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 T X {\displaystyle TX} of a world manifold X {\displaystyle X} and the associated principal frame bundle F X {\displaystyle FX} of linear tangent frames in T X {\displaystyle TX} possess a general linear group structure group G L + ( 4 , R ) {\displaystyle GL^{+}(4,\mathbb {R} )} . A world manifold X {\displaystyle X} is said to be parallelizable if the tangent bundle T X {\displaystyle TX} and, accordingly, the frame bundle F X {\displaystyle FX} are trivial, i.e., there exists a global section (a frame field) of F X {\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 G L + ( 4 , R ) {\displaystyle GL^{+}(4,\mathbb {R} )} of a frame bundle F X {\displaystyle FX} over a world manifold X {\displaystyle X} is always reducible to its maximal compact subgroup S O ( 4 ) {\displaystyle SO(4)} . The corresponding global section of the quotient bundle F X / S O ( 4 ) {\displaystyle FX/SO(4)} is a Riemannian metric g R {\displaystyle g^{R}} on X {\displaystyle X} . Thus, a world manifold always admits a Riemannian metric which makes X {\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 F X {\displaystyle FX} must be reduced to a Lorentz group S O ( 1 , 3 ) {\displaystyle SO(1,3)} . The corresponding global section of the quotient bundle F X / S O ( 1 , 3 ) {\displaystyle FX/SO(1,3)} is a pseudo-Riemannian metric g {\displaystyle g} of signature ( + , − − − ) {\displaystyle (+,---)} on X {\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 F X {\displaystyle FX} is reducible to a Lorentz group, the latter is always reducible to its maximal compact subgroup S O ( 3 ) {\displaystyle SO(3)} . Thus, there is the commutative diagram
G L ( 4 , R ) → S O ( 4 ) {\displaystyle GL(4,\mathbb {R} )\to SO(4)}
↓ ↓ {\displaystyle \downarrow \qquad \qquad \qquad \quad \downarrow }
S O ( 1 , 3 ) → S O ( 3 ) {\displaystyle SO(1,3)\to SO(3)}
of the reduction of structure groups of a frame bundle F X {\displaystyle FX} in gravitation theory. This reduction diagram results in the following. (i) In gravitation theory on a world manifold X {\displaystyle X} , one can always choose an atlas of a frame bundle F X {\displaystyle FX} (characterized by local frame fields { h λ } {\displaystyle \{h^{\lambda }\}} ) with S O ( 3 ) {\displaystyle SO(3)} -valued transition functions. These transition functions preserve a time-like component h 0 = h 0 μ ∂ μ {\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 X {\displaystyle X} . Accordingly, the dual time-like covector field h 0 = h λ 0 d x λ {\displaystyle h^{0}=h_{\lambda }^{0}dx^{\lambda }} also is globally defined, and it yields a spatial distribution
F ⊂ T X {\displaystyle {\mathfrak {F}}\subset TX} on X {\displaystyle X} such that h 0 ⌋ F = 0 {\displaystyle h^{0}\rfloor {\mathfrak {F}}=0} . Then the tangent bundle T X {\displaystyle TX} of a world manifold X {\displaystyle X} admits a space-time decomposition
T X = F ⊕ T 0 X {\displaystyle TX={\mathfrak {F}}\oplus T^{0}X} , where T 0 X {\displaystyle T^{0}X} is a one-dimensional fibre bundle spanned by a time-like vector field h 0 {\displaystyle h_{0}} . This decomposition, is called the g {\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 g {\displaystyle g} and g R {\displaystyle g^{R}} be the corresponding pseudo-Riemannian and Riemannian metrics on X {\displaystyle X} . They form a triple ( g , g R , h 0 ) {\displaystyle (g,g^{R},h^{0})} obeying the relation
g = 2 h 0 ⊗ h 0 − g R {\displaystyle g=2h^{0}\otimes h^{0}-g^{R}} . Conversely, let a world manifold X {\displaystyle X} admit a nowhere vanishing one-form σ {\displaystyle \sigma } (or, equivalently, a nowhere vanishing vector field). Then any Riemannian metric g R {\displaystyle g^{R}} on X {\displaystyle X} yields the pseudo-Riemannian metric
g = 2 g R ( σ , σ ) σ ⊗ σ − g R {\displaystyle g={\frac {2}{g^{R}(\sigma ,\sigma )}}\sigma \otimes \sigma -g^{R}} . It follows that a world manifold X {\displaystyle X} admits a pseudo-Riemannian metric if and only if there exists a nowhere vanishing vector (or covector) field on X {\displaystyle X} . Let us note that a g {\displaystyle g} -compatible Riemannian metric g R {\displaystyle g^{R}} in a triple ( g , g R , h 0 ) {\displaystyle (g,g^{R},h^{0})} defines a g {\displaystyle g} -compatible distance function on a world manifold X {\displaystyle X} . Such a function brings X {\displaystyle X} into a metric space whose locally Euclidean topology is equivalent to a manifold topology on X {\displaystyle X} . Given a gravitational field g {\displaystyle g} , the g {\displaystyle g} -compatible Riemannian metrics and the corresponding distance functions are different for different spatial distributions F {\displaystyle {\mathfrak {F}}}
and F ′ {\displaystyle {\mathfrak {F}}'} . It follows that physical observers associated with these different spatial distributions perceive a world manifold X {\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 F {\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 X {\displaystyle X} whose differential nowhere vanishes. Such a foliation is a fibred manifold X → R {\displaystyle X\to \mathbb {R} } . However, this is not the case of a compact world manifold which can not be a fibred manifold over R {\displaystyle \mathbb {R} } . The stable causality does not provide the simplest causal structure. If a fibred manifold X → R {\displaystyle X\to \mathbb {R} } is a fibre bundle, it is trivial, i.e., a world manifold X {\displaystyle X} is a globally hyperbolic manifold X = R × M {\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.