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 conservedduring fireball evolution

Hence: Look in hadronic phasefor 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

Pb+Pb@CERN 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 explanationto this unexpected observations in high energy physics. Similar results are observed alsoat RHIC -> see the next fits

BRAHMS: Effective temperature vs mass

Hubble diagram of galaxies and of RHIC

Edwin Hubble

Teff(y=0) ~ Tf + m <u>2

Teff(y) = Teff(y=0) / (1 + a y2)

Prediction, Buda-Lund hydro parametr.(Cs. T., B. Lorstad, ‘94-96, axial symmetrynon-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 fieldnormalization

Perfect fluid

EOS 1: energy densityEOS 2: pressure

Non-relativistic limit

Ultrarelativistic gas

Energy and Euler equations

Decomposing 4-momentum conservation

energy equation

rel. Euler equation

general thermodynamicconsiderations E = TS + N - pV -> entrophy conservation

Energy equation + EOS 1 and EOS 2: temperature equation

5 independent variables5 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 solutionsScales 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 onlyon proper time and speed of sound

The temperature is also ellipsoidally symmetric, has a scaling function related to the density

Interesting features

Direction independentHubble flow: same as in the successfull Cracowhydro 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 hydronew families of solutions

Not yet the „final word”:i) lack of accelerationii) directional dependence of flow?iii) temperature and densitycannot 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?