Cosmological Constant and Local Gravity

TL;DR

This paper explores the linearized Einstein equations with a positive cosmological constant, revealing non-spherical solutions with cylindrical symmetry and analyzing the effects of the cosmological constant on local gravity, including stability and potential formulations.

Contribution

It introduces non-spherically symmetric solutions to linearized Einstein equations with a cosmological constant, and compares them with linearized Schwarzschild-de Sitter solutions, highlighting new gravitational potentials.

Findings

Existence of non-spherical, cylindrically symmetric solutions.

Presence of a repulsive tensor potential from pressure density.

Agreement with linearized Schwarzschild-de Sitter metric in Lorentz gauge.

Abstract

We discuss the linearization of Einstein equations in the presence of a cosmological constant, by expanding the solution for the metric around a flat Minkowski space-time. We demonstrate that one can find consistent solutions to the linearized set of equations for the metric perturbations, in the Lorentz gauge, which are not spherically symmetric, but they rather exhibit a cylindrical symmetry. We find that the components of the gravitational field satisfying the appropriate Poisson equations have the property of ensuring that a scalar potential can be constructed, in which both contributions, from ordinary matter and $\Lambda > 0$, are attractive. In addition, there is a novel tensor potential, induced by the pressure density, in which the effect of the cosmological constant is repulsive. We also linearize the Schwarzschild-de Sitter exact solution of Einstein's equations (due to a generalization of Birkhoff's theorem) in the domain between the two horizons. We manage to transform it first to a gauge in which the 3-space metric is conformally flat and, then, make an additional coordinate transformation leading to the Lorentz gauge conditions. We compare our non-spherically symmetric solution with the linearized Schwarzschild-de Sitter metric, when the latter is transformed to the Lorentz gauge, and we find agreement. The resulting metric, however, does not acquire a proper Newtonian form in terms of the unique scalar potential that solves the corresponding Poisson equation. Nevertheless, our solution is stable, in the sense that the physical energy density is positive.