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 \(f^{ab}=(-g)^\alpha g^{ab}\) $f^{\mathit{ab}}={(-g)}^{\alpha }{g}^{\mathit{ab}}$ . 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 \(h^{ij}\) $h^{\mathit{ij}}$ is related with the new one by relation \(a^{ij}=(\det h_{ij})^\beta h^{ij}\) $a^{\mathit{ij}}={(det{h}_{\mathit{ij}})}^{\beta }{h}^{\mathit{ij}}$ and we argue that it corresponds to the gauge fixed version of the General Relativity formulated with the metric \(f^{ab}=(-g)^\alpha g^{ab}\) $f^{\mathit{ab}}={(-g)}^{\alpha }{g}^{\mathit{ab}}$ .