Is Newton-Cartan theory really equivalent to Newton's theory of gravity? It is often said that Newton-Cartan theory is a reformulation or perhaps a generalization of Newton's theory of gravity, and it is said that (given certain conditions/assumptions) the two theories are equivalent. However, in Newton-Cartan theory we have two metrics, a spatial metric $g^{ab}$ of signature $(0, +, +, +)$ and a temporal metric $t_{ab}$ of signature $(+, 0, 0, 0)$.
My question is, does gravity in Newton-Cartan theory cause $g^{ab}$ and $t_{ab}$ to deviate away from what we'd expect them to be in a flat spacetime? If so, wouldn't that mean that Newton-Cartan theory is not empirically equivalent to Newtonian gravity? The reasoning is that if $g^{ab}$ is affected by gravity, then the spatial geometry would be changed, and if $t_{ab}$ is affected, then gravitational time dilation would be predicted. In either case, Newton-Cartan theory would make different predictions than those of Newtonian gravity.
 A: The Newton Cartan theory uses all the concepts of connections, geodesics, covariant derivatives, Riemann and Ricci tensor, without a metric.
While we use a spatial metric and a (separated) temporal metric in Newtonian theory, they don't play any role on the NC theory. The 4-D curved space-time is here a manifold without a metric.
The gravity equation $$\frac{d^2\mathbf {r}}{dt^2} = -\frac{GM\mathbf {r}}{r^3}\implies \frac{d^2\mathbf {r}}{dt^2} + \nabla {\phi} = 0 \implies \frac{d^2 X^j}{dt^2} + \frac{\partial \phi}{\partial X^j} = 0 $$ has the form of a geodesic equation.
A: 
Is Newton-Cartan theory really equivalent to Newton's theory of gravity?

Yes, given certain conditions/assumptions. We can choose constraints on Newton–Cartan (NC) geometry so that it would be precisely equivalent  to Newtonian theory of gravity, and that the suggested effects  would be absent. However, those constraints are external to NC theory, and it is possible (and even natural) to consider theories without those constraints or with another set of constraints in place. As a result we could obtain NC theories with different physical contents. Moreover such NC theories (empirically different from Newtonian) can arise naturally, for example as a small velocities limit of  general relativistic spacetimes or from holographic description of nonrelativistic quantum field theories.

… if $g^{ab}$ is affected by gravity, then the spatial geometry would be changed …

The usual assumption in Newtonian limit is that spatial Ricci curvature is zero: $R^{ab}:=g^{ac}g^{bd}R_{cd}=0$. This is a reasonable assumption for nonrelativistic matter since stress components of stress-energy tensor are small in this case. But in three dimensions  (when considering Riemannian purely spatial geometry) zero Ricci curvature means that space is (locally) flat. Of course, if we drop zero spatial Ricci curvature constraint then we can have curved spatial geometries in NC theories (see here for an example). Also, in higher dimensions even zero Ricci curvature would not imply local flatness since Weyl curvature is locally independent of the Ricci curvature.

… if $t_{ab}$ is affected, then gravitational time dilation would be predicted …

If there is (position-dependent) gravitational time dilation, then NC connection would have torsion: $\bar{Γ}^λ_{
[μν]} = −\hat{v}^λ∂_{[μ}τ_{ν]}\ne 0$, in notations of this paper. Thus by requiring symmetric connection in NC theory we eliminate time dilation effects. On the other hand, if we consider torsional connection, as the linked paper suggests, we can account for the classical tests of general relativity (perihelion precession,
deflection of light and gravitational redshift) within NC theory.
Though not mentioned in the question, Lense–Thirring-type effects are possible in (unconstrained) NC theory and are absent in Newtonian gravity. They could be described using the non-uniform Coriolis field (that contribute to NC connection). A constraint that would eliminate these effects could be called the law of existence of absolute rotation and has several equivalent formulations (in notations of this paper):
$$R ^a{}_{bcd} R ^b{}_a{}_{\bullet}^c{}_e = 0,$$
$$ t_{[a} R^c{}_{b]de} = 0,$$
$$ R ^{[ab]} _{~~~\bullet cd} = 0.$$
An example solution that does not satisfy this constraint (but is compatible with the others mentioned) is the NC limit of the NUT spacetime (see here, sec. 9).
A recent review of various NC theories:

*

*Hartong, J., Obers, N. A., & Oling, G. (2022). Review on Non-Relativistic Gravity. arXiv:2212.11309.

