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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] |
Einstein–Hilbert Action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the action is given as :S = \int R \sqrt \, \mathrm^4x, where g=\det(g_) is the determinant of the metric tensor matrix, R is the Ricci scalar, and \kappa = 8\pi Gc^ is the Einstein gravitational constant (G is the gravitational constant and c is the speed of light in vacuum). If it converges, the integral is taken over the whole spacetime. If it does not converge, S is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler–Lagrange equation of the Einstein–Hilbert action. The action was first proposed by David Hilbert in 1915. Discussion Deriving equations of motion from an action has several advantages. First, it allows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Lanczos Tensor
The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor.Hyôitirô Takeno, "On the spintensor of Lanczos", ''Tensor'', 15 (1964) pp. 103–119. It was first introduced by Cornelius Lanczos in 1949. The theoretical importance of the Lanczos tensor is that it serves as the gauge field for the gravitational field in the same way that, by analogy, the electromagnetic four-potential generates the electromagnetic field.P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", ''EJTP'', 7 (2010) pp. 327–350. Definition The Lanczos tensor can be defined in a few different ways. The most common modern definition is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor. These equations, presented below, were given by Takeno in 1964. The way that Lanczos introduced the tensor originally was as a Lagrange multiplie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein 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 EFE are th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Tensor
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum. Definition The Einstein tensor \mathbf is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as \mathbf=\mathbf-\frac\mathbfR, where \mathbf is the Ricci tensor, \mathbf is the metric tensor and R is the scalar curvature, which is computed as the trace of the Ricci Tensor R_ by R = g^R_ = R_\mu^\mu. In component form, the previous equation reads as G_ = R_ - g_R . The Einstein tensor is symmetric G_ = G_ and, like the on shell stress–energy tensor, and has zero divergence: \nabla_\mu G^ = 0\,. Explicit form The Ricci tensor depends only on the metric ... [...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] |
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] |
Riemann 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 alo ... [...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] |
Holst Action
In the field of theoretical physics, the Holst action is an equivalent formulation of the Palatini action for General Relativity (GR) in terms of vierbeins (4D space-time frame field) by adding a part of a topological term (Nieh-Yan) which does not alter the classical equations of motion as long as there is no torsion, :S = \frac \int e e^_ e^_ (F_^ - \alpha \ast F_^) \equiv \frac \int e e^_ e^_ (F_^ - \frac \epsilon^_ F_^) where e^_ is the tetrad, e its determinant (the space-time metric is recovered from the tetrad by the formula g_=e_^e_^\eta _ where \eta_ the Minkowski metric), F_^ the curvature considered as a function of the connection A_^: :F_^ = ^ = 2 \partial_ ^ + 2^ ^J, \alpha a (complex) parameter, and where we recover the Palatini action when \alpha = 0. It only works in 4D. To be torsion free means the covariant derivative defined by the connection A_^ when acting on the Minkowski metric \eta_ vanishes, implying the connection is anti-symmetric in its interna ... [...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] |
Ashtekar's Variables
In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric q_ (x) on the spatial slice and the metric's conjugate momentum K^ (x), which is related to the extrinsic curvature and is a measure of how the induced metric evolves in time. These are the metric canonical coordinates. In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of an SU(2) gauge field and its complementary variable. Overview Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity and in turn loop quantum gravity and quantum holonomy theory. Let us introduce a set of three vector fields E^a_i, i = 1,2,3 that are orthogonal, that is, :\delta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
