Nonmetricity tensor

In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor.^{[1] [2]} It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes.^[3]

By components, it is defined as follows.^[1]

${\displaystyle Q_{\mu \alpha \beta }=\nabla _{\mu }g_{\alpha \beta }}${\displaystyle Q_{\mu \alpha \beta }=\nabla _{\mu }g_{\alpha \beta }}

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

${\displaystyle \nabla _{\mu }\equiv \nabla _{\partial _{\mu }}}${\displaystyle \nabla _{\mu }\equiv \nabla _{\partial _{\mu }}}

where ${\displaystyle \{\partial _{\mu }\}_{\mu =0,1,2,3}}${\displaystyle \{\partial _{\mu }\}_{\mu =0,1,2,3}} is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional manifold.

We say that a connection ${\displaystyle \Gamma }${\displaystyle \Gamma } is compatible with the metric when its associated covariant derivative of the metric tensor (call it ${\displaystyle \nabla ^{\Gamma }}${\displaystyle \nabla ^{\Gamma }}, for example) is zero, i.e.

${\displaystyle \nabla _{\mu }^{\Gamma }g_{\alpha \beta }=0.}${\displaystyle \nabla _{\mu }^{\Gamma }g_{\alpha \beta }=0.}

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 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 ${\displaystyle g}${\displaystyle g} implies that the modulus of a vector defined on the tangent bundle to a certain point ${\displaystyle p}${\displaystyle p} of the manifold, changes when it is evaluated along the direction (flow) of another arbitrary vector.

• Iosifidis, Damianos; Petkou, Anastasios C.; Tsagas, Christos G. (May 2019). "Torsion/nonmetricity duality in f(R) gravity". General Relativity and Gravitation. 51 (5): 66. arXiv:1810.06602 . Bibcode:2019GReGr..51...66I. doi:10.1007/s10714-019-2539-9. ISSN 0001-7701. S2CID 53554290.

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