Linearity, non self-interacting spherically symmetric gravitational fields, the “sphereland equivalence principle” and Hamiltonian bubbles

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For a large class of spherically symmetric gravitational fields, when matter is dropped in a spherically symmetric way, it is possible to decompose this matter into free-falling shells such that their associated 2+1-dimensional (i.e. surface) energy-momentum tensor is conserved, that is, is not affected by the environment. Further ‘non-interacting’ features can be found for this class of gravitational fields as can be seen by the fact that Einstein's equations are linear in this case. Matter of a shell falling in one of these fields obeys energy momentum conservation from the point of view of the 2+1-dimensional world-sheet of the shell. This means that in the case of a test particle moving in one of these free falling sheets, the motion follows a 2+1-dimensional geodesic equation, or what is the same, its dynamics is governed by a ‘sphereland equivalence principle’. As a result of this, a shell falling in such a gravitational field can be treated as a closed system which can be described by a 2+1-dimensionally defined Hamiltonian.