Analytic solution for relativistic transverse flow at the softest point

We obtain an extension of Bjorken’s 1+1 dimensional scaling relativistic flow solution to relativistic transverse velocities with cylindrical symmetry in 1+3 dimension at constant, homogeneous pressure (vanishing sound velocity). This can be the situation during a first order phase transition converting quark matter into hadron matter in relativistic heavy ion collisions. Hydrodynamics often allow for nonrelativistic scaling solutions. Relativistic flow, however, seems to be an exception: besides Bjorken’s 1+1 dimensional ansatz and the spherically symmetric relativistic expansion, no analytical solution is known [1]. In this paper we present an extension of Bjorken’s ansatz [2] for longitudinally and transversally relativistic flow patterns with cylindrical symmetry in 1+3 dimensions. This is an analytical solution of the flow equations of a perfect fluid for physical situations when the sound velocity is zero, c2s = dp/dǫ = 0, with energy density ǫ and pressure p(ǫ). In particular this happens during a first order phase transition, the pressure is constant while the energy density changes (in heavy ion collisons increases and drops again). This should, in principle, be signalled by a vanishing sound velocity. A remnant of this effect in finite size, finite time transitions might be a softest point of the equation of state, where c2s is minimal. In fact, this has been suggested as a signal of phase transition by Shuryak[3], and investigated numerically in several recent works [4, 5]. In the light of this research, the presentation of an analytical solution including relativistic transverse flow is worthwile. For a general equation of state p(ǫ) the analytical solution given in this paper cannot be extended, only a perturbative expansion in terms of mild (v ≪ 1) transverse velocities can be established. Such an approximation has been recently presented in [6]. Nonrelativistic analytic solution has been also given several times, with respect to heavy ions see [7, 8, 9]. The 1+1 dimensional Bjorken flow four-velocity is a normalized, timelike vector. It is natural to choose this as the first of our comoving frame basis vectors (vierbein). The e-mail: [email protected] ; http://sgi30.rmki.kfki.hu∼tsbiro/