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Homothetic Vector Field
In physics, a homothetic vector field (sometimes homothetic collineation or homothety) is a projective vector field which satisfies the condition: :\mathcal_X g_=2c g_ where c is a real constant. Homothetic vector fields find application in the study of singularities in general relativity. They can also be used to generate new solutions for Einstein equations by similarity reduction. See also * Affine vector field * Conformal Killing vector field * Curvature collineation * Killing vector field * Matter collineation * Spacetime symmetries Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact ... References Mathematical methods in general relativity {{math-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Vector Field
A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) M whose flow preserves the geodesic structure of M without necessarily preserving the affine parameter of any geodesic. More intuitively, the flow of the projective maps geodesics smoothly into geodesics without preserving the affine parameter. Decomposition In dealing with a vector field X on a semi Riemannian manifold (p.ex. in general relativity), it is often useful to decompose the covariant derivative into its symmetric and skew-symmetric parts: :X_=\frach_+ F_ where :h_=(\mathcal_X g)_=X_+X_ and :F_=\frac(X_-X_) Note that X_a are the covariant components of X. Equivalent conditions Mathematically, the condition for a vector field X to be projective is equivalent to the existence of a one-form \psi satisfying :X_\, =R_X^d+2g_\psi_ which is equivalent to :h_\, =2g_\psi_c+g_\psi_b+g_\psi_a The set of all global projective vector fields over a connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Singularity
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the current best theory of gravity, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete. Gravitational singularities are mainly considered in the context of general relativity, where density apparently becomes infinite at the center of a black hole, a ... [...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] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Vector Field
An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter. Mathematically, this is expressed by the following condition: :(\mathcal_X g_)_=0 See also * Conformal vector field * Curvature collineation * Homothetic vector field * Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric tensor, metric. Killing fields are the Lie g ... * Matter collineation * Spacetime symmetries Mathematical methods in general relativity {{math-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Killing Vector Field
In conformal geometry, a conformal Killing vector field on a manifold of dimension ''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) flow defines conformal transformations, that is, preserve g up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative of the flow e.g. \mathcal_g = \lambda g for some function \lambda on the manifold. For n \ne 2 there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to Wilhelm Killing, who first investigated Killing vector fields. Densitized metric tensor and Conformal Killing vectors A vector field X is a Killing vector field if and only if its flow preserves the metric tensor g (strictly speaking for each compact subsets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Collineation
A curvature collineation (often abbreviated to CC) is vector field which preserves the Riemann tensor in the sense that, :\mathcal_X R^a_=0 where R^a_ are the components of the Riemann tensor. The set of all smooth curvature collineations forms a Lie algebra under the Lie bracket operation (if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra). The Lie algebra is denoted by CC(M) and may be infinite-dimensional. Every affine vector field is a curvature collineation. See also * Conformal vector field * Homothetic vector field * Killing vector field * Matter collineation * Spacetime symmetries Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact ... Mathematical methods in general relativity {{math-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector Field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric tensor, metric. Killing fields are the Lie group#The Lie algebra associated to a Lie group, infinitesimal generators of isometry, isometries; that is, flow (geometry), flows generated by Killing fields are Isometry (Riemannian geometry), continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object. Definition Specifically, a vector field ''X'' is a Killing field if the Lie derivative with respect to ''X'' of the metric ''g'' vanishes: :\mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is :g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 \, for all vectors ''Y'' and ''Z''. In local c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matter Collineation
A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition, :\mathcal_X T_=0 where T_ are the energy–momentum tensor components. The intimate relation between geometry and physics may be highlighted here, as the vector field X is regarded as preserving certain physical quantities along the flow lines of X, this being true for any two observers. In connection with this, it may be shown that every Killing vector field is a matter collineation (by the Einstein field equations (EFE), with or without cosmological constant). Thus, given a solution of the EFE, a vector field that preserves the metric necessarily preserves the corresponding energy-momentum tensor. When the energy-momentum tensor represents a perfect fluid, every Killing vector field preserves the energy density, pressure and the fluid flow vector field. When the energy-momentum tensor represents an electromagnetic field, a Killing vector field does ''not nece ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime Symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries. Physical motivation Physical problems are often investigated and solved by noticing features which have some form of symmetry. For example, in the Schwarzschild solution, the role of spherical symmetry is important in deriving the Schwarzschild solution and deducing the physical consequences of this symmetry (such as the nonexistence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry plays a role in the cosmological principle, which restricts the type of universes that are consistent with large-scale observations (e.g. the Friedmann–Lemaître–Robertson–Walker (F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
