In
gravitation theory, a world manifold endowed with some
Lorentzian pseudo-Riemannian metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
and an associated space-time structure is a
space-time
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...
. Gravitation theory is formulated as
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
on
natural bundles over a world manifold.
Topology
A world manifold is a four-dimensional
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
real
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. It is assumed to be a
Hausdorff and
second countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
. Consequently, it is a
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
which is a union of a countable number of compact subsets, a
separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the ...
, a
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
and
completely regular space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
. 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
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
and necessary for the existence of a pseudo-Riemannian metric. A world manifold is assumed to be
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
and, consequently, it is
arcwise connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
.
Riemannian structure
The
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a world manifold
and the associated
principal frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
of linear tangent frames in
possess a
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
structure group
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
. A world manifold
is said to be
parallelizable
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields
\
on the manifold, such that at every point p of M the tangent vectors
\
provide a basis of the tangent space at p. Equiva ...
if the tangent bundle
and, accordingly, the frame bundle
are trivial, i.e., there exists a global section (a
frame field
A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime ...
) 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
In physics, general covariant transformations are symmetries of gravitation theory on a world manifold X. They are gauge transformations whose parameter functions are vector fields on X. From the physical viewpoint, general covariant transfor ...
. 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
of a frame bundle
over a world manifold
is always reducible to its maximal compact subgroup
. The corresponding global section of the quotient bundle
is a Riemannian metric
on
. 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
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
. The corresponding global section of the quotient bundle
is a pseudo-Riemannian metric
of signature
on
. It is treated as a
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
in
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and as a
classical Higgs field in
gauge gravitation theory
In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity.
''Gauge gravitation theory'' should not be confused with th ...
.
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
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
. 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
. Thus, there is the commutative diagram
:
:
:
of the reduction of structure groups of a frame bundle
in
gravitation 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
) with
-valued transition functions. These transition functions preserve a time-like component
of local frame fields which, therefore, is globally defined. It is a nowhere vanishing vector field on
. Accordingly, the dual time-like covector field
also is globally defined, and it yields a spatial
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
on
such that
. Then the tangent bundle
of a world manifold
admits a space-time decomposition
, where
is a one-dimensional fibre bundle spanned by a time-like vector field
. This decomposition, is called the
-compatible ''space-time structure''. It makes a world manifold the
space-time
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...
.
(ii) Given the above-mentioned diagram of reduction of structure groups, let
and
be the corresponding
pseudo-Riemannian and Riemannian metrics on
. They form a triple
obeying the relation
:
.
Conversely, let a world manifold
admit a nowhere vanishing
one-form
(or, equivalently, a nowhere vanishing vector
field). Then any Riemannian metric
on
yields the
pseudo-Riemannian metric
:
.
It follows that a world manifold
admits a pseudo-Riemannian
metric if and only if there exists a nowhere vanishing vector (or covector) field on
.
Let us note that a
-compatible Riemannian metric
in a triple
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 distance
functions are different for different spatial distributions
and
. It follows that physical observers associated with
these 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 In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite rest ...
). 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
is involutive. In this case, its integral manifolds constitute a spatial
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
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 .
However, this is not the case of a compact world manifold which can not be
a fibred manifold over
.
The stable causality does not provide the simplest causal structure. If a fibred manifold
is a fibre bundle, it is trivial, i.e., a world manifold
is a
globally hyperbolic manifold
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold ...
. Since any oriented three-dimensional manifold is parallelizable, a globally
hyperbolic world manifold is parallelizable.
See also
*
Spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
*
Mathematics of general relativity
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian man ...
*
Gauge gravitation theory
In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity.
''Gauge gravitation theory'' should not be confused with th ...
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
*{{cite journal, last1=Sardanashvily, first1=G., author1-link=Gennadi Sardanashvily, title=Classical gauge gravitation theory, journal=International Journal of Geometric Methods in Modern Physics, date=2011, volume=8, issue=8, pages=1869–1895, doi=10.1142/S0219887811005993, arxiv=1110.1176, bibcode=2011IJGMM..08.1869S, s2cid=119711561
Gravity
Theoretical physics