Relativistic Hydrodynamics T. Csörgő (KFKI RMKI Budapest) new solutions with ellipsoidal symmetry Fireball hydrodynamics: Simple models work well at SPS and RHIC Why? Theoretically challenging, difficult problem New classes of simple solutions Non-relativistic as well as relativistic Spherical, cylindrical and ellipsoidal, d=1,2,3 With: S. V. Akkelin, L. P. Csernai, P. Csizmadia, B. Lörstad, F. Grassi, Y. Hama, T. Kodama, Yu. Sinyukov, J. Zimányi, …

In Search of the QGP. Naïve expectations QGP has more degrees of freedom than pion gas, hence it has higher entrophy density Entropy should be conserved during fireball evolution Hence: Look in hadronic phase for signs of: Large spatial size, Long lifetime, Long duration of particle emission

Scaling laws in the observed data  Slope parameter (effective temperature) ~ mass  Effective volumes (horizont radii) scale ~ mass -0.5 SPS: the effective volumes and temperatures are independent of particle type, size, etc, depend only on their mass. Buda-Lund hydro model: gives a natural explanation to this unexpected observations in high energy physics. Similar results are observed also at RHIC -> see the next fits

BRAHMS: Effective temperature vs mass Hubble diagram of galaxies and of RHIC Edwin Hubble T eff (y=0) ~ T f + m 2 T eff (y) = T eff (y=0) / (1 + a y 2 ) Prediction, Buda-Lund hydro parametr. (Cs. T., B. Lorstad, ‘94-96, axial symmetry non-central collisions with Akkelin, Hama, Sinyukov, Csanad and Ster (2003)

Relativistic hydrodynamics A baseline for various theoretical aspects of RHIC physics / part of our folclore “Bjorken” scaling solution (discovered by R. C. Hwa :) A good example of false illusions Phase transitions and observables can be calculated relatively easily Provides an energy density estimate that is easy to measure (good up to a factor of 5 uncertainty). J.D. Bjorken, Phys. Rev. D27, 140 (1983), R. C. Hwa, Phys. Rev. D10, 2260 (1974) Guess, how many citations these papers have received?

Relativistic hydrodynamics Charge conservation 4-momentum conservation 4-velocity field normalization Perfect fluid EOS 1: energy density EOS 2: pressure Non-relativistic limit Ultrarelativistic gas

Energy and Euler equations Decomposing 4-momentum conservation energy equation rel. Euler equation general thermodynamic considerations E = TS + N - pV -> entrophy conservation Energy equation + EOS 1 and EOS 2: temperature equation 5 independent variables 5 equations: Euler (3) continuity + temperature

Self-similar ansatz Direction dependent Hubble profile coordinates physical quantities: assumption of self- similarity Trival volume dependence, additional coordinate dependence only through scaling variable s definition of s why is s a good scaling variable ?

New ellipsoidal solutions Scales depend linearly on time Hence the Hubble flow becomes spherically symmetric but the density profile contains an arbitrary scaling function (s) that limits the solution to ellipsoidal symmetry The pressure depends only on proper time and speed of sound The temperature is also ellipsoidally symmetric, has a scaling function related to the density

Interesting features Direction independent Hubble flow: same as in the successfull Cracow hydro model of Broniowski, Florkowski, Baran et al. Simplest case: the density profile and temperature profile are both constants In non-relativistic limit, goes back to non-rel hydro new families of solutions Not yet the „final word”: i) lack of acceleration ii) directional dependence of flow? iii) temperature and density cannot yet go to 0 at the surface continuosly (cut needed)

Even more generalizations... Natural generalizations for more realistic equations of state Bag model EOS Temperature dependent speed of sound (similar to non-rel case with Akkelin, Hama, Lukacs, Sinyukov, PRC 2003) Description for phase transitions, and sudden timelike deflagrations Accelerating and smooth surface solutions are related, work in progress.

Outlook Search for hydro solutions behind succesfull hydro parameterizations (Cracow, Buda-Lund) is underway Stepping stones are found Generalization is straightforward?