In the theory of
Lorentzian manifolds, particularly in the context of applications to
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 Kretschmann scalar is a quadratic
scalar invariant. It was introduced by
Erich Kretschmann.
Definition
The Kretschmann invariant is
:
where
is 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. ...
(in this equation the
Einstein summation convention was used, and it will be used throughout the article). Because it is a sum of squares of tensor components, this is a ''quadratic'' invariant.
For the use of a computer algebra system a more detailed writing is meaningful:
:
Examples
For a
Schwarzschild black hole of mass
, the Kretschmann scalar is
:
where
is the gravitational constant.
For a general
FRW spacetime with metric
:
the Kretschmann scalar is
:
Relation to other invariants
Another possible invariant (which has been employed for example in writing the gravitational term of the Lagrangian for some ''higher-order gravity'' theories) is
:
where
is the
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 f ...
, the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In
dimensions this is related to the Kretschmann invariant by
:
where
is 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 ...
tensor and
is the Ricci
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 ...
(obtained by taking successive traces of the Riemann tensor). The Ricci tensor vanishes in vacuum spacetimes (such as the Schwarzschild solution mentioned above), and hence there the Riemann tensor and the Weyl tensor coincide, as do their invariants.
Gauge theory invariants
The Kretschmann scalar and the ''Chern-Pontryagin scalar''
:
where
is the ''left dual'' of the Riemann tensor, are mathematically analogous (to some extent, physically analogous) to the familiar invariants of the
electromagnetic field tensor
:
Generalising from the
gauge theory of electromagnetism to general non-abelian gauge theory, the first of these invariants is
:
,
an expression proportional to the
Yang–Mills Lagrangian. Here
is the curvature of a
covariant derivative, and
is a
trace form
In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''.
Definition
Let ''K'' be a field and ''L'' a finite extension (and hence an ...
. The Kretschmann scalar arises from taking the connection to be on the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
.
See also
*
Carminati-McLenaghan invariants, for a set of invariants
*
Classification of electromagnetic fields In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in the study of solutions of Maxwell's equations and has ap ...
, for more about the invariants of the electromagnetic field tensor
*
Curvature invariant, for curvature invariants in Riemannian and pseudo-Riemannian geometry in general
*
Curvature invariant (general relativity) In general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors - which represent curvature, hence the name, - and possibly operations on them such as contraction, covariant differentiation and d ...
*
Ricci decomposition In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. Th ...
, for more about the Riemann and Weyl tensor
References
Further reading
*
*
*
{{DEFAULTSORT:Kretschmann Scalar
Riemannian geometry
Lorentzian manifolds
Tensors in general relativity