대역적 쌍곡 다양체

일반 상대성 이론에서 대역적 쌍곡 다양체(大域的雙曲多樣體, 영어: globally hyperbolic manifold)는 초기 조건 문제가 잘 정의될 수 있는 시공간을 묘사하는 다양체이다.

정의

$M$ 이 경계가 없는 매끄러운 로런츠 다양체라고 하자. 만약 $M$ 이 다음 두 조건을 만족시킨다면, $M$ 을 대역적 쌍곡 다양체라고 한다.

(인과성) 인과적 폐곡선을 갖지 않는다.
(인과 다이아몬드의 콤팩트성) 모든 $x,y\in M$ 에 대하여, $J^{-}(x)\cap J^{+}(y)$ 는 콤팩트하다. 여기서 $J^{\pm }$ 은 인과적 미래와 인과적 과거이다.

만약 첫 번째 조건만 만족시키는 경우, $M$ 을 인과적 다양체(영어: causal manifold)라고 한다. 원래 대역적 쌍곡성은 위 "인과성" 조건 대신 "강한 인과성"(영어: strong causality) 조건으로 정의되었지만, 2007년에는 강한 인과성을 인과성으로 약화시켜도 같은 개념이 정의된다는 것이 증명되었다.^[1]

다양체 $M$ 의 코시 곡면(영어: Cauchy surface)은 다음 두 조건을 만족시키는, 여차원이 1인 초곡면 $S\subset M$ 이다.

(비시간성 영어: achronality) 시간꼴 곡선은 $S$ 를 2번 이상 관통할 수 없다.
모든 점 $x\in M$ 에 대하여, $x$ 를 지나는 모든 인과적 곡선은 $S$ 를 지나게 인과적으로 연장될 수 있다.

코시 곡면의 존재는 대역적 쌍곡성과 동치이다.

성질

대역적 쌍곡 다양체 $M$ 의 코시 곡면 $S$ 들은 모두 미분동형이며, 또 $M$ 은 $S\times \mathbb {R}$ 와 미분동형이다.^[2]

각주

↑ Bernal, Antonio N.; Miguel Sánchez (2007). “Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’”. 《Classical and Quantum Gravity》 (영어) 24 (3): 745–749. arXiv:gr-qc/0611138. Bibcode:2007CQGra..24..745B. doi:10.1088/0264-9381/24/3/N01.
↑ Bernal, Antonio N.; Miguel Sánchez. “On smooth Cauchy hypersurfaces and Geroch's splitting theorem” (영어). arXiv:gr-qc/0306108. Bibcode:2003CMaPh.243..461B. doi:10.1007/s00220-003-0982-6.

참고 문헌

Hawking, Stephen; G. F. R. Ellis (1975년 3월). 《The large scale structure of space-time》. Cambridge Monographs on Mathematical Physics (영어). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511524646. ISBN 978-052109906-6.
Wald, Robert M. (1984년 6월). 《General relativity》 (영어). University of Chicago Press. ISBN 978-022687033-5.
Earman, John (1995). 《Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic spacetimes》 (PDF) (영어). Oxford University Press. ISBN 978-019509591-3.

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