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Van Stockum Dust
In general relativity, the van Stockum dust is an exact solution of the Einstein field equations in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is ''increasing'' with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example. This solution is named after Willem Jacob van Stockum, who rediscovered it in 1937 independently of a much earlier discovery by Cornelius Lanczos in 1924. It is currently recommended that the solution be referred to as the Lanczos–van Stockum dust. Derivation One way of obtaining this solution is to look for a cylindrically symmetric perfect fluid solution in which the fluid exhibits ''rigid rotation''. That is, we demand that the world lines of the fluid particles form a timelike congruence having nonzero vorticity but vanishing expansion and shear. (In ... [...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] |
Abelian Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Timelike Curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949,Stephen Hawking, '' My Brief History'', chapter 11 who discovered a solution to the equations of general relativity (GR) allowing CTCs known as the Gödel metric; and since then other GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. If CTCs exist, their existence would seem to imply at least the theoretical possibility of time travel backwards in time, raising the spectre of the grandfather paradox, although the Novikov self-consistency principle seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of quantum gravity whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Translation
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged (i.e. invariant) under such a transformation. Time translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history. Time translation symmetry is closely connected, via the Noether theorem, to conservation of energy. In mathematics, the set of all time translations on a given system form a Lie group. There are many symmetries in nature besides time translation, such as spatial translation or rotational symmetries. These symmetries can be broken and explain diverse phenomena such as crystals, superconductivity, and the Higgs mechanism. However, it was thought until very recently that time translation symmetry could not be broken. Time crystals, a state of matter first observed in 2017, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thought Experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most ancient pattern of mathematical proof", and existed before Euclidean mathematics, where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment. Johann Witt-Hansen established that Hans Christian Ørsted was the first to use the German term ' (lit. thought experiment) circa 1812. Ørsted was also the first to use the equivalent term ' in 1820. By 1883 Ernst Mach used the term ' in a different way, to denote exclusively the conduct of a experiment that would be subsequently performed as a by his students. Physical and mental experimentation could then be contrasted: Mach asked his students to provide him with explanations whenever the results from their subsequent, real, physical experiment differed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel Metric
The Gödel metric, also known as the Gödel solution or Gödel universe, is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles (dust solution), and the second associated with a negative cosmological constant (see Lambdavacuum solution). This solution has many unusual properties—in particular, the existence of closed timelike curves that would allow time travel in a universe described by the solution. Its definition is somewhat artificial in that the value of the cosmological constant must be carefully chosen to match the density of the dust grains, but this spacetime is an important pedagogical example. This solution was found in 1949 by Kurt Gödel. Definition Like any other Lorentzian spacetime, the Gödel solution presents the metric tensor in terms of some local coordinate chart. It may be easiest to understand the G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Spacetime
In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike. Description and analysis In a stationary spacetime, the metric tensor components, g_, may be chosen so that they are all independent of the time coordinate. The line element of a stationary spacetime has the form (i,j = 1,2,3) : ds^ = \lambda (dt - \omega_\, dy^i)^ - \lambda^ h_\, dy^i\,dy^j, where t is the time coordinate, y^ are the three spatial coordinates and h_ is the metric tensor of 3-dimensional space. In this coordinate system the Killing vector field \xi^ has the components \xi^ = (1,0,0,0). \lambda is a positive scalar representing the norm of the Killing vector, i.e., \lambda = g_\xi^\xi^, and \omega_ is a 3-vector, called the twist vector, which vanishes when the Killing vector is hypersurface orthogonal. The latter arises as the spatial components of the twist 4-vector \omega_ = e_\xi^\nabla^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric tensor, metric. Killing fields are the Lie group#The Lie algebra associated to a Lie group, infinitesimal generators of isometry, isometries; that is, flow (geometry), flows generated by Killing fields are Isometry (Riemannian geometry), continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object. Definition Specifically, a vector field ''X'' is a Killing field if the Lie derivative with respect to ''X'' of the metric ''g'' vanishes: :\mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is :g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 \, for all vectors ''Y'' and ''Z''. In local coo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector Field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object. Definition Specifically, a vector field ''X'' is a Killing field if the Lie derivative with respect to ''X'' of the metric ''g'' vanishes: :\mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is :g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 \, for all vectors ''Y'' and ''Z''. In local coordinates, this amounts to the Killing equation :\nabla_\mu X_\nu + \nabla_ X_\mu = 0 \,. This condition is expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
