Compact elastic objects in general relativity

Abstract

We introduce a rigorous and general framework to study systematically self-gravitating elastic materials within general relativity, and apply it to investigate the existence and viability, including radial stability, of spherically symmetric elastic stars. We present the mass-radius (M − R) diagram for various families of models, showing that elasticity contributes to increasing the maximum mass and the compactness up to ≈22%, thus supporting compact stars with mass well above two solar masses. Some of these elastic stars can reach compactness as high as GM=ðc 2 RÞ ≈ 0.35 while remaining stable under radial perturbations and satisfying all energy conditions and subluminal wave propagation, thus being physically realizable models of stars with a light ring. We provide numerical evidence that radial instability occurs for central densities larger than that corresponding to the maximum mass, as in the perfect-fluid case. Elasticity may be a key ingredient to building consistent models of exotic ultracompact objects and black hole mimickers, and can also be relevant for a more accurate modeling of the interior of neutron stars.

Compactness bounds in General Relativity

“…We focus on spherical symmetry and study static, self-gravitating configurations of elastic matter following the novel formalism developed in Refs. [6,7]. The matter sector is described by the stress-energy tensor T ν µ = diag(ρ, p rad , p tan , p tan ), in terms of the density ρ(r) and radial and tangential pressures p rad (r) and p tan (r), respectively.…”

“…The system of field equations is closed by specifying an equation of state (EoS) that relates the density and the pressures. The latter can be imposed by introducing a stored energy function w(r) that fully describes the properties of the elastic matter [6,7]. The stored energy function can be conveniently written as a functional w(δ, η) that depends on two positive radial functions,…”

“…At variance with the fluid case, elastic materials are generically anisotropic and, in addition to the usual longitudinal matter perturbations, there exist also transverse perturbations. We denote the speeds of longitudinal and transverse perturbations along the radial (tangential) direction as c L (δ, η) and c T (δ, η) (c L (δ, η) and cT (δ, η)), respectively [6,7].…”

“…It is straightforward to check that there is no enough freedom to impose four conditions c L,T = cL,T = 1. Moreover, the elastic generalization of the stability condition c 2 s > 0 implies that [7]…”

“…All this suggests that we should impose c L = cL = 1. These conditions provide two partial differential equations that can be solved analytically for (ρ, p rad , p tan ) in terms of two physical parameters (ρ 0 and defined below) [7].…”