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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.

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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.

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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.

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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).

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