World manifold
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. Topology. 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. 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. Riemannian structure. The tangent bundle formula_1 of a world manifold formula_2 and the associated principal frame bundle formula_3 of linear tangent frames in formula_1 possess a general linear group structure group formula_5. A world manifold formula_2 is said to be parallelizable if the tangent bundle formula_1 and, accordingly , the frame bundle formula_3 are trivial, i.e., there exists a global section (a frame field) of formula_3. 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 formula_5 of a frame bundle formula_3 over a world manifold formula_2 is always reducible to its maximal compact  subgroup formula_13. The corresponding global section of the quotient bundle formula_14 is a Riemannian metric formula_15 on formula_2. Thus, a world manifold always admits a Riemannian metric which makes formula_2 a metric topological space. Lorentzian structure. In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle formula_3 must be reduced to a Lorentz group formula_19. The corresponding global section of the quotient bundle formula_20 is a pseudo-Riemannian metric formula_21 of signature formula_22 on formula_2. 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. Space-time structure. If a structure group of a frame bundle formula_3 is reducible to a  Lorentz group, the latter is always reducible to its maximal compact subgroup formula_25. Thus, there is the commutative diagram of the reduction of structure groups of a frame bundle formula_3 in gravitation theory. This reduction diagram results in the following. (i) In gravitation theory on a world manifold formula_2, one can always choose an atlas of a frame bundle formula_3 (characterized by local frame fields formula_32) with formula_25-valued transition functions. These transition functions preserve a time-like component formula_34 of local frame fields which, therefore, is globally defined. It is a nowhere vanishing vector field on formula_2. Accordingly, the  dual time-like covector field formula_36 also is globally defined, and it yields a spatial distribution formula_37 on formula_2 such that formula_39. Then the tangent bundle formula_1 of a world manifold formula_2 admits a space-time decomposition formula_42, where formula_43 is a one-dimensional fibre bundle spanned by a time-like vector field formula_44. This decomposition, is called the formula_21-compatible "space-time structure". It makes a world manifold the space-time. (ii) Given the above-mentioned diagram of reduction of structure groups, let formula_21 and formula_15 be the corresponding pseudo-Riemannian and Riemannian metrics on formula_2. They form a  triple formula_49 obeying the relation Conversely, let a world manifold formula_2 admit a nowhere vanishing one-form formula_52 (or, equivalently, a nowhere vanishing vector field). Then any Riemannian metric formula_15 on formula_2 yields the pseudo-Riemannian metric It follows that a world manifold formula_2 admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on formula_2. Let us note that a formula_21-compatible Riemannian metric formula_15 in a triple formula_49 defines a formula_21-compatible distance function on a world manifold formula_2. Such a function brings formula_2 into a metric space whose locally  Euclidean topology is equivalent to a manifold topology on formula_2. Given a gravitational field formula_21, the formula_21-compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions formula_67 and formula_68. It follows that physical observers associated with these different spatial distributions perceive a world manifold formula_2 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. Causality conditions. A space-time structure is called integrable if a spatial distribution formula_67 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 formula_2 whose differential nowhere vanishes. Such a foliation is a fibred manifold formula_72. However, this is not the case of a compact world manifold which can not be a fibred manifold over formula_73. The stable causality does not provide the simplest causal structure. If a fibred manifold formula_74 is a fibre bundle, it is trivial, i.e., a world manifold formula_2 is a globally hyperbolic manifold formula_76. Since any oriented three-dimensional manifold is parallelizable, a globally hyperbolic world manifold is parallelizable.