Newtonian Limit for the Curvature of Space Time

Abstract

The curvature of space time expresses the tidal force that a body feels when moving along a geodesic. Harmann Weyl named Weyl¹ curvature tensor which is measure of the curvature of space time. In general relativity, A. Danehkar² studied that the curvature of space time is a solution of vacuum Einstein equation and it governs the propagation of gravitational waves through area of space devoid of matter. Hermann Klaus Hugo Weyl (1955), one of the German Mathematician of 19^th century, published technical and some general work on space, time, matter, philosophy, logic symmetry and visualized general relativity with the laws of electromagnetism.

       In the present manuscript, we have tried to draw our focus on properties of conformal Weyl curvature tensor i.e. curvature of space time and its applications in the modern literature of relativity and cosmology. However, in this note we wish to compliment some recent enhancements in the cosmological literature by implementing notions of Weyl’s conformal curvature tensor and its recurrence properties. In particular, we shall outline some generalized recurrence properties of Weyl’s curvature tensor in the Weyl’s space and then delineate its Newtonian limit. Besides this, we shall discuss some relativistic equations under Newtonian limit. It is shown that for the space-time having dimensions less than 4, needed a tensor (called Cotton tensor), other than Weyl’s tensor to check out the conformal flatness of the space-time and its recurrent nature. Moreover, a relativistic form of Weyl’s tensor and relativistic equation evolved due to its parts (namely, electric and magnetic) has been studied.