Nonmetricity tensor explained

In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can beused to study non-Riemannian spacetimes.

Definition

By components, it is defined as follows.

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

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

where

\{\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

\Gamma is compatible with the metric when its associated covariant derivative of the metric tensor (call it \nabla ^{\Gamma} , for example) is zero, i.e.\nabla ^{\Gamma} _{\mu} 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

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

External links

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