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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 ca ...s belong to a subset of the Kundt spacetimes.. References Lorentzian manifolds {{math-physics-stub ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Scalars
In general relativity, optical scalars refer to a set of three scalar functions \ describing the propagation of a geodesic null congruence.Eric Poisson. ''A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics''. Cambridge: Cambridge University Press, 2004. Chapter 2.Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. ''Exact Solutions of Einstein's Field Equations''. Cambridge: Cambridge University Press, 2003. Chapter 6.Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Oxford: Oxford University Press, 1998. Section 9.(a).Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Section 2.1.3.P Schneider, J Ehlers, E E Falco. ''Gravitational Lenses''. Berlin: Springer, 1999. Section 3.4.2. In fact, these three scalars \ can be defined for both timelike and null geodesic congruences in an identical spirit, but they are called "optic ... [...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] |
Ricci-flat
In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in vacuum with vanishing cosmological constant. In Lorentzian geometry, a number of Ricci-flat metrics are known from works of Karl Schwarzschild, Roy Kerr, and Yvonne Choquet-Bruhat. In Riemannian geometry, Shing-Tung Yau's resolution of the Calabi conjecture produced a number of Ricci-flat metrics on Kähler manifolds. Definition A pseudo-Riemannian manifold is said to be Ricci-flat if its Ricci curvature is zero. It is direct to verify that, except in dimension two, a metric is Ricci-flat if and only if its Einstein tensor is zero. Ricci-flat manifolds are one of three special type of Einstein manifold, arising as the special case of scalar curvature equali ... [...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^ = \lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
