World manifold explained

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

of a world manifold

and the associated principal frame bundle

of linear tangent frames in

possess a general linear group structure group

GL^+(4,R)

. 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

GL^+(4,R)

of a frame bundle

over a world manifold

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

. Thus, a world manifold always admits a Riemannian metric which makes

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

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

of signature

(+,---)

. 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

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,R)\toSO(4)

\downarrow \downarrow

SO(1,3)\toSO(3)

of the reduction of structure groups of a frame bundle

ingravitation 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

\{h^λ\}

) with

SO(3)

-valued transition functions. These transition functions preserve a time-like component

	\mu
h
	0

\partial_\mu

of local frame fields which, therefore, is globally defined. It is a nowhere vanishing vector field on

. Accordingly, the dual time-like covector field

h^0=h

	0

	λ

dx^λ

also is globally defined, and it yields a spatial distribution

akF\subsetTX

such that

h^0\rfloorakF=0

. Then the tangent bundle

of a world manifold

admits a space-time decomposition

TX=akF ⊕ T^0X

, where

T^0X

is a one-dimensional fibre bundle spanned by a time-like vector field

h₀

. 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

g^R

be the correspondingpseudo-Riemannian and Riemannian metrics on

. They form a triple

(g,g^R,h⁰⁾

obeying the relation

g=2h^{0 ⊗}h⁰-g^R

Conversely, let a world manifold

admit a nowhere vanishingone-form

\sigma

(or, equivalently, a nowhere vanishing vectorfield). Then any Riemannian metric

g^R

yields thepseudo-Riemannian metric

g=	2
	g^{R(\sigma,\sigma)}

\sigma ⊗ \sigma-g^R

It follows that a world manifold

admits a pseudo-Riemannianmetric if and only if there exists a nowhere vanishing vector (or covector) field on

Let us note that a

-compatible Riemannian metric

g^R

in a triple

(g,g^R,h⁰⁾

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 distancefunctions are different for different spatial distributions

akF

and

akF'

. It follows that physical observers associated withthese 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.

Causality conditions

A space-time structure is called integrable if a spatial distribution

akF

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

X\toR

.However, this is not the case of a compact world manifold which can not bea fibred manifold over

The stable causality does not provide the simplest causal structure. If a fibred manifold

X\toR

is a fibre bundle, it is trivial, i.e., a world manifold

is a globally hyperbolic manifold

X=R x M

. Since any oriented three-dimensional manifold is parallelizable, a globallyhyperbolic world manifold is parallelizable.

References

S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge Univ. Press, Cambridge, 1973)
C.T.G. Dodson, Categories, Bundles, and Spacetime Topology (Shiva Publ. Ltd., Orpington, UK, 1980)

External links

Sardanashvily. G.. Gennadi Sardanashvily. Classical gauge gravitation theory. International Journal of Geometric Methods in Modern Physics. 2011. 8. 8. 1869–1895. 10.1142/S0219887811005993. 1110.1176. 2011IJGMM..08.1869S. 119711561.