$\Lambda^{mu}_{\nu}$ geometries from the point of view of different observers

TL;DR

This paper explores $ ext{ extsterling}^{ ext{mu}}_{ ext{ extsterling}}$ geometries with variable cosmological terms, analyzing how different observers perceive these configurations, and classifies five types of solutions based on parameters.

Contribution

It introduces a framework for $ ext{ extsterling}^{ ext{mu}}_{ ext{ extsterling}}$ geometries with variable vacuum energy and examines their properties from multiple observer perspectives.

Findings

Five types of $ ext{ extsterling}^{ ext{mu}}_{ ext{ extsterling}}$ geometries identified based on parameters

Descriptions of these geometries from static, Lemaitre, and Kantowski-Sachs observers

Conditions for regularity and finiteness of mass in these geometries

Abstract

$\Lambda^{\mu}_{\nu}$-geometry is a geometry with a variable cosmological term described by a second-rank symmetric tensor $\Lambda^{\mu}_{\nu}$ whose asymptotics are Einstein cosmological term $\Lambda \delta ^{\mu}_{\nu}$ at the origin and $\lambda \delta ^{\mu}_{\nu}$ at infinity (with $\lambda < \Lambda$). It corresponds to extension of the algebraic structure of the Einstein cosmological term $\Lambda \delta ^{\mu}_{\nu}$ in such a way that a scalar $\Lambda$ describing vacuum energy density as $\rho_{vac}=8\pi G \Lambda$ (with $\rho_{vac}$=const by virtue of the Bianchi identities), becomes explicite related to the appropriate component, $\Lambda^0_0$, of an appropriate stress-energy tensor, $T^{\mu}_{\nu}=8\pi G\Lambda^{\mu}_{\nu}$ 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 $q=\sqrt{\Lambda /\lambda}$, $\Lambda^{\mu}_{\nu}$ 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.