Birkhoff’s theorem in f(R) theory of gravity

Birkhoff’s theorem in general relativity states that every spherically symmetric solution of Einstein field equations in vacuum is either static or Schwarzschild. This theorem has been established in the scalar-tensor theories of gravity by Reddy (J. Phys. A 6, 1867 (1973)). In this paper, we prove this theorem in f(R) theory of gravity (where R Ricci scalar) when the scalaron Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi$\end{document}(R) of the theory is time-independent only. However, no attention is given to the order of the f(R) theory. Here it is important to note that the Birkhoff theorem in f(R) gravity states only that if R is time-independent, the spherical solution is static (not Schwarzschild).