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VSI Spacetime
In mathematical physics, vanishing scalar invariant (VSI) spacetimes are Lorentzian manifolds with all polynomial curvature invariants of all orders vanishing. Although the only Riemannian manifold with VSI property is flat space, the Lorentzian case admits nontrivial spacetimes with this property. Distinguishing these VSI spacetimes from Minkowski spacetime requires comparing non-polynomial invariants or carrying out the full Cartan–Karlhede algorithm on non-scalar quantities. All VSI spacetimes are Kundt spacetimes. An example with this property in four dimensions is a pp-wave. VSI spacetimes however also contain some other four-dimensional Kundt spacetimes of Petrov type 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 mos ... N and III. VSI spacetimes in higher dimensions have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentzian Manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible to defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Invariant
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations. Types of curvature invariants The invariants most often considered are ''polynomial invariants''. These are polynomials constructed from contractions such as traces. Second degree examples are called ''quadratic invariants'', and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order ''differential invariants''. The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors. However, it can also be considered a linear operator acting on bivectors, and as such it has a characteristic polynomial, whose coefficients and roots (eigenvalues) are polynomial scalar invariants. Physi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two 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 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 between events.This makes spacetime distance an invariant. Becau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Karlhede Algorithm
The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same dimension, it is not always obvious whether they are locally isometric. Élie Cartan, using his exterior calculus with his method of moving frames, showed that it is always possible to compare the manifolds. Carl Brans developed the method further, and the first practical implementation was presented by in 1980. The main strategy of the algorithm is to take covariant derivatives of the Riemann tensor. Cartan showed that in ''n'' dimensions at most ''n''(''n''+1)/2 differentiations suffice. If the Riemann tensor and its derivatives of the one manifold are algebraically compatible with the other, then the two manifolds are isometric. The Cartan–Karlhede algorithm therefore acts as a kind of generalization of the Petrov classification. The potentially large number of derivatives can be computationally prohibitive. The algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kundt Spacetime
In mathematical physics, Kundt spacetimes are Lorentzian manifolds admitting a geodesic null congruence with vanishing optical scalars (expansion, twist and shear). A well known member of Kundt class is pp-wave. Ricci-flat Kundt spacetimes in arbitrary dimension are algebraically special. In four dimensions Ricci-flat Kundt metrics of Petrov type III and N are completely known. All VSI spacetime In mathematical physics, vanishing scalar invariant (VSI) spacetimes are Lorentzian manifolds with all polynomial curvature invariants of all orders vanishing. Although the only Riemannian manifold with VSI property is flat space, the Lorentzian c ...s belong to a subset of the Kundt spacetimes.. References Lorentzian manifolds {{math-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pp-wave Spacetime
In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. The term ''pp'' stands for ''plane-fronted waves with parallel propagation'', and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt. Overview The pp-waves solutions model radiation moving at the speed of light. This radiation may consist of: * electromagnetic radiation, * gravitational radiation, * massless radiation associated with Weyl fermions, * ''massless'' radiation associated with some hypothetical distinct type relativistic classical field, or any combination of these, so long as the radiation is all moving in the ''same'' direction. A special type of pp-wave spacetime, the plane wave spacetimes, provide the most general analogue in general relativity of the plane waves familiar to students of electromagnetism. In particular, in general relativity, we must take into account the gravitational effects of the energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petrov Type
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] |
