arXiv subjects

M M Akbar

Publications and source records attributed to M M Akbar.

4 recordsLinked to original sources

arXiv2021

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↗

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

On the Existence of Word-representable Line Graphs of Non-word-representable Graphs

An open question in the theory of word-representable graphs for the past decade has been whether the line graph of a non-word-representable graph is always non-word-representable. By formulating an appropriate optimization problem for the decision problem of 3-semi-transitive graphs, we show that the line graph of a non-word-representable graph can be word-representable. Using IBM's CPLEX solver, we demonstrate for several known word-representable and non-word-representable graphs that the line graph of a graph is 3-semi-transitive when there is a solution to the optimization problem. This results in an example where the line graph of a non-word-representable graph is both 3-semi-transitive and semi-transitive and thus is word-representable.

math.CO↗

arXiv2009

Ricci Solitons and Einstein-Scalar Field Theory

B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate to each static Ricci flat spacetime a local Ricci soliton in one higher dimension. As well, solutions of Euclidean-signature Einstein gravity coupled to a free massless scalar field with nonzero cosmological constant are associated to shrinking or expanding Ricci solitons. We exhibit examples, including an explicit family of complete expanding solitons which can be thought of as a Ricci flow for a complete Lorentzian metric. The possible generalization to Ricci-flat stationary metrics leads us to consider an alternative to Ricci flow.

gr-qc↗