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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 transformations are treated as particular ( holonomic) reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles. Mathematical definition Let \pi:Y\to X be a fibered manifold with local fibered coordinates (x^\lambda, y^i)\,. Every automorphism of Y is projected onto a diffeomorphism of its base X. However, the converse is not true. A diffeomorphism of X need not give rise to an automorphism of Y. In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of Y is a projectable vector field : u=u^\lambda(x^\mu)\partial_\lambda + u^i(x^\mu,y^j)\partial_i on Y. This vector field is projecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of phys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (fibred Manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles. Formal definition Let be a fibered manifold. A generalized ''connection'' on is a section , where is the jet manifold of . Connection as a horizontal splitting With the above manifold there is the following canonical short exact sequence of vector bundles over : where and are the tangent bundles of , respectively, is the vertical tangent bundle of , and is the pullback bundle of onto . A connection on a fibered manifold is defined as a linear bundle morphism over which splits the exact sequence . A connection always exists. Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution : \mathrmY=\Gamma\left(Y\times_X \mathrmX \right) \subset \mathrmY ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundles
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 product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadi Sardanashvily
Gennadi Sardanashvily (russian: Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 – September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University. Biography Gennadi Sardanashvily graduated from Moscow State University (MSU) in 1973, he was a Ph.D. student of the Department of Theoretical Physics ( MSU) in 1973–76, where he held a position in 1976. He attained his Ph.D. degree in physics and mathematics from MSU, in 1980, with Dmitri Ivanenko as his supervisor, and his D.Sc. degree in physics and mathematics from MSU, in 1998. Gennadi Sardanashvily was the founder and Managing Editor (2003 - 2013) of the International Journal of Geometric Methods in Modern Physics (IJGMMP). He was a member of Lepage Research Institute (Czech Republic). Research area Gennadi Sardanashvily research area is geometric method in classical and quantum mechanics and field theory, gravitation theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibered Manifold
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ E \to T_B is surjective, or, equivalently, its rank equals \dim B. History In topology, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1932, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which vector bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the similarly-named gauge theory gravity, which is a formulation of (classical) gravitation in the language of geometric algebra. Nor should it be confused with Kaluza–Klein theory, where the gauge fields are used to describe particle fields, but not gravity itself. Overview The first gauge model of gravity was suggested by Ryoyu Utiyama (1916–1990) in 1956 just two years after birth of the gauge theory itself. However, the initial attempts to construct the gauge theory of gravity by analogy with the gauge models of internal symmetries encountered a problem of treating general covariant transformations and establishing the gauge status of a pseudo-Riemannian metric (a tetrad field). In order to overcome this drawback, representing tet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Covariance
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist ''a priori'' in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. While this concept is exhibited by general relativity, which describes the dynamics of spacetime, one should not expect it to hold in less fundamental theories. For matter fields taken to exist independently of the background, it is almost never the case that their equations of motion will take the same form in curved space that they do in flat space. Overview A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems, and is usually expressed in terms of tensor fields. The classical (non-qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 naturally on F(''E'') via a change of basis, giving the frame bundle the structure of a principal GL(''k'', R)-bundle (where ''k'' is the rank of ''E''). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Definition and construction Let ''E'' → ''X'' be a real vector bundle of rank ''k'' over a topological space ''X''. A frame at a point ''x'' ∈ ''X'' is an ordered basis for the vector space ''E''''x''. Equivalently, a frame can be viewed as a linear isomorphism :p : \mathbf^k \to E_x. The set of all frames at ''x'', denoted ''F''''x'', has a natural right action by the general linear group GL(''k'', R) of invertible ''k'' × ''k'' matr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an element of TM can be thought of as a pair (x,v), where x is a point in M and v is a tangent vector to M at x . There is a natural projection : \pi : TM \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Bundle
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors. Definition A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space of the base space, which is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p .... References * * * See also * * * Vector bundles {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
