Spherical inhomogeneous solutions of Einstein and scalar-tensor gravity: a map of the land

Abstract

We review spherical and inhomogeneous analytic solutions of the field equations of Einstein and of scalar-tensor gravity, including Brans-Dicke theory, non-minimally (possibly conformally) coupled scalar fields, and Horndeski gravity. The zoo includes both static and dynamic solutions, asymptotically flat, and asymptotically Friedmann-Lemaître-Robertson-Walker ones. We minimize overlap with existing books and reviews and we place emphasis on scalar field spacetimes and on geometries that are "general" within certain classes. Relations between various solutions, which have largely emerged during the last decade, are pointed out.

A new approach to the thermodynamics of scalar-tensor gravity

“…We look instead for dynamical solutions with timelike ∇ c φ, and we disregard FLRW universes in which the imperfect φ-fluid quantities vanish identically. These criteria exclude most known analytic solutions of scalar-tensor gravity [33], but the following one, reported in Ref. [34] satisfies them.…”

“…It is then easy to see that this tensor vanishes if one considers the ansatz φ(t) = αt + β, with t denoting the coordinate time of the solution of the effective Einstein field equations, and α, β are two positive constants. The linearity in time of the Brans-Dicke scalar field is actually a familiar feature in stealth solutions in Horndeski [57] and beyond Horndeski theories [57][58][59][60][61][62][63] (see also [33]). Furthermore, this ansatz for the scalar field forces a restriction onto the energy-momentum tensor of (ordinary) matter in the theory.…”

“…The conditions ω = 0 and V (φ) = 0 are not necessary to achieve a stealth solution. As an example with ω = 0, V = 0, consider the following solution of vacuum Brans-Dicke theory found in [64] (see also [33]), which satisfies the field equations for ω = −1 and linear potential V (φ) = V 0 φ (with V 0 a positive constant).…”

“…This geometry is a special case of the Campanelli-Lousto static geometry [66], which is the form of the most general solution of the vacuum Brans-Dicke field equation that is static, spherically symmetric and asymptotically flat [31,32], but is expressed in a coordinate system of limited validity [32]. In general, this solution contains only naked singularities or wormhole throats, but not black holes [32,33,67]. This is not a stealth solution (it is straightforward to verify that, for example, the energy density of the effective φ-fluid cannot vanish if m = 0).…”

A quantum state for the late Universe

“…where ρ = ρ(r) is the matter density profile and R s the source radius. In General Relativity, the corresponding geometry, in the exterior of the matter source, is given by the Schwarzschild-de Sitter (SdS) solution [1]…”