Chronal Isomorphism, Singularities and Black Holes

Causal structures and causal boundaries

“…The basic idea, of course, is to define mappings between them which in a precise sense preserve the causal properties and regard those spaces which can be put into correspondence by means of one of these mappings as "causally isomorphic" or sharing the same causality. These sort of mappings have been considered many times in the literature, e. g. [201,105,25,78,116,195,194,141]. Kronheimer and Penrose introduced this subject in [105], and then Budic and Sachs used an improvement in a paper [25] devoted to generalising the construction of the causal boundary (see §6.3.1 for more details) and hence they did not use these mappings to classify spacetimes in terms of their causal properties.…”

“…An important consequence of the above is that, for distinguishing spacetimes, the chronological relation < < determines the Lorentzian metric up to a conformal factor because if two such Lorentzian manifolds are chronally isomorphic then by theorem 4.1 they must be conformally related. Once this is understood, the following statements proving the invariance of the hierarchy of causality conditions under chronal or causal isomorphisms [195,194,141] become rather obvious. Theorem 4.2 Let (V, g) and (V ′ , g ′ ) be chronally isomorphic spacetimes.…”

“…These sorts of mappings have been considered many times in the literature, e.g. [201,105,25,78,116,195,194,141]. Kronheimer and Penrose introduced this subject in [105], and then Budic and Sachs used an improvement in a paper [25] devoted to generalizing the construction of the causal boundary (see subsection 6.3.1 for more details) and hence they did not use these mappings to classify spacetimes in terms of their causal properties.…”

Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples

“…Chronological mappings between causal spaces have been considered many times in the literature (see e.g. [24,8,41,42,28]). …”

“…In general, for non-distinguishing spacetimes, the information provided by the chronology is small and, thus, the concept of causal mapping may provide more useful information. Chronological mappings between causal spaces have been considered many times in the literature (see, e.g., [8,24,28,41,42]).…”