Hausdorff separability of the boundaries for spacetimes and sequential spaces

Abstract

There are several ideal boundaries and completions in General Relativity sharing the topological property of being sequential, i.e., determined by the convergence of its sequences and, so, by some limit operator L. As emphasized in a classical article by Geroch, Liang and Wald, some of them have the property, commonly regarded as a drawback, that there are points of the spacetime M non T 1 -separated from points of the boundary ∂M .Here we show that this problem can be solved from a general topological viewpoint. In particular, there is a canonical minimum refinement of the topology in the completion M which T 2 -separates the spacetime M and its boundary ∂M -no matter the type of completion one chooses. Moreover, we analyze the case of sequential spaces and show how the refined T 2 -separating topology can be constructed from a modification L * of the original limit operator L. Finally, we particularize this procedure to the case of the causal boundary and show how the separability of M and ∂M can be introduced as an abstract axiom in its definition.

Lorentzian surfaces and the curvature of the Schmidt metric

“…In fact this topological behaviour is expected in general spacetimes when the fibre totally degenerates such as in Schwarzschild and Kasner metrics [24]. However, there has been mathematical developments which allows to circumvent this undesirable situation by taking a canonical minimum refinement of the topology in the completion M which T 2-separates the spacetime M and its boundary ∂M [25]. Notice that this result does not guarantee that points which one may consider physically different such as the initial and final singularity in a closed Friedmann-Robertson-Walker scenarios are not identified.…”

Topology of the Misner space and its $$g$$ g -boundary

“…In this approach, M arises as the quotient space {T > |X|}/G θ 0 (or, equivalently {−T > |X|}/G θ 0 ), which will allow us to understand the pathological behaviour of its geodesics. Moreover, by considering the quotient over two adjacent quadrants of R 2 1 plus the line in between them (e.g {T < X}/G θ 0 ), we get an isometric P.1 As H θ is linear, it maps lines (through the origin) to lines (through the origin),…”

“…We think, however, that it is still interesting to verify that in the case of the complete Misner space, the g-boundary is what one expects to be. Besides, for some of these completions, such as for the g-boundary, there exists a canonical way of defining a minimal refining of the topology such that the points of the boundary become T2-separated from all the rest of the points [2] removing some of the worst problems of these constructions; hence, they may regain some interest in the recent future.…”

Spacetime coverings and the casual boundary

“…Most of the results are known (see [6,20]), but we present the concept of first order UTS along some associated results that, as far as we known, are new.…”

“…Among the several constructions proposed (see [2][3][4] for nice reviews on the classical elements and [5,6] for updated progress), two approaches have had a specially important role in general relativity, the conformal and the causal boundaries.…”