi I 度量解

通过使用在局部笛卡尔坐标系的 x 方向上具有圆柱对称性的 Bianchi I 型均匀和各向异性背景度量，我们求解了非最小耦合爱因斯坦-麦克斯韦引力的度量场方程。为此，我们选择长波长 EM 波，其中波的空间依赖性在扩展持续时间可以忽略不计。这项工作的动机是研究电磁矢量势对时空扩展比例因子的各向异性轨迹的方向影响。这个问题只针对德西特暴胀时期进行检查，因为它在宇宙膨胀中很重要。通过应用动力系统方法，我们研究了宇宙系统在两种不同情况下的稳定性性质：(a) 电磁矢量势的方向平行于时空的 x 方向，(b) 它垂直于 x 方向。我们在一些获得的暴胀解的稳定临界点获得了电磁波在情况 a 和 b 中表现为暗能量，其中各向异性通过宇宙膨胀可以忽略不计，而在情况 b 中存在一些具有暴胀稳定解的稳定临界点，其中电磁波表现为具有非消失各向异性的重子可见物质。