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Plebanski Action
General relativity and supergravity in all dimensions meet each other at a common assumption: :''Any configuration space can be coordinatized by gauge fields A^i_a, where the index i is a Lie algebra index and a is a spatial manifold index.'' Using these assumptions one can construct an effective field theory in low energies for both. In this form the action of general relativity can be written in the form of the Plebanski action which can be constructed using the Palatini action to derive Einstein's field equations of general relativity. The form of the action introduced by Plebanski is: :S_ = \int_ \epsilon_ B^ \wedge F^ (A^i_a) + \phi_ B^ \wedge B^ where i, j, l, k are internal indices,F is a curvature on the orthogonal group SO(3, 1) and the connection variables (the gauge fields) are denoted by A^i_a. The symbol \phi_ is the Lagrangian multiplier and :\epsilon_ is the antisymmetric symbol valued over SO(3, 1). The specific definition :B^ = e^i \wedge e^j f ... [...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] |
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] |
Tetradic Palatini Action
The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini. It is called a first order formulation as the variables to vary over involve only up to first derivatives in the action and so doesn't overcomplicate the Euler–Lagrange equations with higher derivative terms. The tetradic Palatini action is another first-order formulation of the Einstein–Hilbert action in terms of a different pair of independent variables, known as frame fields and the spin connection. The use of frame fields and spin connections are essential in the formulation of a generally covariant fermionic action (see the article spin connection for more discussion of this) which couples fermions to gravity when added to the tetradic Palatini action. Not only is this needed to couple fermions to gravity and makes the tetradic actio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrett–Crane Model
The Barrett–Crane model is a model in quantum gravity, first published in 1998, which was defined using the Plebanski action. The B field in the action is supposed to be a so(3, 1)-valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term :B^ \wedge B^ in the action has the same symmetries as it does to provide the Einstein–Hilbert action. But the form of :B^ is not unique and can be posed by the different forms: *\pm e^i \wedge e^j *\pm \epsilon^ e_k \wedge e_l where e^i is the tetrad and \epsilon^ is the antisymmetric symbol of the so(3, 1)-valued 2-form fields. The Plebanski action can be constrained to produce the BF model which is a theory of no local degrees of freedom. John W. Barrett and Louis Crane modeled the analogous constraint on the summation over spin foam. The Barrett–Crane model on spin foam quantizes the Plebanski action, but its path integral amplitude corresponds to the degenerate B field and not the specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein's Field Equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antisymmetric Symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some positive integer . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon or , or less commonly the Latin lower case . Index notation allows one to display permutations in a way compatible with tensor analysis: \varepsilon_ where ''each'' index takes values . There are indexed values of , which can be arranged into an -dimensional array. The key defining property of the symbol is ''total antisymmetry'' in the indices. When any two indices are interchange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. The method can be summarized as follows: in order to find the maximum or minimum of a function f(x) subjected to the equality constraint g(x) = 0, form the Lagrangian function :\mathcal(x, \lambda) = f(x) + \lambda g(x) and find the stationary points of \mathcal considered as a function of x and the Lagrange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerzy Plebański
Jerzy Franciszek Plebański (7 May 1928, Warsaw – 24 August 2005, Mexico) was a Polish theoretical physicist best known for his extensive research into general relativity and supergravity. Biography In 1954, Plebański received his Ph.D. under the direction of Wojciech Rubinowicz at the University of Warsaw. He then went to work at the newly founded Institute of Theoretical Physics of the University of Warsaw. A specialist in the field of general relativity and mathematical physics, his first book with co-author Leopold Infeld was on the problem of motion in general relativity. He was Vice-Dean of the Faculty of Mathematics and Physics at the University of Warsaw from 1958 to 1962. In 1958 Plebański traveled to the United States, and spent two years there, first as an invited professor at the Institute for Advanced Study in Princeton, and then at UCLA in Los Angeles. The year after his return to Poland in 1960, he married Anna Lazarowicz. From 1962 to 1967 the Plebański ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein's Field Equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra a superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes gravity arise in a natural way. Gravitons Like any field theory of gravity, a supergravity theory contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries. History Gauge supersymmetry The first theory of local supersymmetry was proposed by Dick Arnowitt and Pran Nath in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palatini Action
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields called a ''tetrad'' or ''vierbein''. It is a special case of the more general idea of a ''vielbein formalism'', which is set in (pseudo-)Riemannian geometry. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to (pseudo-) Riemannian manifolds in general, and even to spin manifolds. Most statements hold simply by substituting arbitrary n for n=4. In German, "vier" translates to "four", and "viel" to "many". The general idea is to write the metric tensor as the product of two ''vielbeins'', one on the left, and one on the right. The effect of the vielbeins is to change the coordinate system used on the tangent manifold to one that is sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
