On Shearing Fluids with Homogeneous Densities

TL;DR

This paper investigates shearing spherically symmetric homogeneous density fluids in comoving coordinates, revealing relationships between shear, expansion, and density, and exploring solutions with linear mass functions and their historical context.

Contribution

It derives coupled differential equations for such fluids, explores shear-free solutions with anisotropic pressure, and addresses controversies and historical aspects of shear-free solutions.

Findings

Shear is generated by an arbitrary function of time related to mass.

Homogeneous expansion occurs while shear depends on an arbitrary function.

Singularity is closely related to the fluid's shearing motion.

Abstract

In this paper, we study shearing spherically symmetric homogeneous density fluids in comoving coordinates. It is found that the expansion of the four-velocity of a perfect fluid is homogeneous, whereas its shear is generated by an arbitrary function of time M(t), related to the mass function of the distribution. This function is found to bear a functional relationship with density. The field equations are reduced to two coupled first order ordinary differential equations for the metric coefficients, g 11 and g 22. We have explored a class of solutions assuming that M is a linear function of the density. This class embodies, as a subcase, the complete class of shear-free solutions. We have discussed the off quoted work of Kustaanheimo (1947) and have noted that it deals with shear-free fluids having anisotropic pressure. It is shown that the anisotropy of the fluid is characterized by an arbitrary function of time. We have discussed some issues of historical priorities and credentials related to shear-free solutions. Recent controversial claims by Mitra (2011, 2012) have also been addressed. We found that the singularity and the shearing motion of the fluid are closely related. Hence, there is a need for fresh look to the solutions obtained earlier in comoving coordinates. Keywords (separated by '-') Shearing solution - Perfect fluids - Homogeneous density