The Freeze-out Problem in Relativistic Hydrodynamics

Transcription

1 The Freeze-out Problem in Relativistic Hydrodynamics Ky!ll A. Bugaev Bogolyubov ITP, Kiev, Ukraine April

2 Outline What is Relativistic Hydrodynamics? Landau & Bjorken initial conditions Two major aspects of Freeze-out Problem Cut-off formula for particle spectra Hydrodynamics with particle emission A few examples Kinetics of finite domains and Hydrocascade equations Conclusions 2

3 The Beginning: 50-th To describe the reactions of multihadron productions Landau and Fermi suggested to introduce hadronic media and describe it by hydrodynamic equations ( ): µ T µν µν f (x, t) = 0, T f (x, t) = (ɛ f + p f ) u µ f uν f p f g µν, µ N µ f (x, t) = 0, N f ν (x, t) = n f u ν f, For n-conserved charges 4 +n conservation laws: n charges + energy-momentum tensor charge 4-vector 1 energy, 3 momentum. But n variables! 10 components of energymomentum tensor, and 4n components of charge 4-currents. Inevitably one needs some approximations! The simplest one is to use IDEAL FLUID assumption (above). Then 5 + n unknowns. Equation of state (EOS) completes the system. Another assumption: of small deviations from equilibrium leads to Dissipative Fluid Dynamics. (I will not consider) 3

4 In Addition One Needs: The INITIAL CONDITIONS (to solve the Hydro equations), since which the Hydro approach is VALID The BOUNDARY CONDITIONS (to stop solving the Hydro equations where they are NOT VALID!) = Freeze-out criterion Interval square is ds 2 = dt 2! dx 2 Time!like hypersurface ds 2 > 0 Time t Hydro is NOT valid here! Space!like hypersurface ds 2 < 0 Hydro is valid here Initial conditions Boundary conditions at!(x,t) Transverse coordinate X Two criteria of Hydro APPLICABILITY: Mean free path of particles << Specific Size of system: There exists the local thermodynamic equilibrium: λ = 1 σ n 1 cross section part.density typical size Γ = v λ Local reaction rate µu µ mean velocity mean free path expansion rate 4

5 Example: Landau Initial Conditions Historically this was the first example: was suggested for HADRON collisions! Landau & Belenkii, Uspekhi Fiz. Nauk 56 (1955) It requires a FULL STOP of particles and created matter! Hence it was modified for Nuclear Collisions! Comparison with experiments shows: this picture is unrealistic - there is no full stop; also at central region of collision (central rapidity) the baryonic density is zero. Then it works at low colliding energies <1GeVA Here Riemann means the simple wave (or sound wave); afterwards there exists a NON-TRIVIAL solution found analytically by Khalatnikov (1955) v~c 1-dimensional expansion into vacuum 0 t=0!=! 0 v=0 v~c -L+c t 0<t<L/c s L-c t v~c z 0 z -L 0 L z -L-ct 0 L+ct z Riemann -L t Landau L/c s Riemann L s Collision axis z s 5

6 2-nd Example: Bjorken Initial Conditions For v 1 there is NO SCALE! With no full stopping the collective velocity of matter should be of the scaling form v = z/t: suggested and analyzed first by F. Cooper, G. Frye and E. Schonberg, PRD 11 (1975) for HADRONIC Collisions! v~c!=0!=! 0 v=z/t v~c Bjorken, PRD 27 (1983), applied it first to NUCLEAR collisions and gave the physical interpretation! Scaling v = z/t means: ALL particles of velocity v are originated at the collision region. z 0 t z "=" 0 0!! 0 z In absence of scale the Physics at different longitudinal coordinates z MUST be the same, if compared at the same PROPER TIME! pre-equilibrium fluid dynamics 0 Equilibration z 6

7 Bjorken Solution In absence of conserved charges the Hydro equations are t T 00 + z T z0 = 0, t T 0z + z T zz = 0 It is convenient to use not t and z, but proper time τ = t 2 z 2 and ε f τ + ε f + p f η τ = 0, p f η Consequence (i): since d p f s f dt rapidity η = Artanh v = Artanh[z/t]: = 0 τ T = T (τ) only! Consequence (ii): entropy conserves in the form s f τ η + s f τ = 0 s f τ = s 0 τ 0 = const The constant may, in principle, differ for different η, but since the initial state along τ 0 is the same, then it will be the same for all η at τ > τ 0. Thus, entropy density falls as s f 1/τ independently of EOS! 7

8 More on Bjorken Solution It has boost invariance: solution at η=0 gives solution for any η! Is suited for ultrarelativistic nuclear collisions about central rapidity. Therefore, it forms the basis of present Hydro simulations. However, the TRANSVERSAL expansion has to be accounted for! Also, along the constant proper time hypersurface the Bjorken system is INFINITE, but real physical systems are finite! Thus, it has to be TERMINATED somewhere! 8

9 The Freeze-out Problem Here points are causally connected!!! Two Problems: µ p d < 0 µ " These particles return to fluid Free particles No collisions, outgoing particles Time t No EoS Fluid with EoS Hydro is valid!(x,t) Hydro is valid in some domain. There is no Hydro & EoS outside this domain. How to stop solving Hydro Eqs. that they do NOT affect its own solution on time-like parts of FO hypersurface? How to convert fluid into free streaming particles? How to connect domains with EoS and without EoS? p µ d " µ > 0 Transverse coordinate X d "! is external normal 4-vector! Detector 9

10 A bit of History of the FO Problem For Landau (1953) it was EVIDENT: at T = Mpion = 140 MeV the Hydro breaks down and free particles go to detector. Milekhin (1956) wrote the WRONG momentum distribution of secondary particles assuming their distribution is Boltzmann in a moving frame! F. Cooper & G. Frye, PRD 10 (1974), derived the momentum distribution formula for secondary particles which CONSERVES energy! They correctly assumed that distribution is Boltzmann in the rest frame only. dn CF p 0 d 3 = φ (x, t, p) p ν dσ ν p M. Gorenstein & Yu. Sinyukov, ct description Phys. Lett. for the B 142 space-lik (1985), rederived CF formula and showed the presence of NEGATIVE Particle Numbers on time-like parts of FO hypersurface: since there p µ dσ µ < 0 In 1989 Sinyukov, Z. Phys. C 43, derived the momentum distribution formula, which has NO negative particle numbers, but it DOES NOT conserve the angular momentum! 10

11 The FO Problem is Nontrivial This is so because its solution required further assumptions and MODIFICATIONS of the Hydro equations. One can see that it is not sufficient to find the correct momentum distribution on the time-like parts of the FO hypersurface. It is necessary to insert this distribution into Hydro in such a way that it DOES NOT generate a causal paradox due to RECOIL after emission. FURTHER ASSUMPTIONS: the FO happens at some sharp boundary, where the Hydro is not valid, but the distributions are thermal. Kinetic estimates by L. Bravina et al, PRC 60 (1999), show that the WIDTH of FO boundary is between 0.5 to 1.5 fm. Both kinetic simulations and experiments show that DEVIATIONS from thermal equilibrium are about a few percent. This ASSUMPTIONS are reasonable to formulate the FO model. 11

12 t Cooper!Frey formula Fluid Many collisions The Cut-off Distribution Function ( ) On the Boundary with Vacuum x*(t) Transition Region Particles move in both directions! d << L Cut!off formula Free particles No collisions! Vacuum x K.A.B. Nucl. Phys. A606 (1996) }{{} Outgoing! Main Assumptions on the FO hypersurface : There are collisions in the transition region and φ (x, t, p) φ eq (x, t, p), but T g T f, µ g µ f! Small width of the transition region: d L ; No collisions in the gas of free particles! No incoming particles may appear due to rescattering! The number of outgoing particles dn CO with velocity v = p x /p 0 will cross the hypersurface x (t), iff v = p x p0 p 0 dn CO d 3 p dx (t) dt = φ ( x, t, p) [ p x dx (t) p 0 dt ] p 0 dt ds φ ( x, t, p) Θ(p ν dσ ν ) p µ dσ µ, }{{} Outgoing! The Cooper-Frye formula counts ALL (outgoing and incoming) particles! p 0 dn CF d 3 p = φ ( x, t, p) [ Θ(p ν dσ ν ) + negative numbers {}}{ Θ( p ν dσ ν ) } {{ } 1 ] Moving boundary p µ dσ µ, The CO distributions is the CF one, but without returning particles! 12

13 t Cooper!Frey formula Fluid Many collisions The Cut-off Distribution Function On the Boundary with Vacuum x*(t) Transition Region Particles move in both directions! d << L Cut!off formula Free particles No collisions! Vacuum Particles with such velocities come from nowhere! x ( ) }{{} Outgoing! K.A.B. Nucl. Phys. A606 (1996) Main Assumptions on the FO hypersurface : There are collisions in the transition region and φ (x, t, p) φ eq (x, t, p), but T g T f, µ g µ f! Small width of the transition region: d L ; No collisions in the gas of free particles! No incoming particles may appear due to rescattering! The number of outgoing particles dn CO with velocity v = p x /p 0 will cross the hypersurface x (t), iff v = p x p0 p 0 dn CO d 3 p dx (t) dt = φ ( x, t, p) [ p x dx (t) p 0 dt ] p 0 dt ds φ ( x, t, p) Θ(p ν dσ ν ) p µ dσ µ, }{{} Outgoing! The Cooper-Frye formula counts ALL (outgoing and incoming) particles! p 0 dn CF d 3 p = φ ( x, t, p) [ Θ(p ν dσ ν ) + negative numbers {}}{ Θ( p ν dσ ν ) } {{ } 1 ] Moving boundary p µ dσ µ, The CO distributions is the CF one, but without returning particles! 12

14 The } Sun-like {{ Object } Consider a Sun-like, i.e., spherical object with constant radius R, dr dt = 0. Assume: Photon distribution function f γ ( p 0 T ) is isotropic in momentum space and is spherically symmetric in the coordinate space. Gravitation can be ignored. T = T ( R ) is the temperature at the radius R. From the standard expression for the normal 4-vector dσ µ = ( 0, R/ R ) ds dt one finds a zero number of emitted photons by the CF-formula dn CF γ dt ds = d 3 p f γ ( p0 T ) p R p 0 R π 0 dθ p sin(θ p ) cos(θ p ) = +1 1 dx x = 0. according to the Cooper-Frye formula all stars are invisible! The CO-formula is correct because it counts the outgoing particles only! 13

15 Energy-Momentum of the Free Gas Now we can find the energy-momentum tensor of the gas of free particles: T µν g = d 3 p p 0 p µ p ν φ g ( p) Θ (p µ dσ µ ) and N µ g = d 3 p p 0 p µ φ g ( p) Θ (p µ dσ µ ) Consider an ideal gas of the massless particles The charge current for the left (right) hemisphere is L (R) in a special frame: N ν g,l(v σ, T, µ) = n g (T, µ) N ν g,r (v σ, T, µ) = n g (T, µ) ( 1 + vσ 2 ( 1 vσ 2 ; v2 σ 1 ) ; 0; 0 4 ; 1 v2 σ 4 ; 0; 0 ) N ν g,l(+1, T, µ) = n g (T, µ) ( 1; 0; 0; 0) N ν g,r ( 1, T, µ) = n g (T, µ) ( 1; 0; 0; 0) with v σ = dσ0 dσ 1. Energy-momentum tensor for the right hemishere T µν g,r (T, µ, v σ) = ɛ g (T, µ) 1 v σ 2 1 vσ v 2 σ v 3 σ (1 v 0 0 σ ) 2 (2+v σ ) (1 v σ ) 2 (2+v σ ) 12 B#,858;%'4-7<%+'#.'*/+580"%,'.#+ *'&'''''''?'>! "! 187%'5' A*/0%!"89%''()*%+,-+./0% 187%!"89% ()*%+,-+./0% 678,,8#4'.+#7 ()*%+,-+./0%'12'3#4,5! *'&'''''''='>! " here n g (T, µ) and ɛ g (T, µ) are the usual charge and energy densities, respectively.! DEF5G!""#$%&' 1+/4,;%+,%'0##+&84/5%'@ 1(%+%'/+%'4#',-0('7#7%45/'5(/5 :8;%'4%:/58;%'4-7<%+'#.'*/+580"%, This is the simplest case! The general expressions are more involved! 5C 14

16 Why the Cooper-Frye Formula was used? 1. It is correct for the space-like FO hypersurfaces. 2. CF formula (kinetic theory) is consistent with conservation laws of the traditional hydrodynamics! Traditional Hydrodynamics = Equations of motion of the perfect fluid: µ T µν f ( x, t) = 0 and µn µ c.f ( x, t) = 0 + Given Additional Conditions: i) initial conditions on the hypersurface Σ in ( x, t); ii) the equation of state of the fluid; iii) the freeze out criterion, which defines the equation of the FO hypersurface Σ fr ( x, t). Energy-momentum via the distribution function: (kinetic theory) P ν d 3 p p ν d3 N dp 3 Identification: Emitted particles Fluid defines the emitted energy-momentum of the gas of free particles by CF formula: P ν g.fr = d 3 p p 0 pν Σ fr dσ µ p µ φ f ( pρ u ρ f T ) = Σ fr dσ µ T µν f (x, t) due to {}}{ }{{} conservation Σ in dσ µ T µν f (x, t). Thus, energy(momentum) of gas = energy(momentum) of initial fluid! 15

17 Why the Cooper-Frye Formula was used? 1. It is correct for the space-like FO hypersurfaces. 2. CF formula (kinetic theory) is consistent with conservation laws of the traditional hydrodynamics! Traditional Hydrodynamics Traditional = Hydro + CF formula is wrong: since no difference between fluid Equations of motion of the perfect fluid: µ T µν f ( x, and t) emitted = 0 and gas; no effect of the emission on the fluid evolution µn µ c.f ( x, t) = 0 + on the time-like FOHS. Given Additional Conditions: Traditional Hydro + cut-off formula is wrong. i) initial conditions on Modified the hypersurface Hydro + Σ in cut-off ( x, t); ii) formula the equation is correct. of state of the fluid; iii) the freeze out criterion, which defines the equation of the FO hypersurface Σ fr ( x, t). Energy-momentum via the distribution function: (kinetic theory) P ν d 3 p p ν d3 N dp 3 Identification: Emitted particles Fluid defines the emitted energy-momentum of the gas of free particles by CF formula: P ν g.fr = d 3 p p 0 pν Σ fr dσ µ p µ φ f in ( pρ u ρ f T ) = Σ fr dσ µ T µν f (x, t) due to {}}{ }{{} conservation Σ in dσ µ T µν f (x, t). Thus, energy(momentum) of gas = energy(momentum) of initial fluid! fr 15

18 The Correct Equations Must Include Gas! Main idea: for emitting fluid the CONSERVATION LAWS MUST be formulated for BOTH: for FLUID and for GAS of FREE PARTICLES! Assume: The FO criterion is known as F (x, t) = 0. First, we DERIVE equations and, then, from equations we define HOW TO FORMULATE the FO criterion. The full tensor consists of fluid and gas of free particles (with CO distribution) parts (similarly, for charge current): T µν tot (x, t) = Θ f T µν f (x, t) + Θ g T µν g (x, t), N µ tot = Θ f N µ f + Θ g N µ g where Θ f = 1 Θ g, Θ g = Θ(F (x, t)) : Θ g = 1 for gas only! The true equations for emitting fluid are: µ T µν tot (x, t) = 0 and µ N µ tot(x, t) = 0 We included the boundary conditions for the fluid into equations of motion.this is principally different from ALL previous approaches! 16

19 The True Equations for Fluid From the equations of motion µ T µν tot (x, t) = 0 and µ N µ tot(x, t) = 0 and the fact that particles in a free gas move WITHOUT collisions, i.e. along straight lines, and then its equations are: µ T µν g (x, t) 0 The set of COUPLED equations for F luid : Θ f µ T µν f (x, t) = 0, Boundary conditions at Σ fr (x, t ) : dσ µ T µν f (x, t ) = dσ µ T g µν (x, t ). Note two major features: 1. the equations for fluid exit in the FLUID DOMAIN ONLY! 2. The FO hypersurface MUST BE FOUND from equations and CANNOT BE postulated a priory! If the Boundary Conditions are disregarded, i.e. if dσ µ T µν f (x, t ) dσ µ T g µν(x, t ), then the EQUATIONS of fluid motion (WITH SOLUTIONS!) ARE MODIFIED: Θ f µ T µν µν f (x, t) = [T f (x, t ) T g µν (x, t )] δ(f (x, t)) µ F (x, t) Note, this delta-function CANNOT BE ignored, even for small coefficients! 17

20 Analysis of Boundary Conditions Writing it explicitly on the FO hypersurface in a particular frame (post FO): ( d 3 p pρ dσ µ p 0 pµ p ν u ρ ) f d 3 p φ f = dσ µ p µ p ν φ g (p) Θ (p µ dσ µ ) p 0 dσ µ d 3 p p 0 pµ φ c.f T f ( pρ u ρ ) f T f = dσ µ d 3 p p 0 p µ φ c.g (p) Θ (p µ dσ µ ) A. Fluid and Gas have the Same EOS: (a.i) if Θ (p µ dσ µ ) = 1 for p, i.e., on the space-like parts of the FO hypersurface Trivial solution of Cooper-Frye (1974): φ f = φ g ; (a.ii) on the convex time-like parts of the FO hypersurface then there exists a discontinuity, Freeze out shock, with T f T, µ f µ and CO distribution. K.A.B., NPA 606 (1996); (a.iii) on the concave time-like parts of the FO hypersurface there is FEED BACK of particles EMITTED EARLIER. Then one needs an additional term to distribution φ g. It is a special case of shock, Parametric Freeze out shock, K.A.B., PRL 90 (2003) and PRC 70 (2004). B. Fluid and Gas have Different EOS: (b.i) if Θ (p µ dσ µ ) = 1 for p, i.e. on the space-like parts of the FO hypersurface Time-like shock instead of (a.i). Was suggested by L. Csernai et. al. PRD 40 (1989). to describe the transition from supercooled plasma to superhot. It has internal difficulties as shown by kinetic treatment. K.A.B., PRC 70 (2004) 18

21 Difference with Traditional Hydro Essential difference with the traditional approach: FO criterion must be formulated solely for the gas of free particles! Otherwise, if the fluid momentum distribution function is frozen out (no collision!), there is NO EQUILIBRIUM in the gas of free particles on time-like FO hypersurfaces! There is a difference between the fluid and gas on the time-like FO hypersurfaces. Easy to show that energy (momentum, charge) on initial hypersurface is EXACTLY transformed into energy (momentum, charge) of the gas of free particles, and there ARE NO NEGATIVE NUMBERS of PARTICLES! Hydrodynamical parameters of the fluid on the time-like FO hypersurfaces are not known a priori and should be found from the Boundary Conditions. All this makes the treatment rather complicated! Fortunately, the problem is split into two parts: i. Traditional Hydro can be solved independently! ii. Freeze out hypersurface can be found from the Boundary Conditions using the Hydro-solution as an input. 19

22 Equations for the FO Hypersurface Eq. for FO shock is of Renkin-Hugoniot-Taub form, but with CO distribution for gas! Although, it is a highly nonlinear partial differential equation, it can be solved analytically for 1+1 dimensions for a wide class of EOS. It was shown, K. A. B. et al, JPG 25 (1999) and HIP 10 (1999), that the FO shock is thermodynamically stable, i.e. entropy increases by 1% For this class of EOS it was proven that there is NO RECOIL problem due to emission of the gas because the supersonic (or sonic) FO shock propagates inside the fluid faster than the perturbations which can change the state of fluid! The obtained solutions and particle spectra may differ from those of traditional Hydro! 20

23 Example: FO of Simple Wave Simple wave describes the propagation of the perturbations of small and finite amplitudes in the NORMAL MEDIA. Is the simplest solution and it exists in Landau (longitudinal) and Bjorken (transversal) solutions! Hence it must be studied FIRST. In the rest frame of fluid (RFF) the simple wave moves with the velocity of sound c s ; Its characteristics ( isotherms) are just straight lines originating from the initial position; Since wave velocity is subluminar its characteristics are the time-like hypersurfaces; The standard simple wave has a FO problem: fluid density 0 and fluid velocity 1, but Hydro is NOT VALID at MUCH higher densities already! Suppose: µ = 0 and EOS of the fluid is p = 1 3ε, but with different d.o.f. # It is normal media. The FO happens at T g = T Fluid velocity v f and temperature T f in the simple wave v f (T f ) = tanh ( 3 ln ( Tin T f )). Conjugation with the Freeze out Shock is possible, if FO shock velocity = speed of sound in RFF! [ 4 [ The freeze-out temperature of the fluid is T f = T S.W σg ] 1 8 σ g 3σ f 3 σ f ] 1 4 T > T 21

24 Example: FO of Simple Wave Simple wave describes the propagation of the perturbations of SW+FO shosk profile at some t SW profile at some small t! and finite amplitudes! in the NORMAL MEDIA. Initial energy density Is the simplest solution and it exists in Landau (longitudinal) and Bjorken (transversal) solutions! Hence it must be studied FIRST. In the rest frame of fluid (RFF) the simple wave moves with the velocity of sound c s ; Its characteristics ( isotherms) are just straight lines originating from the initial position; Since wave velocity is subluminar its characteristics are the time-like hypersurfaces; R The EOS and Hydro are valid till here!!! Initial energy density Gas density is small Due to CO distribution only! R The EOS and Hydro are valid till here!!! The standard simple wave has a FO problem: fluid density 0 and fluid velocity 1, but Hydro is NOT VALID at MUCH higher densities already! Suppose: µ = 0 and EOS of the fluid is p = 1 3ε, but with different d.o.f. # It is normal media... The FO happens at T g = T Fluid velocity v f and temperature T f in the simple wave v f (T f ) = tanh ( 3 ln ( Tin T f )). Conjugation with the Freeze out Shock is possible, if FO shock velocity = speed of sound in RFF! [ 4 [ The freeze-out temperature of the fluid is T f = T S.W σg ] 1 8 σ g 3σ f 3 σ f ] 1 4 T > T 21

25 EoS p = 1 ε (massless pions) 3 FO temperature is T, initial temperature in the wave is T in. Center of mass rapidity interval is y C.M. [ 2; 2]; T in = 1.1T (dotted line), T in = 1.5T (solid line) and T in = 1.9T (dashed) Transverse momentum spectra supposed to be exponential!$ p_t spectra for the Simple Wave!$ Cooper-Frye Cut-off Lg(dN CF /(p t dp t ds dt T*))!$"#!% Lg(dN CUT /(p t dp t ds dt T*))!$"#!%!%"# Negative particle numbers at work!!!"# $ $"# % p t /T*!%"#!!"# $ $"# % p t /T* 22

26 Rapidity spectra for the Simple Wave Evidence that the negative particle number disappears when rapidity interval is wide. (LEFT) Effect is not small even for rapidity spectra: (Cooper-Frye results are dashed lines) (RIGHT)!!"#, T in = 1.5T, T in = 1.5T "%"' Lg(dN/(p t dp t ds dt T*))!$!$"# dn / (dy CM ds dt T* 3 ) "%"& "%"$ "%"!!%!!"# $ $"# p t /T*!"%"!!! "! # $ y CM 23

27 Despite Some Success at RHIC Hydro describes the elliptic flow v_2 = < cos(2φ) > for hadrons from pi to Omega-baryon; Even it describes the fine structure of scaling relation v_2 (i) ~ M_i y_t^2 of M_i mass hadron There are real problems with Hydro! Hydro predicts the ratio v_4/v_2^2 = 0.5, but it is v_4/v_2^2 = 1.2! find Refs in Shuryak hep-ph/ The transversal HBT radii remain a REAL PUZZLE for usual Hydro and even Hydro-Cascade Models! Imagine the c.m. energy from SPS to RHIC increased by 12 times, but transversal HBT radii remain the SAME! The ONLY possible explanation is in the WRONG Freeze-out procedure! for details see K.A.B., PRC 70 (2004) 24

28 The were hopes that Hydro-Cascade will RESOLVE old Hydro problems! Welcome To Hydro-Cascade! Hidro-Cascade will solve Freeze-out problem! 25

29 Hydro-Cascade problems are similar to old Hydro problems! Welcome To Hydro-Cascade! Freeze-out problem -dynamics Hydro 26

30 Conclusions 1. The correct description of the particle emission from an arbitrary hypersurface is given. 2. Hydrodynamics with the boundary conditions is formulated and the full analysis of all solutions, i.e., time-like shocks, freeze out shock and parametric freeze out shock is given. 3. The equations of motion of the fluid alone are derived. No source term is found, if conservation laws are satisfied! 4. Eqs. for the FO hypersurface are derived and the general scheme to solve them is found. 5. As an application of the general scheme the freeze out of the simple wave is considered. 27

31 Conclusions 1. The correct description of the particle emission from an arbitrary hypersurface is given. 2. Hydrodynamics with the boundary conditions is formulated and the full analysis of all solutions, i.e., time-like shocks, freeze out shock and parametric freeze out shock is given. 3. The equations of motion of the fluid alone are derived. No source term is found, if conservation laws are satisfied! 4. Eqs. for the FO hypersurface are derived and the general scheme to solve them is found. 5. As an application of the general scheme the freeze out of the simple wave is considered. The Story Is NOT Over! 27

32 We need Other Kind of Hydro to Resolve RHIC puzzles! 28