Bounds on M/R for static objects with a positive cosmological constant

TL;DR

This paper derives bounds on the mass-to-radius ratio for static, spherically symmetric objects with a positive cosmological constant, revealing new non-uniqueness phenomena when the cosmological horizon coincides with the black hole horizon.

Contribution

It extends classical bounds on M/R to include positive cosmological constant cases and analyzes the uniqueness of solutions saturating these bounds.

Findings

Derived a new inequality for M/R with positive Lambda.

Identified conditions where solutions saturate the bound.

Showed non-uniqueness of saturating solutions when Lambda R^2=1.

Abstract

We consider spherically symmetric static solutions of the Einstein equations with a positive cosmological constant $\Lambda,$ which are regular at the centre, and we investigate the influence of $\Lambda$ on the bound of M/R, where M is the ADM mass and R is the area radius of the boundary of the static object. We find that for any solution which satisfies the energy condition $p+2p_{\perp}\leq\rho,$ where $p\geq 0$ and $p_{\perp}$ are the radial and tangential pressures respectively, and $\rho\geq 0$ is the energy density, and for which $0\leq \Lambda R^2\leq 1,$ the inequality \frac{M}{R}\leq\frac29-\frac{\Lambda R^2}{3}+\frac29 \sqrt{1+3\Lambda R^2}, holds. If $\Lambda=0$ it is known that infinitely thin shell solutions uniquely saturate the inequality, i.e. the inequality is sharp in that case. The situation is quite different if $\Lambda>0.$ Indeed, we show that infinitely thin shell solutions do not generally saturate the inequality except in the two degenerate situations $\Lambda R^2=0$ and $\Lambda R^2=1$. In the latter situation there is also a constant density solution, where the exterior spacetime is the Nariai solution, which saturates the inequality, hence, the saturating solution is non-unique. In this case the cosmological horizon and the black hole horizon coincide. This is analogous to the charged situation where there is numerical evidence that uniqueness of the saturating solution is lost when the inner and outer horizons of the Reissner-Nordstr\"{o}m solution coincide.