Relativistic virial relation for cosmological structures

Starting with the relativistic Boltzmann equation for a system of particles defined by a distribution function, we have derived the virial relation for a spherical structure within an expanding background in the context of general relativity. This generalized form of the virial relation is then applied to the static case of a spherically symmetric structure to see the difference in the simplest case to the Newtonian relation. A relativistic Mass-Temperature relation for this simple case is also derived which can be applied to compact objects in astrophysics. Our general virial relation is then applied to the non-static case of a structure within an expanding universe where an extra term, usually missed in studies of structures in the presence of the dark energy, appears.Comment: 8 page

Supertranslation-invariant dressed Lorentz charges

Abstract We present an explicit formula for Lorentz boosts and rotations that commute with BMS supertranslations in asymptotically flat spacetimes. Key to the construction is the use of infrared regularizations and of a unitary transformation that makes observables commute with the soft degrees of freedom. We explicitly verify that our charges satisfy the Lorentz algebra and we check that they are consistent with expectations by evaluating them on the supertranslated Minkowski space and on the boosted Kerr black hole.</jats:p