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]

Definition

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

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

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

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

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

Relation to connection

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

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

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

References

1. Hehl, Friedrich W.; McCrea, J. Dermott; Mielke, Eckehard W.; Ne'eman, Yuval (July 1995). "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance". Physics Reports. 258 (1–2): 1–171. arXiv:gr-qc/9402012. doi:10.1016/0370-1573(94)00111-F.
2. Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011), Relativistic Celestial Mechanics of the Solar System, John Wiley & Sons, p. 242, ISBN 9783527408566.
3. Puntigam, Roland A.; Lämmerzahl, Claus; Hehl, Friedrich W. (May 1997). "Maxwell's theory on a post-Riemannian spacetime and the equivalence principle". Classical and Quantum Gravity. 14 (5): 1347–1356. arXiv:gr-qc/9607023. doi:10.1088/0264-9381/14/5/033.

External links

• 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. doi:10.1007/s10714-019-2539-9. ISSN 0001-7701.