World manifold

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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.

Topology[edit]

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.

Riemannian structure[edit]

The tangent bundle TX of a world manifold X and the associated principal frame bundle FX of linear tangent frames in TX possess a general linear group structure group GL^+(4,\mathbb R) . A world manifold X is said to be parallelizable if the tangent bundle TX and, accordingly, the frame bundle FX are trivial, i.e., there exists a global section (a frame field) of FX. 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 GL^+(4,\mathbb R) of a frame bundle FX over a world manifold X is always reducible to its maximal compact subgroup SO(4) . The corresponding global section of the quotient bundle FX/SO(4) is a Riemannian metric g^R on X. Thus, a world manifold always admits a Riemannian metric which makes X a metric topological space.

Lorentzian structure[edit]

In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.e., a structure group of a frame bundle FX must be reduced to a Lorentz group SO(1,3) . The corresponding global section of the quotient bundle FX/SO(1,3) is a pseudo-Riemannian metric g of signature (+,---) on X. 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[edit]

If a structure group of a frame bundle FX is reducible to a Lorentz group, the latter is always reducible to its maximal compact subgroup SO(3) . Thus, there is the commutative diagram

  GL(4,\mathbb R) \to  SO(4)
 \downarrow \qquad \qquad \qquad \quad \downarrow
 SO(1,3) \to SO(3)

of the reduction of structure groups of a frame bundle FX in gravitation theory. This reduction diagram results in the following.

(i) In gravitation theory on a world manifold X, one can always choose an atlas of a frame bundle FX (characterized by local frame fields \{h^\lambda\}) with SO(3) -valued transition functions. These transition functions preserve a time-like component h_0=h^\mu_0 \partial_\mu of local frame fields which, therefore, is globally defined. It is a nowhere vanishing vector field on X. Accordingly, the dual time-like covector field h^0=h^0_\lambda dx^\lambda also is globally defined, and it yields a spatial distribution \mathfrak F\subset TX on X such that h^0\rfloor \mathfrak F=0. Then the tangent bundle TX of a world manifold X admits a space-time decomposition TX=\mathfrak F\oplus T^0X, where T^0X is a one-dimensional fibre bundle spanned by a time-like vector field h_0. This decomposition, is called the g-compatible space-time structure. It makes a world manifold the space-time.

(ii) Given the above mentioned diagram of reduction of structure groups, let g and g^R be the corresponding pseudo-Riemannian and Riemannian metrics on X. They form a triple  (g,g^R,h^0) obeying the relation

g=2h^0\otimes h^0 -g^R.

Conversely, let a world manifold X admit a nowhere vanishing one-form \sigma (or, equivalently, a nowhere vanishing vector field). Then any Riemannian metric g^R on X yields the pseudo-Riemannian metric

g=\frac{2}{g^R(\sigma,\sigma)}\sigma\otimes \sigma -g^R.

It follows that a world manifold X admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on X.

Let us note that a g-compatible Riemannian metric g^R in a triple  (g,g^R,h^0) defines a g-compatible distance function on a world manifold X. Such a function brings X into a metric space whose locally Euclidean topology is equivalent to a manifold topology on X. Given a gravitational field g, the g-compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions \mathfrak F and \mathfrak F'. It follows that physical observers associated with these different spatial distributions perceive a world manifold X 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[edit]

A space-time structure is called integrable if a spatial distribution \mathfrak F 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 X whose differential nowhere vanishes. Such a foliation is a fibred manifold X\to \mathbb R. However, this is not the case of a compact world manifold which can not be a fibred manifold over \mathbb R.

The stable causality does not provide the simplest causal structure. If a fibred manifold X\to\mathbb R is a fibre bundle, it is trivial, i.e., a world manifold X is a globally hyperbolic manifold X=\mathbb R \times M. Since any oriented three-dimensional manifold is parallelizable, a globally hyperbolic world manifold is parallelizable.

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