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 $$TX$$ 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
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
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
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.