Nonmetricity tensor

In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It can be interpreted as the failure of a connection to parallely transport the metric. Physically, this corresponds to the failure of the metric to preseve angles and lengths under parallel transport.

Definition

Let ${\displaystyle M}$ be a manifold equipped with a metric ${\displaystyle g}$, and let ${\displaystyle \nabla }$ be an affine connection on the tangent bundle ${\displaystyle TM}$. The nonmetricity tensor is defined (some authors use the opposite sign convention) as$${\displaystyle Q(X,Y,Z):=(\nabla _{X}g)(Y,Z)}$$for ${\displaystyle X,Y,Z}$ arbitrary vector fields. In abstract index notation, this reads ${\displaystyle Q_{abc}=\nabla _{a}g_{bc}}$.

Properties

It is manifestly symmetric in its latter two indices due to the symmetry of the metric, and carries ${\displaystyle n^{2}(n+1)/2}$ independent components on an ${\displaystyle n}$-dimensional manifold.

One can additionally define the nonmetricity 1-forms either (and equivalently) by contracting the tensor with a basis 1-form on its first index, or by the exterior covariant derivative ${\displaystyle D^{\nabla }}$ associated with the connection ${\displaystyle \nabla }$ as ^[1] $${\displaystyle \mathbf {Q} =D^{\nabla }g}$$We say a connection is metric compatible (or sometimes just "metric") if the nonmetricity tensor associated with that connection vanishes.

The Levi-Civita conneciton is the unique metric compatible connection with vanishing torsion.

Use in Physics

The triple ${\textstyle (M,g,\nabla )}$ are the data for a metric affine spacetime ^[1] .

References

1. Hehl, Friedrich W.; Obukhov, Yuri N. (2003). Foundations of Classical Electrodynamics: Charge, Flux, and Metric. Boston, MA: Birkhäuser Boston. doi:10.1007/978-1-4612-0051-2. ISBN   978-1-4612-6590-0.