AI 中文总结
本文研究静态球对称二次引力中纯二次$f(R)$模型的对称性，发现径向平移与组合缩放为精确Noether对称性，其荷给出守恒曲率通量，并将解空间分为三个扇区，其中标量平坦扇区允许类Reissner--Nordström度规。
AI 中文摘要
本文重新考虑了纯二次模型中静态球对称$f(R)$引力的对称性内容。我们首先在不提前消去径向度规变量所携带的方程的情况下导出径向作用量。随后，可以在保留其相关引力约束的同时施加Schwarzschild型规范。对于所得系统，径向平移和一种组合缩放是精确的离壳Noether对称性。缩放荷在约束面上约化为标量曲率的守恒径向通量。该通量也独立地由四维场方程的迹得出。我们分别检验了Noether不变性、强点Mei判据以及Euler--Lagrange系统的Lie不变性。在总次数至多为二的完整多项式点拟设内，Lie代数恰好包含三个独立生成元。该通量还将解空间划分为非零常曲率Einstein扇区、退化的标量平坦扇区和动力学曲率扇区。特别地，标量平坦扇区允许Reissner--Nordström形式的度规，尽管其平方反比系数在没有Maxwell场的情况下不具有电磁意义。
英文摘要
The symmetry content of static spherical $f(R)$ gravity is reconsidered for the pure quadratic model. We first derive the radial action without prematurely eliminating the equation carried by the radial metric variable. The Schwarzschild-type gauge may then be imposed while retaining its associated gravitational constraint. For the resulting system, radial translations and one combined scaling are exact off-shell Noether symmetries. The scaling charge reduces, on the constraint surface, to a conserved radial flux of the scalar curvature. The same flux follows independently from the trace of the four-dimensional field equations. Noether invariance, the strong point-Mei criterion, and Lie invariance of the Euler--Lagrange system are examined separately. Within the full polynomial point ansatz of total degree at most two, the Lie algebra contains precisely three independent generators. The flux also separates the solution space into a nonzero constant-curvature Einstein sector, a degenerate scalar-flat sector, and a dynamical-curvature sector. In particular, the scalar-flat sector permits a Reissner--Nordström-form metric, although its inverse-square coefficient has no electromagnetic meaning in the absence of a Maxwell field.
Comments15 pages
