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In gravitation theory, a world manifold endowed with some Lorentzian pseudo-Riemannian metric and an associated space-time structure is a space-time. Gravitation theory is formulated as classical field theory on natural bundles over a world manifold.
A world manifold is a four-dimensional orientable real smooth manifold. It is assumed to be a Hausdorff and second countable topological space. Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space, a paracompact and completely regular space. Being paracompact, a world manifold admits a partition of unity by smooth functions. It should be emphasized that paracompactness is an essential characteristic of a world manifold. It is necessary and sufficient in order that a world manifold admits a Riemannian metric and necessary for the existence of a pseudo-Riemannian metric. A world manifold is assumed to be connected and, consequently, it is arcwise connected.
The tangent bundle of a world manifold and the associated principal frame bundle of linear tangent frames in possess a general linear group structure group . A world manifold is said to be parallelizable if the tangent bundle and, accordingly, the frame bundle are trivial, i.e., there exists a global section (a frame field) of . It is essential that the tangent and associated bundles over a world manifold admit a bundle atlas of finite number of trivialization charts.
Tangent and frame bundles over a world manifold are natural bundles characterized by general covariant transformations. These transformations are gauge symmetries of gravitation theory on a world manifold.
By virtue of the well-known theorem on structure group reduction, a structure group of a frame bundle over a world manifold is always reducible to its maximal compact subgroup . The corresponding global section of the quotient bundle is a Riemannian metric on . Thus, a world manifold always admits a Riemannian metric which makes a metric topological space.
In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle must be reduced to a Lorentz group . The corresponding global section of the quotient bundle is a pseudo-Riemannian metric of signature on . It is treated as a gravitational field in General Relativity and as a classical Higgs field in gauge gravitation theory.
A Lorentzian structure need not exist. Therefore a world manifold is assumed to satisfy a certain topological condition. It is either a noncompact topological space or a compact space with a zero Euler characteristic. Usually, one also requires that a world manifold admits a spinor structure in order to describe Dirac fermion fields in gravitation theory. There is the additional topological obstruction to the existence of this structure. In particular, a noncompact world manifold must be parallelizable.
If a structure group of a frame bundle is reducible to a Lorentz group, the latter is always reducible to its maximal compact subgroup . Thus, there is the commutative diagram
of the reduction of structure groups of a frame bundle in gravitation theory. This reduction diagram results in the following.
(i) In gravitation theory on a world manifold , one can always choose an atlas of a frame bundle (characterized by local frame fields ) with -valued transition functions. These transition functions preserve a time-like component of local frame fields which, therefore, is globally defined. It is a nowhere vanishing vector field on . Accordingly, the dual time-like covector field also is globally defined, and it yields a spatial distribution on such that . Then the tangent bundle of a world manifold admits a space-time decomposition , where is a one-dimensional fibre bundle spanned by a time-like vector field . This decomposition, is called the -compatible space-time structure. It makes a world manifold the space-time.
(ii) Given the above mentioned diagram of reduction of structure groups, let and be the corresponding pseudo-Riemannian and Riemannian metrics on . They form a triple obeying the relation
Conversely, let a world manifold admit a nowhere vanishing one-form (or, equivalently, a nowhere vanishing vector field). Then any Riemannian metric on yields the pseudo-Riemannian metric
It follows that a world manifold admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on .
Let us note that a -compatible Riemannian metric in a triple defines a -compatible distance function on a world manifold . Such a function brings into a metric space whose locally Euclidean topology is equivalent to a manifold topology on . Given a gravitational field , the -compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions and . It follows that physical observers associated with these different spatial distributions perceive a world manifold as different Riemannian spaces. The well-known relativistic changes of sizes of moving bodies exemplify this phenomenon.
However, one attempts to derive a world topology directly from a space-time structure (a path topology, an Alexandrov topology). If a space-time satisfies the strong causality condition, such topologies coincide with a familiar manifold topology of a world manifold. In a general case, they however are rather extraordinary.
A space-time structure is called integrable if a spatial distribution is involutive. In this case, its integral manifolds constitute a spatial foliation of a world manifold whose leaves are spatial three-dimensional subspaces. A spatial foliation is called causal if no curve transversal to its leaves intersects each leave more than once. This condition is equivalent to the stable causality of Stephen Hawking. A space-time foliation is causal if and only if it is a foliation of level surfaces of some smooth real function on whose differential nowhere vanishes. Such a foliation is a fibred manifold . However, this is not the case of a compact world manifold which can not be a fibred manifold over .
The stable causality does not provide the simplest causal structure. If a fibred manifold is a fibre bundle, it is trivial, i.e., a world manifold is a globally hyperbolic manifold . Since any oriented three-dimensional manifold is parallelizable, a globally hyperbolic world manifold is parallelizable.
- S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge Univ. Press, Cambridge, 1973) ISBN 0-521-20016-4
- C.T.G. Dodson, Categories, Bundles, and Spacetime Topology (Shiva Publ. Ltd., Orpington, UK, 1980) ISBN 0-906812-01-1