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GRTensorII
GRTensorII is a Maple package designed for tensor computations, particularly in general relativity. This package was developed at Queen's University in Kingston, Ontario by Peter Musgrave, Denis Pollney and Kayll Lake. While there are many packages which perform tensor computations (including a ''standard Maple package''), GRTensorII is particularly well suited for carrying out routine computations of useful quantities when working with (or searching for) exact solutions in general relativity. Its principal advantages include *convenience of definition of new spacetimes and tensor expression *efficient computation with frames *efficient computation of Ricci and Weyl spinor components and of Petrov classification *efficient computation of the Carminati-McLenaghan invariants and other curvature invariant In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maple Computer Algebra System
Maple is a symbolic and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis, data processing, visualization, and others. A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation. Maple's capacity for symbolic computing include those of a general-purpose computer algebra system. For instance, it can manipulate mathematical expressions and find symbolic solutions to certain problems, such as those arising from ordinary and partial differential equations. Maple is developed commercially by the Canadian software company Maplesoft. The name 'Maple' is a reference to the software's Canadian heritage. Overview Core functionality Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...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] |
Queen's University At Kingston
Queen's University at Kingston, commonly known as Queen's University or simply Queen's, is a public university, public research university in Kingston, Ontario, Kingston, Ontario, Canada. Queen's holds more than of land throughout Ontario and owns Herstmonceux Castle in East Sussex, England. Queen's is organized into eight faculties and schools. The Church of Scotland established Queen's College in October 1841 via a royal charter from Queen Victoria. The first classes, intended to prepare students for the ministry, were held 7 March 1842, with 15 students and two professors. In 1869, Queen's was the first Canadian university west of the The Maritimes, Maritime provinces to admit women. In 1883, a women's college for medical education affiliated with Queen's University was established after male staff and students reacted with hostility to the admission of women to the university's medical classes. In 1912, Queen's ended its affiliation with the Presbyterian Church, and adopted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kingston, Ontario
Kingston is a city in Ontario, Canada, on the northeastern end of Lake Ontario. It is at the beginning of the St. Lawrence River and at the mouth of the Cataraqui River, the south end of the Rideau Canal. Kingston is near the Thousand Islands, a tourist region to the east, and the Prince Edward County, Ontario, Prince Edward County tourist region to the west. Kingston is nicknamed the "Limestone City" because it has many heritage buildings constructed using local limestone. Growing European exploration in the 17th century and the desire for the Europeans to establish a presence close to local Native occupants to control trade led to the founding of a New France, French trading post and military fort at a site known as "Cataraqui" (generally pronounced ) in 1673. The outpost, called Fort Cataraqui, and later Fort Frontenac, became a focus for settlement. After the Conquest of New France (1759–1763), the site of Kingston was relinquished to the British. Cataraqui was renamed K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ontario
Ontario is the southernmost Provinces and territories of Canada, province of Canada. Located in Central Canada, Ontario is the Population of Canada by province and territory, country's most populous province. As of the 2021 Canadian census, it is home to 38.5% of the country's population, and is the second-largest province by total area (after Quebec). Ontario is Canada's fourth-largest jurisdiction in total area of all the Canadian provinces and territories. It is home to the nation's capital, Ottawa, and its list of the largest municipalities in Canada by population, most populous city, Toronto, which is Ontario's provincial capital. Ontario is bordered by the province of Manitoba to the west, Hudson Bay and James Bay to the north, and Quebec to the east and northeast. To the south, it is bordered by the U.S. states of (from west to east) Minnesota, Michigan, Ohio, Pennsylvania, and New York (state), New York. Almost all of Ontario's border with the United States follows riv ... [...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] |
Spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive ''where'' and ''when'' events occur. Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, space and time took on new meanings with the Lorentz transformation and special theory of relativity. In 1908, Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions into a single four-dimensional continuum now known as Minkowski space. This interpretation proved vital t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frame Fields In General Relativity
In general relativity, a frame field (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by \vec_0 and the three spacelike unit vector fields by \vec_1, \vec_2, \, \vec_3. All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field. Frame fields were introduced into general relativity by Albert Einstein in 1928 and by Hermann Weyl in 1929.Hermann Weyl "Elektron und Gravitation I", ''Zeitschrift Physik'', 56, p330–352, 1929. The index notation for tetrads is explained in tetrad (index notation). Physical interpretation Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal transformation, infinitesimal) rotation, but unlike Euclidean vector, geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of Section (fiber bundle), sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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^ = \la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
