Editing Nonmetricity tensor

{{Short description|Constant derivative of the metric tensor}}
In [[mathematics]], the '''nonmetricity tensor''' in [[differential geometry]] is the [[covariant derivative]] of the [[metric tensor]].<ref name="Hehl1995" /><ref name="KopeikinEK2011" /> It is therefore a [[tensor field]] of [[Tensor order|order]] three. It vanishes for the case of [[Riemannian geometry]] and can be
used to study non-Riemannian spacetimes.<ref name="Puntigam1997" />

== Definition ==
By components, it is defined as follows.<ref name="Hehl1995" />

:<math> Q_{\mu\alpha\beta}=\nabla_{\mu}g_{\alpha\beta} </math>

It measures the rate of change of the components of the metric tensor along the flow of a given vector field, since

:<math>\nabla_{\mu}\equiv\nabla_{\partial_{\mu}} </math>

where <math>\{\partial_{\mu}\}_{\mu=0,1,2,3}</math> is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional [[manifold]].

==Relation to connection==
We say that a [[connection (mathematics)|connection]] <math>\Gamma</math> is compatible with the metric when its associated covariant derivative of the [[metric tensor]] (call it <math>\nabla^{\Gamma}</math>, for example) is zero, i.e.

:<math> \nabla^{\Gamma}_{\mu}g_{\alpha\beta}=0 .</math>

If the connection is also torsion-free (i.e. totally symmetric) then it is known as the [[Levi-Civita connection]], which is the only one without [[torsion tensor|torsion]] and compatible with the metric tensor. If we see it from a geometrical point of view, a non-vanishing nonmetricity tensor for a metric tensor <math>g</math> implies that the modulus of a vector defined on the [[tangent bundle]] to a certain point <math>p</math> of the manifold, ''changes'' when it is evaluated along the direction (flow) of another arbitrary vector.

==References==
{{reflist|refs=

<ref name="KopeikinEK2011">{{citation | last1 = Kopeikin | first1 = Sergei | last2 = Efroimsky | first2 = Michael | last3 = Kaplan | first3 = George | isbn = 9783527408566 | page = 242 | publisher = John Wiley & Sons | title = Relativistic Celestial Mechanics of the Solar System | url = https://books.google.com/books?id=RfR2GawB-xcC&pg=PA242 | url-access = limited | year = 2011}}.</ref>

<ref name="Hehl1995">{{cite journal | last1 = Hehl | first1 = Friedrich W.  | last2 = McCrea | first2 = J. Dermott | last3 = Mielke | first3 = Eckehard W.  | last4 = Ne'eman | first4 = Yuval | title = Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance | journal = [[Physics Reports]] | volume = 258 | issue = 1–2 | pages = 1–171 | date = July 1995  | doi = 10.1016/0370-1573(94)00111-F  | arxiv=gr-qc/9402012 | bibcode = 1995PhR...258....1H | s2cid = 119346282 }}</ref>

<ref name="Puntigam1997">{{cite journal | last1 = Puntigam | first1 = Roland A.  | last2 = Lämmerzahl | first2 = Claus | last3 = Hehl | first3 = Friedrich W.  | title = Maxwell's theory on a post-Riemannian spacetime and the equivalence principle | journal = [[Classical and Quantum Gravity]] | volume = 14 | issue = 5 | pages = 1347–1356 | date = May 1997 | doi = 10.1088/0264-9381/14/5/033 | arxiv= gr-qc/9607023 | bibcode = 1997CQGra..14.1347P | s2cid = 44439510 }}</ref>

}}

{{Tensors}}

==External links==
*{{Cite journal|last1=Iosifidis|first1=Damianos|last2=Petkou|first2=Anastasios C.|last3=Tsagas|first3=Christos G.|date=May 2019|title=Torsion/nonmetricity duality in f(R) gravity|url=http://link.springer.com/10.1007/s10714-019-2539-9|journal=General Relativity and Gravitation|language=en|volume=51|issue=5|pages=66|doi=10.1007/s10714-019-2539-9|issn=0001-7701|arxiv=1810.06602|bibcode=2019GReGr..51...66I|s2cid=53554290}}

[[Category:Differential geometry]]


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