# Perturbative Accelerating Solutions of Relativistic Hydrodynamics

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## Abstract

**:**

## 1. Introduction

## 2. Perturbative Solutions of Hydrodynamics

## 3. Perturbations on Top of Hubble-Flow

- The scale variable S fulfills ${u}_{\mu}{\partial}^{\mu}S=0$ with the original flow field.

## 4. A Selected Sub-Class of Perturbative Solutions

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Adcox, K.; Adler, S.S.; Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Al-Jamel, A.; Alexander, J.; Amirikas, R.; Aoki, K.; et al. Formation of dense partonic matter in relativistic nucleus nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration. Nucl. Phys.
**2005**, 757, 184–283. [Google Scholar] [CrossRef] - Adams, J.; Aggarwal, M.M.; Ahammed, Z.; Amonett, J.; Anderson, B.D.; Arkhipkin, D.; Averichev, G.S.; Badyal, S.K.; Bai, Y.; Balewski, J.; et al. Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR collaboration’s critical assessment of the evidence from RHIC collisions. Nucl. Phys.
**2005**, 757, 102–183. [Google Scholar] [CrossRef] - Aamodt, K.; Abrahantes Quintana, A.; Adamová, D.; Adare, A.M.; Aggarwal, M.M.; Aglieri Rinella, G.; Agocs, A.G.; Aguilar Salazar, S.; Ahammed, Z.; Ahmad, N.; et al. Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb–Pb Collisions at $\sqrt{{s}_{NN}}$ = 2.76 TeV. Phys. Lett.
**2011**, 696, 30–39. [Google Scholar] [CrossRef] - Aamodt, K.; Abelev, B.; Abrahantes Quintana, A.; Adamová, D.; Adare, A.M.; Aggarwal, M.M.; Aglieri Rinella, G.; Agocs, A.G.; Aguilar Salazar, S.; Ahammed, Z.; et al. Elliptic flow of charged particles in Pb-Pb collisions at 2.76 TeV. Phys. Rev. Lett.
**2010**, 105, 252302. [Google Scholar] [CrossRef] [PubMed] - Chatrchyan, S.; Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Erö, J.; Fabjan, C.; Friedl, M.; et al. Study of high-pT charged particle suppression in PbPb compared to pp collisions at $\sqrt{{s}_{NN}}$ = 2.76 TeV. Eur. Phys. J. C
**2012**, 72, 1945. [Google Scholar] [CrossRef] - Chatrchyan, S.; Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Erö, J.; Fabjan, C.; Friedl, M.; et al. Measurement of the elliptic anisotropy of charged particles produced in PbPb collisions at nucleon-nucleon center-of-mass energy = 2.76 TeV. Phys. Rev.
**2013**, 87, 014902. [Google Scholar] - Csanád, M.; Nagy, M.; Lökös, S. Exact solutions of relativistic perfect fluid hydrodynamics for a QCD equation of state. Eur. Phys. J. A
**2012**, 48, 173. [Google Scholar] [CrossRef] - Landau, L.D. On the multiparticle production in high-energy collisions. Izv. Akad. Nauk SSSR Ser. Fiz.
**1953**, 17, 51–64. [Google Scholar] - Khalatnikov, I.M. Some problems of relativistic hydrodynamics. Zh. Eksp. Teor. Fiz.
**1954**, 27, 529. [Google Scholar] - Hwa, R.C. Statistical Description of Hadron Constituents as a Basis for the Fluid Model of High-Energy Collisions. Phys. Rev. D
**1974**, 10, 2260. [Google Scholar] [CrossRef] - Bjorken, J.D. Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region. Phys. Rev. D
**1983**, 27, 140–151. [Google Scholar] [CrossRef] - Shen, C.; Qiu, Z.; Song, H.; Bernhard, J.; Bass, S.; Heinz, U. The iEBE-VISHNU code package for relativistic heavy-ion collisions. Comput. Phys. Commun.
**2016**, 199, 61–85. [Google Scholar] [CrossRef] - Pang, L.G.; Petersen, H.; Wang, Q.; Wang, X.N. Vortical Fluid and Λ Spin Correlations in High-Energy Heavy-Ion Collisions. Phys. Rev. Lett.
**2016**, 117, 192301. [Google Scholar] [CrossRef] [PubMed] - Weller, R.D.; Romatschke, P. One fluid to rule them all: Viscous hydrodynamic description of event-by-event central p+p, p+Pb and Pb+Pb collisions at $\sqrt{s}$ = 5.02 TeV. Phys. Lett. B
**2017**, 774, 351–356. [Google Scholar] [CrossRef] - Csörgo, T.; Csernai, L.P.; Hama, Y.; Kodama, T. Simple solutions of relativistic hydrodynamics for systems with ellipsoidal symmetry. Heavy Ion Phys. A
**2004**, 21, 73–84. [Google Scholar] [CrossRef] - Csörgo, T.; Nagy, M.I.; Csanád, M. A new family of simple solutions of perfect fluid hydrodynamics. Phys. Lett. B
**2008**, 663, 306–311. [Google Scholar] [CrossRef] - Nagy, M.I.; Csörgo, T.; Csanád, M. Detailed description of accelerating, simple solutions of relativistic perfect fluid hydrodynamics. Phys. Rev. C
**2008**, 77, 024908. [Google Scholar] [CrossRef] - Borshch, M.S.; Zhdanov, V.I. Exact Solutions of the Equations of Relativistic Hydrodynamics Representing Potential Flows. Symmetry Integrab. Geom. Methods Appl.
**2007**, 3, 116. [Google Scholar] [CrossRef] - Pratt, S. A co-moving coordinate system for relativistic hydrodynamics. Phys. Rev. C
**2007**, 75, 024907. [Google Scholar] [CrossRef] - Gubser, S.S. Symmetry constraints on generalizations of Bjorken flow. Phys. Rev. D
**2010**, 82, 085027. [Google Scholar] [CrossRef] - Csanád, M.; Szabó, A. Multipole solution of hydrodynamics and higher order harmonics. Phys. Rev. C
**2014**, 90, 054911. [Google Scholar] [CrossRef] - Csanád, M.; Vargyas, M. Observables from a solution of 1+3 dimensional relativistic hydrodynamics. Eur. Phys. J. A
**2010**, 44, 473–478. [Google Scholar] [CrossRef] - Csanád, M.; Májer, I. Equation of state and initial temperature of quark gluon plasma at RHIC. Central Eur. J. Phys.
**2012**, 10, 850–857. [Google Scholar] [CrossRef] - Shi, S.; Liao, J.; Zhuang, P. “Ripples” on a relativistically expanding fluid. Phys. Rev. C
**2014**, 90, 064912. [Google Scholar] [CrossRef]

**Figure 1.**The perturbed flow field component (${u}_{x}+\delta {u}_{x}$) is shown in the left plot as a function of x, for $\tau =6$ fm/c (the other parameters are given in the text). The right plot indicates the relative change $({u}_{x}+\delta {u}_{x})/{u}_{x}$ for various $\delta $ and c values. The bottom plot shows the flow perturbation field $(\delta {u}_{x},\delta {u}_{y})$ in the transverse plane, for various proper-time values.

**Figure 2.**The perturbed pressure $p+\delta p$ is shown in the left plot as a function of x, for $\tau =6$ fm/c (the other parameters are given in the text). The right plot indicates the relative change $(p+\delta p)/p$ for various $\delta $ and c values. The bottom plot shows the pressure perturbation $\delta p$ in the transverse plane, for various proper-time values.

**Figure 3.**The perturbed density $n+\delta n$ is shown in the left plot as a function of x, for $\tau =6$ fm/c (the other parameters are given in the text). The right plot indicates the relative change $(n+\delta n)/n$ for various $\delta $ and c values.

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Kurgyis, B.; Csanád, M.
Perturbative Accelerating Solutions of Relativistic Hydrodynamics. *Universe* **2017**, *3*, 84.
https://doi.org/10.3390/universe3040084

**AMA Style**

Kurgyis B, Csanád M.
Perturbative Accelerating Solutions of Relativistic Hydrodynamics. *Universe*. 2017; 3(4):84.
https://doi.org/10.3390/universe3040084

**Chicago/Turabian Style**

Kurgyis, Bálint, and Máté Csanád.
2017. "Perturbative Accelerating Solutions of Relativistic Hydrodynamics" *Universe* 3, no. 4: 84.
https://doi.org/10.3390/universe3040084