“Causality conditions”的意思、由来-开放百科全书

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Causality conditions

释义

The hierarchy
Non-totally vicious
Chronological
Causal
Distinguishing
Past-distinguishing Future-distinguishing
Strongly causal
Stably causal
Globally hyperbolic
See also
References

{{About|the classification of Lorentzian manifolds according to the types of causal structures they admit|a basic treatment of the possible causal relationships among points in a Lorentzian manifold, including the definitions of terms used in this article|Causal structure}}

In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.^[1]

The weaker the causality condition on a spacetime, the more unphysical the spacetime is. Spacetimes with closed timelike curves, for example, present severe interpretational difficulties. See the grandfather paradox.

It is reasonable to believe that any physical spacetime will satisfy the strongest causality condition: global hyperbolicity. For such spacetimes the equations in general relativity can be posed as an initial value problem on a Cauchy surface.

The hierarchy

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 . Where two or more are given they are equivalent.

Notation:

denotes the chronological relation.
denotes the causal relation.

(See causal structure for definitions of , and , .)

Non-totally vicious

For some points we have .

Chronological

There are no closed chronological (timelike) curves.
The chronological relation is irreflexive: for all .

Causal

There are no closed causal (non-spacelike) curves.
If both and then

Distinguishing

Past-distinguishing

Two points which share the same chronological past are the same point:

For any neighborhood of there exists a neighborhood such that no past-directed non-spacelike curve from intersects more than once.

Future-distinguishing

Two points which share the same chronological future are the same point:

For any neighborhood of there exists a neighborhood such that no future-directed non-spacelike curve from intersects more than once.

Strongly causal

For any there exists a neighborhood of such that there exists no timelike curve that passes through more than once.
For any neighborhood of there exists a neighborhood such that is causally convex in (and thus in ).
The Alexandrov topology agrees with the manifold topology.

Stably causal

A manifold satisfying any of the weaker causality conditions defined above may fail to do so if the metric is given a small perturbation. A spacetime is stably causal if it cannot be made to contain closed causal curves by arbitrarily small perturbations of the metric. Stephen Hawking showed^[2] that this is equivalent to:

There exists a global time function on . This is a scalar field on whose gradient 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).

Globally hyperbolic

is strongly causal and every set (for points ) is compact.

Robert Geroch showed^[3] that a spacetime is globally hyperbolic if and only if there exists a Cauchy surface for . This means that:

is topologically equivalent to for some Cauchy surface (Here denotes the real line).

References

1. ^E. Minguzzi and M. Sanchez, The causal hierarchy of spacetimes in H. Baum and D. Alekseevsky (eds.), vol. Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., (Eur. Math. Soc. Publ. House, Zurich, 2008), pp. 299–358, {{ISBN|978-3-03719-051-7}}, gr-qc/0609119
2. ^S.W. Hawking, [https://www.jstor.org/stable/2416157 The existence of cosmic time functions] Proc. R. Soc. Lond. (1969), A308, 433
3. ^R. Geroch, Domain of Dependence {{webarchive|url=https://archive.is/20130224130421/http://link.aip.org/link/?JMAPAQ/11/437/1 |date=2013-02-24 }} J. Math. Phys. (1970) 11, 437–449

{{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}}
{{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}}

4 : Lorentzian manifolds|Theory of relativity|General relativity|Theoretical physics

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