In
differential geometry, the Weyl curvature tensor, named after
Hermann Weyl, is a measure of the
curvature of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
or, more generally, a
pseudo-Riemannian 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 ...
. Like the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
, the Weyl tensor expresses the
tidal force
The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
that a body feels when moving along a
geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
, or
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the
traceless
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace o ...
component of the Riemann tensor. This
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free:
metric contraction
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.
In
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 ...
, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the
vacuum Einstein equation—and it governs the propagation of
gravitational waves
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
through regions of space devoid of matter.
More generally, the Weyl curvature is the only component of curvature for
Ricci-flat manifolds and always governs the
characteristics of the field equations of an
Einstein manifold.
In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locally
conformally flat: there exists a
local coordinate system
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
in which the metric tensor is proportional to a constant tensor. This fact was a key component of
Nordström's theory of gravitation, which was a precursor of
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 ...
.
Definition
The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a (0,4) valence tensor (by contracting with the metric). The (0,4) valence Weyl tensor is then
:
where ''n'' is the dimension of the manifold, ''g'' is the metric, ''R'' is the Riemann tensor, ''Ric'' is the
Ricci tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
, ''s'' is the
scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
, and
denotes the
Kulkarni–Nomizu product of two symmetric (0,2) tensors:
:
In tensor component notation, this can be written as
:
The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric.
The decomposition () expresses the Riemann tensor as an
orthogonal direct sum, in the sense that
:
This decomposition, known as the
Ricci decomposition, expresses the Riemann curvature tensor into its
irreducible components under the action of the
orthogonal group . In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the
special orthogonal group, the self-dual and antiself-dual parts ''C''
+ and ''C''
−.
The Weyl tensor can also be expressed using the
Schouten tensor, which is a trace-adjusted multiple of the Ricci tensor,
:
Then
:
In indices,
:
where
is the Riemann tensor,
is the Ricci tensor,
is the Ricci scalar (the scalar curvature) and brackets around indices refers to the
antisymmetric part. Equivalently,
:
where ''S'' denotes the
Schouten tensor.
Properties
Conformal rescaling
The Weyl tensor has the special property that it is invariant under
conformal changes to the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
. That is, if
for some positive scalar function
then the (1,3) valent Weyl tensor satisfies
. For this reason the Weyl tensor is also called the conformal tensor. It follows that a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for a Riemannian manifold to be
conformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is
sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
as well. In dimension 3 the vanishing of the
Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat. Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence of
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
.
Indeed, the existence of a conformally flat scale amounts to solving the overdetermined partial differential equation
:
In dimension ≥ 4, the vanishing of the Weyl tensor is the only
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
for this equation; in dimension 3, it is the
Cotton tensor instead.
Symmetries
The Weyl tensor has the same symmetries as the Riemann tensor. This includes:
:
In addition, of course, the Weyl tensor is trace free:
:
for all ''u'', ''v''. In indices these four conditions are
:
Bianchi identity
Taking traces of the usual second Bianchi identity of the Riemann tensor eventually shows that
:
where ''S'' is the
Schouten tensor. The valence (0,3) tensor on the right-hand side is the
Cotton tensor, apart from the initial factor.
See also
*
Curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigoro ...
*
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
provides a coordinate expression for the Weyl tensor.
*
Lanczos tensor
*
Peeling theorem In general relativity, the peeling theorem describes the asymptotic behavior of the Weyl tensor as one goes tnull infinity Let \gamma be a null geodesic in a spacetime (M, g_) from a point p to null infinity, with affine parameter \lambda. Then the ...
*
Petrov classification
*
Plebanski tensor
*
Weyl curvature hypothesis
*
Weyl scalar
Notes
References
*
*.
* .
*
*
*
{{DEFAULTSORT:Weyl Tensor
Riemannian geometry
Tensors in general relativity