causality conditions

There is a hierarchy of causality conditions, each one of which is strictly stronger than the previous. This is sometimes called the causal ladder. The conditions, from weakest to strongest, are:

• Non-totally vicious
• Chronological
• Causal
• Distinguishing
• Strongly causal
• Stably causal
• Causally continuous
• Causally simple
• Globally hyperbolic

Given are the definitions of these causality conditions for a Lorentzian manifold $(M,g)$. Where two or more are given they are equivalent.

Notation:

• $p\; \backslash ll\; q$ denotes the chronological relation.
• $p\; \backslash prec\; q$ denotes the causal relation.

(See causal structure for definitions of $\backslash ,I^+(x)$, $\backslash ,I^-(x)$ and $\backslash ,J^+(x)$, $\backslash ,J^-(x)$.)

• For some points $p\; \backslash in\; M$ we have $p\; \backslash not\backslash ll\; p$.

• There are no closed chronological (timelike) curves.
• The chronological relation is irreflexive: $p\; \backslash not\backslash ll\; p$ for all $p\; \backslash in\; M$.

• There are no closed causal (non-spacelike) curves.
• If both $p\; \backslash prec\; q$ and $q\; \backslash prec\; p$ then $p\; =\; q$

• Two points $p,\; q\; \backslash in\; M$ which share the same chronological past are the same point:

:: $I^-(p)\; =\; I^-(q)\; \backslash implies\; p\; =\; q$

• Equivalently, for any neighborhood $U$ of $p\; \backslash in\; M$ there exists a neighborhood $V\; \backslash subset\; U,\; p\; \backslash in\; V$ such that no past-directed non-spacelike curve from $p$ intersects $V$ more than once.

• Two points $p,\; q\; \backslash in\; M$ which share the same chronological future are the same point:

:: $I^+(p)\; =\; I^+(q)\; \backslash implies\; p\; =\; q$

• Equivalently, for any neighborhood $U$ of $p\; \backslash in\; M$ there exists a neighborhood $V\; \backslash subset\; U,\; p\; \backslash in\; V$ such that no future-directed non-spacelike curve from $p$ intersects $V$ more than once.

• For every neighborhood $U$ of $p\; \backslash in\; M$ there exists a neighborhood $V\; \backslash subset\; U,\; p\; \backslash in\; V$ through which no timelike curve passes more than once.
• For every neighborhood $U$ of $p\; \backslash in\; M$ there exists a neighborhood $V\; \backslash subset\; U,\; p\; \backslash in\; V$ that is causally convex in $M$ (and thus in $U$).
• The Alexandrov topology agrees with the manifold topology.

For each of the weaker causality conditions defined above, there are some manifolds satisfying the condition which can be made to violate it by arbitrarily small perturbations of the metric. A spacetime is stably causal if it cannot be made to contain closed causal curves by any perturbation smaller than some arbitrary finite magnitude. Stephen Hawking showedS.W. Hawking, [https://www.jstor.org/stable/2416157 The existence of cosmic time functions] Proc. R. Soc. Lond. (1969), A308, 433 that this is equivalent to:

• There exists a global time function on $M$. This is a scalar field $t$ on $M$ whose gradient $\backslash nabla^a\; t$ is everywhere timelike and future-directed. This global time function gives us a stable way to distinguish between future and past for each point of the spacetime (and so we have no causal violations).

{{Main|Globally hyperbolic manifold}}

• $\backslash ,M$ is strongly causal and every set $J^+(x)\; \backslash cap\; J^-(y)$ (for points $x,y\; \backslash in\; M$) is compact.

Robert Geroch showedR. Geroch, [http://link.aip.org/link/?JMAPAQ/11/437/1 Domain of Dependence] {{webarchive|url=https://archive.today/20130224130421/http://link.aip.org/link/?JMAPAQ/11/437/1 |date=2013-02-24 }} J. Math. Phys. (1970) 11, 437–449 that a spacetime is globally hyperbolic if and only if there exists a Cauchy surface for $M$. This means that:

• $M$ is topologically equivalent to $\backslash mathbb\{R\}\; \backslash times\backslash !\backslash ,\; S$ for some Cauchy surface $S.$ (Here $\backslash mathbb\{R\}$ denotes the real line).

• Spacetime
• Lorentzian manifold
• Causal structure
• Globally hyperbolic manifold
• Closed timelike curve

• {{cite book | author=S.W. Hawking, G.F.R. Ellis | title=The Large Scale Structure of Space-Time | location=Cambridge | publisher=Cambridge University Press | year=1973 | isbn=0-521-20016-4| title-link=The Large Scale Structure of Space-Time }}
• {{cite book | author = S.W. Hawking, W. Israel | title = General Relativity, an Einstein Centenary Survey| publisher = Cambridge University Press | year =1979 | isbn=0-521-22285-0| title-link = General Relativity, an Einstein Centenary Survey}}

Category:Lorentzian manifolds

Category:Theory of relativity

Category:General relativity

Category:Theoretical physics