The gravitational sector of classical Lagrangian theories can generally be expressed in the form of a power series  L = \sqrt {-g} [-1 \over 2 \kappa ^-2 R+ \sum _n=2^\infty  (a_n Rⁿ+\tilde a_n \partial ² Rⁿ ) ], where \kappa ² is the gravitational coupling and R is the Ricci scalar. By means of a metric field-redefinition g_ij \rightarrow (1+\beta R)g_ij + \gamma R_ij+ \delta R_ikR_j^k + . . . , the quadratic terms R² can be removed completely (due to the Gauss–Bonnet identity) and the cubic and higher-order terms  Rⁿ partially, only those terms constructed solely from the Riemann tensor R_ijkl remaining invariant. It has been shown by Lawrence, however, that the implementation of this procedure at a specific order n inevitably gives rise to ghosts at the next and higher orders n'\ge n+1, in the sense that a term Rⁿ in L is replaced by terms R^n-m(\partial ²R)^m, for example. Classically, these ghosts may lead to instabilities, and it is therefore necessary to investigate the stability of the theory to linear perturbations, both before and after the metric has been transformed. In the cosmological Friedmann space-time ds²=dt²-a₀² e^{2\alpha (t)} dx² which describes the Universe, where t is comoving time and a₀ e^{\alpha (t)} is the radius function of the three-space dx², assumed flat, we find, by examining the characteristic equation, that the low-energy solution invariably possesses exponentially growing (and decaying) modes, after carrying out the field redefinition, irrespective of whether such modes were present initially. Therefore, it is not expedient to redefine the metric in this background, which, rather, should be considered as fixed. We discuss the relevance of this result for the heterotic superstring theory, particularly with regard to the vacuum solutions obtained previously from the effective Lagrangian including terms n \le 4, and to the terms R².

PACS numbers: 04.50.+h, 11.25.Db, 11.30.Pb, 98.80.Hw