Abstract
The topology-changing transition between black strings and black holes localized in a Kaluza-Klein circle is investigated in an expansion in the inverse of the number of dimensions D. Performing a new kind of large-D scaling reduces the problem to a Ricci flow of the near-horizon geometry as it varies along the circle direction. The flows of interest here simplify to a non-linear logarithmic diffusion equation, with solutions known in the literature which are interpreted as the smoothed conifold geometries involved in the transition, namely, split and fused cones, which connect to black holes and non-uniform black strings away from the conical region. Our study demonstrates the adaptability of the 1/D expansion to deal with all the regimes and aspects of the static black hole/black string system, and provides another instance of the manner in which the large D limit reduces the task of solving Einstein’s equations to a simpler but compelling mathematical problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Kol, The phase transition between caged black holes and black strings: a review, Phys. Rept.422 (2006) 119 [hep-th/0411240] [INSPIRE].
T. Harmark, V. Niarchos and N.A. Obers, Instabilities of black strings and branes, Class. Quant. Grav.24 (2007) R1 [hep-th/0701022] [INSPIRE].
G.T. Horowitz ed., Black holes in higher dimensions, Cambridge Univ. Press, Cambridge, U.K. (2012) [INSPIRE].
R. Gregory, The Gregory-Laflamme instability, in Black holes in higher dimensions, G.T. Horowitz ed., (2012), pg. 29 [arXiv:1107.5821] [INSPIRE].
L. Lehner and F. Pretorius, Final state of Gregory-Laflamme instability, in Black holes in higher dimensions, G.T. Horowitz ed., (2012), pg. 44 [arXiv:1106.5184] [INSPIRE].
G.T. Horowitz and T. Wiseman, General black holes in Kaluza-Klein theory, in Black holes in higher dimensions, G.T. Horowitz ed., (2012), pg. 69 [arXiv:1107.5563] [INSPIRE].
T. Wiseman, Numerical construction of static and stationary black holes, in Black holes in higher dimensions, G.T. Horowitz ed., (2012), pg. 233 [arXiv:1107.5513] [INSPIRE].
R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett.70 (1993) 2837 [hep-th/9301052] [INSPIRE].
L. Lehner and F. Pretorius, Black strings, low viscosity fluids and violation of cosmic censorship, Phys. Rev. Lett.105 (2010) 101102 [arXiv:1006.5960] [INSPIRE].
R. Emparan, M. Martínez and M. Zilhao, Black hole fusion in the extreme mass ratio limit, Phys. Rev.D 97 (2018) 044004 [arXiv:1708.08868] [INSPIRE].
B. Kol, Topology change in general relativity and the black hole black string transition, JHEP10 (2005) 049 [hep-th/0206220] [INSPIRE].
R. Emparan and N. Haddad, Self-similar critical geometries at horizon intersections and mergers, JHEP10 (2011) 064 [arXiv:1109.1983] [INSPIRE].
B. Kol and T. Wiseman, Evidence that highly nonuniform black strings have a conical waist, Class. Quant. Grav.20 (2003) 3493 [hep-th/0304070] [INSPIRE].
H. Kudoh and T. Wiseman, Connecting black holes and black strings, Phys. Rev. Lett.94 (2005) 161102 [hep-th/0409111] [INSPIRE].
V. Asnin, B. Kol and M. Smolkin, Analytic evidence for continuous self similarity of the critical merger solution, Class. Quant. Grav.23 (2006) 6805 [hep-th/0607129] [INSPIRE].
R. Emparan, P. Figueras and M. Martínez, Bumpy black holes, JHEP12 (2014) 072 [arXiv:1410.4764] [INSPIRE].
M. Kalisch and M. Ansorg, Pseudo-spectral construction of non-uniform black string solutions in five and six spacetime dimensions, Class. Quant. Grav.33 (2016) 215005 [arXiv:1607.03099] [INSPIRE].
M. Kalisch, S. Möckel and M. Ammon, Critical behavior of the black hole/black string transition, JHEP08 (2017) 049 [arXiv:1706.02323] [INSPIRE].
B. Cardona and P. Figueras, Critical Kaluza-Klein black holes and black strings in D = 10, JHEP11 (2018) 120 [arXiv:1806.11129] [INSPIRE].
M. Ammon, M. Kalisch and S. Moeckel, Notes on ten-dimensional localized black holes and deconfined states in two-dimensional SYM, JHEP11 (2018) 090 [arXiv:1806.11174] [INSPIRE].
V. Asnin, D. Gorbonos, S. Hadar, B. Kol, M. Levi and U. Miyamoto, High and low dimensions in the black hole negative mode, Class. Quant. Grav.24 (2007) 5527 [arXiv:0706.1555] [INSPIRE].
R. Emparan, R. Suzuki and K. Tanabe, The large D limit of general relativity, JHEP06 (2013) 009 [arXiv:1302.6382] [INSPIRE].
R. Emparan, D. Grumiller and K. Tanabe, Large-D gravity and low-D strings, Phys. Rev. Lett.110 (2013) 251102 [arXiv:1303.1995] [INSPIRE].
R. Emparan, R. Suzuki and K. Tanabe, Decoupling and non-decoupling dynamics of large D black holes, JHEP07 (2014) 113 [arXiv:1406.1258] [INSPIRE].
R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe and T. Tanaka, Effective theory of black holes in the 1/D expansion, JHEP06 (2015) 159 [arXiv:1504.06489] [INSPIRE].
R. Emparan, R. Suzuki and K. Tanabe, Evolution and end point of the black string instability: large D solution, Phys. Rev. Lett.115 (2015) 091102 [arXiv:1506.06772] [INSPIRE].
S. Bhattacharyya, A. De, S. Minwalla, R. Mohan and A. Saha, A membrane paradigm at large D, JHEP04 (2016) 076 [arXiv:1504.06613] [INSPIRE].
S. Bhattacharyya, M. Mandlik, S. Minwalla and S. Thakur, A charged membrane paradigm at large D, JHEP04 (2016) 128 [arXiv:1511.03432] [INSPIRE].
R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom.17 (1982) 255.
P. Topping, Lectures on the Ricci flow, http://homepages.warwick.ac.uk/~maseq/RFnotes.html, Cambridge University Press, Cambridge, U.K. (2006).
R. Emparan, R. Luna, M. Martínez, R. Suzuki and K. Tanabe, Phases and stability of non-uniform black strings, JHEP05 (2018) 104 [arXiv:1802.08191] [INSPIRE].
J.R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Roy. Soc. LondonA 343 (1993) 337.
J.L. Vazquez, J.R. Esteban and A. Rodriguez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Adv. Diff. Eq.1 (1996) 21.
P.G. de Gennes, Spreading laws for microscopic droplets, C. R. Acad. Sci. Paris II298 (1984) 475.
J.R. King, Exact polynomial solutions to some nonlinear diffusion equations, PhysicaD 64 (1993) 35.
J.R. King, Asymptotic results for nonlinear outdiffusion, Eur. J. Appl. Math.5 (1994) 359.
P. Rosenau, Fast and superfast diffusion processes, Phys. Rev. Lett.74 (1995) 1056.
M. Headrick, S. Kitchen and T. Wiseman, A new approach to static numerical relativity and its application to Kaluza-Klein black holes, Class. Quant. Grav.27 (2010) 035002 [arXiv:0905.1822] [INSPIRE].
P. Daskalopoulos, R. Hamilton and N. Šešum, Classification of ancient compact solutions to the Ricci flow on surfaces, J. Diff. Geom.91 (2012) 171 [arXiv:0902.1158].
O. Aharony, E.Y. Urbach and M. Weiss, Generalized Hawking-Page transitions, arXiv:1904.07502 [INSPIRE].
D. Marolf, M. Rangamani and T. Wiseman, Holographic thermal field theory on curved spacetimes, Class. Quant. Grav.31 (2014) 063001 [arXiv:1312.0612] [INSPIRE].
G.T. Horowitz, N. Iqbal, J.E. Santos and B. Way, Hovering black holes from charged defects, Class. Quant. Grav.32 (2015) 105001 [arXiv:1412.1830] [INSPIRE].
B. Kol, Choptuik scaling and the merger transition, JHEP10 (2006) 017 [hep-th/0502033] [INSPIRE].
M.W. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett.70 (1993) 9 [INSPIRE].
M. Rozali and B. Way, Gravitating scalar stars in the large D limit, JHEP11 (2018) 106 [arXiv:1807.10283] [INSPIRE].
D. Gorbonos and B. Kol, A dialogue of multipoles: matched asymptotic expansion for caged black holes, JHEP06 (2004) 053 [hep-th/0406002] [INSPIRE].
D. Gorbonos and B. Kol, Matched asymptotic expansion for caged black holes: regularization of the post-Newtonian order, Class. Quant. Grav.22 (2005) 3935 [hep-th/0505009] [INSPIRE].
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
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.01062
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Emparan, R., Suzuki, R. Topology-changing horizons at large D as Ricci flows. J. High Energ. Phys. 2019, 94 (2019). https://doi.org/10.1007/JHEP07(2019)094
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2019)094