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 parallelly transport the metric. Physically, this corresponds to the failure of the metric to preserve angles and lengths under parallel transport.

Definition

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

Properties

It is manifestly symmetric in its latter two indices due to the symmetry of the metric, and carries $n^{2}(n+1)/2$ independent components on an $n$-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 $D^{\nabla }$ associated with the connection $\nabla$ as$$\mathbf {Q} =D^{\nabla }g$$We say a connection is metric compatible (or sometimes just "metric") if the nonmetricity tensor associated with that connection vanishes.

The Levi-Civita connection 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.