Template:Short description 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 order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes.<ref name="Puntigam1997" />
DefinitionEdit
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 connectionEdit
We say that a 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 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.