Conformal flatness selects a universal isothermal attractor in pure Lovelock gravity

Abstract

We determine the complete solution set of the conformally flat isotropic pure Lovelock field equations for every order $N\ge 2$ and every admissible dimension $d\ge 2N+1$. Conformal flatness and pressure isotropy combine into a single identity that factorises exactly, splitting the solutions into two branches. The first is the constant-density Schwarzschild interior, which persists at every order with both metric potentials keeping their Einstein form. The second has no Einstein counterpart and is governed by the single dimension--order parameter $k=(d-2N)/[4(1-N)]$; we obtain its physically admissible orbit in explicit closed parametric form, both potentials included. A phase-space analysis on the projective line shows that no solution of this branch has a pressure-free boundary at finite radius, so only the Schwarzschild branch can describe a bounded star. The second instead loses all memory of its central data and relaxes onto a pure Lovelock isothermal sphere with $\rho \propto r^{-2N}$, a higher-curvature analogue of the singular isothermal halo, which we show to be the attractor of the whole family, approached at the closed-form rate ${\lambda }_{\ast }=-(d-2)/(d-2N)$ and with limiting equation of state fixed by $(d,N)$ alone. We prove that $p/\rho$ increases monotonically outward there, and that of the two regular-centre orientations only one is admissible. Since $k$ is rational the spatial potential is algebraic of degree $u+v$, where $2k=-u/v$; radical inversion is guaranteed for degree at most four, while four representative cases have full symmetric Galois group.

43 pages, 7 figures, 2 tables