Symmetry and conserved quantities in $$ f(R) $$-gravity: Mei vs. Noether approaches

Abstract We study the symmetries and conserved quantities in $$f(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity for the static, spherically symmetric Reissner–Nordström spacetime using two complementary frameworks: Noether symmetries and Mei symmetries. Starting from a canonical Lagrangian for radial metric functions and the curvature scalar $$R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , 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 $$f(R)=R^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> 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 $$f(R)=R^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , we find eight independent Mei generators and construct the corresponding conserved currents, some with no direct Noether analogue. 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 $$f(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> models and to rotating spacetimes.