Abstract
This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds à la Boucher–Gibbons–Horowitz, respectively. Building on the related rigidity statements, we also deduce a uniqueness result for the Schwarzschild–de Sitter spacetime.
Highlights
Introduction and Statement of the MainResultsIn this paper we continue the study started in [14] about the notion of virtual mass of a static metric with positive cosmological constant
We prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds à la Boucher–Gibbons–Horowitz, respectively
To introduce our first main result, we focus on this simple version of the Riemannian Penrose Inequality and we observe that, using the definition of the Schwarzschild radius given below formula (1.16), it can be rephrased as follows
Summary
In this paper we continue the study started in [14] about the notion of virtual mass of a static metric with positive cosmological constant. We discuss a characterization of both the Schwarzschild–de Sitter and the Nariai solution, which is in some ways reminiscent of the well known Black Hole Uniqueness Theorem proved in different ways by Israel [35], Zum Hagen et al [59], Robinson [48], Bunting and Masood-ul Alam [20] and recently by the second author in collaboration with Agostiniani [3] This classical result states that when the cosmological constant is zero, the only asymptotically flat static solutions with nonempty boundary are the Schwarzschild triples described in (1.16).
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