Axially symmetric equilibrium regions of Friedmann–Lemaître–Robertson–Walker universes

Abstract

Abstract. The study of the matching of stationary and axisymmetric spacetimes with Friedmann-Lemaître-Robertson-Walker spacetimes preserving the axial symmetry is presented. We show, in particular, that any orthogonally transitive stationary and axisymmetric region in FLRW must be static, irrespective of the matter content. Therefore, previous results on static regions in FLRW cosmologies apply. As a result, the only stationary and axisymmetric vacuum region that can be matched to a (nonstatic) FLRW spacetime is a spherically symmetric region of Schwarzschild. This constitutes another uniqueness result for the Einstein-Straus model (as well as its Oppenheimer-Snyder counterpart), and hence another indication of its unsuitability as an answer to the influence of the cosmic expansion on local physics.

Review on exact and perturbative deformations of the Einstein–Straus model: uniqueness and rigidity results

“…The study of axially symmetric equilibrium regions in FLRW universe was dealt with in [52], and, in short, the main result found was that those stationary regions must, in fact, be static, and therefore the results of the previous section apply.…”

“…With these assumptions at hand, in [52], it is proven, first, that the stationary (timelike) Killing vector field ξ is nowhere tangent to Ω SX . As explained in the previous section, this serves in particular to construct a neighbourhood of Ω SX by dragging it along the orbits of ξ, in which the geometry is thus fully determined by the information in Ω SX .…”

“…As explained in the previous section, this serves in particular to construct a neighbourhood of Ω SX by dragging it along the orbits of ξ, in which the geometry is thus fully determined by the information in Ω SX . The main result in [52] is that if a OT stationary and axisymmetric region (W SX , g SX ) can be matched to a FLRW region through a hypersurface Ω SX preserving the axial symmetry, then the region (W SX , g SX ) must be, in fact, static on a neighbourhood of Ω SX .…”

First order perturbations of the Einstein-Straus and Oppenheimer-Snyder models

“…The background matching conditions are obtained simply by particularising the equations (13) (which correspond to (2) in spherical symmetry) to the Schwarzschild region and the FLRW region. The second equation in (13) implies…”

“…Regarding the assumption of staticity, it has recently been shown by Nolan & Vera [13] that if a stationary and axially symmetric region is to be matched to a non-static FLRW region across a hypersurface preserving the axial symmetry, then the stationary region must be static. Hence, the results in [11,12] can be applied to any stationary and axisymmetric region.…”