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

${\displaystyle GL(4,\mathbb {R} )\to SO(4)}$

${\displaystyle \downarrow \qquad \qquad \qquad \quad \downarrow }$

${\displaystyle SO(1,3)\to SO(3)}$

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

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

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

${\displaystyle g=2h^{0}\otimes h^{0}-g^{R}}$.

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

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

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

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

The stable causality does not provide the simplest causal structure. If a fibred manifold ${\displaystyle X\to \mathbb {R} }$ is a fibre bundle, it is trivial, i.e., a world manifold ${\displaystyle X}$ is a globally hyperbolic manifold ${\displaystyle X=\mathbb {R} \times M}$. 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. arXiv:1110.1176. Bibcode:2011IJGMM..08.1869S. doi:10.1142/S0219887811005993. S2CID 119711561.