Canonial analysis of general relativity formulated with the new metric $$f^{ab}=(-g)^{\alpha }g^{ab}$$

In this short note we investigate canonical formalism for General Relativity which is formulated with the metric \(f^{ab}=(-g)^\alpha g^{ab}\). We find corresponding Hamiltonian and we show that constraint structure is the same as in the standard formulation. We also analyze another model when the spatial part of metric \(h^{ij}\) is related with the new one by relation \(a^{ij}=(\det h_{ij})^\beta h^{ij}\) and we argue that it corresponds to the gauge fixed version of the General Relativity formulated with the metric \(f^{ab}=(-g)^\alpha g^{ab}\).