Symmetry and Conserved Quantities in $f(R)$-Gravity: Mei vs. Noether Approaches

Abstract

We study the symmetries and conserved quantities in $f(R)$ gravity for the static, spherically symmetric Reissner--Nordstr\"om spacetime using two complementary frameworks: Noether symmetries and Mei symmetries. Starting from a canonical Lagrangian for radial metric functions and the curvature scalar $R$, we derive the associated Hamiltonian and show that the Legendre map is regular whenever both the first derivative of $f(R)$ with respect to $R$ and the second derivative with respect to $R$ is non-zero. Within Noether's approach (variational and Lie-derivative forms), we obtain general, canonical, and internal symmetry classes and identify explicit generators; for the quadratic model $f(R)=R^{2}$ these include radial translations and scaling symmetries. We then formulate Mei symmetry conditions as invariance of the Euler--Lagrange equations under the first prolongation, which yields an overdetermined partial differential equation (PDE) system for the generator components. Solving this system for $f(R)=R^{2}$, we find eight independent Mei generators and construct the corresponding conserved currents, some without a direct Noether analog. The analysis demonstrates that Mei symmetries extend the standard Noether framework for higher-order Lagrangians and provide additional conserved quantities relevant to black-hole dynamics in modified gravity. We conclude with a comparison of the two symmetry schemes and outline applications to broader $f(R)$ models and to rotating spacetimes.