|
Kretschmann Invariant
In the theory of pseudo-Riemannian manifold, Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic curvature invariant (general relativity), scalar invariant. It was introduced by Erich Kretschmann. Definition The Kretschmann invariant is : K = R_ \, R^ where R_ is the Riemann curvature tensor (in this equation the Einstein summation convention was used, and it will be used throughout the article). Because it is a sum of squares of tensor components, this is a ''quadratic'' invariant. For the use of a computer algebra system a more detailed writing is meaningful: : K = R_ \, R^ =\sum_^\sum_^3 \sum_^3\sum_^3 R_ \, R^ \text R^ =\sum_^3 g^\,\sum_^3 g^\,\sum_^3 g^\,\sum_^3 g^\, R_. \, Examples For a Schwarzschild metric, Schwarzschild black hole of mass M, the Kretschmann scalar is : K = \frac \,. where G is the gravitational constant. For a general Friedmann–Lemaître–Robertson–Walker metric, FRW spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-Riemannian Manifold
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 requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang–Mills Theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics. History and theoretical description In 1953, in a private correspondence, Wolfgang Pauli formulated a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Decomposition
In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry. Definition of the decomposition Let (''M'',''g'') be a Riemannian or pseudo-Riemannian ''n''-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention :R_=g_\Big(\partial_i\Gamma_^p-\partial_j\Gamma_^p+\Gamma_^p\Gamma_^q-\Gamma_^p\Gamma_^q\Big); written multilinearly, this is the convention :\operatorname(W,X,Y,Z)=g\Big(\nabla_W\nabla_XY-\nabla_X\nabla_WY-\nabla_Y,Z\Big). With this convention, the Ricci tensor is a (0,2)-tensor field defined by ''Rjk''=''gilRijkl'' and the scalar curvature is defined by ''R''=''gjkRjk.'' Define the traceless Ricci tensor :Z_=R_-\fracRg_, and then define three (0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Invariant (general Relativity)
In general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors - which represent curvature, hence the name, - and possibly operations on them such as contraction, covariant differentiation and dualisation. Certain invariants formed from these curvature tensors play an important role in classifying spacetimes. Invariants are actually less powerful for distinguishing locally non- isometric Lorentzian manifolds than they are for distinguishing Riemannian manifolds. This means that they are more limited in their applications than for manifolds endowed with a positive definite metric tensor. Principal invariants The principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants (i.e., sums of squares of components). The principal invariants of the Riemann tensor of a four-dimensional Lorentzian manifold are #the '' Kretschmann scalar'' K_1 = R_ \, R^ #the ''Chern–Pontryagin scalar'' K_2 = _ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Invariant
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations. Types of curvature invariants The invariants most often considered are ''polynomial invariants''. These are polynomials constructed from contractions such as traces. Second degree examples are called ''quadratic invariants'', and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order ''differential invariants''. The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors. However, it can also be considered a linear operator acting on bivectors, and as such it has a characteristic polynomial, whose coefficients and roots (eigenvalues) are polynomial scalar invariants. Physi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Electromagnetic Fields
In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in the study of solutions of Maxwell's equations and has applications in Einstein's theory of relativity. The classification theorem The electromagnetic field at a point ''p'' (i.e. an event) of a Lorentzian spacetime is represented by a real bivector defined over the tangent space at ''p''. The tangent space at ''p'' is isometric as a real inner product space to E1,3. That is, it has the same notion of vector magnitude and angle as Minkowski spacetime. To simplify the notation, we will assume the spacetime ''is'' Minkowski spacetime. This tends to blur the distinction between the tangent space at ''p'' and the underlying manifold; fortunately, nothing is lost by this specialization, for reasons we discuss as the end of the article. The classification theorem for electromagnetic fields charac ... [...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'' matrices: a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras. History and name The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term ''"Killing form"'' first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term ''" Cartan-Killing form"''. At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetic Field Tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below. Definition The electromagnetic tensor, conventionally labelled ''F'', is defined as the exterior derivative of the electromagnetic four-potential, ''A'', a differential 1-form: :F \ \stackrel\ \mathrmA. Therefore, ''F'' is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form, :F_ = \partial_\mu A_\nu - \partial_\nu A_\mu. where \partial is the four-gradient and A is the fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
