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

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Abstract
For Einstein's gravitational equations with fourth‐order corrections being proportional to the square of an elementary lengthl, 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 αl2are 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 ≳lthe 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 ofm= 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].