en>Yobot: WP:CHECKWIKI error fixes (22) using AWB (9842)

2014-01-08T12:11:58Z

WP:CHECKWIKI error fixes (22) using AWB (9842)

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			{{Merge from\|Curvature invariant (general relativity)\|date=November 2010}}
			In [[Riemannian geometry]] and [[pseudo-Riemannian geometry]], '''curvature invariants''' are [[scalar (mathematics)\|scalar]] quantities constructed from tensors that represent [[curvature]]. These tensors are usually the [[Riemann tensor]], the [[Weyl tensor]], the [[Ricci tensor]] and tensors formed from these by the operations of taking [[dual]]s,<!-- [[Dual]] is (as you would expect) a disambiguation page. Please disambiguate this link. --> [[contraction (geometry)\|contraction]]s and [[covariant differentiation]]s.

			==Types of curvature invariants==

			The invariants most often considered are ''polynomial invariants''. These are polynomials constructed from contractions such as traces. Second degree examples are called ''quadratic invariants'', and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order ''differential invariants''.

			The Riemann tensor is a [[multilinear operator]] of fourth rank acting on [[tangent vector]]s. However, it can also be considered a [[linear operator]] acting on [[bivector]]s, and as such it has a [[characteristic polynomial]], whose coefficients and roots ([[eigenvalue]]s) are polynomial scalar invariants.

			==Physical applications==

			In [[metric theory of gravitation\|metric theories of gravitation]] such as [[general relativity]], curvature scalars play an important role in telling distinct spacetimes apart.

			Two of the most basic curvature invariants in general relativity are the [[Curvature invariant (general relativity)#Principal invariants\|Kretschmann scalar]]

			:<math>R_{abcd} \, R^{abcd}</math>

			and the ''Chern–Pontryagin scalar'',

			:<math>R_{abcd} \, {{}^\star \!R}^{abcd}</math>

			These are analogous to two familiar quadratic invariants of the [[electromagnetic field tensor]] in classical electromagnetism.

			An important unsolved problem in general relativity is to give a [[basis (linear algebra)\|basis]] (and any [[syzygy (mathematics)\|syzygies]]) for the zero-th order invariants of the Riemann tensor.

			They have limitations because many distinct spacetimes cannot be distinguished on this basis. In particular, so called [[VSI spacetime]]s (including pp-waves as well as some other Petrov type N and III spacetimes) cannot be distinguished from [[Minkowski spacetime]] using any number of polynomial curvature invariants (of any order).

			==See also==
			*[[Cartan–Karlhede algorithm]]
			*[[Carminati–McLenaghan invariants]]
			*[[Curvature invariant (general relativity)]]
			*[[Ricci decomposition]]

			==References==
			{{Reflist}}
			*{{cite book\|last=Stephani\|first=Hans\|title=Exact Solutions of Einstein's Field Equations\|chapter=9. Invariants and the characterization of geometries \|year=2009\|publisher=Cambridge Univ Pr\|location=Cambridge [u.a.]\|isbn=978-0521467025\|edition=2. ed., 1. paperback}}

			{{DEFAULTSORT:Curvature Invariant}}
			[[Category:Riemannian geometry]]


			{{relativity-stub}}
			{{differential-geometry-stub}}

94.30.38.169: Reverted change as bold face not compatible with \hat

2011-01-04T17:21:05Z

Reverted change as bold face not compatible with \hat

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en>Yobot: WP:CHECKWIKI error fixes (22) using AWB (9842)

94.30.38.169: Reverted change as bold face not compatible with \hat