$Λ^{mu}_ν$ geometries from the point of view of different observers

Abstract

$Λ^μ_ν$-geometry is a geometry with a variable cosmological term described by a second-rank symmetric tensor $Λ^μ_ν$ whose asymptotics are Einstein cosmological term $Λδ^μ_ν$ at the origin and $λδ^μ_ν$ at infinity (with $λ< Λ$). It corresponds to extension of the algebraic structure of the Einstein cosmological term $Λδ^μ_ν$ in such a way that a scalar $Λ$ describing vacuum energy density as $ρ_{vac}=8πG Λ$ (with $ρ_{vac}$=const by virtue of the Bianchi identities), becomes explicite related to the appropriate component, $Λ^0_0$, of an appropriate stress-energy tensor, $T^μ_ν=8πGΛ^μ_ν$ whose vacuum properties follow from its symmetry, $T_0^0=T_1^1$, and whose variability follows from the contracted Bianchi identities. In the spherically symmetric case existence of such geometries in frame of GR follows from imposing on Einstein equations requirements of finiteness of the ADM mass $m$, and of regularity of density and pressures. Dependently on parameters $m$ and $q=\sqrt{Λ/λ}$, $Λ^μ_ν$ geometry describes five types of configurations. We summarize here the results which tell us how these configurations look from the point of view of different observers: a static observer, a Lemaitre co-moving observer, and a Kantowski-Sachs observer.