Higher derivative gravity’s anti-Newtonian limit and the Caldirola–Kanai oscillator

Abstract

Fourth order gravity in dimensions is investigated in its anti-Newtonian limit where lightcones shrink to lines. The limiting theory is fourth order in time derivatives, still fully diffeomorphism invariant, and retains the original number of physical degrees of freedom. In an unimodular Hamiltonian formulation the dynamics can, however, be reduced to a second order one by regarding the unimodular metric as a composite field. In a gauge where the logarithm of the volume element is ‘time’ a Hamiltonian proper arises, which (for each spatial point) is a matrix generalization of a Caldirola–Kanai oscillator with inverted quartic potential. In an alternative ‘Wick-flipped’ Hamiltonian formulation an analogous reduced system arises with non-inverted quartic potential.