In the standard Einstein’s theory the exterior gravitational field of any static and axially symmetric stellar object can be described by means of a single function from which we obtain a metric into a four-dimensional space–time. In this work we present a generalization of those so called Weyl solutions to a space–time–matter metric in a five-dimensional manifold within a non-compactified Kaluza–Klein theory of gravity. The arising field equations reduce to those of vacuum Einstein’s gravity when the metric function associated to the fifth dimension is considered to be constant. The calculation of the geodesics allows to identify the existence or not of different behaviours of test particles, in orbits on a constant plane, between the two metrics. In addition, static solutions on the hypersurface orthogonal to the added dimension but with time dependence in the five-dimensional metric are also obtained. The consequences on the variation of the rest mass, if the fifth dimension is identified with it, are studied.

在标准爱因斯坦理论中，任何静态轴对称恒星物体的外部引力场都可以通过单个函数来描述，从中我们可以获得四维时空的度量。在这项工作中，我们提出了非紧致卡鲁扎-克莱因引力理论中五维流形中时空物质度量的所谓韦尔解的概括。当与第五维相关的度量函数被认为是常数时，所产生的场方程就简化为真空爱因斯坦引力场方程。测地线的计算允许识别测试粒子在恒定平面上的轨道上、在两个度量之间是否存在不同行为。此外，还获得了与添加的维度正交但在五维度量中具有时间依赖性的超曲面上的静态解。如果将第五维与静止质量等同起来，则研究对静止质量变化的影响。