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Birkhoff's Theorem (electromagnetism)
In physics, in the context of electromagnetism, Birkhoff's theorem concerns spherically symmetric static solutions of Maxwell's field equations of electromagnetism. The theorem is due to George D. Birkhoff. It states that any spherically symmetric solution of the source-free Maxwell equations is necessarily static. Pappas (1984) gives two proofs of this theorem, using Maxwell's equations and Lie derivatives. It is a limiting case of Birkhoff's theorem (relativity) by taking the flat metric without backreaction. Derivation from Maxwell's equations The source-free Maxwell's equations state that \begin \nabla \times \mathbf &= -\frac,\\ \nabla \times \mathbf &= \mu\varepsilon \frac. \end Since the fields are spherically symmetric, they depend only on the radial distance in spherical coordinates. The field is purely radial as non-radial components cannot be invariant under rotation, which would be necessary for symmetry. Therefore, we can rewrite the fields as \begin \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electricity and magnetism, two distinct but closely intertwined phenomena. In essence, electric forces occur between any two charged particles, causing an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs exclusively between ''moving'' charged particles. These two effects combine to create electromagnetic fields in the vicinity of charge particles, which can exert influence on other particles via the Lorentz force. At high energy, the weak force and electromagnetic force are unified as a single electroweak force. The electromagnetic force is responsible for many o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxwell's Equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.''Electric'' and ''magnetic'' fields, according to the theory of relativity, are the components of a single electromagnetic field. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George D
George may refer to: People * George (given name) * George (surname) * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Washington, First President of the United States * George W. Bush, 43rd President of the United States * George H. W. Bush, 41st President of the United States * George V, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1910-1936 * George VI, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1936-1952 * Prince George of Wales * George Papagheorghe also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George Harrison, an English musician and singer-songwriter Places South Africa * George, Western Cape ** George Airport United States * George, Iowa * George, Missouri * George, Washington * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Characters * George (Peppa Pig), a 2-year-old pig ... [...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] |
Birkhoff's Theorem (relativity)
In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution (i.e. the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric. The converse of the theorem is true and is called Israel's theorem. The converse is not true in Newtonian gravity. The theorem was proven in 1923 by George David Birkhoff (author of another famous '' Birkhoff theorem'', the ''pointwise ergodic theorem'' which lies at the foundation of ergodic theory). However, Nils Voje Johansen, Finn Ravndal, Stanley Deser recently pointed out that it was published two years earlier by a little-known Norwegian physicist, Jørg Tofte Jebsen.J.T. Jebsen, ''Uber die allgemeinen kugelsymmetrischen Lösungen der Einsteinschen Gravitationsgleichungen im Vakuum'', Arkiv för matematik, astronomi och fysik, 15 (18), 1 - 9 (1921).J.T. Jebsen, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Manifold
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by . Examples The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the 2-dimensional torus is not the metric induced by its usual embedding into \mathbb^3). Dimension 1 Every one-dimensional Riemannian manifold is flat. Conversely, given that every connected one-dimensional smooth manifold is diffeomorphic to either \mathbb or S^1, it is straightforward to see that every connected one-dimensional Riemannian mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Back-reaction
In theoretical physics, back-reaction (or backreaction) is often necessary to calculate the self-consistent behaviour of a particle or an object in an external field. Intuitive definition When a particle is considered to have no mass or to have an infinitesimal charge, this can be described as saying that we deal with a probe and that back-reaction is neglected. However, a real object also carries (in general) a mass and a charge itself. These properties imply that the model of the original environment needs to be modified to reach self-consistency. For example, a particle can be described as helping to curve the space in general relativity. Taking into account the constraints implied on the model by the particle's properties – the back-reaction – is one way of reaching a more accurate model than if those constraints are ignored. Cosmology In inhomogeneous cosmology, in which structure formation is taken into account in a general-relativistic model of the Universe, the term "b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinate System
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measured from a fixed zenith direction, and the ''azimuthal angle'' of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called '' colatitude'', ''zenith angle'', '' normal angle'', or ''inclination angle''. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates. The use of symbols and the order of the coordinates differs among sources and disciplines. This article will us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to each fibre is a linear functional on the tangent space. Symbolically, \alpha : TM \rightarrow ,\quad \alpha_x = \alpha, _: T_xM \rightarrow , where \alpha_x is linear. Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates: \alpha_x = f_1(x) \, dx_1 + f_2(x) \, dx_2 + \cdots + f_n(x) \, dx_n , where the f_i are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field. Examples The most basic non-trivial differential one-form is the "change in angle" form d\theta. This is defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Star Operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients \tbinom nk = \tbinom. The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a ps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
