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Dual Graviton
In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions. The dual graviton was first hypothesized in 1980. It was theoretically modeled in 2000s, which was then predicted in eleven-dimensional mathematics of SO(8) supergravity in the framework of electric-magnetic duality. It again emerged in the ''E''11 generalized geometry in eleven dimensions, and the ''E''7 generalized vielbein-geometry in eleven dimensions. While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a BF model as non-local gravitational fields in extra dimensions. A ''massive'' dual gravity of Ogievetsky–Polubarinov model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor. The previously mentioned theories of dual gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, antiquarks, and antileptons, which generally are matter particles and antimatter particles), as well as the fundamental bosons ( gauge bosons and the Higgs boson), which generally are force particles that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle. Ordinary matter is composed of atoms, once presumed to be elementary particles – ''atomos'' meaning "unable to be cut" in Greek – although the atom's existence remained controversial until about 1905, as some leading physicists regarded molecules as mathematical illusions, and matter as ultimately composed of energy. Subatomic constituents of the atom were first identified in the early 1930s; the electron and the proto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anti-de Sitter Space
In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe. Manifolds of constant curvature are most familiar in the case of two dimensions, where the elliptic plane or surface of a sphere is a surface of constant positive curvature, a flat (i.e., Euclidean) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature. Einstein's general theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Scalar
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] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a \mathfrak g-valued one-form on ''P''). Then the curvature form is the \mathfrak g-valued 2-form on ''P'' defined by :\Omega=d\omega + omega \wedge \omega= D \omega. (In another convention, 1/2 does not appear.) Here d stands for exterior derivative, cdot \wedge \cdot/math> is defined in the article "Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, :\,\Omega(X, Y)= d\omega(X,Y) + omega(X),\omega(Y)/math> where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: \sigma\Omega(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space V^ obtained from the action of S_n on V^ by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young. Definition Given a finite symmetric group ''S''''n'' and specific Young tableau λ corresponding to a numbered partition of ''n'', and consider the action of S_n given by permuting the boxes of \lambda. Define two permutation subgroups P_\lambda and Q_\lambda of ''S''''n'' as follows: :P_\lambda=\ and :Q_\lambda=\. Corresponding to these two subgroups, define two vectors in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massive Gravity
In theoretical physics, massive gravity is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light. Massive gravity has a long and winding history, dating back to the 1930s when Wolfgang Pauli and Markus Fierz first developed a theory of a massive spin-2 field propagating on a flat spacetime background. It was later realized in the 1970s that theories of a massive graviton suffered from dangerous pathologies, including a ghost mode and a discontinuity with general relativity in the limit where the graviton mass goes to zero. While solutions to these problems had existed for some time in three spacetime dimensions, they were not solved in four dimensions and higher until the work of Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT model) in 2010. One of the very early massive gravity theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horndeski's Theory
Horndeski's theory is the most general theory of gravity in four dimensions whose Lagrangian is constructed out of the metric tensor and a scalar field and leads to second order equations of motion. The theory was first proposed by Gregory Horndeski in 1974 and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy. Horndeski's theory contains many theories of gravity, including General relativity, Brans-Dicke theory, Quintessence, Dilaton, Chameleon and covariant Galileon as special cases. Action Horndeski's theory can be written in terms of an action as S _,\phi= \int\mathrm^x\,\sqrt\left sum_^\frac\mathcal_[g_,\phi,+\mathcal_[g_,\psi_">_,\phi.html" ;"title="sum_^\frac\mathcal_[g_,\phi">sum_^\frac\mathcal_[g_,\phi,+\mathcal_[g_,\psi_right] with the Lagrangian (field theory), Lagrangian densities \mathcal_ = G_(\phi,\, X) \mathcal_ = G_(\phi,\,X)\Box\phi \mathcal_ = G_(\phi,\,X)R+G_(\phi,\,X)\left left(\Box ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as follows: \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \vdots & \vdots & \ddots & \vdots \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \end, or, by stating an equation for the coefficients using indices i and j, (\mathbf H_f)_ = \fra ... [...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] |
Ricci Curvature
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] |
Riemann Curvature Tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is ''flat'', i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
