Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds

Abstract

Recently, the old notion of causal boundary for a spacetime V has been redefined consistently. The computation of this boundary ∂V on any standard conformally stationary spacetime V = R × M , suggests a natural compactification MB associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary ∂BM is constructed in terms of Busemann-type functions. Roughly, ∂BM represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions.This Busemann boundary ∂BM is related to two classical boundaries: the Cauchy boundary ∂C M and the Gromov boundary ∂GM . In a natural way ∂C M ⊂ ∂BM ⊂ ∂GM , but the topology in ∂BM is coarser than the others. Strict coarseness reveals some remarkable possibilities -in the Riemannian case, either ∂C M is not locally compact or ∂GM contains points which cannot be reached as limits of ray-like curves in M .In the non-reversible Finslerian case, there exists always a second boundary associated to the reverse metric, and many additional subtleties appear. The spacetime viewpoint interprets the asymmetries between the two Busemann boundaries, ∂ + B M (≡ ∂BM ), ∂ − B M , and this yields natural relations between some of their points.Our aims are: (1) to study the subtleties of both, the Cauchy boundary for any generalized (possibly non-symmetric) distance and the Gromov compactification for any (possibly incomplete) Finsler manifold, (2) to introduce the new Busemann compactification MB, relating it with the previous two completions, and (3) to give a full description of the causal boundary ∂V of any standard conformally stationary spacetime.

Foundations of Finsler Spacetimes from the Observers’ Viewpoint

“…4.10), supported by some mathematical properties pointed out above. The first consideration is that Postulate 2 should be regarded now as an approximate symmetry at each point, in a similar way as the affine structure of Postulate 1 has been regarded as an aproximate symmetry to the structure of a relativistic spacetime 22 . This means that, now, one cannot find a set of coordinate charts such that the relations (3) occur at each p; however, one would expect that we will not be far from this situation (at least in regions of spacetime free of extremely exotic or violent situations).…”

“…The name and a thorough study of Aanisotropic connections were given in [37,38]; see also [56,57] for a study of connections on fiber bundles from a more general viewpoint. 22 Even though we focus on the relativistic case, (disregarding the Leibnizian case and the other possibilities), one could also consider a Leibniz-Finsler structure (Ω, h) on a manifold M , where h would be now a Finsler metric on Ker(Ω) instead of a Riemannian one, according to Table 1. the sets S p of linear bases at each T p M playing the role of (linear) IFR at p. However, one would expect that the set of observers O introduced in Def. 3.1 will still make sense and will be "close" to the space of observers for a relativistic spacetime.…”

Spacetime coverings and the casual boundary

“…Observe that such metrics are generalizations of RobertsonWalker models to the standard stationary settings. In fact, the theory developed in [31] for the stationary case is enough to study their c-completion (see [20, section 3]). The c-completion of the standard stationary case presents remarkable differences with respect to the Static one, mainly because its causality is no longer determined by a (regular) distance but by a (non-symmetric) generalized distance.…”

Finsler metrics and relativistic spacetimes

“…So, one will find a correspondence between the conformal properties of the elements in this class of metrics (V, g L ) and the geometric properties of Randers spaces (M, F ). This has been carried out at different levels (see [9,10,11,12,13,18,20,24] or [23] for a review), and we will focus here in three of them, with clear physical applications.…”

“…The causal boundary is a conformally invariant alternative, which is intrinsic and can be constructed systematically in any strongly causal spacetime (see [17] for a comprehensive study of this boundary). The computation of the causal boundary and completion of a stationary spacetime (4) has been carried out in full generality in [18]. It must be emphasized that this boundary has motivated the definition of a new Busemann boundary in any Finslerian manifold.…”