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Degenerate Extensions of General Relativity

General relativity has previously been extended to incorporate degenerate metrics using Ashtekar's hamiltonian formulation of the theory. In this letter, we show that a natural alternative choice for the form of the hamiltonian constraints leads to a theory which agrees with GR for non-degenerate metrics, but differs in the degenerate sector from Ashtekar's original degenerate extension. The Poisson bracket algebra of the alternative constraints closes in the non-degenerate sector, with structure functions that involve the {\it inverse} of the spatial triad. Thus, the algebra does {\it not} close in the degenerate sector. We find that it must be supplemented by an infinite number ofsecondary constraints, which are shown to be first class (although their explicit form is not worked out in detail). All of the constraints taken together are implied by, but do not imply, Ashtekar's original form of constraints. Thus, the alternative constraints give rise to a different degenerate extension of GR. In the corresponding quantum theory, the single loop and intersecting loop holonomy states found in the connection representation satisfy {\it all} of the constraints. These states are therefore exact (formal) solutions to this alternative degenerate extension of quantum gravity, even though they are {\it not} solutions to the usual vector constraint.