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Principle Of General Covariance
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist ''a priori'' in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. While this concept is exhibited by general relativity, which describes the dynamics of spacetime, one should not expect it to hold in less fundamental theories. For matter fields taken to exist independently of the background, it is almost never the case that their equations of motion will take the same form in curved space that they do in flat space. Overview A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems, and is usually expressed in terms of tensor fields. The classical (non-quantum) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concer ... [...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] |
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 str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate-free
A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system. Benefits Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater mathematical elegance at the cost of some abstraction from the detailed formulae needed to evaluate these equations within a particular system of coordinates. In addition to elegance, coordinate-free treatments are crucial in certain applications for proving that a given definition is well formulated. For example, for a vector space V with basis v_1, ..., v_n, it may be tempting to construct the dual space V^* as the formal span of the symbols v_1^*, ..., v_n^* with bracket \langle \sum \alpha_i v_i , \sum \beta_i v_i^* \rangle := \sum \alpha_i \beta_i, but it is not immediately clear that this construction is independent of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Conditions
In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the description of the world as given by the laws of physics does not depend on our choice of coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions. A coordinate condition selects such coordinate system(s). Indeterminacy in general relativity The Einstein field equations do not determine the metric uniquely, even if one knows what the metric tensor equals everywhere at an initial time. This situation is analogous to the failure of the Maxwell equations to determine the potentials uniquely. In both cases, the ambiguity can be removed by gauge fixing. Thus, coordinate conditions are a type of gauge condition. No coordinate condition is generally covariant, but many coordinate conditions are Lorentz covariant or rotationally covariant. Naively, one might think that coord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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