Nonmetricity tensor

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

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 $D^{\mathrm{\nabla }}$ associated with the connection $\mathrm{\nabla }$ as$$\mathbf{𝐐}=D^{\mathrm{\nabla }}g$$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 $(M,g,\mathrm{\nabla })$ are the data for a metric affine spacetime.