Linear and Riccati equations in generating functions for stellar models in general relativity

Abstract

It is shown that the expressions for the tangential pressure, the anisotropy factor and the radial pressure in the Einstein equations may serve as generating functions for stellar models. The latter can incorporate an equation of state when the expression for the energy density is also used. Other generating functions are based on the condition for the existence of conformal motion (conformal flatness in particular) and the Karmarkar condition for embedding class one metrics. In all these cases the equations are linear first order differential equations for one of the metric components and Riccati equations for the other. The latter may be always transformed into second order homogenous linear differential equations. These conclusions are illustrated by numerous particular examples from the study of stellar models.

Arrow of time and gravitational entropy in collapse

“…A detailed study on heat conducting shear-free fluid solutions was done by Sussman in [25]. For recent treatments of the geometrical properties of the metric (7) in the context of the radiating stars see Paliathanasis et al [26] and Ivanov [27].…”

Strange star with Krori–Barua potential in the presence of anisotropy

“…Herrera et al [67] extended this work by introducing locally anisotropic fluids and proved that two functions instead of one is required to generate all possible solutions for anisotropic fluid. Very recently Ivanov [68] also obtained the generating functions based on the condition for the existence of conformal motion (conformal flatness in particular) and the Karmarkar's [35] condition for embedding class one metrics. Now by introducing DB [69] transformation x = r 2 , V (x) = 1 B 2 , and y(x) = A 2 , and using the notation, ∆ = p t − p r , from eqns.…”