Static anisotropic stars in Lovelock gravity: Universal closed-form equations

Abstract

A characteristic-polynomial formulation is already known for static perfect fluids in Lovelock gravity [Phys. Rev. D 113, 084008 (2026)]. We extend it to the complete anisotropic matter sector. For arbitrary dimension and independently coupled curvature orders, we derive closed, summation-free expressions for the density, radial pressure, tangential pressure and anisotropy in terms of a single polynomial \(W\), its derivatives and two derived polynomials \(V\) and \(U\). The dimension-dependent combinatorial coefficients are obtained directly from the generalized-delta antisymmetrisation and independently confirmed through an inductive argument. Imposing pressure isotropy recovers the known total-derivative equation, which we resolve into an explicit operator affine in Lovelock order and linear in dimension. This structure proves that, apart from the branch of vanishing angular sectional curvature, the interior Schwarzschild geometry is the unique isotropic interior common to arbitrary Lovelock couplings. It also identifies the fixed-sign obstruction to bounded pure Lovelock spheres in \(d=2N+1\) with the polynomial identity \(V\equiv0\), and shows how a lower-order term or cosmological term removes this obstruction. Finally, we integrate the resulting arbitrary-order stellar equations with all admissible orders active in \(d=10\) and \(d=11\). For the positive single-parameter hierarchy of Lovelock couplings considered, the general-relativistic sequence topology persists while higher orders increase the maximum mass cumulatively.