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Palatini Variation
In general relativity and gravitation the Palatini variation is nowadays thought of as a variation of a Lagrangian with respect to the connection. In fact, as is well known, the Einstein–Hilbert action for general relativity was first formulated purely in terms of the spacetime metric . In the Palatini variational method one takes as independent field variables not only the ten components but also the forty components of the affine connection , assuming, a priori, no dependence of the from the and their derivatives. The reason the Palatini variation is considered important is that it means that the use of the Christoffel connection in general relativity does not have to be added as a separate assumption; the information is already in the Lagrangian. For theories of gravitation which have more complex Lagrangians than the Einstein–Hilbert Lagrangian of general relativity, the Palatini variation sometimes gives more complex connections and sometimes tensorial equations. At ... [...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 gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian (field Theory)
Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set ** Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem * Lagrangian, a functional whose extrema are to be determined in the calculus of variations * Lagrangian submanifold, a class of submanifolds in symplectic geometry * Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density Physics * Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ..., ... [...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 all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor (general Relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past. Notation and conventions Throughout this article we work with a metric signature that is mostly positive (); see sign convention. The gravitation constant G will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over. Definition Mathematically, spacetime is represented by a four-dimensional differentiable manifold M and the metric tensor is given as a covariant, second- degree, symmetric tensor on M, conventionally denoted by g. Moreover, the metric is required to be nondeg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attilio Palatini
Attilio Palatini (18 November 1889 – 24 August 1949) was an Italian mathematician born in Treviso. Biography Palatini was the seventh of the eight children of Michele (1855-1914) and Ilde Furlanetto (1856-1895). In 1900, during the celebrations for the election of his father to Parliament, he was blinded by a young man from Treviso, losing the use of one eye. He completed his secondary studies in Treviso. He graduated in mathematics in 1913 at the University of Padua, where he was a student of Ricci-Curbastro and of Levi-Civita. He taught rational mechanics at the Universities of Messina, Parma and Pavia. He was mainly involved in absolute differential calculus and in general relativity. Within this latter subject he gave a sound generalization of the variational principle. In 1919, Palatini wrote an important article where he proposed a new approach to the variational formulation of Einstein's gravitational field equations. In the same paper, Palatini also showed that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Padova
The University of Padua ( it, Università degli Studi di Padova, UNIPD) is an Italian university located in the city of Padua, region of Veneto, northern Italy. The University of Padua was founded in 1222 by a group of students and teachers from Bologna. Padua is the second-oldest university in Italy and the world's fifth-oldest surviving university. In 2010, the university had approximately 65,000 students. In 2021, it was ranked second "best university" among Italian institutions of higher education with more than 40,000 students according to Censis institute, and among the best 200 universities in the world according to ARWU. History The university is conventionally said to have been founded in 1222 when a large group of students and professors left the University of Bologna in search of more academic freedom ('Libertas scholastica'). The first subjects to be taught were law and theology. The curriculum expanded rapidly, and by 1399 the institution had divided in two: a ''Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tullio Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation) and hydrodynamics. Biography Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Mecha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the calculus of tensors, signing it as Gregorio Ricci. This appears to be the only time that Ricci-Curbastro used the shortened form of his name in a publication, and continues to cause confusion. Ricci-Curbastro also published important works in other fields, including a book on higher algebra and infinitesimal analysis, and papers on the theory of real numbers, an area in which he extended the research begun by Richard Dedekind. Early life and education Completing privately his high school studies at only 16 years of age, he enrolled on the course of philosophy-mathematics at Rome University (1869). The following year the Papal State fell and so Gregorio was called by his father to the city of his birth, Lugo di Romagna. Subsequently he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palatini Formalism , elite regiments, literally "Palace troops"
{{disambiguation ...
Palatini may refer to: * Attilio Palatini (1889–1949), Italian mathematician * (1855–1914), Italian politician * Palatini identity * Palatini variation * Latin plural of Palatine * Palatini (Roman military) The ''palatini'' ( Latin for "palace troops") were elite units of the Late Roman army mostly attached to the ''comitatus praesentales'', or imperial escort armies. In the elaborate hierarchy of troop-grades, the ''palatini'' ranked below the ''sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palatini Identity
In general relativity and tensor calculus, the Palatini identity is: : \delta R_ = \nabla_\rho (\delta \Gamma^\rho_) - \nabla_\nu (\delta \Gamma^\rho_), where \delta \Gamma^\rho_ denotes the variation of Christoffel symbols and \nabla_\rho indicates covariant differentiation. A proof can be found in the entry Einstein–Hilbert action. The "same" identity holds for the Lie derivative \mathcal_ R_. In fact, one has: : \mathcal_ R_ = \nabla_\rho (\mathcal_ \Gamma^\rho_) - \nabla_\nu (\mathcal_ \Gamma^\rho_), where \xi = \xi^\partial_ denotes any vector field on the spacetime manifold M. See also *Einstein–Hilbert action *Palatini variation * Ricci calculus *Tensor calculus *Christoffel symbols *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. ... Note ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
