Spacetime of a 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, \(ds^2=e^{a}\,dt^2-e^{b}(d\rho^2+dz^2)-\rho^2e^{c}d\phi^2\), with the frame-dragging function \(\omega\) 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 \(b_\rho,b_z\) valid wherever (∇W)2≠0, \(W=\rho e^{(a+c)/2}\). The integrability of the resulting quadrature is not automatic: we prove that the compatibility condition \(\partial_zb_\rho=\partial_\rho b_z\) 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 \(\beta=2GM/c^2\) it becomes the linear Poisson-type problem \(\mathcal{L}c_2=4\kappa P_m\). A barotropy criterion shows that a prescribed flattened potential with strictly isotropic pressure is inconsistent already 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, \(\kappa P_m=\beta^2\varepsilon^2(A+\zeta)^2/(4\zeta^2S^3)\) and \(\kappa(P_\phi-P_m)=A\beta^2\varepsilon^2\rho^2/(2\zeta^3S^3)\), all of definite sign, so that \(P_\phi>P_m\) 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 \(v_\phi/v_c\) in closed form, 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 (\(\sim10^{-3}\,\mu\)as yr−1) is astrometrically hopeless, but the parity-odd rotational asymmetry of time delays between opposite-side images of strongly lensed transients, \(\Delta t\sim 8GJ/c^4b\sim10^3\)~s, is detectable in principle given VLBI-grade (\(\sim10\,\mu\)as) astrometry, the threshold being \(M\gtrsim2\times10^{9}M_\odot\) at disc-galaxy velocities; in the compact regime the model yields falsifiable energy-condition bounds (\(\beta_{\rm DEC}\approx3.8\,\varepsilon\), \(\beta_{\rm WEC}\approx6.8\,\varepsilon\)) and dragging frequencies in the QPO band.