Are there any models with homogeneous energy density?

Abstract

By applying a recent method -based on a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor-to the simple spherical static case, we found that the only static solution with homogeneous energy density is the Schwarzschild solution and that there are no spherically symmetric dynamic solutions consistent with the homogeneous energy density assumption. Finally, a circular equivalence is shown among the most frequent conditions considered in the spherical symmetric case: homogeneous density, isotropy in pressures, conformally flatness and shear-free conditions. We demonstrate that, due to the regularity conditions at the center of the matter distribution, the imposition of two conditions necessarily leads to the static case.

Karmarkar scalar condition

“…The uniform density spherical matter configuration is the standard entry point in all textbooks of General Relativity and Relativistic Astrophysics [34,35,36,37]. We have recently shown [2] that, despite its simplicity and pedagogical interest, this widespread assumption is very restricted. Any dynamic homogeneous density profile satisfying the Karmarkar condition will lead to the Schwarzschild solution.…”

“…Most of them are recent and devoted to describing compact objects with anisotropic equations of state. On the right plate, the publications are grouped, attending to the commonly cited references in each paper case, showing that it is possible to obtain relevant information from self-gravitating systems [1,2]. This formalism provides coordinate-free results expressed in terms of structure scalars closely related to the kinematical and physical properties of the fluid.…”

“…[24] 1 + αr 2 tanh βr 2 + δ −2α r 2 (β−α/2)(tanh(βr 2 +δ)) 2 − 3 2 tanh(βr 2 +δ)−βr 2 (1+αr 2 tanh(βr 2 +δ)) 2 Table 1: In this table, we present a non-exhaustive list of two dozen static metric functions that appeared in the literature between 2016 and 2017. These functions used to generate anisotropic equations of state via the Karmarkar differential condition (39), lead to a surprising number of mostly descriptive work done by small changes, and thus obtaining various models with negligible or no discussion in their interrelations.…”

All analytic solutions for geodesic motion in axially symmetric space-times

“…This formalism provides coordinate-free results expressed in terms of structure scalars related to the kinematical and physical properties of the fluid. We showed that it is possible to obtain relevant information for self-gravitating systems in a simple form from these scalars [14,15]. Based on the formalism above, we provide a method to integrate all possible orbits for any stationary axially symmetric solutions.…”

On general spherical fluid collapse

“…Chan studied numerically its solutions in separated variables [20][21][22][23]. The questions of cavity evolution and constant energy density were discussed in [24,25]. Other authors realized that the main junction condition is a Riccati equation in time for the metric component g rr [26][27][28].…”