Gravitational Fields of Conical Mass Distributions

The gravitational field of conical mass distributions is formulated using the general theory of relativity. The gravitational metric tensor is constructed and applied to the motion of test particles and photons in this gravitational field. The expression for gravitational time dilation is found to have the same form as that in spherical, oblate spheroidal, and prolate spheroidal gravitational fields and hence confirms an earlier assertion that this gravitational phenomena is invariant in form with various mass distributions. It is shown using the pure radial equation of motion that as a test particle moves closer to the conical mass distribution along the radial direction, its radial speed decreases. 1. Introduction In recent articles [1–4], we introduced an approach of studying gravitational fields of various mass distributions as extensions of Schwarzschild’s method. Of interest in this article is the gravitational field of conical mass distributions placed in empty space. Sputnik III, the third Soviet satellite launched on May 15, 1958 has a conical shape. This study is aimed at studying the behaviour of test particles and photons in the vicinity of conically shaped objects placed in empty space such as Sputnik III. 2. Gravitational Metric Tensors It is well known [5, 6] that the general relativistic metric tensor for flat space-time (empty space without mass) is invariant (invariance of the line element) and can be obtained in any orthogonal curvilinear coordinate by the transformation . It is worth noting that the metric tensor obtained through this standard procedure (i)satisfies Einstein’s field equations a priori and(ii)yields the expected equations of motion for test particles and photons in flat space-time. Using these crucial facts, we now realize that choosing the particular orthogonal curvilinear coordinate corresponding to the geometry of the body facilitates the formulation of boundary conditions on the universal gravitational scalar potential which is always expected to be a part of the metric tensor. Thus, transforming Schwarzschild’s metric into the particular orthogonal curvilinear coordinate using the invariance of the line element subject to the fact that the arbitrary function in the metric tensor transforms as yields the metric tensor for the mass distribution of the orthogonal curvilinear coordinate. Thus, the arbitrary function is determined by the mass or pressure distribution and hence possess symmetries imposed by the latter a priori. In approximate gravitational fields, the arbitrary function is equal to the gravitational