Scalar field equation in Lemaitre-Tolman-Bondi cosmological models

The scalar field equation is studied in the context of the Lemaitre-Tolman- Bondi cosmological models. A class of models is determined where the equati on can be separated. The separated time equation can be integrated in the m assless case while the separated radial equation still depends on an arbitr ary function relative to the cosmological background. Some examples where t he separated radial equation is explicitly integrated are given. In view of a quantization of the theory the problem of the normal modes of the scalar field is discussed. The normal modes are explicitly determined or their ex istence is proved by the property of the Wronskian of the time equation and from general results of one-dimensional differential operators.