Conformal geodesics in spherically symmetric vacuum spacetimes with cosmological constant

Abstract

An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschildde Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the case of the Schwarzschild-anti de Sitter family, the natural parameter of the curves only covers a restricted time span so that these global conformal Gaussian systems do not exist. *

Type D conformal initial data

“…For example, an existence result in the case of the Kerr solution of this kind of data would provide an important insight in the open problem of the non-linear stability of this solution. At this point we recall the work of [16,20] where the existence of congruences of conformal geodesics is proven in globally hyperbolic domains of vacuum type D solutions with a null or timelike conformal boundary. Also interesting in this regard is the work in [3] where conformal hyperboloidal data for vacuum solutions with a timelike conformal boundary are studied.…”

“…The initial data so constructed could be used as the starting point in the analysis of the conformal boundary for members of this important class of exact solutions. In this sense there are already results for the Schwarzschild [16] and the Kottler family of solutions [20] where the construction of congruences of conformal geodesics enables us to determine geometric properties of the conformal boundary without carrying out the actual conformal extension.…”

On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

“…Let ( M, g) denote an Einstein spacetime. Suppose that (x(τ ), β(τ )) is a solution to the conformal geodesic equations (10a)-(10b) and that {e a } is a g-orthonormal frame propagated along the curve according to equation (11). If Θ satisfies (12), then one has that…”

“…A class of non-intersecting conformal geodesics which cover the whole maximal extension of the sub-extremal Schwarzschild-de Sitter spacetime has been studied in [11]. The main outcome of the analysis in that reference is that the resulting congruence covers the whole maximal analytic extension of the spacetime and, accordingly, provides a global system of coordinates -modulo the usual difficulties with the prescription of coordinates on S 2 .…”

“…Since the cosmological constant takes a de Sitter-like value, the conformal boundary of the spacetime is spacelike and moreover, there exists a conformal representation in which the induced 3-metric on the conformal boundary I is homogeneous. Thus, it is possible to integrate the extended conformal field equations along single conformal geodesics -see [11].…”