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Gibbons–Hawking–York Boundary Term
In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined. However, the use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold \mathcal is closed, i.e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary \partial\mathcal, the action should be supplemented by a boundary term so that the variational principle is well-defined. The necessity of such a boundary term was first realised by York and later refined in a minor way by Gibbons and Hawking. For a manifold that is not closed, the appropriate action is :\mathcal_\mathrm + \mathcal_\mathrm = \frac \int_\mathcal \mathrm^4 x \, \sqrt R + \frac \int_ \mathrm^3 y ... [...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 gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Tensor
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum. Definition The Einstein tensor \mathbf is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as \mathbf=\mathbf-\frac\mathbfR, where \mathbf is the Ricci tensor, \mathbf is the metric tensor and R is the scalar curvature, which is computed as the trace of the Ricci Tensor R_ by R = g^R_ = R_\mu^\mu. In component form, the previous equation reads as G_ = R_ - g_R . The Einstein tensor is symmetric G_ = G_ and, like the on shell stress–energy tensor, and has zero divergence: \nabla_\mu G^ = 0\,. Explicit form The Ricci tensor depends only on the metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Quantum Gravity
Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation rather than the treatment of gravity as a force. As a theory LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network. These networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale above the order of a Planck length, approximately 10−35 meters, and smaller scales are meaningless. Consequently, not just matter, but space itself, prefers an atomic structure. The areas of research, which involves about 30 research groups worldwide, share the basic physical assumptions and the mathematical description of quantum space. Research has ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the quantum state vector of a system and the results of observations of that system, a link was first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measureme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton's Principal Function
Buck Meadows (formerly Hamilton's and Hamilton's Station) is a census-designated place in Mariposa County, California, United States. It is located east-northeast of Smith Peak, at an elevation of . The population was 21 at the 2020 census. Buck Meadows lies just south of the Tuolumne County line. It is on State Route 120, east of Groveland. The ZIP Code for this community is shared with Groveland (95321), and wired telephones work out of Groveland's telephone exchange with numbers following the format ( 209) 962-xxxx. The official U.S. Geological Survey coordinates for the community are . The area is named for Buck's Meadow which lies at the corner of SR120 and Smith Station Road. According to ''The Big Oak Flat Road'', a variant name for the area was Hamilton's Station. This may have referred to the name of a stagecoach stop. east of U.S.F.S. Buck Meadows Fire Station, "Rim of the World" overlooks the canyon containing the South Fork of the Tuolumne River. Further east ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Derivative Gravity
Infinite derivative gravity is a theory of gravity which attempts to remove cosmological and black hole singularities by adding extra terms to the Einstein–Hilbert action, which weaken gravity at short distances. History In 1987, Krasnikov considered an infinite set of higher derivative terms acting on the curvature terms and showed that by choosing the coefficients wisely, the propagator would be ghost-free and exponentially suppressed in the ultraviolet regime. Tomboulis (1997) later extended this work. By looking at an equivalent scalar-tensor theory, Biswas, Mazumdar and Siegel (2005) looked at bouncing FRW solutions. In 2011, Biswas, Gerwick, Koivisto and Mazumdar demonstrated that the most general infinite derivative action in 4 dimensions, around constant curvature backgrounds, parity invariant and torsion free, can be expressed by: :S = \int \mathrm^4x \sqrt \left(M^2_P R+ R F_1 (\Box) R + R^ F_2 (\Box) R_ + C^ F_3 (\Box) C_ \right) where the F_i (\Box)=\sum^\infty_ f_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nathalie Deruelle
Nathalie Deruelle (born 1952) is a French physicist specializing in general relativity and known for her research on the two-body problem in general relativity and on cosmological perturbation theory. Education and career Deruelle began her studies at the École normale supérieure in 1971, earned an agrégation in 1975, then, after visiting positions at the European Space Agency and the University of Cambridge, completed a doctorate in 1982 at Pierre and Marie Curie University. Formerly a director of research for the French National Centre for Scientific Research, associated with Paris Diderot University Paris Diderot University, also known as Paris 7 (french: Université Paris Diderot), was a French university located in Paris, France. It was one of the inheritors of the historic University of Paris, which was split into 13 universities in 197 ..., she is now listed as a researcher emeritus. Books Deruelle is the author of books including: *''Relativity in Modern Physics'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ADM Formalism
The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959. The comprehensive review of the formalism that the authors published in 1962 has been reprinted in the journal '' General Relativity and Gravitation'', while the original papers can be found in the archives of ''Physical Review''. Overview The formalism supposes that spacetime is foliated into a family of spacelike surfaces \Sigma_t, labeled by their time coordinate t, and with coordinates on each slice given by x^i. The dynamic variables of this theory are taken to be the metric tensor of three-dimensional spatial slices \gamma_(t,x^k) and their conjugate momenta \pi^(t,x^k). Using these variables it is possible to define a Hamiltonian, and thereby write the equations of motion for general relativity in the fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F(R) Gravity
() is a type of modified gravity theory which generalizes Einstein's general relativity. () gravity is actually a family of theories, each one defined by a different function, , of the Ricci scalar, . The simplest case is just the function being equal to the scalar; this is general relativity. As a consequence of introducing an arbitrary function, there may be freedom to explain the accelerated expansion and structure formation of the Universe without adding unknown forms of dark energy or dark matter. Some functional forms may be inspired by corrections arising from a quantum theory of gravity. () gravity was first proposed in 1970 by Hans Adolph Buchdahl (although was used rather than for the name of the arbitrary function). It has become an active field of research following work by Starobinsky on cosmic inflation. A wide range of phenomena can be produced from this theory by adopting different functions; however, many functional forms can now be ruled out on observati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotically Flat
An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime. While this notion makes sense for any Lorentzian manifold, it is most often applied to a spacetime standing as a solution to the field equations of some metric theory of gravitation, particularly general relativity. In this case, we can say that an asymptotically flat spacetime is one in which the gravitational field, as well as any matter or other fields which may be present, become negligible in magnitude at large distances from some region. In particular, in an asymptotically flat vacuum solution, the gravitational field (curvature) becomes negligible at large distances from the source of the field (typically some isolated massive object such as a star). Intuitive significance The condition of asymptotic flatness is analogous to sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extrinsic Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stokes Theorem
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, and Stokes' theorem is the case of a surface in \R^3. Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus. Stokes' theorem says that the integral of a differential form \omega over the boundary \partial\Omega of some orientable manifold \Omega is equal to the integral of its exterior derivative d\omega over the whole of \Omega, i.e., \int_ \omega = \int_\Omega d\omega\,. Stokes' theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
