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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). Israel's theorem was proved by Werner Israel. Intuitive rationale The intuitive idea of Birkhoff's theorem is that a spherically symmetric gravitational field should be produced by some massive object at the origin; if there were another concentration of mass–energy somewhere else, this would disturb the spherical symmetry, so we can expect the solution to represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stellar Surface
Stellar means anything related to one or more stars (''stella''). The term may also refer to: Arts, entertainment, and media * ''Stellar'' (magazine), an Irish lifestyle and fashion magazine * Stellar Loussier, a character from ''Mobile Suit Gundam SEED Destiny'' * Dr. Stellar, a Big Bang Comics superhero * '' Stellar 7'', a game for the Apple II computer system * ''Stellar'' (film), a Canadian film Music * Stellar (group), a South Korean girl group * Stellar (New Zealand band), a New Zealand-based rock band * Stellar (musical artist), an American singer, songwriter, and producer * "Stellar" (song), a 2000 song by Incubus * Stellar Awards, awards for the gospel music industry Brands and enterprises * Stellar (payment network), a system for sending money through the internet * Stellar Group (construction company), a construction company in Florida, United States * Hasselblad Stellar, a compact digital camera * Hyundai Stellar, an automobile model * O2 XDA Stellar, an HTC m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University Press
Harvard University Press (HUP) is an academic publishing house established on January 13, 1913, as a division of Harvard University. It is a member of the Association of University Presses. Its director since 2017 is George Andreou. The press maintains offices in Cambridge, Massachusetts, near Harvard Square, and in London, England. The press co-founded the distributor TriLiteral LLC with MIT Press and Yale University Press. TriLiteral was sold to LSC Communications in 2018. Notable authors published by HUP include Eudora Welty, Walter Benjamin, E. O. Wilson, John Rawls, Emily Dickinson, Stephen Jay Gould, Helen Vendler, Carol Gilligan, Amartya Sen, David Blight, Martha Nussbaum, and Thomas Piketty. The Display Room in Harvard Square, dedicated to selling HUP publications, closed on June 17, 2009. Related publishers, imprints, and series HUP owns the Belknap Press imprint (trade name), imprint, which it inaugurated in May 1954 with the publication of the ''Harvard Guide to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clarendon Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 400 years, OUP has focused primarily on the publication of pedagogic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrupole Formula
In general relativity, the quadrupole formula describes the gravitational waves that are emitted from a system of masses in terms of the (mass) quadrupole moment. The formula reads : \bar_(t,r) = \frac \ddot_(t-r/c), where \bar_ is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. G is the gravitational constant, c the speed of light in vacuum, and I_ is the mass quadrupole moment. It is useful to express the gravitational wave strain in the transverse traceless gauge, by replacing the mass quadrupole moment I_ with the transverse traceless projection I_^, which is defined as: : _^ = \int \rho(\mathbf) \left _i r_j - r_n(r_i n_j + r_j n_i) + \frac r_n^2 (n_i n_j + \delta_ ) + \frac r^2 (n_i n_j - \delta_ ) \right d^3 r where \mathbf is a unit vector in the direction of the observer, r_n \equiv \mathbf\cdot\mathbf, and r^2 \equiv \mathbf\cdot\mathbf. The total energy carried away by gravitational waves can be expressed as: : \fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shell Theorem
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ... body. This theorem has particular application to astronomy. Isaac Newton proved the shell theorem and stated that: # A sphere, spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point mass, point at its center. # If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell. A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Field Equations
In the General relativity, 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#In general relativity and cosmology, matter within it. The equations were published by Albert Einstein in 1915 in the form of a Tensor, 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 Charge (physics), charges and Electric current, 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 (general relativity), metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Solutions In General Relativity
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field. Background and definition These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor T^. (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers. Formal definition Let V be a real vector space. The of is defined by taking the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. The subscript, \R, on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, V^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newman–Janis Algorithm
In general relativity, the Newman–Janis algorithm (NJA) is a complexification technique for finding exact solutions to the Einstein field equations. In 1964, Newman and Janis showed that the Kerr metric could be obtained from the Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ... by means of a coordinate transformation and allowing the radial coordinate to take on complex values. Originally, no clear reason for why the algorithm works was known. In 1998, Drake and Szekeres gave a detailed explanation of the success of the algorithm and proved the uniqueness of certain solutions. In particular, the only perfect fluid solution generated by NJA is the Kerr metric and the only Petrov type D solution is the Kerr–Newman metric. The algorithm works well on '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Theorem (electromagnetism)
Birkhoff's theorem may refer to several theorems named for the American mathematician George David Birkhoff: * Birkhoff's theorem (relativity) * Birkhoff's theorem (electromagnetism) * Birkhoff's ergodic theorem It may also refer to theorems named for his son, Garrett Birkhoff: *Doubly_stochastic_matrix, Birkhoff–von Neumann theorem for doubly stochastic matrices * Birkhoff's HSP theorem, concerning the closure operations of homomorphism, subalgebra and product * Birkhoff's representation theorem for distributive lattices * Birkhoff's theorem (equational logic), stating that syntactic and semantic consequence coincide {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reissner–Nordström Metric
In physics and astronomy, the Reissner–Nordström metric is a Static spacetime, static solution to the Einstein–Maxwell equations, Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass ''M''. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric. The metric was discovered between 1916 and 1921 by Hans Reissner, Hermann Weyl, Gunnar Nordström and George Barker Jeffery independently. Metric In spherical coordinates , the Reissner–Nordström metric (i.e. the line element) is : ds^2 = c^2\, d\tau^2 = \left( 1 - \frac + \frac \right) c^2\, dt^2 -\left( 1 - \frac + \frac \right)^ \, dr^2 - ~ r^2 \, d\theta^2 - ~ r^2\sin^2\theta \, d\varphi^2 , where * c is the speed of light * \tau is the proper time * t is the time coordinate (measured by a stationary clock at infinity). * r is the radial coordinate * (\theta, \varphi) are the spherical angles * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
