Research Article

General Relativity and Gravitation

, Volume 43, Issue 12, pp 3355-3366

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

On cosmological mass with positive Λ

  • Roger PenroseAffiliated withMathematical InstituteDepartment of Physics, University of Leiden Email author 


For asymptotically flat space-times, a very satisfactory expression for the total mass/energy of a system defined at future null infinity was provided by Bondi and Sachs, in the early 1960s. A generalization of this to space-times that are asymptotically de Sitter now has particular relevance in view of observational evidence, from 1998 onwards, indicating the presence of a positive cosmological constant Λ. In this article, some of the issues involved in such a definition are examined, showing that a somewhat different attitude to mass/energy must be taken, from that which was appropriate for asymptotically flat space-times. Two tentative suggestions are put forward for a retarded mass/energy definition with positive Λ, one based on a conformally invariant integral expression whose advanced time-derivative gives the Bondi–Sachs definition in the asymptotically flat case and the other, on the author’s 1982 approach to quasi-local energy. Such expressions could have some direct relevance to the proposal of Conformal Cyclic Cosmology, for which recent analysis of the CMB has provided some striking support.


Cosmological constant Energy in cosmology Bondi–Sachs mass Conformal cyclic cosmology