Spacetime of Slowly Rotating Miyamoto–Nagai Source and Its Observability

We construct a systematic general-relativistic model of the spacetime generated by a slowly rotating, axially symmetric, self-gravitating clump of collisionless matter whose gravitational potential is of Miyamoto–Nagai type. The metric is taken in a three-function cylindrical gauge, ds2=eadt2−eb(dρ2+dz2)−ρ2ecdϕ2, with the frame-dragging function ω added at the stationary level. We prove a no-go lemma of explicitly delimited scope (no pointwise relation c=F(a,b) can annihilate the off-diagonal Ricci component) and reduce the field equations, for a diagonal source, to an algebraic system for bρ,bz valid wherever (∇W)2≠0, W=ρe(a+c)/2. The integrability of the resulting quadrature is not automatic: we prove that the compatibility condition ∂zbρ=∂ρbz is exactly equivalent to the meridional matter-conservation equations—one combination of which is an identity, the other being the equation that determines the remaining metric function—and that at second order in β=2GM/c2 it becomes the linear Poisson-type problem Lc2=4κPm. A barotropy criterion shows that a prescribed flattened potential with strictly isotropic pressure is already inconsistent at the Newtonian level, whereas the Miyamoto–Nagai potential admits a globally physical anisotropic solution: density, meridional pressure and stress anisotropy follow in closed form, κPm=β2ε2(A+ζ)2/(4ζ2S3) and κ(Pϕ−Pm)=Aβ2ε2ρ2/(2ζ3S3), all of definite sign, so that Pϕ>Pm everywhere off the axis and the anisotropy vanishes identically in the spherical limit. Rotation is included to Hartle order: the dragging obeys a linear equation at all orders in our gauge, the second-order diagonal metric is degenerate under the dispersion–rotation partition of the azimuthal support, and the degeneracy is broken only by the gravitomagnetic sector; the fully rotational realization gives vϕ/vc in closed form, which is near-circular for thin discs. On this basis we assess detectability: the flattening layer is routinely observable, with a deflection anisotropy decaying anomalously slowly (as A/b); galactic frame dragging (∼10−3μas yr−1) is astrometrically hopeless, but the parity-odd rotational asymmetry of time delays between opposite-side images of strongly lensed transients, Δt∼8GJ/c4b∼103 s, is detectable in principle given VLBI-grade (∼10 μas) astrometry, the threshold being M≳2×109M⊙ at disc-galaxy velocities; in the compact regime the model yields falsifiable energy-condition bounds (βDEC≈3.8ε, βWEC≈6.8ε) and dragging frequencies in the QPO band.