Physics: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 [math]\displaystyle{ TX }[/math] of a world manifold [math]\displaystyle{ X }[/math] and the associated principal frame bundle [math]\displaystyle{ FX }[/math] of linear tangent frames in [math]\displaystyle{ TX }[/math] possess a general linear group structure group [math]\displaystyle{ GL^+(4,\mathbb R) }[/math]. A world manifold [math]\displaystyle{ X }[/math] is said to be parallelizable if the tangent bundle [math]\displaystyle{ TX }[/math] and, accordingly, the frame bundle [math]\displaystyle{ FX }[/math] are trivial, i.e., there exists a global section (a frame field) of [math]\displaystyle{ FX }[/math]. 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 [math]\displaystyle{ GL^+(4,\mathbb R) }[/math] of a frame bundle [math]\displaystyle{ FX }[/math] over a world manifold [math]\displaystyle{ X }[/math] is always reducible to its maximal compact subgroup [math]\displaystyle{ SO(4) }[/math]. The corresponding global section of the quotient bundle [math]\displaystyle{ FX/SO(4) }[/math] is a Riemannian metric [math]\displaystyle{ g^R }[/math] on [math]\displaystyle{ X }[/math]. Thus, a world manifold always admits a Riemannian metric which makes [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ FX }[/math] must be reduced to a Lorentz group [math]\displaystyle{ SO(1,3) }[/math]. The corresponding global section of the quotient bundle [math]\displaystyle{ FX/SO(1,3) }[/math] is a pseudo-Riemannian metric [math]\displaystyle{ g }[/math] of signature [math]\displaystyle{ (+,---) }[/math] on [math]\displaystyle{ X }[/math]. 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 [math]\displaystyle{ FX }[/math] is reducible to a Lorentz group, the latter is always reducible to its maximal compact subgroup [math]\displaystyle{ SO(3) }[/math]. Thus, there is the commutative diagram

[math]\displaystyle{ GL(4,\mathbb R) \to SO(4) }[/math]

[math]\displaystyle{ \downarrow \qquad \qquad \qquad \quad \downarrow }[/math]

[math]\displaystyle{ SO(1,3) \to SO(3) }[/math]

of the reduction of structure groups of a frame bundle [math]\displaystyle{ FX }[/math] in gravitation theory. This reduction diagram results in the following.

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

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

[math]\displaystyle{ g=2h^0\otimes h^0 -g^R }[/math].

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

[math]\displaystyle{ g=\frac{2}{g^R(\sigma,\sigma)}\sigma\otimes \sigma -g^R }[/math].

It follows that a world manifold [math]\displaystyle{ X }[/math] admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on [math]\displaystyle{ X }[/math].

Let us note that a [math]\displaystyle{ g }[/math]-compatible Riemannian metric [math]\displaystyle{ g^R }[/math] in a triple [math]\displaystyle{ (g,g^R,h^0) }[/math] defines a [math]\displaystyle{ g }[/math]-compatible distance function on a world manifold [math]\displaystyle{ X }[/math]. Such a function brings [math]\displaystyle{ X }[/math] into a metric space whose locally Euclidean topology is equivalent to a manifold topology on [math]\displaystyle{ X }[/math]. Given a gravitational field [math]\displaystyle{ g }[/math], the [math]\displaystyle{ g }[/math]-compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions [math]\displaystyle{ \mathfrak F }[/math] and [math]\displaystyle{ \mathfrak F' }[/math]. It follows that physical observers associated with these different spatial distributions perceive a world manifold [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ \mathfrak F }[/math] 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 [math]\displaystyle{ X }[/math] whose differential nowhere vanishes. Such a foliation is a fibred manifold [math]\displaystyle{ X\to \mathbb R }[/math]. However, this is not the case of a compact world manifold which can not be a fibred manifold over [math]\displaystyle{ \mathbb R }[/math].

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

See also

• Spacetime
• Mathematics of general relativity
• Gauge gravitation theory

References

• 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

External links

• Sardanashvily, G. (2011). "Classical gauge gravitation theory". International Journal of Geometric Methods in Modern Physics 8 (8): 1869–1895. doi:10.1142/S0219887811005993. Bibcode: 2011IJGMM..08.1869S.