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Contorsion Tensor
The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) 11-dimensional supergravity. That is, the contorsion tensor, along with the connection, becomes one of the dynamical objects of the theory, demoting the metric to a secondary, derived role. The elimination of torsion in a connection is referred to as the ''absorption of torsion'', and is one of the steps of Cartan's equivalence method for establishing the equivalence of geometric structures. Definition in metric geometry In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol \Gamm ... [...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] |
Vertical And Horizontal Bundles
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the vertical bundle VE and horizontal bundle HE are subbundles of the tangent bundle TE of E whose Whitney sum satisfies VE\oplus HE\cong TE. This means that, over each point e\in E, the fibers V_eE and H_eE form complementary subspaces of the tangent space T_eE. The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle. To make this precise, define the vertical space V_eE at e\in E to be \ker(d\pi_e). That is, the differential d\pi_e\colon T_eE\to T_bB (where b=\pi(e)) is a linear surjection whose kernel has the same dimension as the fibers of \pi. If we write F=\pi^(b), then V_eE consists of exactly the vectors in T_eE which are also tangent to F. The name is motivated by low-dimensional examples like the tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensors
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belinfante–Rosenfeld Stress–energy Tensor
In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved. In a classical or quantum local field theory, the generator of Lorentz transformations can be written as an integral : M_ = \int \mathrm^3x \, _ of a local current : _= (x_\nu _\lambda - x_\lambda _\nu)+ _. Here _\lambda is the canonical Noether energy–momentum tensor, and _ is the contribution of the intrinsic (spin) angular momentum. Local conservation of angular momentum : \partial_\mu _=0 \, requires that : \partial_\mu _=T_-T_. Thus a source of spin-current implies a non-symmetric canonical energy–momentum tensor. The Belinfante–Rosenfeld tensor is a modification of the energy momentum tensor : T_B^ = T^ +\frac 12 \partial_\lambda(S^+S^-S^) that is constructed from the canonical energy momentum tensor and the spin curre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bryce DeWitt
Bryce Seligman DeWitt (January 8, 1923 – September 23, 2004), was an American theoretical physicist noted for his work in gravitation and quantum field theory. Life He was born Carl Bryce Seligman, but he and his three brothers, including the noted ichthyologist Hugh Hamilton DeWitt, added "DeWitt" from their mother's side of the family, at the urging of their father, in 1950. In the early-1970s, this change of name so angered Felix Bloch that he blocked DeWitt's appointment to Stanford University and DeWitt and his wife Cecile DeWitt-Morette, a mathematical physicist, accepted faculty positions at the University of Texas at Austin. DeWitt served in World War II as a naval aviator. He died September 23, 2004 from pancreatic cancer at the age of 81. He is buried in France, and was survived by his wife and four daughters. Work He pioneered work in the quantization of general relativity and, in particular, developed canonical quantum gravity, manifestly covariant methods, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below. Informal definition An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by :(x,\theta,\bar) where ''x'' is the ( real-number-valued) spacetime coordinate, and \theta\, and \bar are Grassmann-valued spatial "directions". The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for supersymmetry have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weitzenböck Connection
Teleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field. Teleparallel spacetimes The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set of four vector fields defined on ''all'' of such that for every the set is a basis of , where denotes the fiber over of the tangent vector bundle . Hence, the four-dimensional spacetime manifold must be a parallelizable manifold. The tetrad field was introduced to allow the distant comparison of the direction of tangent vectors at different points of the manifold, hence the name distant parallelism. His attempt failed because there was no Schwarzs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teleparallelism
Teleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field. Teleparallel spacetimes The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set of four vector fields defined on ''all'' of such that for every the set is a basis of , where denotes the fiber over of the tangent vector bundle . Hence, the four-dimensional spacetime manifold must be a parallelizable manifold. The tetrad field was introduced to allow the distant comparison of the direction of tangent vectors at different points of the manifold, hence the name distant parallelism. His attempt failed because there was no Schwarzs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Equation
In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equation are mathematical functions which correspond directly to the field, as functions of time and space. Since the field equation is a partial differential equation, there are families of solutions which represent a variety of physical possibilities. Usually, there is not just a single equation, but a set of coupled equations which must be solved simultaneously. Field equations are not ordinary differential equations since a field depends on space and time, which requires at least two variables. Whereas the "wave equation", the " diffusion equation", and the " continuity equation" all have standard forms (and various special cases or generalizations), there is no single, special equation referred to as "the field equation". The topic broa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehresmann Connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action. Introduction A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of a vector bundle in a covariant manner. It also allows one to formulate a notion of a parallel section of a bundle in the direction of a vector: a section ''s'' is parallel along a vector ''X'' if \nabla_X s = 0. So a covariant derivative provides at least two t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological One-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics (on the manifold Q). The exterior derivative of this form defines a symplectic form giving T^Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. To define the tautological one-form, select a coordinate chart U on T^*Q and a canonical coordinate system on U. Pick an arbitrary point m \in T^*Q. By definition of cotangent bundl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Form
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet–Serret frame about the tangent vector). In the geometry of surfaces, the ''geodesic torsion'' describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting". More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
