I think my question is naive but I would like to ask why why volume preservation is important for MCMC and specifically Metropolis update.I'm reading the following paper https://arxiv.org/pdf/1206.1901.pdf where it says

The significance of volume preservation for MCMC is that we needn’t account for any change in volume in the acceptance probability for Metropolis updates. If we proposed new states using some arbitrary, non-Hamiltonian, dynamics, we would need to compute the determinant of the Jacobian matrix for the mapping the dynamics defines, which might well be infeasible.

where the Metropolis update for HMC is defined by

$\alpha=min\left \{1,exp\big(-H(q^{*},p^{*})+H(q,p)\big) \right \}$

where $q$ denotes the position and $p$ denotes the momenta and $(q^{*},p^{*})$ is the proposed state.

What would be the problem for $\alpha$ if we had different volume for the proposed $(q^{*},p^{*})$.