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.