A study of vorticity formation in high energy nuclear collisions
We present a quantitative study of vorticity formation in peripheral ultrarelativistic heavy ion collisions at GeV by using the ECHO-QGP numerical code, implementing relativistic dissipative hydrodynamics in the causal Israel-Stewart framework in 3+1 dimensions with an initial Bjorken flow profile. We consider and discuss different definitions of vorticity which are relevant in relativistic hydrodynamics. After demonstrating the excellent capabilities of our code, which proves to be able to reproduce Gubser flow up to 8 fm/, we show that, with the initial conditions needed to reproduce the measured directed flow in peripheral collisions corresponding to an average impact parameter fm and with the Bjorken flow profile for a viscous Quark Gluon Plasma with fixed, a vorticity of the order of some /fm can develop at freezeout. The ensuing polarization of baryons does not exceed 1.4% at midrapidity. We show that the amount of developed directed flow is sensitive to both the initial angular momentum of the plasma and its viscosity.
The hydrodynamical model has by now become a paradigm for the study of the QCD plasma formed in nuclear collisions at ultrarelativistic energies. There has been a considerable advance in hydrodynamics modeling and calculations of these collisions over the last decade. Numerical simulations in 2+1D vishnu () and in 3+1 D music (); echoqgp (); karpe (); molnar (); japan (); bozekcode () including viscous corrections are becoming the new standard in this field and existing codes are also able to handle initial state fluctuations.
An interesting issue is the possible formation of vorticity in peripheral collisions beca0 (); csernai1 (); csernai2 (). Indeed, the presence of vorticity may provide information about the (mean) initial state of the hydrodynamical evolution which cannot be achieved otherwise, and it is related to the onset of peculiar physics in the plasma at high temperature, such as the chiral vortical effect chiral (). Furthermore, it has been shown that vorticity gives rise to polarization of particles in the final state, so that e.g. baryon polarization - if measurable - can be used to detect it beca1 (); beca2 (). Finally, as we will show, numerical calculation of vorticity can be used to make stringent tests of numerical codes, as the T-vorticity (see sect. II for the definition) is expected to vanish throughout under special initial conditions in the ideal case.
Lately, vorticity has been the subject of investigations in refs. csernai1 (); csernai2 () with peculiar initial conditions in cartesian coordinates, ideal fluid approximation and isochronous freezeout. Instead, in this work, we calculate different kinds of vorticity with our 3+1D ECHO-QGP 111The code is publicly available at the web site http://theory.fi.infn.it/echoqgp code echoqgp (), including dissipative relativistic hydrodynamics in the Israel-Stewart formulation with Bjorken initial conditions for the flow (i.e. with ), henceforth denoted as BIC. It should be pointed out from the very beginning that the purpose of this work is to make a general assessment of vorticity at top RHIC energy and not to provide a precision fit to all the available data. Therefore, our calculations do not take into account effects such as viscous corrections to particle distribution at the freezeout and initial state fluctuations, that is we use smooth initial conditions obtained averaging over many events.
In this paper we use the natural units, with .
The Minkowskian metric tensor is ; for the Levi-Civita symbol we use the convention .
We will use the relativistic notation with repeated indices assumed to be summed over, however contractions of indices will be sometimes denoted with dots, e.g. . The covariant derivative is denoted as (hence ), the exterior derivative by , whereas is the ordinary derivative.
Ii Vorticities in relativistic hydrodynamics
Unlike in classical hydrodynamics, where vorticity is the curl of the velocity field , several vorticities can be defined in relativistic hydrodynamics which can be useful in different applications (see also the review gourg ()).
ii.1 The kinematical vorticity
This is defined as:
where is the four-velocity field. This tensor includes both the acceleration and the relativistic extension of the angular velocity pseudo-vector in the usual decomposition of an antisymmetric tensor field into a polar and pseudo-vector fields:
where is the Levi-Civita symbol. Using of the transverse (to ) projector:
and the usual definition of the orthogonal derivative
where , it is convenient to define also a transverse kinematical vorticity as:
Using the above definition in the decomposition (II.1) it can be shown that:
that is is the tensor formed with the angular velocity vector only. As we will show in the next subsection, only shares the “conservation” property of the classical vorticity for an ideal barotropic fluid.
ii.2 The T-vorticity
This is defined as:
and it is particularly useful for a relativistic uncharged fluid, such as the QCD plasma formed in nuclear collisions at very high energy. This is because from the basic thermodynamic relations when the temperature is the only independent thermodynamic variable, the ideal relativistic equation of motion can be recast in the simple form (see e.g. stephanov ()):
The above (6) is also known as Carter-Lichnerowicz equation gourg () for an ideal uncharged fluid and it entails conservation properties which do not hold for the kinematical vorticity. This can be better seen in the the language of differential forms, rewriting the definition of the T-vorticity as the exterior derivative of a the vector field (1-form) , that is . Indeed, the eq. (6) implies - through the Cartan identity - that the Lie derivative of along the vector field vanishes, that is
because is itself the external derivative of the vector field and . The eq. (7) states that the T-vorticity is conserved along the flow and, thus, if it vanishes at an initial time it will remain so at all times. This can be made more apparent by expanding the Lie derivative definition in components:
The above equation is in fact a differential equation for precisely showing that if at the initial time then . Thereby, the T-vorticity has the same property as the classical vorticity for an ideal barotropic fluid, such as the Kelvin circulation theorem, so the integral of over a surface enclosed by a circuit comoving with the fluid will be a constant.
One can write the relation between T-vorticity and kinematical vorticity by expanding the definition (5):
implying that the double-transverse projection of :
Hence, the tensor shares the same conservation properties of , namely it vanishes at all times if it is vanishing at the initial time. Conversely, the mixed projection of the kinematical vorticity:
does not. It then follows that for an ideal uncharged fluid with at the initial time, the kinematical vorticity is simply:
ii.3 The thermal vorticity
This is defined as beca2 ():
where is the temperature four-vector. This vector is defined as once a four-velocity , that is a hydrodynamical frame, is introduced, but it can also be taken as a primordial quantity to define a velocity through becaframe (). The thermal vorticity features two important properties: it is adimensional in natural units (in cartesian coordinates) and it is the actual constant vorticity at the global equilibrium with rotation becacov () for a relativistic system, where is a Killing vector field whose expression in Minkowski spacetime is being and constant. In this case the magnitude of thermal vorticity is - with the natural constants restored - simply where is a constant angular velocity. In general, (replacing with the classical vorticity defined as the curl of a proper velocity field) it can be readily realized that the adimensional thermal vorticity is a tiny number for most hydrodynamical systems, though it can be significant for the plasma formed in relativistic nuclear collisions.
Furthermore, the thermal vorticity is responsible for the local polarization of particles in the fluid according to the formula beca1 ():
which applies to spin 1/2 fermions, being the Fermi-Dirac-Juttner distribution function.
Similarly to the previous subsection, one can readily obtain the relation between T-vorticity and thermal vorticity:
Again, the double transverse projection of is proportional to the one of :
whereas the mixed projection turns out to be, using eq. (13)
Again, for an ideal uncharged fluid with at the initial time, by using the equations of motion (6), one has the above projection is just and that the thermal vorticity is simply:
A common feature of the kinematical and thermal vorticity is that their purely spatial components can be non-vanishing if the acceleration and velocity field are non-parallel, even though velocity is vanishing at the beginning.
Iii High energy nuclear collisions
In nuclear collisions at very large energy, the QCD plasma is an almost uncharged fluid. Therefore, according to previous section’s arguments, in the ideal fluid approximation, if the transversely projected vorticity tensor initially vanishes, so will the transverse projection and and the kinematical and thermal vorticities will be given by the formulae (9) and (14) respectively. Indeed, the T-vorticity will vanish throughout because also its longitudinal projection vanishes according to eq. (6). This is precisely what happens for the usually assumed BIC for the flow at , that is , where one has at the beginning as it can be readily realized from the definition (1). On the other hand, for a viscous uncharged fluid, transverse vorticities can develop even if they are zero at the beginning.
It should be noted though, that even if the space-space components ( indices) of the kinematical vorticity tensor vanish at the initial Bjorken time , they can develop at later times even for an ideal fluid if the spatial parts of the acceleration and velocity fields are not parallel, according to eq. (9). The equation makes it clear that the onset of spatial components of the vorticity is indeed a relativistic effect as, with the proper dimensions, it goes like (.
In the full longitudinally boost invariant Bjorken picture, that is throughout the fluid evolution, in the ideal case, as , the only allowed components of the kinematical vorticity are and from the first eq. (II.1). The component, at , because of the reflection symmetry (see fig. 1) in both the and axes, can be different from zero but it ought to change sign by moving clockwise (or counterclockwise) to the neighbouring quadrant of the plane; for central collisions it simply vanishes.
However, in the viscous case, more components of the vorticities can be non-vanishing. Furthermore, in more realistic 3+1 D hydrodynamical calculations, a non-vanishing can develop because of the asymmetries of the initial energy density in the and planes at finite impact parameter. The asymmetry is essential to reproduce the observed directed flow coefficient in a 3+1D ideal hydrodynamic calculation with BIC, as shown by Bozek bozek (), and gives the plasma a total angular momentum, as it will be discussed later on.
In this work, we calculate the vorticities, and especially the thermal vorticity by using basically the same parametrization of the initial conditions in ref. bozek (). Those initial conditions are a modification of the usual BIC to take into account that the plasma, in peripheral collisions, has a relatively large angular momentum (see Appendix A). They are a minimal modifications of the BIC in that the initial flow velocity Bjorken components are still zero, but the energy density longitudinal profile is changed and no longer symmetric by the reflection . They are summarized hereinafter. Given the usual thickness function expression:
where , and are the nuclear density, the width and the radius of the nuclear Fermi distribution respectively, the following functions are defined:
where is the inelastic NN cross section, the mass number of the colliding nuclei, and:
where is the vector of the transverse plane coordinates and is the impact parameter vector, connecting the centers of the two nuclei. In our conventional cartesian reference frame, the vector is oriented along the positive axis and the two nuclei have initial momentum along the axis (whence the reaction plane is the plane) and their momenta are directed so as to make the initial total angular momentum oriented along the negative axis (see fig. 1). The wounded nucleons weight function is then defined:
Finally, the initial proper energy density distribution is assumed to be:
where the total weight function is defined as:
In the eq. (21) is the mean number of binary collisions:
and is the collision hardness parameter, which can vary between 0 and 1.
This parametrization, and especially the chosen forms of the functions , are certainly not unique as a given angular momentum can be imparted to the plasma in infinitely many ways. Nevertheless, as has been mentioned, it proved to reproduce correctly the directed flow in a 3+1D hydrodynamical calculation of peripheral Au-Au collisions at high energy bozek (), thus we took it as a good starting point. A variation of this initial condition will be briefly discussed in sect. VII. Besides, the parametrization (20) essentially respects the causality constraint that the plasma cannot extend beyond . Indeed, at GeV while the 3 point in the gaussian profile in eq. (22) lies at .
We have run the ECHO-QGP code in both the ideal and viscous modes with the parameters reported in table 1 and the equation of state reported in ref. laine (). The impact parameter value was chosen as, in the optical Glauber model, it corresponds to the mean value of the 40-80% centrality class ( fm steinberg ()) used by the STAR experiment for the directed flow measurement in ref. star08 (). The initial flow velocities were set to zero, according to BIC. The freezeout hypersurface - isothermal at MeV - is determined with the methods described in refs. echoqgp (); rolando ().
Iv Qualification of the ECHO-QGP code
To show that our code is well suited to model the evolution of the matter produced in heavy-ion collisions and hence to carry out our study on the development of vorticity in such an environment, we have performed two calculations, referring to an ideal and viscous scenario respectively, providing a very stringent numerical test.
Before describing these tests, it should be pointed out that the vorticities components are to be calculated in Bjorken coordinates, whose metric tensor is , hence they do not all have the same dimension nor they are adimensional as it is desirable (except the thermal vorticity, as it has been emphasized in Sect. II). For a proper comparison it is better to use the orthonormal basis, which involves a factor when the components are considered. Moreover, the cumulative contribution of all components is well described by the invariant modulus, which, for a generic antisymmetric tensor is:
Furthermore, we have always rescaled the T-vorticity by in order to have an adimensional number. Since the T-vorticity has always been determined at the isothermal freezeout, in order to get its actual magnitude, one just needs to multiply it by .
iv.1 T-vorticity for an ideal fluid
Since the fluid is assumed to be uncharged and the initial T-vorticity is vanishing with the BIC, it should be vanishing throughout, according to the discussion in sect. II). However, the discretization of the hydrodynamical equations entails a numerical error, thus the smallness of in an ideal run is a gauge of the quality of the computing method. In fig. 2 we show the mean of the absolute values of the six independent Bjorken components at the freeze-out hypersurface, of the T-vorticity divided by to make it adimensional, as a function of the grid resolution (the boundaries in being fixed) 222It should be pointed out that, throughout this work, by mean values of the vorticities we mean simple averages of the (possibly rescaled by ) Bjorken components over the freezeout hypersurface without geometrical cell weighting. Therefore, the plotted mean values have no physical meaning and they should be taken as descriptive numbers which are related to the global features of vorticity components at the freeze-out. As it is expected, the normalized T-vorticity decreases as the resolution improves.
Because of the relation (13), the residual value at our best spatial resolution of 0.15 fm can be taken as a numerical error for later calculations of the thermal vorticity.
iv.2 Gubser flow
A very useful test for the validation of a numerical code of relativistic dissipative hydrodynamics is the explicit solution of Israel-Stewart theory of a Bjorken flow with an azimuthally symmetric radial expansion gub1 (); gub2 (); den1 (); den2 (), the so-called Gubser flow. Indeed, this solution provides a highly non-trivial theoretical benchmark.
For the sake of clarity, we briefly summarize the main steps leading to the analytical solution, to be compared with the numerical computation. In the case of a conformal fluid, with EOS, the invariance for scale transformations sets the terms entering the second-order viscous hydrodynamic equations. The additional requests of azimuthal and longitudinal-boost invariance, constrain the solution of the hydrodynamic equations, which has to be invariant under transformations. To start with, one defines a modified space-time metric as follows (with usual Bjorken coordinates, being the spacetime rapidity):
which can be viewed as a rescaling of the metric tensor:
It can be shown that is the invariant spacetime interval of , where is the three-dimensional de Sitter space and refers to the rapidity coordinate. It is then convenient to perform a coordinate transformation ( is an arbitrary parameter setting an energy scale for the solution once one goes back to physical dimensionful coordinates)
after which the rescaled spacetime element reads
The full symmetry of the problem is now manifest. and refer to the usual invariance for longitudinal boosts and inversion, while reflects the spherical symmetry of the rescaled metric tensor in the new coordinates. In Gubser coordinates the fluid is at rest:
The corresponding flow in Minkowski space can be obtained taking into account both the rescaling of the metric and the change of coordinates
where and . Other quantities such as the temperature or the viscous tensors require the solution of the following set of hydrodynamic equations (their most general form actually admits further terms that were derived for a system of massless particles in refs. jai1 (); jai2 ()), valid for the case of a conformal fluid with :
In the case of the Gubser flow in Eq. (28), due to the traceless and transverse conditions and , one has simply to solve the two equations ()
and (, being the ratio dimensionless)
The solution can be then mapped back to Minkowski space through the formulae:
In fig. 4 we show the comparison between the Gubser analytical solution and our numerical computation for the temperature and the components , and of the viscous stress tensor respectively, at different times. The initial energy density profile was taken from the exact Gubser solution at the time fm/c. The simulation has been performed with a grid of 0.025 fm in space and 0.001 fm in time. The shear viscosity to entropy density ratio was set to , while the shear relaxation time is . The energy scale is set to fm. As it can be seen, the agreement is excellent up to late times.
iv.3 T-vorticity for a viscous fluid
Unlike for an ideal uncharged fluid, T-vorticity can be generated in a viscous uncharged fluid even if it is initially vanishing. Thus, the T-vorticity can be used as a tool to estimate the numerical viscosity of the code in the ideal mode by extrapolating the viscous runs.
A comment is in order here. In general, in addition to standard truncation errors due to finite-difference interpolations, all shock-capturing upwind schemes are known to introduce numerical approximations that behave roughly as a dissipative effects, especially in the simplified solution to the Riemann problems at cell interfaces leveque (). It is therefore important to check whether the code is not introducing, for a given resolution, numerical errors which are larger than the effects induced by the physics. We refer to the global numerical errors generically as numerical viscosity.
We have thus calculated the T-vorticity for different physical viscosities (in fact ratios), in order to provide an upper bound for the numerical viscosity of ECHO-QGP in the ideal mode. The mean value of the T-vorticity is shown in fig. 5 and its extrapolation to zero occurs when which is a very satisfactory value, comparable with the one obtained in ref. karpe (). The good performance is due to the use of high-order reconstruction methods that are able to compensate for the highly diffusive two-wave Riemann solver employed echoqgp ().
V Directed flow, angular momentum and thermal vorticity
With the initial conditions reported at the end of the Sect. III we have calculated the directed flow of pions (both charged states) at the freezeout and compared it with the STAR data for charged particles collected in the centrality interval 40-80% star08 (). Directed flow is an important observable for several reasons. Recently, it has been studied at lower energy steinh () with a hybrid fluid- transport model (see also ref. ivanov ()). At GeV, it has been calculated with an ideal 3+1D hydro code first by Bozek bozek (). Herein, we extend the calculation to the viscous regime.
The amount of generated directed flow at the freezeout depends of course on the initial conditions, particularly on the parameter (see Sect. III), as shown in fig. 6. The directed flow also depends on as shown in fig. 7 and could then be used to measure the viscosity of the QCD plasma along with other azimuthal anisotropy coefficients. It should be pointed out that, apparently, the directed flow can be reproduced by our hydrodynamical calculation only for .
The dependence of on and makes it possible to adjust the parameter for a given value. This adjustment cannot be properly called a precision fit because, as we have mentioned in the Introduction, several effects in the comparison between data and calculations have been deliberately neglected in this work. However, since our aim was to obtain a somewhat realistic evaluation of the vorticities, we have chosen the value of for which we obtain the best agreement between our calculated pion and the measured for charged particles in the central rapidity region. For the fixed (approximately twice the conjectured universal lower bound) the corresponding best value of turns out to be (see fig. 8).
It is worth discussing more in detail an interesting relationship between the value of the parameter and that of a conserved physical quantity, the angular momentum of the plasma, which, for BIC is given by the integral (see Appendix A for the derivation):
Since controls the asymmetry of the energy density distribution in the plane, one expects that will vary as a function of . Indeed, if the energy density profile is symmetric in , the integral in eq. (32) vanishes. Yet, for any finite , the profile (20) is not symmetric and (looking at the definition of and it can be realized that only in the limit the energy density profile becomes symmetric). The dependence of the angular momentum on with all the initial parameters kept fixed is shown in fig. 9. For the value it turns out to be around in units.
It is also interesting to estimate an upper bound on the angular momentum of the plasma by evaluating the angular momentum of the overlap region of the two colliding nuclei. This can be done by trying to extend the simple formula for two sharp spheres. In our conventional reference frame, the initial angular momentum of the nuclear overlap region is directed along the axis with negative value and can be written as:
where are the thickness functions like in eq. 18 and
is the function which extends the simple product of two functions used for the overlap of two sharp spheres. Note that the is 1 for full overlap (=0) and implies a vanishing angular momentum for very large (see fig. 10) (see also ref. vovchenko ()).
At fm the above angular momentum is about in units. This means that, with the current parametrization of the initial conditions, for that impact parameter about 89% of the angular momentum is retained by the hydrodynamical plasma while the rest is possibly taken away by the corona particles.
With the final set of parameters, we have calculated the thermal vorticity . As it has been mentioned in Sect. II, this vorticity is adimensional in cartesian coordinates) and it is constant at global thermodynamical equilibrium becacov (), e.g. for a globally rotating fluid with a rigid velocity field. In relativistic nuclear collisions we are far from such a situation, nevertheless some thermal vorticity can be generated, both in the ideal and viscous case. This is shown in figs. 11 and 12.
It can be seen that the generated amount of thermal vorticity has some non-trivial dependence on the viscosity. Particularly, as it is apparent from fig. 12, the component - which is directed along the initial angular momentum - has a non-vanishing mean value whose magnitude significantly increases with increasing viscosity. Its map at the freezeout, for a fixed value of the coordinate , is shown in fig. 13 where it can be seen that it attains a top (negative) value of about 0.05 corresponding to a kinematical vorticity, at the freezeout temperature of 130 MeV, of about 0.033 /fm . In this respect, the Quark Gluon Plasma would be the fluid with the highest vorticity ever made in a terrestrial laboratory. However, the mean value of this component at the same value of is of the order of , that is about ten times less than its peak value, as shown in fig. 12. This mean thermal vorticity is the consistently lower than the one estimated in ref. beca2 () (about 0.05) with the model described in refs. csernai1 (); csernai2 () implying an initial non-vanishing transverse kinematical and thermal vorticity . This reflects in a quite low value of the polarization of baryons, as it will be shown in the next section.
As it has been mentioned in the Introduction, vorticity can result in the polarization of particles in the final state. The relation between the polarization vector of a spin particle and thermal vorticity in a relativistic fluid was derived in ref. beca1 () and reads:
where is the Fermi-Dirac-Juttner distribution function (12) and the integration is over the freeze-out hypersurface . The interesting feature of this relation is that it makes it possible to obtain an indirect measurement of the mean thermal vorticity at the freezeout by measuring the polarization of some hadron. For instance, the polarization of baryons, as it is well known, can be determined with the analysis of the angular distribution of its decay products, because of parity violation. The polarization pattern depends on the momentum of the decaying particle, as it is apparent from eq. (34).
The formula (34) makes sense only if the components of the integrand are Minkowskian, as an integrated vector field yields a vector only if the tangent spaces are the same at each point. Before summing over the freezeout hypersurface we have then transformed the components of the thermal vorticity from Bjorken coordinates to Minkowskian by using the known rules. The thus obtained polarization vector is the one in the collision frame. However, the polarization vector which is measurable is the one in the decaying particle rest frame which can be obtained by means of the Lorentz transformations:
In figure 14 we show the polarization vector components, as well as its modulus, as a function of the transverse momentum for expected under the assumptions of local thermodynamical equilibrium for the spin degrees of freedom maintained till kinetic freezeout. It can be seen that the polarization vector has quite an assorted pattern, with an overall magnitude (see fig. 14, panel (a)) hardly exceeding 1% at momenta around 4 GeV. As expected, the component is predominantly negative, oriented along the initial angular momentum vector and a magnitude of the order of 0.1%. Indeed, the main contribution to the polarization stems from the longitudinal component , with a maximum and minumum along the bisector .
The obtained polarization values are - as expected - consistently smaller than those estimated in ref. beca2 () (of the order of several percent with a top value of 8-9%) with the already mentioned initial conditions used in refs. csernai1 (); csernai2 (). This is a consequence of the much lower value of the implied thermal vorticity, as discussed in the previous section. Also, the pattern is remarkably different, with different location of maxima and minima.
Vii Conclusions, discussion and outlook
To summarize, we have calculated the vorticities developed in peripheral ( fm) nuclear collisions at GeV ( fm) with the most commonly used initial conditions in the Bjorken hydrodynamical scheme, by using the code ECHO-QGP implementing second-order, causal, relativistic dissipative hydrodynamics. An extensive testing of the high accuracy and very low numerical diffusion properties of the code has been carried out, followed by long-time simulations (up to fm/c) of the so-called viscous Gubser flow, a stringent test of numerical implementations of Israel-Stewart theory in Bjorken coordinates.
We have found that the magnitude of the component of the thermal vorticity at freezeout can be as large as and yet its mean value is not large enough to produce a polarization of hyperons much larger than 1%, which is a consistently lower estimate in comparison with other recent calculations based on different initial conditions. We have found that the magnitude of directed flow, at this energy, has an interestingly sizeable dependence on both the shear viscosity and the longitudinal energy density profile asymmetry parameter which in turn governs the amount of initial angular momentum retained by the plasma.
The fact that in 3+1D the plasma needs to have an initial angular momentum in order to reproduce the observed directed flow raises the question whether the Bjorken initial condition is a compelling one or, instead, the same angular momentum can be obtained with a non trivial and with a suitable change of the energy density profile. For a testing purpose, we have run ECHO-QGP with an initial profile:
which meets the causality constraint (see Appendix B). It is found that the directed flow is very sensitive to an initial . For a small positive value of the parameter fm corresponding to a , keeping all other parameters fixed, the directed flow exhibits two slight wiggles around midrapidity (see fig. 15) which are not seen in the data. For a very small negative value of the parameter fm, corresponding to , the directed flow increases while approximately keeping the same shape as for around midrapidity. However, more detailed studies are needed to determine whether a non-vanishing initial flow velocity is compatible with the experimental observables.
We plan to extend this kind of calculation to different centralities, different energies and with initial state fluctuations in order to determine the possibly best conditions for vorticity formation in relativistic nuclear collisions.
We are grateful to P. Bozek, L. Csernai and Y. Karpenko for very useful comments and suggestions.
APPENDIX A - Angular momentum
The calculation of the total angular momentum of the plasma can be done provided that initial conditions are such that energy density falls off rapidly at large . This condition, which is met by the profile in eq. (20), indeed implies that a boundary exists where the angular momentum density tensor (that is the integrand below) vanishes and the following integral is conserved:
where is any spacelike hypersurface extending over the region where the angular momentum density vanishes. The obvious choice for Bjorken-type initial conditions is the hypersurface .
It should be stressed that a vector (or tensor) integral is meaningful in flat spacetime only if the components are the cartesian ones. Hence, for the hypersurface , the integration variables are conveniently chosen to be the Bjorken ones, but the components of the stress-energy tensor as well as the vector will be cartesian. Since the only non vanishing component of the angular momentum in our conventional reference frame is , orthogonal to the reaction plane, we can write:
Finding the hypersurface measure in cartesian components, but expressed through Bjorken variable, requires some reasoning. First, one has to remind that:
where is the unit vector normal to the hypersurface which is readily found to be (cartesian covariant components):
Now, since Bjorken coordinates are time-orthogonal () and with , the invariant spacetime measure can be factorized into the product of the infinitesimal “time” and the infinitesimal measure of the orthogonal hypersurface :
At the same time:
At the time , the stress-energy tensor is supposedly the ideal one and there is no transverse velocity, so that , while and . Pluggin these expressions into the (42) along with the transformation equation:
one finally gets:
being . In the case of Bjorken initial conditions with and , the eq. (44) boils down to:
APPENDIX B - Causality constraints
The inequality expressing the causality constraint in the hydrodynamical picture of relativistic heavy ion collisions is that the initial longitudinal flow velocity must not exceed the velocity of beam protons (assuming vanishing initial transverse velocity):
By using the transformation rules (APPENDIX A - Angular momentum):
the inequality (47) becomes:
which can be solved for :
The form (36) fulfills the above inequality.