Areal-radius form and circular geodesics of a proposed MOND vacuum metric in general relativity

We examine the geometry and circular geodesics of the metric family [Formula: see text] used in proposed general-relativistic descriptions of MOND phenomenology. We use the invariant areal radius [Formula: see text] to compare the proposed geometries and assess their vacuum and geodesic properties. The rotational Killing fields establish spherical symmetry, and the vacuum equations yield the Schwarzschild family. An explicit coordinate transformation shows that a later proposed exact vacuum solution also belongs to this family. For the lapse [Formula: see text] with radial coefficient [Formula: see text], a quadratic curvature invariant demonstrates nonzero Ricci curvature even when the scalar curvature vanishes. The timelike circular geodesics of this metric are radially unstable, and their locally measured speed decreases as [Formula: see text] at large radius. We also clarify the relation between the conserved azimuthal charge and the tangential speed measured by a static observer. Finally, for a stationary, isolated weak-field exterior, we show that the logarithmic MOND potential requires a nonzero effective Poisson source. These conclusions concern the specified metrics, ordinary metric geodesics, and the stated asymptotic assumptions.