Skip to main content

Shock waves in Lifshitz-like spacetimes

A preprint version of the article is available at arXiv.

Abstract

We construct shock waves for Lifshitz-like geometries in four- and fivedimensional effective theories as well as in D3-D7 and D4-D6 brane systems. The solutions to the domain wall profile equations are found. Further, the study makes a connection with the implications for the quark-gluon plasma formation in heavy-ion collisions. According to the holographic approach, the multiplicity of particles produced in heavy-ion collisions can be estimated by the area of the trapped surface formed in shock wave collisions. We calculate the areas of trapped surfaces in the geometry of two colliding Lifshitz domain walls. Our estimates show that for five-dimensional cases with certain values of the critical exponent the dependence of multiplicity on the energy of colliding ions is rather close to the experimental data \( \mathrm{\mathcal{M}} \)s 0.15 observed at RHIC and LHC.

References

  1. [1]

    J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  3. [3]

    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions, arXiv:1101.0618 [INSPIRE].

  5. [5]

    I.Ya. Aref’eva, Holographic approach to quark-gluon plasma in heavy ion collisions, Phys. Usp. 57 (2014) 527.

    Article  Google Scholar 

  6. [6]

    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].

    ADS  Article  Google Scholar 

  7. [7]

    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  8. [8]

    R.M. Hornreich, The Lifshitz point: Phase diagrams and critical behavior, J. Magn. Magn. Mater. 15-18 (1980) 387.

    Article  Google Scholar 

  9. [9]

    H.W. Diehl, Critical behavior at M-Axial Lifshitz points, Act. phys. slov. 52 (2002) 271 [arXiv:cond-mat/0205284].

    Google Scholar 

  10. [10]

    I.R. Yukhnovskii, Phase Transitions of the Second Order - Collective Variables Method, World Scientific, 1987.

  11. [11]

    L.D. Landau, E.M. Lifshitz, Statistical Physics, vol. 5, 3rd ed., Butterworth-Heinemann, 1980.

  12. [12]

    C. Hoyos and P. Koroteev, On the Null Energy Condition and Causality in Lifshitz Holography, Phys. Rev. D 82 (2010) 084002 [Erratum ibid. D 82 (2010) 109905] [arXiv:1007.1428] [INSPIRE].

  13. [13]

    K. Copsey and R. Mann, Pathologies in Asymptotically Lifshitz Spacetimes, JHEP 03 (2011) 039 [arXiv:1011.3502] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    G.T. Horowitz and B. Way, Lifshitz Singularities, Phys. Rev. D 85 (2012) 046008 [arXiv:1111.1243] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    N. Bao, X. Dong, S. Harrison and E. Silverstein, The Benefits of Stress: Resolution of the Lifshitz Singularity, Phys. Rev. D 86 (2012) 106008 [arXiv:1207.0171] [INSPIRE].

    ADS  Google Scholar 

  16. [16]

    S. Harrison, S. Kachru and H. Wang, Resolving Lifshitz Horizons, JHEP 02 (2014) 085 [arXiv:1202.6635] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    A. Donos and J.P. Gauntlett, Lifshitz Solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala, Lifshitz solutions in supergravity and string theory, JHEP 12 (2010) 047 [arXiv:1009.3445] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].

  21. [21]

    U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. [22]

    R.B. Mann, Lifshitz Topological Black Holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  23. [23]

    G. Bertoldi, B.A. Burrington and A. Peet, Black Holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  24. [24]

    E.J. Brynjolfsson, U.H. Danielsson, L. Thorlacius and T. Zingg, Holographic Superconductors with Lifshitz Scaling, J. Phys. A 43 (2010) 065401 [arXiv:0908.2611] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  25. [25]

    J. Tarrio and S. Vandoren, Black holes and black branes in Lifshitz spacetimes, JHEP 09 (2011) 017 [arXiv:1105.6335] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    S.A. Hartnoll, Horizons, holography and condensed matter, arXiv:1106.4324 [INSPIRE].

  27. [27]

    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. [28]

    S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  29. [29]

    C. Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis and R. Meyer, Effective Holographic Theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  30. [30]

    B. Gouteraux and E. Kiritsis, Generalized Holographic Quantum Criticality at Finite Density, JHEP 12 (2011) 036 [arXiv:1107.2116] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  31. [31]

    L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang, Aspects of holography for theories with hyperscaling violation, JHEP 06 (2012) 041 [arXiv:1201.1905] [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    S.S. Pal, Anisotropic gravity solutions in AdS/CMT, arXiv:0901.0599 [INSPIRE].

  34. [34]

    T. Azeyanagi, W. Li and T. Takayanagi, On String Theory Duals of Lifshitz-like Fixed Points, JHEP 06 (2009) 084 [arXiv:0905.0688] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    D. Mateos and D. Trancanelli, The anisotropic N = 4 super Yang-Mills plasma and its instabilities, Phys. Rev. Lett. 107 (2011) 101601 [arXiv:1105.3472] [INSPIRE].

    ADS  Article  Google Scholar 

  36. [36]

    D. Mateos and D. Trancanelli, Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma, JHEP 07 (2011) 054 [arXiv:1106.1637] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  37. [37]

    A. Rebhan and D. Steineder, Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma, Phys. Rev. Lett. 108 (2012) 021601 [arXiv:1110.6825] [INSPIRE].

    ADS  Article  Google Scholar 

  38. [38]

    K.B. Fadafan and H. Soltanpanahi, Energy loss in a strongly coupled anisotropic plasma, JHEP 10 (2012) 085 [arXiv:1206.2271] [INSPIRE].

    ADS  Article  Google Scholar 

  39. [39]

    M. Chernicoff, D. Fernandez, D. Mateos and D. Trancanelli, Drag force in a strongly coupled anisotropic plasma, JHEP 08 (2012) 100 [arXiv:1202.3696] [INSPIRE].

    ADS  Article  Google Scholar 

  40. [40]

    D. Giataganas, Probing strongly coupled anisotropic plasma, JHEP 07 (2012) 031 [arXiv:1202.4436] [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    M. Chernicoff, D. Fernandez, D. Mateos and D. Trancanelli, Jet quenching in a strongly coupled anisotropic plasma, JHEP 08 (2012) 041 [arXiv:1203.0561] [INSPIRE].

    ADS  Article  Google Scholar 

  42. [42]

    K.B. Fadafan, D. Giataganas and H. Soltanpanahi, The Imaginary Part of the Static Potential in Strongly Coupled Anisotropic Plasma, JHEP 11 (2013) 107 [arXiv:1306.2929] [INSPIRE].

    ADS  Article  Google Scholar 

  43. [43]

    S. Chakrabortty, S. Chakraborty and N. Haque, Brownian motion in strongly coupled, anisotropic Yang-Mills plasma: A holographic approach, Phys. Rev. D 89 (2014) 066013 [arXiv:1311.5023] [INSPIRE].

    ADS  Google Scholar 

  44. [44]

    D. Giataganas and H. Soltanpanahi, Heavy Quark Diffusion in Strongly Coupled Anisotropic Plasmas, JHEP 06 (2014) 047 [arXiv:1312.7474] [INSPIRE].

    ADS  Article  Google Scholar 

  45. [45]

    L. Cheng, X.-H. Ge and S.-J. Sin, Anisotropic plasma with a chemical potential and scheme-independent instabilities, Phys. Lett. B 734 (2014) 116 [arXiv:1404.1994] [INSPIRE].

    ADS  Article  Google Scholar 

  46. [46]

    A. Rebhan and D. Steineder, Probing Two Holographic Models of Strongly Coupled Anisotropic Plasma, JHEP 08 (2012) 020 [arXiv:1205.4684] [INSPIRE].

    ADS  Article  Google Scholar 

  47. [47]

    R.A. Janik and P. Witaszczyk, Towards the description of anisotropic plasma at strong coupling, JHEP 09 (2008) 026 [arXiv:0806.2141] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. [48]

    S. Pal, More gravity solutions of AdS/CMT, arXiv:0809.1756 [INSPIRE].

  49. [49]

    S.S. Gubser, S.S. Pufu and A. Yarom, Entropy production in collisions of gravitational shock waves and of heavy ions, Phys. Rev. D 78 (2008) 066014 [arXiv:0805.1551] [INSPIRE].

    ADS  Google Scholar 

  50. [50]

    J.L. Albacete, Y.V. Kovchegov and A. Taliotis, Modeling Heavy Ion Collisions in AdS/CFT, JHEP 07 (2008) 100 [arXiv:0805.2927] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  51. [51]

    L. Álvarez-Gaumé, C. Gomez, A. Sabio Vera, A. Tavanfar and M.A. Vazquez-Mozo, Critical formation of trapped surfaces in the collision of gravitational shock waves, JHEP 02 (2009) 009 [arXiv:0811.3969] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  52. [52]

    P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 102 (2009) 211601 [arXiv:0812.2053] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  53. [53]

    S. Lin and E. Shuryak, Grazing Collisions of Gravitational Shock Waves and Entropy Production in Heavy Ion Collision, Phys. Rev. D 79 (2009) 124015 [arXiv:0902.1508] [INSPIRE].

    ADS  Google Scholar 

  54. [54]

    I.Y. Aref’eva, A.A. Bagrov and E.A. Guseva, Critical Formation of Trapped Surfaces in the Collision of Non-expanding Gravitational Shock Waves in de Sitter Space-Time, JHEP 12 (2009) 009 [arXiv:0905.1087] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  55. [55]

    P.M. Chesler and L.G. Yaffe, Holography and colliding gravitational shock waves in asymptotically AdS 5 spacetime, Phys. Rev. Lett. 106 (2011) 021601 [arXiv:1011.3562] [INSPIRE].

    ADS  Article  Google Scholar 

  56. [56]

    I.Y. Arefeva, A.A. Bagrov and E.O. Pozdeeva, Holographic phase diagram of quark-gluon plasma formed in heavy-ions collisions, JHEP 05 (2012) 117 [arXiv:1201.6542] [INSPIRE].

    ADS  Article  Google Scholar 

  57. [57]

    E. Kiritsis and A. Taliotis, Multiplicities from black-hole formation in heavy-ion collisions, JHEP 04 (2012) 065 [arXiv:1111.1931] [INSPIRE].

    ADS  Article  Google Scholar 

  58. [58]

    I.Y. Aref’eva, E.O. Pozdeeva and T.O. Pozdeeva, Holographic estimation of multiplicity and membranes collision in modified spaces AdS 5, Theor. Math. Phys. 176 (2013) 861 [arXiv:1401.1180] [INSPIRE].

    Article  MATH  Google Scholar 

  59. [59]

    V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].

    ADS  Google Scholar 

  60. [60]

    J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  61. [61]

    R. Callan, J.-Y. He and M. Headrick, Strong subadditivity and the covariant holographic entanglement entropy formula, JHEP 06 (2012) 081 [arXiv:1204.2309] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  62. [62]

    I.Y. Arefeva and I.V. Volovich, On Holographic Thermalization and Dethermalization of quark-gluon Plasma, arXiv:1211.6041 [INSPIRE].

  63. [63]

    V. Keranen, E. Keski-Vakkuri and L. Thorlacius, Thermalization and entanglement following a non-relativistic holographic quench, Phys. Rev. D 85 (2012) 026005 [arXiv:1110.5035] [INSPIRE].

    ADS  Google Scholar 

  64. [64]

    K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [hep-th/9408169] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  65. [65]

    I.Y. Aref’eva, A. A. Bagrov, Formation of trapped surfaces in the collision of nonexpanding gravitational shock waves in an AdS4 spacetime, Theor. Math. Phys. 161 (2010) 1643 [INSPIRE].

    Google Scholar 

  66. [66]

    M. Hotta and M. Tanaka, Shock wave geometry with nonvanishing cosmological constant, Class. Quant. Grav. 10 (1993) 307 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  67. [67]

    I.Ya. Aref’eva, A.A. Bagrov, L.V. Joukovskaya, Several aspects of applying distributions to analysis of gravitational shock waves in general relativity, St. Petersburg Math. J. 22 (2011)337.

    MathSciNet  Article  MATH  Google Scholar 

  68. [68]

    D.S. Ageev and I.Y. Aref’eva, Holographic Thermalization in Quark Confining Background, arXiv:1409.7558 [INSPIRE].

  69. [69]

    T. Ortin, Gravity and Strings, Cambridge University, Cambridge University Press, 2004.

  70. [70]

    J. Gath, J. Hartong, R. Monteiro and N.A. Obers, Holographic Models for Theories with Hyperscaling Violation, JHEP 04 (2013) 159 [arXiv:1212.3263] [INSPIRE].

    ADS  Article  Google Scholar 

  71. [71]

    K. Narayan, On Lifshitz scaling and hyperscaling violation in string theory, Phys. Rev. D 85 (2012) 106006 [arXiv:1202.5935] [INSPIRE].

    ADS  Google Scholar 

  72. [72]

    H. Braviner, R. Gregory and S.F. Ross, Flows involving Lifshitz solutions, Class. Quant. Grav. 28 (2011) 225028 [arXiv:1108.3067] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  73. [73]

    J.T. Liu and Z. Zhao, Holographic Lifshitz flows and the null energy condition, arXiv:1206.1047 [INSPIRE].

  74. [74]

    J. Bhattacharya, S. Cremonini and B. Goutéraux, Intermediate scalings in holographic RG flows and conductivities, JHEP 02 (2015) 035 [arXiv:1409.4797] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  75. [75]

    C. Hoyos, B.S. Kim and Y. Oz, Lifshitz Hydrodynamics, JHEP 11 (2013) 145 [arXiv:1304.7481] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anastasia A. Golubtsova.

Additional information

ArXiv ePrint: 1410.4595

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aref’eva, I.Y., Golubtsova, A.A. Shock waves in Lifshitz-like spacetimes. J. High Energ. Phys. 2015, 11 (2015). https://doi.org/10.1007/JHEP04(2015)011

Download citation

Keywords

  • Holography and quark-gluon plasmas
  • Gauge-gravity correspondence
  • Intersecting branes models