\documentstyle[prl,multicol,aps,psfig]{revtex}

\addtolength{\topmargin}{2.cm}
\def\bvec#1{{\rm\bf #1}}
\def\meanflow{\langle u_t\rangle}
\def\mf{\langle u_t\rangle}
\def\mi{\langle u_i\rangle}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\eps{\epsilon}
\def\dst{\displaystyle\phantom{|}}
\def\ov{\over\displaystyle}
\def\p{\prime}

\begin{document}
\title{Observables and initial conditions 
for self-similar ellipsoidal flows}
\author{T. Cs{\"o}rg{\H o}$^{1,2}$, S.V. Akkelin$^{3}$,  
Y. Hama$^{2}$, B. Luk\'acs$^1$ and Yu.M.  Sinyukov$^{2,3}$}
\address{$^1$MTA KFKI RMKI, H-1525 Budapest 114, POB 49, Hungary\\
$^2$ IF-USP, C.P. 66318, 05389-970 S\~ao Paulo, SP, Brazil,\\
$^3$ Bogolyubov Institute for Theoretical Physics, Kiev 03143,
Metrologichna 14b,Ukraine} 
\maketitle

\begin{abstract}
Single-particle spectra and two-particle Bose-Einstein correlation 
functions are determined analytically utilizing a self-similar 
solution of non-relativistic hydrodynamics for 
ellipsoidally-symmetric, expanding fireballs, by assuming that the 
symmetry axes of the ellipsoids are tilted in the frame of the 
observation. The directed, elliptic and third flows are calculated 
analytically. The mass dependences of the slope parameters in the 
principal directions of the expansion, together with the mass and 
angular dependences of the HBT radius parameters, reflect directly 
the ellipsoidal properties of the flow. 
\end{abstract}

\begin{multicols}{2}
{\it Introduction} --- The equations of hydrodynamics are scale 
invariant, reflecting the local conservation of matter, momentum 
and energy. These equations are well suited to study problems 
related to flows in various fields ranging from evolution of 
galaxies in astrophysics to heavy-ion and elementary-particle 
collisions in high-en\-ergy physics.  The finding of exact 
self-similar hydro solutions sometimes represents essential progress 
in physics, as the discovery of the Hubble flow of our Universe or 
the Bjorken flow of ultrarelativistic heavy-ion collisions. 

In this Letter we consider the case of a non-relativistic hydrodynamical problem with ellipsoidal symmetry. Our goal is to 
demonstrate the influence of initial conditions on the final state 
observables, utilizing an explicit, exact  and simple analytic 
solution of fireball hydrodynamics. In particular, we attempt to 
understand the relationship between  the initial conditions (the 
ellipsoidal asymmetry  and the tilt of the major axis) and the final 
observables. 


{\it The new family of self-similar ellipsoidal solutions} --- 
A self-similar solution with a generalized (direction-dependent) 
Hubble flow, a three-dimensional ellipsoidal Gaussian density 
profile and a homogeneous, space-indepenent temperature profile has 
just been found in ref.~\cite{ellsol}. This solution has many 
interesting properties, e.g. the partial differential equations of 
hydrodynamics are reduced to ordinary differential equations 
corresponding to a Hamiltonian motion of a massive particle in a 
non-central repulsive potential. 

These results correspond to the generalization of earlier, data 
motivated hydrodynamical parameterizations and/or solutions of 
refs.~\cite{jnr,nr,cspeter,cssol,nrt,3d,jde} to ellipsoidal 
symmetry and non-central heavy-ion collisions with homogeneous 
temperature profile. All the observables of non-central collisions 
can be calculated analytically from our hydrodynamical solution, 
and can be directly connected to the initial conditions. 
However, we do not aim here to apply directly the new hydro 
solution to data fitting in high-energy heavy-ion physics. In order 
to reach the level of data fitting, generalizations to relativistic 
flow patterns, non-ideal equations of state and more realistic 
temperature profiles are needed. Some of these generalizations seem 
to be straight forward and are in progress~\cite{nrtgen,1drel}. 

Consider the non-relativistic hydrodynamical problem, as given by 
the continuity, Euler and energy equations: 
\begin{eqnarray} 
{\partial_t n} + {\rm{\bf\nabla}} ({\rm{\bf v}} n) & = & 0\,,
\label{e:cont} \\ 
{\partial_t {\rm{\bf v}}} + ({\rm{\bf v}}{\rm{\bf\nabla}})
{\rm{\bf v}}
& = & - ({\rm {\bf \nabla }} p) / (m n)\, ,  \label{e:Eu} \\
{\partial_t \epsilon} + {\rm{\bf\nabla}}(\epsilon{\rm{\bf v}}) & =
& - p {\rm {\bf \nabla }} {\rm {\bf v }}\, ,  \label{e:en}
\end{eqnarray}
where $n$ denotes the particle number density, ${\rm {\bf v}}$ 
stands for the non-relativistic (NR) flow velocity field, 
$\epsilon$ for the energy density, $p$ for the pressure and in the 
following the temperature field is denoted by $T$. This set of 
equations are closed by the equations of state (EoS), for example 
\begin{eqnarray}
p & = & n T\,, \qquad
\epsilon  =  \kappa p \, = \, \kappa n T\,.  \label{e:eos}
\end{eqnarray}
The parameter $\kappa$ characterizes the EoS for a broad variety of 
materials: e.g. a non-relativistic ideal gas yields  $\kappa=3/2$. 
Softening of the EoS can be modelled by increasing the values of 
$\kappa$ in certain temperature domains. We choose $n$, 
${\rm{\bf v}}$ and $T$ as the independent variables. 

These NR hydro equations are solved~\cite{ellsol,nrtgen} by the 
following self-similar, ellipsoidal flows: 
\begin{eqnarray} 
n(t,{\rm {\bf r}^\p}) & = & n_0 {\frac{V_0} {V}} 
    \mathrm{exp}\left({ -{\frac{r_x^{\p 2} }{2X^2}} 
                 -{\frac{r_y^{\p 2}}{2Y^2}} 
                 -{\frac{r_z^{\p 2}}{2Z^2}} }\right),  \label{n} \\
{\rm {\bf v}^\p}(t,{\rm {\bf r}^\p}) &=&
    \left({\frac{\dot{X}}{X}}\, r_x^\p,
     {\frac{\dot{Y}}{Y}}\, r_y^\p, 
     {\frac{\dot{Z}}{Z}}\, r_z^\p \right),  \label{v} \\
T(t) & = & T_0 \left( {\frac{\displaystyle\phantom{|}V_0}{\displaystyle%
\phantom{|}V}} \right)^{1/\kappa},  \label{T}
\end{eqnarray}
where the variables are defined in a center of mass frame $K^\p$,
but with the axes pointing to the principial directions of the 
expansion. The time dependent scale parameters are denoted by 
$(X,Y,Z)=(X(t),Y(t),Z(t))$, the  typical volume of the expanding 
system is $V = XYZ$, and the initial temperature and volume are 
$T_0=T(t_0)$ and $V_0=V(t_0)$, and $n_0$ is a constant. The time 
evolution of the radius parameters $X$, $Y$ and $Z$ is equivalent 
to the classical motion of a particle in a non-central potential, 
see refs.~\cite{ellsol,nrtgen}. 


{\it Observables from the new solution} --- 
In order to evaluate the measurable quantities, any hydrodynamical 
solution has to be supplemented with an additional freeze-out 
criterion, that specifies  the end of the hydrodynamical evolution. 
Here we assume sudden particle freeze-out at a constant temperature 
$T(t_f,{\bf r}) = T_f\,$. This freeze-out condition is reached 
everywhere at the same time in the considered  class of exact 
hydrodynamical solutions and it is motivated by the simplicity of 
the results. Then, the emission function is proportional to 
\be
S(t,{\rm \bf r}^\p,{\rm \bf k}^\p) \propto 
{\rm e}^{- \frac{ ({\rm \bf k}^\p - m {\rm \bf v}^\p)^2}{2 m T_f} 
         -{\frac{r_x^{\p 2}}{2X_f^2}} 
         -{\frac{r_y^{\p 2}}{2Y_f^2}} 
         -{\frac{r_z^{\p 2}}{2Z_f^2}}}\, \delta(t - t_f)\,. 
\label{e:sxp} 
\ee


{\it Single particle spectrum } --- 
 The single-particle spectrum and the two-particle correlation 
function can be evaluated similarly to that of 
ref.~\cite{nr,cspeter}: 
\begin{eqnarray} 
E\frac{d^3n}{d{\bf k}^\p} & \propto & 
      E\exp\left(-\frac{k_x^{\p 2}}{ 2 m T_x^\p} 
		     -\frac{k_y^{\p 2}}{ 2 m T_y^\p} 
		     -\frac{k_z^{\p 2}}{ 2 m T_z^\p} 
           \right),               \label{e:ellsp}\\
T_x^\p & = & T_f + m \dot X_f^2\ , \label{e:txp}\\
T_y^\p & = & T_f + m \dot Y_f^2\ , \label{e:typ}\\
T_z^\p & = & T_f + m \dot Z_f^2\ . \label{e:tzp}
\end{eqnarray}
Here $E=m+{\bf k}^{\p 2}/(2m)$ in the non-relativistic limit we are 
considering, ${\bf k}^\p = ( k_x^\p, k_y^\p, k_z^\p)$ stands for 
the momentum vector in $K^\p$,  $X_f = X(t_f)$, etc. In the 
spherically symmetric case of $X = Y = Z = R$, we recover the 
earlier results~\cite{nr,cspeter}, with $\langle u \rangle = \dot R$ 
and $T_{\rm eff} = T_f + m \langle u \rangle^2$. 

The observables are determined in the center of mass frame of the 
collision, $K$, where the $r_z$ axis points to the direction of the 
beam and the $r_x$ axis to that of the impact parameter. 
In this frame, the coordinates and the momenta are denoted by 
${\bf x}$ and ${\bf k}$.
We assume that the initial state of the hydrodynamic evolution 
corresponds to a rotated ellipsoid in $K$. The tilt angle $\theta$ 
represents the rotation of the major (longitudinal) direction of 
expansion, $r_z^\p$ from the beam axis $r_z$. Hence the event plane 
is the $(r_x^\p, r_z^\p)$ plane, which is the same as the 
$(r_x,r_z)$ plane. The (zenithal) angle between directions $r_z$ and 
$r_z^\p$ is $\theta$, while the (azimuthal) angle between the 
transverse momentum ${\mathbf k}_t$ and the event plane is $\phi\,$. 

The ellipsoidal spectrum of eq.~(\ref{e:ellsp}) generates the 
following $\phi$ averaged single-particle spectrum in the $K$ frame: 
\begin{eqnarray} 
 \frac{d^2n}{2\pi k_t dk_t dk_z} & \propto &  
     \exp\left(-\frac{k_t^2}{ 2 m T_{\rm eff}} 
               -\frac{k_z^2}{ 2 m T_z}
	   \right) f(v,w)\ ,  \label{e:ellsptr}\\
 \frac{1}{T_z} & = & 
           \frac{\cos^2 \theta}{T_z^\p} +
           \frac{\sin^2 \theta}{T_x^\p}\ ,\\
 \frac{1}{T_x} & = & 
           \frac{\cos^2 \theta}{T_x^\p} +
           \frac{\sin^2 \theta}{T_z^\p}\ ,\\
 \frac{1}{T_{\rm eff}} & = & 
           \frac{1}{2}
           \left( \frac{1}{T_x} + \frac{1}{T_y^\p} \right),\\
 w  & = &  \frac{k_t^2}{4 m} 
           \left( \frac{1}{T_y^\p} - \frac{1}{T_x} \right),\\
 v & = & - \frac{k_t k_z} {2m} \sin(2\theta)
           \left(\frac{1}{T_x^\p} - \frac{1}{T_z^\p} \right),\\
 f(v,w)& \approx & I_0(w) + frac{v^2}{4}\left[I_0(w)+I_1(w)\right]
         \ , 
\end{eqnarray}
where $f(v,w)$ is calculated for $|v| \ll 1$ and 
$I_n(w) = \frac{1}{\pi} \int_0^\pi dz \cos(n z) \exp[w \cos(z)]$ 
is the modified Bessel function of order $n$ ($n=0,1,...)$. For 
small ellipsoidal asymmetries, $w \ll 1$, $I_0(w) \simeq 1$ and 
the effective temperature parameter in the transverse direction is
the harmonic mean of the temperature parameters of the principal
directions of expansion (projected to the transverse plane). As 
$T_z^\p \ge T_x^\p \ge T_y^\p$ is expected from the initial 
conditions, we obtain $T_x \ge T_x^\p$ and $T_z\le T_z^\p\,$.

The flow coefficients $v_n$ are defined as
\begin{equation}
 \frac{d^3n}{dk_z k_t dk_t d\phi} = 
 \frac{d^2n}{2 \pi dk_z k_t dk_t}
      \left[1 + 2 \sum_{n=1}^{\infty} v_n \cos(n\phi)\right].  \label{e:harm} 
\end{equation}
Here $v_1$ is called the directed flow, $v_2$ the elliptic flow and 
$v_3$ the third flow. The transverse- and longitudinal-momentum 
dependence of the $v_n$ flow components can be written in terms of 
$v$ and $w$. Assuming that the tilt angle $\theta$ or the 
anisotropy is small, $|v| \ll 1$, the directed, elliptic and third 
flow components are evaluated as
\begin{eqnarray}
 v_1 & = &  \frac{v}{2}\left[1+\frac{I_1(w)}{I_0(w)}\right],
            \label{e:v1} \\
 v_2 & = &  \frac{I_1(w)}{I_0(w)} 
      + \frac{v^2}{8} 
	  \left[ 1 + \frac{I_1(w)}{I_0(w)} -
	      2\left(\frac{I_1(w)}{I_0(w)}\right)^2 
        \right], \label{e:v2} \\
 v_3 & = & \frac{v}{2}\,\frac{I_2(w) + I_1(w)}{I_0(w)}\ . \label{e:v3} 
\end{eqnarray} 
An angular tilt $\theta \ne 0$ is evidenced by the rise of the 
directed and third flows as a function of  rapidity 
$y = 0.5 \ln [(E + k_z)/(E-k_z)]$ and by a minimum of the elliptic 
flow at mid-rapidity, see Fig.~1. This and other features are in 
qualitative agreement with most of the data on intermediate- and 
high-energy heavy-ion 
collisions~\cite{dflow,na49flow,starflow,3flow}, suggesting that in 
non-central collisions the dominant longitudinal direction of 
expansion is slightly deviating from the beam direction. A more 
straightforward proof of the ellipsoidal nature of the flow can be 
obtained by determining the mass dependence of the parameters 
$T_x^\p\,$, $T_y^\p$ and $T_z^\p\,$, cf. eqs.~(\ref{e:txp}-\ref{e:tzp}) 
and Fig.~2. 


{\it Two-particle correlations} ---  
The two-particle Bose-Einstein correlation function (BECF) is 
related to a Fourier-transform of the emission (or source) function 
$S(t,{\bf r}^\p,{\bf k}^\p)$ of eq.~(\ref{e:sxp}), see e.g.  
refs.~\cite{nr,cspeter,nrt}. If the core-halo picture~\cite{chalo} 
is valid, an effective intercept parameter 
$\lambda\equiv\lambda({\bf k}) = [N_c({\bf k})/N({\bf k})]^2$ 
appears, that measures the fraction of particles emitted directly 
from the {\underline c}ore. The two-particle BECF is diagonal in 
$K^\p$, as 
\begin{eqnarray}
 C({\bf K}^\p,{\bf q}^\p) & = & 1 + 
	\lambda \exp\left( - q_x^{\p 2} R_x^{\p 2} 
	                   - q_y^{\p 2} R_y^{\p 2}
                         - q_z^{\p 2} R_z^{\p 2} 
			\right), \label{e:ellbecf}\\
 {\bf K}^\p & = & {\bf K}^\p_{12}\, = 
                  \, 0.5({\bf k}^\p_1 + {\bf k}^\p_2)\,, \\
 {\bf q}^\p & = & {\bf q}^\p_{12} \, = 
                  \, {\bf k}^\p_1 - {\bf k}^\p_2 \, = 
                  \, (q_x^\p\,, q_y^\p\,, q_z^\p)\,, \\
 R_x^{\p -2} & = & X_f^{-2} \left( 1 + \frac{m}{T_f} \dot X_f^2 
                            \right), \label{e:rxp}\\ 
 R_y^{\p -2} & = & Y_f^{-2} \left( 1 + \frac{m}{T_f} \dot Y_f^2 
                            \right), \label{e:ryp}\\
 R_z^{\p -2} & = & Z_f^{-2} \left( 1 + \frac{m}{T_f} \dot Z_f^2
                            \right). 
\label{e:rzp}
\end{eqnarray}
These radius parameters measure the lengths of 
homogeneity~\cite{sinyukov}. They are dominated by the shorter of 
the geometrical scales $(X_f, Y_f, Z_f)$ and the corresponding  
thermal scales defined by $(X_T, Y_T, Z_T ) = \sqrt{\frac{T_f}{m}} 
(\frac{X_f}{\dot X_f}, \frac{Z_f}{\dot Z_f}, \frac{Z_f}{\dot Z_f})$,  
generalizing the results of refs.~\cite{nr,nrt,3d} to ellipsoidal 
flows. The geometrical scales characterize the spatial variation of 
the fugacity term $\mu(t, {\bf r}^\p)/T(t, {\bf r}^\p)$, while the 
thermal scales characterize the spatial variations of the Boltzmann 
term $E_{{\rm loc}}^\p(t,{\bf r}^\p)/T(t,{\bf r}^\p)$, both 
evaluated at the point of maximal emittivity. In the $K^\p$ frame, 
cross-terms~\cite{xterm} vanish, $R^2_{i\ne j} = 0$, if the 
emission is sudden.

If the particle emission is not sudden, but it happens in a narrow 
interval $\Delta t$ centered at $t_f\,$, then the BECF can be 
evaluated using the replacement 
$\delta(t-t_f)\rightarrow(2\pi\Delta t^2)^{-1/2}\exp[-(t-t_f)^2/2 
\Delta t^2]$ in eq.~(\ref{e:sxp}), if $\Delta t \dot X_f << X_f\,$, 
etc. Hence all the previous radius components, including the 
cross-terms, are extended with an additional term 
$\delta R^{\p 2}_{ij} = \beta_i^\p \beta_j^\p \Delta t^2$, where 
$\mbox{\boldmath$\beta$}^\p = ({{\mathbf k}^\p_{1} + 
{\mathbf k}^\p_2})/({E^\p_1 + E^\p_2})$ is the velocity of the pair 
in $K^\p$. 

The BECF-s are usually given in the side-out-longitudinal or 
Bertsch-Pratt (BP) parameterization. The longitudinal direction, 
$r_{\rm long}\equiv r_{\rm l}$ in BP coincides with the beam 
direction. The plane orthogonal to the beam is decomposed to a 
direction parallel to the mean transverse momentum of the pair, 
$r_{{\rm out}}\equiv r_{\rm o}\,$, and the one perpendicular both to 
this and the beam direction, $r_{\rm side} = r_{\rm s}\,$. The mean 
velocity of the particle pair can be written in BP as 
$\mbox{\boldmath $\beta$} = (\beta_{\rm o},0,\beta_{\rm l})$, where
$\beta_{\rm o} = \beta_t\,$. Let $\phi$ denote the angle of the event 
plane and the mean transverse momentum of the measured pair. 
The result is 
\begin{eqnarray}
 C_{2}({\bf K},{\bf q}) &=&1+\lambda \exp 
      \left(-\sum_{i,j={\rm s,o,l}}q_i q_j R_{i,j}^2\right), \\ 
 R_{\rm s}^{2} &=& R_{y}^{\p 2} \cos^{2}\phi 
                 + R_{x}^{2} \sin^{2}\phi\,, \\
 R_{\rm o}^{2} &=& R_{x}^{2} \cos^{2}\phi 
                 + R_{y}^{\p 2}\sin^{2}\phi 
                 + \beta_{t}^{2}\Delta t^{2}, \\
 R_{\rm l}^{2} &=&R_{z}^{\p 2} \cos^2\theta 
                 + R_{x}^{\p 2} \sin^2\theta
                 + \beta _{\rm l}^{2}\Delta t^{2}, \\
 R^{2}_{\rm o,l} &=& (R_x^{\p 2}- R_z^{\p 2})
                       \cos\theta\sin\theta\cos\phi
                   +\beta _{t}\beta _{\rm l}\Delta t^{2}, \\
 R^{2}_{\rm o,s} &=& (R_{x}^{2}-R_{y}^{\p 2}) 
                       \cos \phi \sin \phi\,, \\
 R^{2}_{\rm s,l} &=&(R_x^{\p 2} - R_z^{\p 2}) 
                       \cos\theta\sin\theta\sin\phi\,,
\end{eqnarray}
where an auxiliary quantity is introduced as 
\begin{equation}
R_x^{2} = R_{x}^{\p 2} \cos^2\theta + R_{z}^{\p 2} \sin^2\theta\ .
\end{equation} 
These results imply that all the radius parameters oscillate in the 
$K$ frame. In particular, a $\phi$ dependent oscillation appears in 
the radius parameters indexed either by the {\it side} or the 
{\it out} direction, as illustrated in Fig. 3. These oscillations 
are similar to those obtained in ref.~\cite{hhflow}, corresponding 
to $\theta =0$. We find that the radius parameters indexed by the 
{\it longitudinal} direction depend also on the zenithal angle 
$\theta$. A toy model for tilted ($\theta \ne 0$) ellipsoidal static 
pion sources was introduced in refs.~\cite{lisa}, to understand the 
$\phi$ dependent oscillations of measured HBT radii at AGS energies. 
In our case, the amplitude of the oscillations is reduced for 
heavier particles due to the hydrodynamic expansion, which results 
in a decrease of the lengths of homogeneity with increasing mass. 
The oscillations of the radius parameters were not related before 
either to an exact solution of ellipsoidal, self-similar  
hydrodynamical flow, or with the observable directed and elliptic 
flows, and were not connected to the initial conditions of a 
hydrodynamic expansion. 

A check of the applicability of our hydrodynamic solution is that 
the BECF and the single particle spectrum become diagonal (after 
removing a term of $\beta_i\beta_j \Delta t^2$ from all the HBT 
radius parameters) in the {\it same} frame, see 
eqs.~(\ref{e:ellsp},\ref{e:txp}-\ref{e:tzp}) and 
eqs.~(\ref{e:ellbecf},\ref{e:rxp}-\ref{e:rzp}). This frame is 
$K^\p$, the natural System of Ellipsoidal Expansion or SEE. 


{\it Summary} ---  We have analytically evaluated the 
single-particle spectrum and the two-particle Bose-Einstein 
correlation function for a self-similarly expanding, exact, 
ellipsoidal solution of the non-relativistic hydrodynamical 
equations, assuming a constant freeze-out temperature. 

The parameters of the hydro solution at freeze-out time, $T_f$, 
$(X_f, Y_f, Z_f)$ and $(\dot X_f, \dot Y_f, \dot Z_f)$, can be 
reconstructed from the measurement of single particle spectrum and 
two-particle correlation functions. The direction of the major axis 
of expansion in the center of mass frame of the collision is 
characterized by the polar angles $(\theta,0)$. With the exception 
of $R_{\rm l}$, all the radius parameters oscillate as function of 
$\phi$, while the radius parameters $R^2_{\rm l}\,$, 
$R^2_{\rm o,l}\,$, $R_{\rm s,l}^2$ depend also on $\theta$. All the 
radius parameters decrease with increasing mass. If $\theta \neq 0$, 
the effective temperature in the transverse direction is increased 
by a contribution from the longitudinal expansion. 

The initially more compressed longitudinal and impact parameter 
directions (the $r_z$ and $r_x$ directions) expand more 
dynamically~\cite{ellsol,nrtgen}, that implies 
$T_z^\p\geq T_x^\p \geq T_y^\p\,$. The initial time $t_i$ can be 
identified from the requirement that $\dot Y(t_i) = 0$. {\it The 
initial conditions  for this hydrodynamical system can be uniquely 
reconstructed from  final state measurements}, given the parameter 
$\kappa$ of the equation of state. This  parameter influences only 
the time evolution of the scales $(X,Y,Z)$. Solutions with different 
$\kappa$ can be matched if $\kappa$ jumps, to model the softening of 
EoS. 

We have deliberately chosen the presentation as simple as possible, 
which limits the direct applicability of our results in 
high-energy heavy-ion collisions only to sufficiently small 
transverse momentum, $ p_{t} \ll m$, at mid-rapidity. But the 
scheme permits generalization in many points and still the 
qualitative features of our results may survive even in the 
relativistic regime. A generalization to relativistic longitudinal 
flows has been  described in ref.~\cite{1drel}. Ref.~\cite{nrtgen} 
includes another extension to an arbitrary, inhomogeneous, 
ellipsoidally symmetric initial temperature profile, which does not 
change the time evolution of the scale parameters. Also, the shape 
of the solution is independent of $\kappa$. It turns out that each 
of these generalizations is essentially straightforward. 
 
Although we considered a non-relativisitic problem, our results 
provide generic insight into the time evolution of non-central 
heavy-ion collisions  and relate initial conditions to final-state 
observables in a simple and straightforward manner impossible 
before. 


{\it Acknowledgments:} This work has been supported by the grants 
FAPESP 00/04422-7, 99/09113-3, 01/05122-0 of S\~ao Paulo, Brazil, 
by the Hungarian - Ukrainian S\&T grant 45014 (2M/125-199), the 
OTKA grant T026435 and the NWO - OTKA  grant N 25487. Y. S. has 
been supported by  CNRS and the DLR grant 2M-141- 2000.

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\bibitem{hhflow} U.A. Wiedemann, 
                 Phys. Rev. {\bf C57} (1998) 266; 
                 H. Heiselberg and A. Levy,
		     Phys. Rev. {\bf C59} (1999) 2716. 

\bibitem{lisa}
                 M.A. Lisa, U. Heinz and U.A. Wiedemann,
                  Phys. Lett. {\bf B489} 
                 (2000) 287; M. A. Lisa et al, E895 Collaboration,
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\end{references}

\begin{figure}[tbp]
\vspace*{5.0cm}
\begin{center}
\special{psfile=v123y.eps hscale=70 vscale=70 angle=0 hoffset=20}
\end{center}
\vspace{-1.3cm}
\caption{The directed, elliptic and third flows $v_1$, $v_2$, $v_3$ 
are illustrated, respectively with solid, long-dashed and 
short-dashed lines, as a function of rapidity, for $m$ = 940 MeV, 
$T_x^\p = 200$ MeV, $T_y^\p = 150$ MeV, $T_z^\p =700$ MeV, at a 
fixed $k_t = 500$ MeV  and $\theta = \pi/5$, see 
eqs.~(\ref{e:v1}-\ref{e:v3}).}
\end{figure}
\begin{figure}[tbp]
\vspace{6.4cm}
\begin{center}
\special{psfile=txyeff.eps hscale=90 vscale=90 angle=0 hoffset=-10}
\end{center}
\vspace{-2.5cm}
\caption{The linear mass dependence of the effective temperatures 
in the transverse directions, $T_x^\p$, $T_y^\p$ and their 
(harmonic) average $T_{\rm eff}$ (shown with solid, long-dashed and 
short-dashed lines) for non-central heavy-ion collisions, if 
$T_f = 120$ MeV, $\dot X = 0.4$, and $\dot Y = 0.25$ and 
$\theta = 0$.}
\end{figure}
\begin{figure}[tbp]
\vspace{6.0cm}
\begin{center}
\special{psfile=rso2.eps hscale=85 vscale=85 angle=0 hoffset=-20}
\end{center}
\vspace{-2.0cm}
\caption{
The out-side and the side-long cross terms are plotted as a function 
of the polar angle $\phi$, for $R_x^\p = 5$ fm, $R_y^\p = 4$ fm, 
$R_z^\p$ = 8 fm and $\theta  = \pi/5$. Note that
$R^2_{\rm os} \propto \sin(\phi)$ while 
$R^2_{sl} \propto \sin(2\phi)$.} 
\end{figure}
\end{multicols}
\end{document}
