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| In the theory of [[pseudo-Riemannian manifold|Lorentzian manifolds]], particularly in the context of applications to [[general relativity]], the '''Kretschmann scalar''' is a quadratic [[curvature invariant (general relativity)|scalar invariant]]. It was introduced by [[Erich Kretschmann]].<ref name="Henry"/>
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| ==Definition==
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| The Kretschmann invariant is<ref name="Henry">{{cite journal|title=Kretschmann Scalar for a Kerr-Newman Black Hole |author=Richard C. Henry |journal=[[The Astrophysical Journal]] |publisher=The American Astronomical Society |year=2000 |url=http://iopscience.iop.org/0004-637X/535/1/350/fulltext/40794.text.html |pages=350–353 |volume=535 |arxiv=astro-ph/9912320v1|bibcode = 2000ApJ...535..350H |doi = 10.1086/308819 }}</ref><ref>{{Harvnb|Grøn|Hervik|2007 |loc=p 219}}</ref>
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| :<math> K = R_{abcd} \, R^{abcd}</math>
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| where <math>R_{abcd}</math> is the [[Riemann curvature tensor]]. Because it is a sum of squares of tensor components, this is a ''quadratic'' invariant.
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| For [[Schwarzschild metric|Schwarzschild black hole]], the Kretschmann scalar is<ref name="Henry"/>
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| :<math> K = \frac{48 G^2 M^2}{c^4 r^6} \,.</math>
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| ==Relation to other invariants==
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| Another possible invariant (which has been employed for example in writing the gravitational term of the Lagrangian for some ''higher-order gravity'' theories) is
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| :<math>C_{abcd} \, C^{abcd}</math> | |
| where <math>C_{abcd}</math> is the [[Weyl tensor]], the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In <math>d</math> dimensions this is related to the Kretschmann invariant by<ref name="CherubiniBini2002">{{cite journal|last1=Cherubini|first1=Christian|last2=Bini|first2=Donato|last3=Capozziello|first3=Salvatore|last4=Ruffini|first4=Remo|title=Second Order Scalar Invariants of the Riemann Tensor: Applications to Black Hole Spacetimes|journal=International Journal of Modern Physics D|volume=11|issue=06|year=2002|pages=827–841|issn=0218-2718|doi=10.1142/S0218271802002037|arxiv=gr-qc/0302095v1|bibcode = 2002IJMPD..11..827C }}</ref>
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| :<math>R_{abcd} \, R^{abcd} = C_{abcd} \, C^{abcd} +\frac{4}{d-2} R_{ab}\, R^{ab} - \frac{2}{(d-1)(d-2)}R^2</math>
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| where <math>R^{ab}</math> is the [[Ricci curvature]] tensor and <math>R</math> is the Ricci [[scalar curvature]] (obtained by taking successive traces of the Riemann tensor).
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| The Kretschmann scalar and the ''Chern-Pontryagin scalar''
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| :<math>R_{abcd} \, {{}^\star \! R}^{abcd}</math>
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| where <math>{{}^\star R}^{abcd}</math> 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]]
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| :<math>F_{ab} \, F^{ab}, \; \; F_{ab} \, {{}^\star \! F}^{ab}</math>
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| ==See also==
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| *[[Carminati-McLenaghan invariants]], for a set of invariants.
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| *[[Classification of electromagnetic fields]], for more about the invariants of the electromagnetic field tensor.
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| *[[Curvature invariant]], for curvature invariants in Riemannian and pseudo-Riemannian geometry in general.
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| *[[Curvature invariant (general relativity)]].
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| *[[Ricci decomposition]], for more about the Riemann and Weyl tensor.
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{Citation|last=Grøn|first=Øyvind |authorlink=Øyvind Grøn| coauthors=Hervik, Sigbjørn|title=Einstein's General Theory of Relativity|location=New York|publisher=Springer|year=2007|isbn=978-0-387-69199-2|ref=harv}}
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| *{{Citation|author=B. F. Schutz|authorlink=Bernard F. Schutz|title=A First Course in General Relativity (Second Edition)|publisher=Cambridge University Press| year=2009|isbn=978-0-521-88705-2|ref=harv}}
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| *{{Citation|first=Charles W.|last=Misner|authorlink=Charles W. Misner|first2=Kip. S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler|author3-link=John A. Wheeler|title=[[Gravitation (book)|Gravitation]]|publisher= W. H. Freeman|year=1973|isbn=0-7167-0344-0|ref=harv}}
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| {{DEFAULTSORT:Kretschmann Scalar}}
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| [[Category:Riemannian geometry]]
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| [[Category:Lorentzian manifolds]]
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| [[Category:Tensors in general relativity]]
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She is recognized by the name of Myrtle Shryock. North Dakota is her birth location but she will have to move one day or another. What I adore performing is performing ceramics but I haven't made a dime with it. I am a meter reader but I strategy on changing it.
Also visit my web site: www.ddaybeauty.com