High-frequency Waves in Gravitational Theories with Fourth-order Derivative Equations

For Einstein's gravitational equations with fourth-order corrections being proportional to the square of an elementary length l, we discuss the behaviour of high-frequency waves. It is shown that (1) only waves with lengths λ ? can generate a macroscopic avarage background (for λ < l, only the terms αl² are decisive such that one has the same situation as in a pure fourth-order theory without Einstein term which cannot be interpreted as gravitational theory), (2) for λ ? l the background metric is purely determined via the second-order derivative Einstein tensor (formally one obtains the same equations for the background as in the non-modified Einsteinian theory), and (3) only waves corresponding to the massless and the massive spin-two gravitons contribute to background curvature; in the geometrical-optics approximation, these both particle sorts are moving independent of each other and satisfy a conservation law for the total number of m = 0 and massive spin-two gravitons, respectively. The results obtained in this paper corroborate partly the conclusions drawn in the weak-field approximation [11, 15, 18].