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| {{Merge to|Curvature invariant|date=November 2010}}
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| In [[physics]], '''curvature invariants in general relativity''' are a set of [[Scalar (mathematics)|scalar]]s called [[curvature invariant]]s that arise in [[general relativity]]. They are formed from the [[Riemann tensor|Riemann]], [[Weyl tensor|Weyl]] and [[Ricci tensor|Ricci]] tensors - which represent [[curvature]] - and possibly operations on them such as [[Tensor contraction|contraction]], [[covariant differentiation]] and [[Dual tensor|dualisation]].
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| Certain invariants formed from these curvature tensors play an important role in classifying [[spacetime]]s. Invariants are actually less powerful for distinguishing locally non-[[isometry|isometric]] [[Lorentzian manifold]]s than they are for distinguishing [[Riemannian manifold]]s. This means that they are more limited in their applications than for manifolds endowed with a [[positive definite]] [[metric tensor]].
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| ==Principal invariants==
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| The principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants (i.e., sums of squares of components).
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| The principal invariants of the [[Riemann tensor]] of a four-dimensional Lorentzian manifold are
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| #the ''[[Kretschmann scalar]]'' <math>K_1 = R_{abcd} \, R^{abcd}</math>
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| #the ''Chern-Pontryagin scalar'' <math>K_2 = {{}^\star R}_{abcd} \, R^{abcd}</math>
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| #the ''Euler scalar'' <math>K_3 = {{}^\star R^\star}_{abcd} \, R^{abcd}</math>
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| 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.)
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| 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 number]] and [[Euler characteristic]] respectively, they have some justification.
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| The principal invariants of the [[Weyl tensor]] are
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| # <math>I_1 = C_{abcd} \, C^{abcd}</math>
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| # <math>I_2 = {{}^\star C}_{abcd} \, C^{abcd}</math>
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| (Because <math>{{}^\star C^\star}_{abcd} = -C_{abcd}</math>, there is no need to define a third principal invariant for the Weyl tensor.)
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| ===Relation with Ricci decomposition===
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| As one might expect from the [[Ricci decomposition]] of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed from the second rank [[Ricci tensor]] and from the [[Ricci scalar]], these two sets of invariants are related:
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| :<math>K_1 = I_1 + 2 \, R_{ab} \, R^{ab} - \frac{1}{3} \, R^2 </math>
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| :<math>K_3 = -I_1 + 2 \, R_{ab} \, R^{ab} - \frac{2}{3} \, R^2</math>
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| ===Relation with Bel decomposition===
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| Recall that (in four dimensions), the [[Bel decomposition]] of the Riemann tensor, with respect to a timelike unit vector field <math>\vec{X}</math>, not necessarily geodesic or hypersurface orthogonal, consists of three pieces
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| #the ''[[electrogravitic tensor]]'' <math>E[\vec{X}]_{ab} = R_{ambn} \, X^m \, X^n</math>
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| #the ''[[magnetogravitic tensor]]'' <math>B[\vec{X}]_{ab} = {{}^\star R}_{ambn} \, X^m \, X^n</math>
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| #the ''[[topogravitic tensor]]'' <math>L[\vec{X}]_{ab} = {{}^\star R^\star}_{ambn} \, X^m \, X^n</math>
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| 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]], 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:
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| *<math>K_1/4</math> is the trace of '''E'''<sup>2</sup> + '''L'''<sup>2</sup> - 2 '''B''' '''B'''<sup>T</sup>,
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| *<math>-K_2/8</math> is the trace of '''B''' ( '''E''' - '''L''' ),
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| *<math>K_3/8</math> is the trace of '''E''' '''L''' - '''B'''<sup>2</sup>.
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| ===Expression in Newman-Penrose formalism===
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| In terms of the [[Weyl scalar]]s in the [[Newman-Penrose formalism]], the principal invariants of the Weyl tensor may be obtained by taking the real and imaginary parts of the expression
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| :<math>I_1 - i \, I_2 = 16 \, \left( 3 \Psi_2^2 + \Psi_0 \, \Psi_4 - 4 \, \Psi_1 \Psi_3 \right)</math>
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| (But note the minus sign!)
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| The principal quadratic invariant of the [[Ricci tensor]], <math>R_{ab} \, R^{ab}</math>, may be obtained as a more complicated expression involving the ''Ricci scalars'' (see the paper by Cherubini et al. cited below).
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| ==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]] is non-compact.
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| 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 spacetime]]s.
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| ==See also==
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| *[[Bach tensor]], for a sometimes useful tensor generated by <math>I_2</math> via a variational principle.
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| *[[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.
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| *[[Curvature invariant]], for curvature invariants in a more general context.
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| ==References==
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| *{{cite journal | author=Cherubini, C.; Bini, D.; Capozziello, S.; and Ruffini R. | title=Second order scalar invariants of the Riemann tensor: applications to black hole spacetimes | journal=Int. J. Mod. Phys. D | year=2002 | volume=11 | pages=827–841 | doi=10.1142/S0218271802002037|arxiv = gr-qc/0302095 |bibcode = 2002IJMPD..11..827C | issue=6 }} See also the [http://www.arxiv.org/abs/gr-qc/0302095 eprint version].
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| *{{cite journal|last=Coley|first=A.|coauthors=Hervik, S. and Pelavas, N.|title=Spacetimes characterized by their scalar curvature invariants|journal=Class. Quantum Grav.|year=2009|volume=26|pages=025013|doi=10.1088/0264-9381/26/2/025013|arxiv=0901.0791}}
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| {{DEFAULTSORT:Curvature Invariant (General Relativity)}}
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| [[Category:Tensors in general relativity]]
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