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Bach Tensor
In differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension . Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor.P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 304, No. 1476 (Apr. 2, 1968), pp113€“122 In abstract indices the Bach tensor is given by :B_ = P_^d+\nabla^c\nabla_cP_-\nabla^c\nabla_aP_ where ''W'' is the Weyl tensor, and ''P'' the Schouten tensor given in terms of the Ricci tensor ''R_'' and scalar curvature ''R'' by :P_=\frac\left(R_-\fracg_\right). See also *Cotton tensor In differential geometry, the Cotton tensor on a (pseudo)- Riemannian manifold of dimension ''n'' is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for is necessary and sufficient condition for the manifold ... * Obstruction tensor Ref ... [...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 structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the ' is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
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, cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformally Invariant
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the natural logarithm, logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a Surface (topology), surface), but it contains more structure (specifically a Complex Manifold, complex stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically independent over K if and only if \alpha is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S. Example The two real numbers \sqrt and 2\pi+1 are each transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, each of the two singleton sets \ and \ are algebraically independent over the field \mathbb of rational numbers. However, the set \ is ''not'' algebraically independent over the rational numbers, because the nontrivial polynomial :P(x,y)=2x^2-y+1 is zero when x=\sqrt and y=2\pi+1. Algebraic independence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Index Notation
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved. Let V be a vector space, and V^* its dual space. Consider, for example, an order-2 covariant tensor h \in V^*\otimes V^*. Then h can be identified with a bilinear form on V. In other words, it is a function of two arguments in V which can be represented as a pair of ''slots'': : h = h(-,-). Abstract in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schouten Tensor
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by: :P=\frac \left(\mathrm -\frac g\right)\, \Leftrightarrow \mathrm=(n-2) P + J g \, , where Ric is the Ricci tensor (defined by contracting the first and third indices of the Riemann tensor), ''R'' is the scalar curvature, ''g'' is the Riemannian metric, J=\fracR is the trace of ''P'' and ''n'' is the dimension of the manifold. The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni–Nomizu product of the Schouten tensor with the metric. In an index notation :R_=W_+g_ P_-g_ P_-g_ P_+g_ P_\, . The Schouten tensor often appears in conformal geometry because of its relatively simple conformal transformation law :g_\mapsto \Omega^2 g_ \Rightarrow P_\mapsto P_-\nabla_i \Upsilon_j + \Upsilon_i \Upsilon_j -\frac12 \Upsilon_k \Upsilon^k g_\, , where \Upsilon_i := \Omega^ \partial_i \Omega\, . Further reading *Arthur L. Besse, ''Einste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotton Tensor
In differential geometry, the Cotton tensor on a (pseudo)- Riemannian manifold of dimension ''n'' is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for is necessary and sufficient condition for the manifold to be conformally flat. By contrast, in dimensions , the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For the Cotton tensor is identically zero. The concept is named after Émile Cotton. The proof of the classical result that for the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by Eisenhart using a standard integrability argument. This tensor density is uniquely characterized by its conformal properties coupled with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obstruction Tensor
In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension ''n'' is realized (''ambiently'') as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian manifold. The ambient construction is canonical in the sense that it is performed only using the conformal class of the metric: it is conformally invariant. However, the construction only works asymptotically, up to a certain order of approximation. There is, in general, an obstruction to continuing this extension past the critical order. The obstruction itself is of tensorial character, and is known as the (conformal) obstruction tensor. It is, along with the Weyl tensor, one of the two primitive invariants in conformal differential geometry. Aside from the obstruction tensor, the ambient construction can be used to define a class of conformally invariant differential ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
