Non-Zero Coriolis Field in Ehlers’ Frame Theory

Ehlers’ Frame Theory is a class of geometric theories parameterized by 𝜆:=1/𝑐2 and identical to the General Theory of Relativity for 𝜆≠0. The limit 𝜆→0 does not recover Newtonian gravity, as one might expect, but yields the so-called Newton–Cartan theory of gravity, which is characterized by a second gravitational field 𝝎 called the Coriolis field. Such a field encodes at a non-relativistic level the dragging feature of general spacetimes, as we show explicitly for the case of the (𝜂,𝐻) geometries. Taking advantage of the Coriolis field, we apply Ehlers’ theory to an axially symmetric distribution of matter, mimicking, for example, a disc galaxy, and show how its dynamics might reproduce a flattish rotation curve. In the same setting, we further exploit the formal simplicity of Ehlers’ formalism in addressing non-stationary cases, which are remarkably difficult to treat with the General Theory of Relativity. We show that the time derivative of the Coriolis field gives rise to a tangential acceleration which allows for studying a possible formation in time of the rotation curve’s flattish feature.