Journal of High Energy Physics

, 2018:181 | Cite as

Evolution of magnetic fields from the 3 + 1 dimensional self-similar and Gubser flows in ideal relativistic magnetohydrodynamics

  • M. Shokri
  • N. SadooghiEmail author
Open Access
Regular Article - Theoretical Physics


Motivated by the recently found realization of the 1 + 1 dimensional Bjorken flow in ideal and nonideal relativistic magnetohydrodynamics (MHD), we use appropriate symmetry arguments, and determine the evolution of magnetic fields arising from the 3 + 1 dimensional self-similar and Gubser flows in an infinitely conductive relativistic fluid (ideal MHD). In the case of the 3 + 1 dimensional self-similar flow, we arrive at a family of solutions, that are related through a differential equation arising from the corresponding Euler equation. To find the magnetic field evolution from the Gubser flow, we solve the MHD equations of a stationary fluid in a conformally flat dS3 × E1 spacetime. The results are then Weyl transformed back into the Minkowski spacetime. In this case, the temporal evolution of the resulting magnetic field is shown to exhibit a transition between an early time 1/t decay to a 1/t3 decay at a late time. Here, t is the time coordinate. Transverse and longitudinal components of the magnetic fields arising from these flows are also found. The latter turns out to be sensitive to the transverse size of the fluid. In contrast to the result arising from the Gubser flow, the radial domain of validity of the magnetic field arising from the self-similar flow is highly restricted. A comparison of the results suggests that the (conformal) Gubser MHD may give a more appropriate qualitative picture of the magnetic field decay in the plasma of quarks and gluons created in heavy ion collisions.


Quark-Gluon Plasma Conformal and W Symmetry Space-Time Symmetries 


Open Access

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  1. [1]
    U. Gürsoy, D. Kharzeev, E. Marcus, K. Rajagopal and C. Shen, Charge-dependent Flow Induced by Magnetic and Electric Fields in Heavy Ion Collisions, Phys. Rev. C 98 (2018) 055201 [arXiv:1806.05288] [INSPIRE].
  2. [2]
    X.-G. Huang, Electromagnetic fields and anomalous transports in heavy-ion collisions — A pedagogical review, Rept. Prog. Phys. 79 (2016) 076302 [arXiv:1509.04073] [INSPIRE].
  3. [3]
    D.E. Kharzeev, L.D. McLerran and H.J. Warringa, The Effects of topological charge change in heavy ion collisions: ‘Event by event P and CP-violation’, Nucl. Phys. A 803 (2008) 227 [arXiv:0711.0950] [INSPIRE].
  4. [4]
    V. Skokov, A. Yu. Illarionov and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A 24 (2009) 5925 [arXiv:0907.1396] [INSPIRE].
  5. [5]
    B.G. Zakharov, Electromagnetic response of quark-gluon plasma in heavy-ion collisions, Phys. Lett. B 737 (2014) 262 [arXiv:1404.5047] [INSPIRE].
  6. [6]
    K. Yagi, T. Hatsuda and Y. Miake, Quark-gluon plasma: from big bang to little bang, Cambridge University Press, Cambridge U.K. (2005).Google Scholar
  7. [7]
    P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium — Ten Years of Progress in Theory and Numerical Simulations of Nuclear Collisions, arXiv:1712.05815 [INSPIRE].
  8. [8]
    V. Roy, S. Pu, L. Rezzolla and D. Rischke, Analytic Bjorken flow in one-dimensional relativistic magnetohydrodynamics, Phys. Lett. B 750 (2015) 45 [arXiv:1506.06620] [INSPIRE].
  9. [9]
    S. Pu, V. Roy, L. Rezzolla and D.H. Rischke, Bjorken flow in one-dimensional relativistic magnetohydrodynamics with magnetization, Phys. Rev. D 93 (2016) 074022 [arXiv:1602.04953] [INSPIRE].
  10. [10]
    J.D. Bjorken, Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region, Phys. Rev. D 27 (1983) 140 [INSPIRE].
  11. [11]
    M. Shokri and N. Sadooghi, Novel self-similar rotating solutions of nonideal transverse magnetohydrodynamics, Phys. Rev. D 96 (2017) 116008 [arXiv:1705.00536] [INSPIRE].
  12. [12]
    E. Stewart and K. Tuchin, Magnetic field in expanding quark-gluon plasma, Phys. Rev. C 97 (2018) 044906 [arXiv:1710.08793] [INSPIRE].
  13. [13]
    V. Roy, S. Pu, L. Rezzolla and D.H. Rischke, Effect of intense magnetic fields on reduced-MHD evolution in \( \sqrt{s_{\mathrm{NN}}}= 200 \) GeV Au+Au collisions, Phys. Rev. C 96 (2017) 054909 [arXiv:1706.05326] [INSPIRE].
  14. [14]
    A. Das, S.S. Dave, P.S. Saumia and A.M. Srivastava, Effects of magnetic field on plasma evolution in relativistic heavy-ion collisions, Phys. Rev. C 96 (2017) 034902 [arXiv:1703.08162] [INSPIRE].
  15. [15]
    L.-G. Pang, G. Endrödi and H. Petersen, Magnetic-field-induced squeezing effect at energies available at the BNL Relativistic Heavy Ion Collider and at the CERN Large Hadron Collider, Phys. Rev. C 93 (2016) 044919 [arXiv:1602.06176] [INSPIRE].
  16. [16]
    S. Pu and D.-L. Yang, Transverse flow induced by inhomogeneous magnetic fields in the Bjorken expansion, Phys. Rev. D 93 (2016) 054042 [arXiv:1602.04954] [INSPIRE].
  17. [17]
    P.F. Kolb and U.W. Heinz, Hydrodynamic description of ultrarelativistic heavy ion collisions, in R.C. Hwa et al. eds., Quark gluon plasma, pp. 634-714, [nucl-th/0305084] [INSPIRE].
  18. [18]
    J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions, Cambridge University Press, Cambridge U.K. (2014), [arXiv:1101.0618] [INSPIRE].
  19. [19]
    W. Busza, K. Rajagopal and W. van der Schee, Heavy Ion Collisions: The Big Picture and the Big Questions, Ann. Rev. Nucl. Part. Sci. 68 (2018) 339 [arXiv:1802.04801] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    T. Csörgö, F. Grassi, Y. Hama and T. Kodama, Simple solutions of relativistic hydrodynamics for cylindrically symmetric systems, Acta Phys. Hung. A 21 (2004) 63 [hep-ph/0204300] [INSPIRE].
  21. [21]
    T. Csörgö, L.P. Csernai, Y. Hama and T. Kodama, Simple solutions of relativistic hydrodynamics for systems with ellipsoidal symmetry, Acta Phys. Hung. A 21 (2004) 73 [nucl-th/0306004] [INSPIRE].
  22. [22]
    T. Csörgö, F. Grassi, Y. Hama and T. Kodama, Simple solutions of relativistic hydrodynamics for longitudinally expanding systems, Acta Phys. Hung. A 21 (2004) 53 [hep-ph/0203204] [INSPIRE].
  23. [23]
    S.S. Gubser, Symmetry constraints on generalizations of Bjorken flow, Phys. Rev. D 82 (2010) 085027 [arXiv:1006.0006] [INSPIRE].
  24. [24]
    S.S. Gubser and A. Yarom, Conformal hydrodynamics in Minkowski and de Sitter spacetimes, Nucl. Phys. B 846 (2011) 469 [arXiv:1012.1314] [INSPIRE].
  25. [25]
    J.D. Bekenstein and E .Oron, New conservation laws in general-relativistic magnetohydrodynamics, Phys. Rev. D 18 (1978) 1809.Google Scholar
  26. [26]
    L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, Oxford U.K. (2013).Google Scholar
  27. [27]
    R. Peschanski and E.N. Saridakis, On an exact hydrodynamic solution for the elliptic flow, Phys. Rev. C 80 (2009) 024907 [arXiv:0906.0941] [INSPIRE].
  28. [28]
    S.-J. Sin, S. Nakamura and S.P. Kim, Elliptic Flow, Kasner Universe and Holographic Dual of RHIC Fireball, JHEP 12 (2006) 075 [hep-th/0610113] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Y. Hatta, B.-W. Xiao and D.-L. Yang, Non-boost-invariant solution of relativistic hydrodynamics in 1+3 dimensions, Phys. Rev. D 93 (2016) 016012 [arXiv:1512.04221] [INSPIRE].
  30. [30]
    Y. Hatta, Analytic approaches to relativistic hydrodynamics, Nucl. Phys. A 956 (2016) 152 [arXiv:1601.04128] [INSPIRE].
  31. [31]
    Y. Hatta and B.-W. Xiao, Building up the elliptic flow: analytical insights, Phys. Lett. B 736 (2014) 180 [arXiv:1405.1984] [INSPIRE].
  32. [32]
    L.D. Landau and E.M. Lifshitz, Fluid mechanics, second edition, Elsevier, Amsterdam Netherlands (1987).Google Scholar
  33. [33]
    A. Zee, Einstein gravity in a nutshell, Princeton University Press, Princeton New Jersey U.S.A. (2013).Google Scholar
  34. [34]
    S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity, John Wiley & Sons, Inc., New York U.S.A. (1972).Google Scholar
  35. [35]
    C. Misner, K.S. Throne, J.A. Wheeler and D.I. Kaiser, Gravitation, W.H. Freeman and Co., San Fransisco U.S.A. (1973).Google Scholar
  36. [36]
    J. Hernandez and P. Kovtun, Relativistic magnetohydrodynamics, JHEP 05 (2017) 001 [arXiv:1703.08757] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    R. Romero, J.M. Marti, J.A. Pons, J.M. Ibáñez and J.A. Miralles, The Exact solution of the Riemann problem in relativistic MHD with tangential magnetic fields, J. Fluid Mech. 544 (2005) 323 [astro-ph/0506527] [INSPIRE].
  38. [38]
    O. DeWolfe, S.S. Gubser, C. Rosen and D. Teaney, Heavy ions and string theory, Prog. Part. Nucl. Phys. 75 (2014) 86 [arXiv:1304.7794] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    M. Gedalin, Linear waves in relativistic anisotropic magnetohydrodynamics Phys. Rev. E 47 (1993)4354.Google Scholar
  40. [40]
    T. Csörgö, G. Kasza, M. Csanád and Z. Jiang, New exact solutions of relativistic hydrodynamics for longitudinally expanding fireballs, Universe 4 (2018) 69 [arXiv:1805.01427] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    V. Roy and S. Pu, Event-by-event distribution of magnetic field energy over initial fluid energy density in \( \sqrt{s_{\mathrm{NN}}} = 200 \) GeV Au-Au collisions, Phys. Rev. C 92 (2015) 064902 [arXiv:1508.03761] [INSPIRE].
  42. [42]
    V. Roy, S. Pu, L. Rezzolla and D.H. Rischke, Effect of intense magnetic fields on reduced-MHD evolution in \( \sqrt{s_{\mathrm{NN}}} = 200 \) GeV Au+Au collisions, Phys. Rev. C 96 (2017) 054909 [arXiv:1706.05326] [INSPIRE].
  43. [43]
    G. Inghirami, L. Del Zanna, A. Beraudo, M. Haddadi Moghaddam, F. Becattini and M. Bleicher, Magneto-hydrodynamic simulations of Heavy Ion Collisions with ECHO-QGP, J. Phys. Conf. Ser. 1024 (2018) 012043 [INSPIRE].
  44. [44]
    G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands and J.-I. Skullerud, Electrical conductivity and charge diffusion in thermal QCD from the lattice, JHEP 02 (2015) 186 [arXiv:1412.6411] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    H. Bantilan, T. Ishii and P. Romatschke, Holographic Heavy-Ion Collisions: Analytic Solutions with Longitudinal Flow, Elliptic Flow and Vorticity, Phys. Lett. B 785 (2018) 201 [arXiv:1803.10774] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsSharif University of TechnologyTehranIran

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