Asymptotics with a positive cosmological constant: I. Basic framework

TL;DR

This paper reviews the well-established asymptotic structure of gravitational fields when the cosmological constant is zero and highlights the challenges and lack of useful frameworks when extending these concepts to a positive cosmological constant, emphasizing the need for new approaches.

Contribution

It demonstrates that the standard asymptotic framework for gravitational fields does not extend straightforwardly to the case of a positive cosmological constant, identifying key conceptual gaps.

Findings

Standard framework fails for positive cosmological constant

Lack of positive energy theorems in this context

No invariant gravitational wave notion for Λ > 0

Abstract

The asymptotic structure of the gravitational field of isolated systems has been analyzed in great detail in the case when the cosmological constant $\Lambda$ is zero. The resulting framework lies at the foundation of research in diverse areas in gravitational science. Examples include: i) positive energy theorems in geometric analysis; ii) the coordinate invariant characterization of gravitational waves in full, non-linear general relativity; iii) computations of the energy-momentum emission in gravitational collapse and binary mergers in numerical relativity and relativistic astrophysics; and iv) constructions of asymptotic Hilbert spaces to calculate $S$-matrices and analyze the issue of information loss in the quantum evaporation of black holes. However, by now observations have established that $\Lambda$ is positive in our universe. In this paper we show that, unfortunately, the standard framework does not extend from the $\Lambda =0$ case to the $\Lambda >0$ case in a physically useful manner. In particular, we do not have positive energy theorems, nor an invariant notion of gravitational waves in the non-linear regime, nor asymptotic Hilbert spaces in dynamical situations of semi-classical gravity. A suitable framework to address these conceptual issues of direct physical importance is developed in subsequent papers.