The algebraic structure of a cosmological term in spherically symmetric solutions

Abstract

We propose to describe the dynamics of a cosmological term in the spherically symmetric case by an r-dependent second rank symmetric tensor Λµν invariant under boosts in the radial direction. This proposal is based on the Petrov classification scheme and Einstein field equations in the spherically symmetric case. The inflationary equation of state p = −ρ is satisfied by the radial pressure, p Developments in particles and quantum field theory, as well as the confrontation of models with observations in cosmology [1], compellingly favour treating the cosmological constant Λ as a dynamical quantity.The Einstein equations with a cosmological term readwhere G µν is the Einstein tensor, T µν is the stress-energy tensor of a matter, and Λ is the cosmological constant.In the absence of matter described by T µν , Λ must be constant, since the Bianchi identities guarantee vanishing covariant divergence of the Einstein tensor, G µν ;ν = 0. In quantum field theory, the vacuum stress-energy tensor has the form < T µν >=< ρ vac > g µν which behaves like a cosmological term with Λ = 8πGρ vac .The idea that Λ might be variable has been studied for more than two decades (see [2,3] and references therein). In a recent paper on Λ-variability, Overduin and Cooperstock distinguish three approaches [4]. In the first approach Λg µν is shifted onto the right-hand side of the field equations (1) and treated as part of the matter content. This approach, characterized by Overduin and Cooperstock as being connected to dialectic materialism, goes back to Gliner who interpreted Λg µν as corresponding to vacuum stress-energy tensor with the equation of state p = −ρ [5], to Zel'dovich who connected Λ with the gravitational interaction of virtual particles [6], and to Linde who suggested that Λ can vary [7]. In the ref.[8] a cosmological model was proposed with the equation of state varying from p = −ρ to p = ρ/3. In contrast, idealistic approach prefers to keep Λ on the left-hand side of the Eq.(1) and treat it as a constant of nature. The third approach, allowing Λ to vary while keeping it on the lefthand side as a geometrical entity, was first applied by Dolgov in a model in which a classically unstable scalar field, non-minimally coupled to gravity, develops a negative energy density cancelling the initial positive value of a cosmological constant Λ [9]. Whenever variability of Λ is possible, it requires the presence of some matter source other than T µν = (8πG) −1 Λg µν , since the conservation equation G µν ;ν = 0 implies Λ = const in this case. This requirement makes it impossible to introduce a cosmological term as variable in itself. However, it is possible for a stress-energy tensor other than Λg µν .The aim of this letter is to show what the algebraic structure of a cosmological term can be in the spherically symmetric case, as suggested by the Petrov classification scheme [10] and by the Einstein field equations.In the spherically symmetric static case a line element can be written in the form [11]where dΩ 2 is the line element on ...

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