Static (not stationary) solutions of the Einstein–Klein–Gordon (EKG) equations including matter are obtained for real scalar fields. The scalar field interaction with matter is considered. The introduced coupling allows the existence of static solutions in contraposition with the case of the simpler EKG equations for real scalar fields and gravity. Surprisingly, when the considered matter is a photon-like gas, it turns out that the gravitational field intensity at large radial distances becomes nearly a constant, exerting an approximately fixed force to small bodies at any distance. The effect is clearly related with the massless character of the photon-like field. It is also argued that the gravitational field can generate a bounding attraction, that could avoid the unlimited increase in mass with the radius of the obtained here solution. This phenomenon, if verified, may furnish a possible mechanism for explaining how the increasing gravitational potential associated to dark matter, finally decays at large distances from the galaxies. A method for evaluating these photon bounding effects is just formulated in order to be further investigated.

中文翻译：

---

包括物质在内的 EKG 方程的实标量场解

包含物质的爱因斯坦-克莱因-戈登 (EKG) 方程的静态（非静态）解是针对实标量场获得的。考虑了标量场与物质的相互作用。引入的耦合允许存在与实标量场和重力的​​更简单 EKG 方程的情况相反的静态解。令人惊讶的是，当所考虑的物质是类光子气体时，大径向距离处的引力场强度几乎变为常数，对任何距离处的小物体施加近似固定的力。这种效应显然与类光子场的无质量特性有关。也有人认为，引力场可以产生边界吸引力，这可以避免质量随着此处获得的解的半径而无限增加。这一现象如果得到证实，可能会提供一种可能的机制来解释与暗物质相关的不断增加的引力势如何最终在距离星系很远的地方衰变。为了进一步研究，刚刚制定了一种评估这些光子边界效应的方法。