arXiv subjects

R Solanki

Publications and source records attributed to R Solanki.

3 recordsLinked to original sources

arXiv2021

Generating Spherically Symmetric Static Anisotropic Fluid Solutions of Einstein's Equations from Hydrostatic Equilibrium

For static fluid spheres, the condition of hydrostatic equilibrium is given by the generalized Tolman--Oppenheimer--Volkoff (TOV) equation, a Riccati equation in the radial pressure. For a perfect fluid source, it is known that finding a new solution from an existing solution requires solving a Bernoulli equation, if the density profile is kept the same. In this paper, we consider maps between static (an)isotropic fluid spheres with the same (arbitrary) density profile and present solution-generating techniques to find new solutions from existing ones. The maps, in general, require solving an associated Riccati equation, which, unlike the Bernoulli equation, cannot be solved by quadrature. In any case, it can be shown that the output solution is not, in general, regular for a given regular input solution. However, if pressure anisotropy is kept the same, the new solution is both regular and can be found by solving a Bernoulli equation. We give a few examples where the generalized TOV equation, under algebraic constraints, can be converted into a Bernoulli equation and thus, solved exactly. We discuss the physical significance of these Bernoulli equations. Since the density profile remains the same in our approach, the spatial line element is identical for all solutions, which facilitates direct comparison between various equilibrium configurations using fluid variables as functions of the radial coordinate. Finally, combining with the previous study on generation algorithms, we show how this study leads us to a new three-parameter family of exact solutions that satisfy all desirable physical conditions.

gr-qc↗

arXiv2021

Kottler Spacetime in Isotropic Static Coordinates

The Kottler spacetime in isotropic coordinates is known where the metric is time-dependent. In this paper, the Kottler spacetime is given in isotropic static coordinates (i.e., the metric components are time-independent). The metric is found in terms of the Jacobian elliptic functions through coordinate transformations from the Schwarzschild-(anti-)de Sitter metric. In canonical coordinates, it is known that the unparameterized spatially projected null geodesics of the Kottler and Schwarzschild spacetimes coincide. We show that in isotropic static coordinates, the refractive indices of Kottler and Schwarzschild are not proportional, yielding spatially projected null geodesics that are different.

gr-qc↗

arXiv2020

Algorithms for Generating All Static Spherically Symmetric (An)isotropic Fluid Solutions of Einstein's Equations

We study the Einstein equations of the static spherically symmetric anisotropic fluid system in curvature coordinates to find algorithms that generate all solutions and all solutions that are regular at the center. All possible combinations of input functions from the set of four functions that characterize the anisotropic system are considered and all equivalent conditions for central regularity are determined (for both isotropic and anisotropic systems). We provide the first regularity analysis of the known algorithm that uses the potential function and anisotropy as inputs. For three other choices of input function pairs (any two of the potential function, density, or radial pressure), a remarkably straightforward algorithm follows, which is very efficient in generating regular anisotropic solutions. This is because the equivalency of the three pairs in this algorithm arises precisely from the same algebraic relation that made the different equivalent sets of regularity conditions possible. In addition, the choice of functions makes this algorithm very suitable for finding particular solutions that admit other desirable physical properties; we construct three examples. This algorithm does not admit an isotropic limit although all isotropic solutions are produced as part of the anisotropic system. The remaining two choices of input function pairs (anisotropy with the radial pressure or density) lead to the old barriers one encounters in the isotropic system: Riccati and Abel equations. However, with any solution generated by the new and existing algorithms, one can now construct the general solution of the corresponding Riccati equation to obtain a one-parameter family of geometries for each input solution. We discuss the regularity of the resulting solutions.

gr-qc↗