McVittie solution with a negative cosmological constant

Abstract

Whereas current cosmological observations suggest that the universe is dominated by a positive cosmological constant (Λ > 0), the AdS/CFT correspondence tells us that the case Λ < 0 is still worthy of consideration. In this paper we study the McVittie solution with Λ < 0. Following a related study, the solution is understood here by way of a systematic construction of conformal diagrams based on detailed numerical integrations of the null geodesic equations. As in the pure Robertson -Walker case, we find that Λ < 0 ensures collapse to a Big Crunch, a feature which completely dominates the global structure.

Cosmological and Black Hole Apparent Horizons

“…[78,95] for details). It can be proved that the geodesic deviation vector Á a satisfies the geodesic deviation equation…”

“…Let Á a be the geodesic deviation vector and define the tensor field [78,95] B ab Á r b l a ; (2.11) which satisfies l b r b Á a D B a b Á b and is orthogonal to the null geodesics, B ab l a D B ab l b D 0. The transverse part of the deviation vector ("relative velocity" of two neighbouring geodesics) is…”

“…The transverse tensor Q B ab is decomposed into its symmetric and antisymmetric parts, and the symmetric part is further decomposed into its trace and trace-free parts as [78,95] …”

Inhomogeneities in Cosmological “Backgrounds” in Einstein Theory

“…(4.68) can be solved explicitly at different times t giving the location of the apparent horizons. The dynamics of the apparent horizon radii in comoving time is qualitatively the same that we have seen for the uncharged McVittie spacetime [55,56,85,94,95,130,131]. The innermost apparent horizon is located in the inner disconnected region, which is separated from the external geometry by the singularity (4.60) (which converged to R D 0 at the Big Bang).…”

“…The charged McVittie solution was rediscovered by Gao and Zhang [64], who also generalized it to higher dimensions [65], and was studied in [56,116,117]. Conformal diagrams of the McVittie spacetime for various "backgrounds" were obtained in [38,94,95,100]. Restricting again to a spatially flat FLRW "background", the charged McVittie line element and the only nonzero component of the Maxwell tensor are…”