AI 中文总结
该研究确定纯Lovelock引力中所有阶数及容许维数下共形平坦各向同性场方程的完整解，发现共形平坦性选择的通用等温吸引子为纯Lovelock等温球，仅史瓦西分支可描述有界恒星。
AI 中文摘要
我们确定了共形平坦各向同性纯Lovelock场方程在所有阶数N≥2及所有容许维数d≥2N+1下的完整解集。共形平坦性与压力各向同性结合为一个恒等式，该恒等式可精确因式分解，将解分为两个分支。第一个分支是常数密度的史瓦西内部解，它在所有阶数下均存在，且两个度规势均保持爱因斯坦形式。第二个分支无爱因斯坦对应解，由单一的维数-阶参数k=(d-2N)/[4(1-N)]支配；我们以显式闭合参数形式得到了其物理上可容许的轨道，包含两个势。对射影线的相空间分析表明，该分支无有限半径处的无压力边界，因此仅史瓦西分支可描述有界恒星。第二个分支则丧失了所有中心数据的记忆，弛豫为纯Lovelock等温球，满足ρ∝r^{-2N}，这是奇异等温晕的高阶曲率类似物，我们证明它是整个族的吸引子，以闭合形式速率λ_*=-(d-2)/(d-2N)趋近，且其极限状态方程仅由(d,N)固定。我们证明此处p/ρ随径向向外单调递增，且两个规则中心取向中仅一个可容许。由于k为有理数，空间势为u+v次代数，其中2k=-u/v；次数至多为四时可保证根式可逆，而四个代表性案例具有完全对称的伽罗瓦群。
英文摘要
We determine the complete solution set of the conformally flat isotropic pure Lovelock field equations for every order $N\ge2$ and every admissible dimension $d\ge2N+1$. 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 $ρ\propto r^{-2N}$, 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 $λ_*=-(d-2)/(d-2N)$ and with limiting equation of state fixed by $(d,N)$ alone. We prove that $p/ρ$ 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.
Comments43 pages, 7 figures, 2 tables
