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Weak-field Approximation
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing. Weak-field approximation The Einstein field equation (EFE) describing the geometry of spacetime is given as (using natural units) :R_ - \fracRg_ = 8\pi GT_ where R_ is the Ricci tensor, R is the Ricci scalar, T_ is the energy–momentum tensor, and g_ is the spacetime metric tensor that represent the solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric which render the prospect of finding exact solutions impractical in most systems. Howeve ... [...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] |
Minkowski Metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (relativity), events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Albert Einstein, Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correspondence Principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations. The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom. The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter. Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1893 and then later by Henri Poincaré in 1905 as waves similar to electromagnetic waves but the gravitational equivalent. Gravitational waves were later predicted in 1916 by Albert Einstein on the basis of his general theory of relativity as ripples in spacetime. Later he refused to accept gravitational waves. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation. Newton's law of universal gravitation, part of classical mechanics, does not provide for their existence, since that law is predicated on the assumption that physical interactions propagate instantaneously (at infinite speed)showing one of the ways the methods of Newtonian physics are unable to explain ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Coordinate Condition
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions ''x''α (regarded as scalar fields) satisfies d'Alembert's equation. The parallel notion of a harmonic coordinate system in Riemannian geometry is a coordinate system whose coordinate functions satisfy Laplace's equation. Since d'Alembert's equation is the generalization of Laplace's equation to space-time, its solutions are also called "harmonic". Motivation The laws of physics can be expressed in a generally invariant form. In other words, the real world does not care about our coordinate systems. However, for us to be able to solve the equations, we must fix upon a particular coordinate system. A coordinate condition selects one (or a smaller set of) such coordinate system(s). The Cartesian coordinates used in spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1893 and then later by Henri Poincaré in 1905 as waves similar to electromagnetic waves but the gravitational equivalent. Gravitational waves were later predicted in 1916 by Albert Einstein on the basis of his general theory of relativity as ripples in spacetime. Later he refused to accept gravitational waves. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation. Newton's law of universal gravitation, part of classical mechanics, does not provide for their existence, since that law is predicated on the assumption that physical interactions propagate instantaneously (at infinite speed)showing one of the ways the methods of Newtonian physics are unable to explain ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X(T). The differential operator T \mapsto \mathcal_X(T) is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback (differential Geometry)
Suppose that is a smooth map between smooth manifolds ''M'' and ''N''. Then there is an associated linear map from the space of 1-forms on ''N'' (the linear space of sections of the cotangent bundle) to the space of 1-forms on ''M''. This linear map is known as the pullback (by ''φ''), and is frequently denoted by ''φ''∗. More generally, any covariant tensor field – in particular any differential form – on ''N'' may be pulled back to ''M'' using ''φ''. When the map ''φ'' is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from ''N'' to ''M'' or vice versa. In particular, if ''φ'' is a diffeomorphism between open subsets of R''n'' and R''n'', viewed as a change of coordinates (perhaps between different charts on a manifold ''M''), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphisms
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighborhood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called '' gauge bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
