In General Relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

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

Scalar (physics)

In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

,
,

describing the curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 of a four-dimensional spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. They are the expression of the ten independent degrees of freedom of the Weyl tensor

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 force that a body feels when moving along a geodesic...

  in the Newman-Penrose Formalism

Newman-Penrose Formalism

The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for General Relativity. Their notation is an effort to treat General Relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR.The NP formalism is itself a...

 for general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. Given a null tetrad (), the scalars are given (up to an overall conventional sign) by

Physical Interpretation

Szekeres (1965) gave an interpretation of the different Weyl scalars at large distances:
is a "Coulomb" term, representing the gravitational monopole of the source; & are ingoing and outgoing "longitudinal" radiation terms; & are ingoing and outgoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I), & 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 .
It can be shown to describe outgoing gravitational radiation (in an asymptotically flat spacetime) as
Here, and 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

Newman-Penrose Formalism

The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for General Relativity. Their notation is an effort to treat General Relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR.The NP formalism is itself a...

.