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Physics
Mathematical Physics

Causally continuous spacetimes

Profile image of Rainer SachsRainer Sachs

1974, Communications in Mathematical Physics

https://doi.org/10.1007/BF01646350
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Abstract

Causally continuous general relativistic spacetimes are defined and analyzed. In a causally continuous spacetime, the past and future of a local observer behave continuously under small perturbations of the metric or small changes in his location. Causally simple spacetimes are causally continuous; causally continuous spacetimes are causally stable. Possible reasons for taking causal continuity as a basic postulate in macrophysics are briefly discussed.

Figures (2)
Fig. 1.2. The spacetime 4 is reflecting; the spacetime B is not Let SEM be a subset. The chronological past I~ [S] of S consists of those events which can signal to some observer in S a speed of light. Formally, J” [S]={zeM: there a speed less than the is a future-directec timelike curve from z to some se S}. Similarly, the causal past J~ [S] of S is defined by J” [S] = {ze M: zeS or there is a future-directed curve from z to some se S$}. If S is open, its (chronological) common past | S is the largest open set each event of which can signal o each observer in S$ at a speed less than that of light. Formally, | S = Interior {z¢ M: for all séS there is a future-directed timelike curve from z to s}. Unless S is empty, I~ [S]2[S. J*[S], J*[S], and ¢S are d efined dually; dual results will often be taken for granted. If z is an event, we abbreviate I* [{z}] by I*(z) and similarly for J*.
Fig. 1.2. The spacetime 4 is reflecting; the spacetime B is not Let SEM be a subset. The chronological past I~ [S] of S consists of those events which can signal to some observer in S a speed of light. Formally, J” [S]={zeM: there a speed less than the is a future-directec timelike curve from z to some se S}. Similarly, the causal past J~ [S] of S is defined by J” [S] = {ze M: zeS or there is a future-directed curve from z to some se S$}. If S is open, its (chronological) common past | S is the largest open set each event of which can signal o each observer in S$ at a speed less than that of light. Formally, | S = Interior {z¢ M: for all séS there is a future-directed timelike curve from z to s}. Unless S is empty, I~ [S]2[S. J*[S], J*[S], and ¢S are d efined dually; dual results will often be taken for granted. If z is an event, we abbreviate I* [{z}] by I*(z) and similarly for J*.

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References (15)

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