# Contravariant metric in Newton-Cartan spacetime

I'm interested in the geometrized Newtonian gravitation or Newton-Cartan theory. In every reference that I have found begins saying that a Newton-Cartan spacetime is a manifold $$M$$ with some structures. Among then, is always pointed a contravariant metric $$g^{ab}$$ that represents the spatial distances.

My question is: why is contravariant? Should it not be a covariant metric to measure the length of vectors? I understand that a contravariant metric measures lengths and angles of covectors or 1-forms.

The metric structure in Newton-Cartan geometry is given by two elements (in d+1 spacetime dimensions):

• A contravariant metric $$h^{\mu\nu}$$ of rank d
• A one-form $$\psi_\mu$$ spanning the radical of $$h$$, namely $$h^{\mu\nu}\psi_{\nu}=0$$.

The 1-form $$\psi$$ allows to distinguish between timelike ($$\psi_\mu X^\mu\neq0$$) and spacelike ($$\psi_\mu X^\mu=0$$) vector fields (there are no light-like vectors).

Consistently with usual Newtonian theory, the notion of distance should only makes sense to measure spatial distances (as opposed to space-time distances as in general relativity).

Now, one can show that the contravariant metric $$h$$ provides exactly what is needed as in can be shown that the above definition of $$h$$ is in fact equivalent to defining a d-dimensional Riemannian metric $$\gamma$$ acting on the kernel of $$\psi$$, namely $$\gamma$$ acts on spacelike vector fields and thus provides a notion of spatial distance.

The situation is even clearer when the distribution of spacelike vector fields is involutive (i.e. if $$[X,Y]$$ is spacelike for all spacelike vector fields X and Y or equivalently if $$\psi$$ satisfies the Frobenius integrability condition $$d\psi\wedge\psi=0$$). In this case, the $$d+1$$-dimensional spacetime is foliated by $$d$$-dimensional hypersurfaces (absolute spaces) corresponding to leaves of equal time, each of which is endowed with a $$d$$-dimensional Riemannian metric $$\gamma$$ allowing to measure spatial distances within this instantaneous $$d$$-dimensional space.

The signature is (+++) and the metric has rank 3. See e.g.

https://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_R_Andringa.pdf

in which this is motivated by calculating which metrics are kept invariant under the Galilei group.

Both Newton and Einsteins theory of gravity are already geometric, in differing ways. In GR, we typically show that there is a unique connection, called the Levi-Civita connection, that is compatible with the metric and has vanishing torsion. In Einstein-Cartan theory, they allow torsionful connections.

My question is: why is contravariant? Should it not be a covariant metric to measure the lenght of vectors? I understand that a contravariant metric measures lenghts and angles of covectors or 1-forms...

The metric is used to measure length and angles; it can actually be written in covariant form ($$g_{\alpha\beta}$$), in contravariant form ($$g^{\alpha\beta}$$) or mixed ($$g_{\alpha}^{\beta}$$ or $$g^{\alpha}_{\beta}$$ - here, the 'beta' index should be in the 'second' position, but I don't know enough about latex to get that to work).