  We highlight some connections between the final state hadronic observables
and the initial conditions using a recently found new exact family of solutions
of relativistic hydrodynamics. These relations provide explicit examples of the
scaling behaviour in relativistic hydrodynamics and may provide an advanced
estimate of the lifetime and the initial energy density in $\sqrt{s_{NN}} =
62.4$, $130$, and $200$ GeV Au+Au collisions at RHIC and $\sqrt{s_{NN}} = 5.0$
TeV Pb+Pb and $5.44$ TeV Xe+Xe as well as $\sqrt{s} = 7$, $8$ and $13$ TeV p+p
collisions at LHC energies. A surprising result is that these advanced
estimates yield a non-monotonic increase of the initial energy density with
increasing collision energy at the RHIC energy range.
