Weyl scalar

In General Relativity, the Weyl scalars are a set of five complex scalar quantities,

: $Psi_0, ldots, Psi_4$ ,

describing the curvature of a four-dimensional spacetime. They are the expression of the ten independent degrees of freedom of the Weyl tensor $C_{abcd}$ in the Newman-Penrose Formalism for general relativity. Given a null tetrad ( $l^a, n^a, m^a, ar{m}^a$ ), the scalars are given (up to an overall conventional sign) by

: $Psi_0 = - C_{alphaetagammadelta} l^alpha m^eta l^gamma m^delta ,$

: $Psi_1 = -C_{alphaetagammadelta} l^alpha n^eta l^gamma m^delta ,$

: $Psi_2 = -C_{alphaetagammadelta} l^alpha m^eta ar{m}^gamma n^delta ,$

: $Psi_3 = -C_{alphaetagammadelta} l^alpha n^eta ar{m}^gamma n^delta ,$

: $Psi_4 = -C_{alphaetagammadelta} n^alpha ar{m}^eta n^gamma ar{m}^delta .$

Physical Interpretation

: $Psi_2$ is a "Coulomb" term, representing the gravitational monopole of the source;: $Psi_1$ & $Psi_3$ are ingoing and ougoing "longitudinal" radiation terms;: $Psi_0$ & $Psi_4$ are ingoing and ougoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I), $Psi_1$ & $Psi_3$ can be transformed to zero by an appropriate choice of null tetrad. Thus these can be viewed as gauge quantities.

A particularly important case is the Weyl scalar $Psi_4$ .It can be shown to describe outgoing gravitational radiation (in an asymptotically flat spacetime) as: $Psi_4 = frac{1}{2}left( ddot{h}_{hat{ heta} hat{ heta - ddot{h}_{hat{phi} hat{phi
ight) + i ddot{h}_{hat{ heta}hat{phi = -ddot{h}_+ + i ddot{h}_ imes .$ Here, $h_+$ and $h_ imes$ are the "plus" and "cross" polarizations of gravitational radiation, and the double dots represent double time-differentiation.

For more details, see the article on the Newman-Penrose Formalism.

References

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

Weyl tensor — In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal… … Wikipedia
Weyl-Brauer matrices — In mathematics, particularly in the theory of spinors, the Weyl Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra. They generalize to n dimensions the Pauli matrices. They are named for Richard Brauer and… … Wikipedia
Weyl transformation — See also Weyl quantization, for another definition of the Weyl transform. In theoretical physics, the Weyl transformation is a local rescaling of the metric tensor::g {ab} ightarrow g {ab} exp(2omega(x))The invariance of a theory or an expression … Wikipedia
List of topics named after Hermann Weyl — This is a list of topics named after Hermann Weyl, the influential German mathematician. * Weyl algebra * Weyl basis of the gamma matrices * Weyl chamber * Weyl character formula * Weyl s criterion * Weyl curvature: see Weyl tensor * Weyl… … Wikipedia
Kretschmann scalar — In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann. DefinitionThe Kretschmann invariant is: K =… … Wikipedia
Newman-Penrose formalism — The Newman Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose [cite journal | author= Ezra T. Newman and Roger Penrose | title=An Approach to Gravitational Radiation by a Method of Spin Coefficients |… … Wikipedia
Newman–Penrose formalism — The Newman Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose[1] for General Relativity. Their notation is an effort to treat General Relativity in terms of spinor notation, which introduces complex forms of the… … Wikipedia
Spinor — In mathematics and physics, in particular in the theory of the orthogonal groups (such as the rotation or the Lorentz groups), spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the… … Wikipedia
Curvature of Riemannian manifolds — In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous… … Wikipedia
Nordström's theory of gravitation — In theoretical physics, Nordström s theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913… … Wikipedia

Academic Dictionaries and Encyclopedias

Weyl scalar

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Weyl scalar

Look at other dictionaries:

Share the article and excerpts

Direct link