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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] |
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] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations Of Physics
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Bergmann
Peter Gabriel Bergmann (24 March 1915 – 19 October 2002) was a German-American physicist best known for his work with Albert Einstein on a unified field theory encompassing all physical interactions. He also introduced primary and secondary constraints into mechanics. Early life and education Bergmann was born into a Jewish family of Max Bergmann, a biochemistry professor and Emmy Bergmann, a pediatrician in Berlin. His father would later be a professor of chemistry at the Rockefeller Institute for Medical Research. He began college in 1931, at the age of 16, at ''Technische Hochschule'' (now TU Dresden) under the mentorship of Harry Dember. Bergmann obtained his PhD at the age of 21 from the German University in Prague in 1936 under the direction of Philipp Frank. Bergmann's family scattered all over the world during Nazi rule; his sister Clara stayed behind and ultimately was murdered at Auschwitz. Career Bergmann's association with Einstein began without his knowled ... [...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] |
Tensor Calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his General relativity, general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a Manifest covariance, form that is independent of the coordinate chart, choice of coordinates on the manifold. Tensor calculus has many applications in physics, engineering and computer science including Elasticity (physics), elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning. Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century. A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. Tensors and tensor fields can be expressed in terms of their components, and operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based a work on special relativity on two postulates: * The laws of physics are invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his General relativity, general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a Manifest covariance, form that is independent of the coordinate chart, choice of coordinates on the manifold. Tensor calculus has many applications in physics, engineering and computer science including Elasticity (physics), elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning. Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X(T). The differential operator T \mapsto \mathcal_X(T) is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
