jats:titleAbstract</jats:title> jats:pWe investigate recent claims that gravitomagnetic effects in linearised general relativity can explain flat and rising rotation curves, such as those observed in galaxies, without the need for dark matter. If one models a galaxy as an axisymmetric, stationary, rotating, non-relativistic and pressureless ‘dust’ of stars in the gravitoelectromagnetic (GEM) formalism, we show that gravitomagnetic effects on the circular velocity jats:italicv</jats:italic> of a star are jats:inline-formula jats:tex-math</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:miO</mml:mi> <mml:mo stretchy="false">(</mml:mo> mml:msup mml:mn10</mml:mn> mml:mrow mml:mo−</mml:mo> mml:mn6</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cqgacef8bieqn1.gif" xlink:type="simple" /> </jats:inline-formula> smaller than the standard Newtonian (gravitoelectric) effects and thus any modification of galaxy rotation curves must be negligible, as might be expected. Moreover, we find that gravitomagnetic effects are jats:inline-formula jats:tex-math</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:miO</mml:mi> <mml:mo stretchy="false">(</mml:mo> mml:msup mml:mn10</mml:mn> mml:mrow mml:mo−</mml:mo> mml:mn6</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cqgacef8bieqn2.gif" xlink:type="simple" /> </jats:inline-formula> too small to provide the vertical support necessary to maintain the dynamical equilibrium assumed in such a model. These issues are obscured if one constructs a single equation for jats:italicv</jats:italic>, as considered previously. We nevertheless solve this equation for a galaxy having a Miyamoto–Nagai density profile since this allows for both an exact numerical integration and an accurate analytic approximation. We show that for the values of the mass, jats:italicM</jats:italic>, and semi-major and semi-minor axes, jats:italica</jats:italic> and jats:italicb</jats:italic>, typical for a dwarf galaxy, the rotation curve depends only very weakly on jats:italicM</jats:italic>, and becomes independent of it for larger jats:italicM</jats:italic> values. Moreover, for aspect ratios jats:inline-formula jats:tex-math</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:mia</mml:mi> mml:mrow mml:mo/</mml:mo> </mml:mrow> mml:mib</mml:mi> mml:mo></mml:mo> mml:mn2</mml:mn> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cqgacef8bieqn3.gif" xlink:type="simple" /> </jats:inline-formula>, the rotation curves are concave over their entire range, which does not match observations in any galaxy. Most importantly, we show that for the poloidal gravitomagnetic flux jats:italicψ</jats:italic> to provide the necessary vertical support, it must become singular at the origin and have extremely large values near to it. This originates from the unwitting, but forbidden, inclusion of free-space solutions of the Poisson-like equation that determines jats:italicψ</jats:italic> and also clearly contradicts the linearised treatment implicit in the GEM formalism, hence ruling out the methodology in the form used as a means of explaining flat galaxy rotation curves. We further show that recent deliberate attempts to leverage such free-space solutions against the rotation curve problem yield no deterministic modification outside the thin disk approximation, and that, in any case, the homogeneous contributions to jats:italicψ</jats:italic> are ruled out by the boundary value problem posed by any physical axisymmetric galaxy.</jats:p>