Can Relativistic Effects explain Galactic Dynamics without Dark Matter?

TL;DR

This work investigates whether General Relativity can replace dark matter in galactic dynamics by combining an exact stationary-spacetime analysis with gravitomagnetic and gravitoelectric fields. Using the metric $ds^2 = -e^{2\Phi}(dt - A_i dx^i)^2 + h_{ij} dx^i dx^j$, it derives an exact geodesic equation $\tilde{D}\mathbf{U}/d\tau = \mathbf{G} + \mathbf{v} \times \mathbf{H}$ and shows that, to affect rotation curves, the gravitomagnetic field would have to be unrealistically large, which would produce lensing far exceeding observations. The paper also demonstrates that nonlinear terms $G^2$ and $H^2/2$ enter as negative energy sources, counteracting gravity and thus cannot enhance attraction to mimic dark matter; at 1PN order the effect is negligible. Together, these results reinforce the standard dark matter paradigm under the assumption of stationarity and clarify the limitations of GR in explaining galactic dynamics without unseen matter.

Abstract

We show that, contrary to some recent claims, relativistic effects cannot mimic dark matter in the galactic rotation curves and gravitational lensing.