- Source: Nonmetricity tensor
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 be
used to study non-Riemannian spacetimes.
Definition
By components, it is defined as follows.
Q
μ
α
β
=
∇
μ
g
α
β
{\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
∇
μ
≡
∇
∂
μ
{\displaystyle \nabla _{\mu }\equiv \nabla _{\partial _{\mu }}}
where
{
∂
μ
}
μ
=
0
,
1
,
2
,
3
{\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
Γ
{\displaystyle \Gamma }
is compatible with the metric when its associated covariant derivative of the metric tensor (call it
∇
Γ
{\displaystyle \nabla ^{\Gamma }}
, for example) is zero, i.e.
∇
μ
Γ
g
α
β
=
0.
{\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
g
{\displaystyle g}
implies that the modulus of a vector defined on the tangent bundle to a certain point
p
{\displaystyle p}
of the manifold, changes when it is evaluated along the direction (flow) of another arbitrary vector.
References
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. Bibcode:2019GReGr..51...66I. doi:10.1007/s10714-019-2539-9. ISSN 0001-7701. S2CID 53554290.
Kata Kunci Pencarian:
- Nonmetricity tensor
- Tensor product
- Tensor
- Tensor product of modules
- Tensor field
- Riemann curvature tensor
- Cauchy stress tensor
- Tensor (intrinsic definition)
- Ricci curvature
- Ricci calculus
