Four-volumes ($V_4=$ spatial-3-volume$\times$lifetime) are calculated within the model of three-fluid dynamics (3FD) and compared with those of the jet AA microscopic transport model (JAM). The calculations are performed for central $\mathrm{Au}+\mathrm{Au}$ collisions at energies of $\sqrt{s_{NN}}=3–19.6\mathrm{GeV}$. These $V_4$'s indicate optimal collision-energy ranges for realizing macroscopic high-baryon-density matter. It is found that the 3FD four-volumes noticeably exceed those in the JAM, which indicates a stronger baryon stopping in the 3FD model as compared to that in the JAM. It is argued that this difference in the baryon stopping correlates with stiffness of the equation of state implemented in these models. Contrary to the JAM, the four-volume, where a baryon density $\left(n_B\right)$ exceeds three times the normal nuclear density $\left(n_0\right)$, does not exhibit a maximum as a function of $\sqrt{s_{NN}}$. It decreases monotonically with increasing $\sqrt{s_{NN}}$, remaining at a fairly macroscopic level (i.e., $V_4\gtrsim 5.5^4{\mathrm{fm}}^4/c$). For higher baryon densities, $V_4$ exhibits maxima in its dependence on $\sqrt{s_{NN}}$. The optimal energy range for densities $n_B/n_0>4$ is located at $\sqrt{s_{NN}}=3.2–8$ GeV. Even for $n_B/n_0>6$, the four-volume remains quite macroscopic ($V_4\gtrsim 4^4{\mathrm{fm}}^4/c$) at $\sqrt{s_{NN}}=4.5–9$ GeV, contrary to the JAM.

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• Relativistic heavy-ion collisions
• Hydrodynamic models