A spherically symmetric gravitational solution of nearly conformally flat metric measure space

Abstract

In this manuscript, we study the nearly flat approximation of a conformally invariant gravitational theory in metric measure space (MMS). In addition, we investigate the transformation of the energy–momentum tensor in this context and obtain the vacuum solution of MMS and its weak field limit in the spherically symmetric coordinates. We show that while it is already a vacuum solution, it can simulate dark matter when restricted to the framework of general relativity, i.e., a symmetry-broken conformal frame. This is done by employing a density function which is an essential part of MMS. We derive an equation for the density function for a general profile of a rotation curve obtained from observations. Specifically, the density function corresponding to two well-known profiles PSS and NFW are provided.