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]
- [math]\displaystyle{ 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]\displaystyle{ \nabla_{\mu}\equiv\nabla_{\partial_{\mu}} }[/math]
where [math]\displaystyle{ \{\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 [math]\displaystyle{ \Gamma }[/math] is compatible with the metric when its associated covariant derivative of the metric tensor (call it [math]\displaystyle{ \nabla^{\Gamma} }[/math], for example) is zero, i.e.
- [math]\displaystyle{ \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 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]\displaystyle{ g }[/math] implies that the modulus of a vector defined on the tangent bundle to a certain point [math]\displaystyle{ p }[/math] of the manifold, changes when it is evaluated along the direction (flow) of another arbitrary vector.
References
- ↑ 1.0 1.1 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. doi:10.1016/0370-1573(94)00111-F. Bibcode: 1995PhR...258....1H.
- ↑ Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011), Relativistic Celestial Mechanics of the Solar System, John Wiley & Sons, p. 242, ISBN 9783527408566, https://books.google.com/books?id=RfR2GawB-xcC&pg=PA242.
- ↑ 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. doi:10.1088/0264-9381/14/5/033. Bibcode: 1997CQGra..14.1347P.
External links
- Iosifidis, Damianos; Petkou, Anastasios C.; Tsagas, Christos G. (May 2019). "Torsion/nonmetricity duality in f(R) gravity" (in en). General Relativity and Gravitation 51 (5): 66. doi:10.1007/s10714-019-2539-9. ISSN 0001-7701. Bibcode: 2019GReGr..51...66I. http://link.springer.com/10.1007/s10714-019-2539-9.
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