Curvature invariant (general relativity)
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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 ...
, curvature invariants are a set of scalars formed from the
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
,
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
and
Ricci Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment * Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin * Christina Ricci (born 1980), Ameri ...
tensors - which represent curvature, hence the name, - and possibly operations on them such as contraction,
covariant differentiation In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...
and dualisation. Certain invariants formed from these curvature tensors play an important role in classifying
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 ...
s. Invariants are actually less powerful for distinguishing locally non- isometric
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 ...
s than they are for distinguishing Riemannian manifolds. This means that they are more limited in their applications than for manifolds endowed with a positive definite metric tensor.


Principal invariants

The principal invariants of the Riemann and Weyl tensors are certain quadratic
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
invariants (i.e., sums of squares of components). The principal invariants of the
Riemann 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. ...
of a four-dimensional Lorentzian manifold are #the '' Kretschmann scalar'' K_1 = R_ \, R^ #the ''Chern–Pontryagin scalar'' K_2 = _ \, R^ #the ''Euler scalar'' K_3 = _ \, R^ These are quadratic polynomial invariants (sums of squares of components). (Some authors define the Chern–Pontryagin scalar using the right dual instead of the left dual.) The first of these was introduced by Erich Kretschmann. The second two names are somewhat anachronistic, but since the integrals of the last two are related to the
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
number and Euler characteristic respectively, they have some justification. The principal invariants of 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 ...
are # I_1 = C_ \, C^ # I_2 = _ \, C^ (Because _ = -C_, there is no need to define a third principal invariant for the Weyl tensor.)


Relation with Ricci decomposition

As one might expect from the
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 ...
of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed from the second rank
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 ...
and from the
Ricci scalar 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 ...
, these two sets of invariants are related (in d=4): :K_1 = I_1 + 2 \, R_ \, R^ - \tfrac \, R^2 :K_3 = -I_1 + 2 \, R_ \, R^ - \tfrac \, R^2


Relation with Bel decomposition

In four dimensions, the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field \vec, not necessarily geodesic or hypersurface orthogonal, consists of three pieces #the '' electrogravitic tensor'' E vec = R_ \, X^m \, X^n #the ''
magnetogravitic tensor In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric fi ...
'' B vec = _ \, X^m \, X^n #the '' topogravitic tensor'' L vec = _ \, X^m \, X^n Because these are all ''transverse'' (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three-dimensional vectors, or as three by three real matrices. They are respectively symmetric,
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 ...
, and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows: *K_1/4 is the trace of E2 + L2 - 2 B BT, *-K_2/8 is the trace of B ( E - L ), *K_3/8 is the trace of E L - B2.


Expression in Newman–Penrose formalism

In terms of the Weyl scalars in the
Newman–Penrose formalism The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...
, the principal invariants of the Weyl tensor may be obtained by taking the real and imaginary parts of the expression :I_1 - i \, I_2 = 16 \, \left( 3 \Psi_2^2 + \Psi_0 \, \Psi_4 - 4 \, \Psi_1 \Psi_3 \right) (But note the minus sign!) The principal quadratic invariant of 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 ...
, R_ \, R^, may be obtained as a more complicated expression involving the ''Ricci scalars'' (see the paper by Cherubini et al. cited below).


Distinguishing Lorentzian manifolds

An important question related to Curvature invariants is when the set of polynomial curvature invariants can be used to (locally) distinguish manifolds. To be able to do this is necessary to include higher-order invariants including derivatives of the Riemann tensor but in the Lorentzian case, it is known that there are spacetimes which cannot be distinguished; e.g., the VSI spacetimes for which all such curvature invariants vanish and thus cannot be distinguished from flat space. This failure of being able to distinguishing Lorentzian manifolds is related to the fact that the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
is non-compact. There are still examples of cases when we can distinguish Lorentzian manifolds using their invariants. Examples of such are fully general Petrov type I spacetimes with no Killing vectors, see Coley ''et al.'' below. Indeed, it was here found that the spacetimes failing to be distinguished by their set of curvature invariants are all Kundt spacetimes.


See also

* Bach tensor, for a sometimes useful tensor generated by I_2 via a variational principle. * Carminati-McLenaghan invariants, for a set of polynomial invariants of the Riemann tensor of a four-dimensional Lorentzian manifold which is known to be ''complete'' under some circumstances. * Curvature invariant, for curvature invariants in a more general context.


References

* See also th
eprint version
* {{DEFAULTSORT:Curvature Invariant (General Relativity) Tensors in general relativity