The Hamiltonian of f(R) gravity

Abstract

We derive conjugate momenta variable tensors and the Hamiltonian equation of the source-free f(R) gravity from first principles using the Legendre transformation of these conjugate momenta variable tensors, conjugate coordinates variables — fundamental metric tensor and its first ordinary partial derivatives with respect to space–time coordinates and second ordinary partial derivatives with respect to space–time coordinates — and the Lagrangian of the f(R) gravity. Interpreting the derived Hamiltonian as the energy of the f(R) gravity we have shown that it vanishes for linear Lagrangians in Ricci scalar curvature without source (e.g., Einstein–Hilbert Lagrangian without matter fields), which is the same result obtained using the stress–energy tensor equation derived from variation of the matter field Lagrangian density. The resulting Hamiltonian equation forbids any negative power law model in the dependence of the f(R) gravity on Ricci scalar curvature: f(R) = αR–r, where r and α are positive real numbers; it also forbids any polynomial equation that contains terms with negative powers of the Ricci scalar curvature including a constant term, in which cases the Hamiltonian function in the Ricci scalar and therefore the energy of the f(R) gravity would attain a negative value and would not be bounded from below. The restrictions imposed by the non-negative Hamiltonian have far-reaching consequences as a result of applying f(R) gravity to the study of black holes and the Friedmann–Lemaître–Robertson–Walker model in cosmology.