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Petrov Classification
In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is most often applied in studying exact solutions of Einstein's field equations, but strictly speaking the classification is a theorem in pure mathematics applying to any Lorentzian manifold, independent of any physical interpretation. The classification was found in 1954 by A. Z. Petrov and independently by Felix Pirani in 1957. Classification theorem We can think of a fourth rank tensor such as the Weyl tensor, ''evaluated at some event'', as acting on the space of bivectors at that event like a linear operator acting on a vector space: : X^ \rightarrow \frac \, _ X^ Then, it is natural to consider the problem of finding eigenvalues \lambda and eigenvectors (which are now referred to as eigenbivectors) X^ such that :\frac \, _ \, X^ = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value). History Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it couldn't be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book '' Ars Magna'' (1545). The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newtonian Gravity
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.It was shown separately that separated spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the " first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors. This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newton's work ''Philosophiæ Naturalis Principia Mathematica'' ("the ''Principia''"), first published on 5 July 1687. When Newton presented Book 1 of the unpublished text in April 1686 to the Roya ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrogravitic Tensor
In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric field and magnetic field. Such a decomposition was partially described by Alphonse Matte in 1953 and by Lluis Bel in 1958. This decomposition is particularly important in general relativity. This is the case of four-dimensional Lorentzian manifolds, for which there are only three pieces with simple properties and individual physical interpretations. Decomposition of the Riemann tensor In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field \vec, not necessarily geodesic or hypersurface orthogonal, consists of three pieces: # the ''electrogravitic tensor'' E vec = R_ \, X^m \, X^n #* Also known as the tidal tensor. It can be physically interpreted as giving the tidal stresses on sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Congruence
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation. Types of congruences Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called ''timelike'', ''null'', or ''spacelike'' respectively. A congruence is called a '' geodesic congruence'' if it admits a tangent vector field \vec with vanishing covariant derivative, \nabla_ \vec = 0. Relation with vector fields The integral curves of the vector field are a family of ''non-intersecting'' parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the ''same'' congru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multipole Moments
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular features ... [...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 gravitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric (general Relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past. Notation and conventions Throughout this article we work with a metric signature that is mostly positive (); see sign convention. The gravitation constant G will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over. Definition Mathematically, spacetime is represented by a four-dimensional differentiable manifold M and the metric tensor is given as a covariant, second- degree, symmetric tensor on M, conventionally denoted by g. Moreover, the metric is required to be nondeg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Scalar
In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars \ which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime. Definitions Given a complex null tetrad \ and with the convention \, the Weyl-NP scalars are defined byJeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 2.Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998. Appendix E.Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. ''Isolated horizons: Hamiltonian evolution and the first law''. Physical Review D, 2000, 62(10): 104025. Appendix Bgr-qc/0005083/ref> :\Psi_0 := C_ l^\alpha m^\beta l^\gamma m^\delta\ , :\Psi_1 := C_ l^\alpha n^\beta l^\gamma m^\delta\ , :\Psi_2 := C_ l^\alpha m^\beta \bar^\gamma n^\delta\ , :\Psi_3 := C_ l^\alpha n^\beta \bar^\gamma n^\d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrads
Tetrad ('group of 4') or tetrade may refer to: * Tetrad (area), an area 2 km x 2 km square * Tetrad (astronomy), four total lunar eclipses within two years * Tetrad (chromosomal formation) * Tetrad (general relativity), or frame field ** Tetrad formalism, an approach to general relativity * Tetrad (geometry puzzle), a set of four simply connected disjoint planar regions in the plane * Tetrad (meiosis), the four cells produced by meiotic cell division * Tetrad (music), a set of four notes ** Tetrad (chord), a series of four notes * Tetrad (symbol), or tetractys, a triangular figure of ten points arranged in four rows, and mystical symbol * Medical tetrad, a group of four signs or symptoms which characterise a specific medical condition * Nibble, or tetrade, a 4-bit group * a tuple of length 4 * Tetrad Islands, in the Antarctic See also * * 4 * Triad (other) ('group of 3') * Pentad (other) ('group of 5') * Dark tetrad, group of four undesirable personality t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newman–Penrose Formalism
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Journal of Mathematical Physics, 1963, 4(7): 998. is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformally Flat
A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the geodesics maintain in all points of M the angles by moving from one to the other, as well as keeping the null geodesics unchanged, that means exists a function \lambda(x) such that g(x) = \lambda^2(x)\, \eta, where \lambda(x) is known as the conformal factor and x is a point on the manifold. More formally, let (M,g) be a pseudo-Riemannian manifold. Then (M,g) is conformally flat if for each point x in M, there exists a neighborhood U of x and a smooth function f defined on U such that (U,e^ g) is flat (i.e. the curvature of e^ g vanishes on U). The function f need not be defined on all of M. Some authors use the definition of locally conformally flat when referred to just some point x on M and reserve the definition of ''conformally fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
