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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 James W. York and later refined in a minor way by Gary Gibbons and Stephen Hawking. For a manifold that is not closed, the appropriate action is :\mathcal_\mathrm + \mathcal_\mathrm = \frac \int_\mathcal \mathrm^4 x \, \sqrt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, 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 in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. 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 gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac=(\frac, \frac, ... , \frac) where ‖u‖ is the Norm (mathematics), norm (or length) of u and \, \mathbf\, = (u_1, u_2, ..., u_n). The proof is the following: \, \mathbf\, =\sqrt=\sqrt=\sqrt=1 A unit vector is often used to represent direction (geometry), directions, such as normal directions. Unit vectors are often chosen to form the basis (linear algebra), basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors. Orthogonal coordinates ... [...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. The community 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 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. History Buck Meadows started as a stage stop called Hamilton's establish ... [...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 (Paris), 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, she is now listed as a researcher emeritus. Books Deruelle is the author of books including: *''Relativity in Modern Physics'' (with Jean-Philippe Uzan, 2018) (translated by Patricia de Forcrand-Millard) *''Les ondes gravitationnelles'' (with Jean-Pierre Lasota, 2018) *''De Pythagore à Einstein, tout est nombre : la relativité général ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ADM Formalism
The Arnowitt–Deser–Misner (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 g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F(R) Gravity
In physics, ''f''(''R'') is a type of modified gravity theory which generalizes Einstein's general relativity. ''f''(''R'') 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. ''f''(''R'') 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 Alexei Starobinsky on cosmic inflation. A wide range of phenomena can be produced from this theory by adopting different functions; however, many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotically Flat
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extrinsic Curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined ''extrinsically'' relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined ''intrinsically'' without reference to a larger space. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form \mathrmV = \rho(u_1,u_2,u_3)\,\mathrmu_1\,\mathrmu_2\,\mathrmu_3 where the u_i are the coordinates, so that the volume of any set B can be computed by \operatorname(B) = \int_B \rho(u_1,u_2,u_3)\,\mathrmu_1\,\mathrmu_2\,\mathrmu_3. For example, in spherical coordinates \mathrmV = u_1^2\sin u_2\,\mathrmu_1\,\mathrmu_2\,\mathrmu_3, and so \rho = u_1^2\sin u_2. The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stokes Theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: : The line integral of a vector field over a loop is equal to the surface integral of its '' curl'' over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem. In particular, a vector field on \R^3 can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form. Theorem Let \Sigma be a smooth oriented surface in \R^3 with boundary \partial \Sigma \equiv \Gamma . If a vector field \mathbf(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z)) is defined and has continuous fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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. Proof The Riemann curvature tensor is defined in terms of the Levi-Civita connection \Gamma^\lambda_ as : _ = \partial_\mu\Gamma^\rho_ - \partial_\nu\Gamma^\rho_ + \Gamma^\rho_ \Gamma^\lambda_ - \Gamma^\rho_\Gamma^\lambda_. Its variation is : \delta_ = \partial_\mu \delta\Gamma^\rho_ - \partial_\nu \delta\Gamma^\rho_ + \delta\Gamma^\rho_ \Gamma^\lambda_ + \Gamma^\rho_ \delta\Gamma^\lambda_ - \delta\Gamma^\rho_ \Gamma^\lambda_ - \Gamma^\rho_ \delta\Gam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
