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We have proposed previously a method for constructing self-conjugateHamiltonians H_eta in the eta-representation with a flat scalar product todescribe the dynamics of Dirac particles in arbitrary gravitational fields. Inthis paper, we prove that, for block-diagonal metrics, the Hamiltonians H_etacan be obtained, in particular, using "reduced" parts of Dirac Hamiltonians,i.e. expressions for Dirac Hamiltonians derived using tetrad vectors in theSchwinger gauge without or with a few summands with bispinor connectivities.Based on these results, we propose a modified method for constructingHamiltonians in the eta-representation with a significantly smaller amount ofrequired calculations. Using this method, here we for the first time findself-conjugate Hamiltonians for a number of metrics, including the Kerr metricin the Boyer-Lindquist coordinates, the Eddington-Finkelstein,Finkelstein-Lemaitre, Kruskal, Clifford torus metrics and for non-stationarymetrics of open and spatially flat Friedmann models.

A Modified Method for Deriving Self-Conjugate Dirac Hamiltonians in  Arbitrary Gravitational Fields and Its Application to Centrally and Axially  Symmetric Gravitational Fields