Abstract
In this paper, we study static vacuum solutions of quantum gravity at a fixed Lifshitz point in (2+1) dimensions, and present all the diagonal solutions in closed forms in the infrared limit. The exact solutions represent spacetimes with very rich structures: they can represent generalized BTZ black holes, Lifshitz space-times or Lifshitz solitons, in which the spacetimes are free of any kind of space-time singularities, depending on the choices of the free parameters of the solutions. We also find several classes of exact static non-diagonal solutions, which represent similar space-time structures as those given in the diagonal case. The relevance of these solutions to the non-relativistic Lifshitz-type gauge/gravity duality is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge U.K. (2002).
S. Sachdev, Quantum phase transitions, second edition, Cambridge University Press, Cambridge U.K. (2013).
O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
J. Maldacena, The gauge/gravity duality, arXiv:1106.6073 [INSPIRE].
J. Polchinski, Introduction to gauge/gravity duality, arXiv:1010.6134 [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE].
G.T. Horowitz, Introduction to holographic superconductors, Lect. Notes Phys. 828 (2011) 313 [arXiv:1002.1722] [INSPIRE].
S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Ann. Rev. Condens. Matter Phys. 3 (2012) 9.
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].
A. Donos and J.P. Gauntlett, Lifshitz solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].
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].
P. Dey and S. Roy, From AdS to Schrödinger/Lifshitz dual space-times without or with hyperscaling violation, JHEP 11 (2013) 113 [arXiv:1306.1071] [INSPIRE].
K. Copsey and R. Mann, Pathologies in asymptotically Lifshitz spacetimes, JHEP 03 (2011) 039 [arXiv:1011.3502] [INSPIRE].
G.T. Horowitz and B. Way, Lifshitz singularities, Phys. Rev. D 85 (2012) 046008 [arXiv:1111.1243] [INSPIRE].
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].
T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].
S. Harrison, S. Kachru and H. Wang, Resolving Lifshitz horizons, JHEP 02 (2014) 085 [arXiv:1202.6635] [INSPIRE].
G. Knodel and J.T. Liu, Higher derivative corrections to Lifshitz backgrounds, JHEP 10 (2013) 002 [arXiv:1305.3279] [INSPIRE].
S. Kachru, N. Kundu, A. Saha, R. Samanta and S.P. Trivedi, Interpolating from Bianchi attractors to Lifshitz and AdS spacetimes, JHEP 03 (2014) 074 [arXiv:1310.5740] [INSPIRE].
R.B. Mann, Lifshitz topological black holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [INSPIRE].
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].
K. Balasubramanian and J. McGreevy, An analytic Lifshitz black hole, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263] [INSPIRE].
E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Lifshitz black hole in three dimensions, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347] [INSPIRE].
R.-G. Cai, Y. Liu and Y.-W. Sun, A Lifshitz black hole in four dimensional R 2 gravity, JHEP 10 (2009) 080 [arXiv:0909.2807] [INSPIRE].
M. Setare and D. Momeni, Plane symmetric solutions in Hořava-Lifshitz theory, Int. J. Mod. Phys. D 19 (2010) 2079 [arXiv:0911.1877] [INSPIRE].
Y.S. Myung, Lifshitz black holes in the Hořava-Lifshitz gravity, Phys. Lett. B 690 (2010) 534 [arXiv:1002.4448] [INSPIRE].
D.-W. Pang, On charged Lifshitz black holes, JHEP 01 (2010) 116 [arXiv:0911.2777] [INSPIRE].
E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Analytic Lifshitz black holes in higher dimensions, JHEP 04 (2010) 030 [arXiv:1001.2361] [INSPIRE].
M. Dehghani and R.B. Mann, Lovelock-Lifshitz black holes, JHEP 07 (2010) 019 [arXiv:1004.4397] [INSPIRE].
M. Dehghani, R. Mann and R. Pourhasan, Charged Lifshitz black holes, Phys. Rev. D 84 (2011) 046002 [arXiv:1102.0578] [INSPIRE].
W. Brenna, M. Dehghani and R. Mann, Quasi-topological Lifshitz black holes, Phys. Rev. D 84 (2011) 024012 [arXiv:1101.3476] [INSPIRE].
J. Matulich and R. Troncoso, Asymptotically Lifshitz wormholes and black holes for Lovelock gravity in vacuum, JHEP 10 (2011) 118 [arXiv:1107.5568] [INSPIRE].
I. Amado and A.F. Faedo, Lifshitz black holes in string theory, JHEP 07 (2011) 004 [arXiv:1105.4862] [INSPIRE].
L. Barclay, R. Gregory, S. Parameswaran, G. Tasinato and I. Zavala, Lifshitz black holes in IIA supergravity, JHEP 05 (2012) 122 [arXiv:1203.0576] [INSPIRE].
H. Lü, Y. Pang, C. Pope and J.F. Vazquez-Poritz, AdS and Lifshitz black holes in conformal and Einstein-Weyl gravities, Phys. Rev. D 86 (2012) 044011 [arXiv:1204.1062] [INSPIRE].
S.H. Hendi and B. Eslam Panah, Asymptotically Lifshitz black hole solutions in F(R) gravity, Can. J. Phys. 92 (2013) 1 [arXiv:1309.2135] [INSPIRE].
M.-I. Park, The rotating black hole in renormalizable quantum gravity: the three-dimensional Hořava gravity case, Phys. Lett. B 718 (2013) 1137 [arXiv:1207.4073] [INSPIRE].
M. Gutperle, E. Hijano and J. Samani, Lifshitz black holes in higher spin gravity, arXiv:1310.0837 [INSPIRE].
H.-S. Liu and H. Lu, Lifshitz and Schrödinger vacua, superstar resolution in gauged maximal supergravities, JHEP 02 (2014) 122 [arXiv:1310.8348] [INSPIRE].
M. Bravo-Gaete and M. Hassaine, Lifshitz black holes with arbitrary dynamical exponent in Horndeski theory, arXiv:1312.7736 [INSPIRE].
D.O. Devecioglu, Lifshitz black holes in Einstein-Yang-Mills theory, arXiv:1401.2133 [INSPIRE].
U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [INSPIRE].
R. Mann, L. Pegoraro and M. Oltean, Lifshitz solitons, Phys. Rev. D 84 (2011) 124047 [arXiv:1109.5044] [INSPIRE].
H.A. Gonzalez, D. Tempo and R. Troncoso, Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes, JHEP 11 (2011) 066 [arXiv:1107.3647] [INSPIRE].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
D. Blas, O. Pujolàs and S. Sibiryakov, Models of non-relativistic quantum gravity: the good, the bad and the healthy, JHEP 04 (2011) 018 [arXiv:1007.3503] [INSPIRE].
S. Mukohyama, Hořava-Lifshitz cosmology: a review, Class. Quant. Grav. 27 (2010) 223101 [arXiv:1007.5199] [INSPIRE].
P. Hořava, General covariance in gravity at a Lifshitz point, Class. Quant. Grav. 28 (2011) 114012 [arXiv:1101.1081] [INSPIRE].
T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys. Rept. 513 (2012) 1 [arXiv:1106.2476] [INSPIRE].
T. Pavlopoulos, Breakdown of Lorentz invariance, Phys. Rev. 159 (1967) 1106 [INSPIRE].
S. Chadha and H.B. Nielsen, Lorentz invariance as a low-energy phenomenon, Nucl. Phys. B 217 (1983) 125 [INSPIRE].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Lifshitz gravity for Lifshitz holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].
P. Hořava and C.M. Melby-Thompson, Anisotropic conformal infinity, Gen. Rel. Grav. 43 (2011) 1391 [arXiv:0909.3841] [INSPIRE].
A. Wang, Stationary and slowly rotating spacetimes in Hořava-Lifshitz gravity, Phys. Rev. Lett. 110 (2013) 091101 [arXiv:1212.1876] [INSPIRE].
A. Borzou, K. Lin and A. Wang, Static electromagnetic fields and charged black holes in general covariant theory of Hořava-Lifshitz gravity, JCAP 02 (2012) 025 [arXiv:1110.1636] [INSPIRE].
J. Greenwald, J. Lenells, J. Lu, V. Satheeshkumar and A. Wang, Black holes and global structures of spherical spacetimes in Hořava-Lifshitz theory, Phys. Rev. D 84 (2011) 084040 [arXiv:1105.4259] [INSPIRE].
E.B. Kiritsis and G. Kofinas, On Hořava-Lifshitz ’black holes’, JHEP 01 (2010) 122 [arXiv:0910.5487] [INSPIRE].
T. Suyama, Notes on matter in Hořava-Lifshitz gravity, JHEP 01 (2010) 093 [arXiv:0909.4833] [INSPIRE].
D. Capasso and A.P. Polychronakos, Particle kinematics in Hořava-Lifshitz gravity, JHEP 02 (2010) 068 [arXiv:0909.5405] [INSPIRE].
J. Alexandre, K. Farakos, P. Pasipoularides and A. Tsapalis, Schwinger-Dyson approach for a Lifshitz-type Yukawa model, Phys. Rev. D 81 (2010) 045002 [arXiv:0909.3719] [INSPIRE].
J.M. Romero, V. Cuesta, J.A. Garcia and J.D. Vergara, Conformal anisotropic mechanics and the Hořava dispersion relation, Phys. Rev. D 81 (2010) 065013 [arXiv:0909.3540] [INSPIRE].
S.K. Rama, Particle motion with Hořava-Lifshitz type dispersion relations, arXiv:0910.0411 [INSPIRE].
L. Sindoni, A note on particle kinematics in Hořava-Lifshitz scenarios, arXiv:0910.1329 [INSPIRE].
S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1973).
F.J. Tipler, Black holes in closed universes, Nature 270 (1977) 500.
S. Hayward, General laws of black hole dynamics, Phys. Rev. D 49 (1994) 6467 [INSPIRE].
S.A. Hayward, Gravitational waves, black holes and cosmic strings in cylindrical symmetry, Class. Quant. Grav. 17 (2000) 1749 [gr-qc/9909070] [INSPIRE].
A. Wang, Critical collapse of cylindrically symmetric scalar field in four-dimensional Einstein’s theory of gravity, Phys. Rev. D 68 (2003) 064006 [gr-qc/0307071] [INSPIRE].
A. Wang, Comment on ‘Absence of trapped surfaces and singularities in cylindrical collapse’, Phys. Rev. D 72 (2005) 108501 [gr-qc/0309003] [INSPIRE].
A. Wang, No-go theorem in spacetimes with two commuting spacelike Killing vectors, Gen. Rel. Grav. 37 (2005) 1919 [INSPIRE].
A.Y. Miguelote, N. Tomimura and A. Wang, Gravitational collapse of selfsimilar perfect fluid in 2 + 1 gravity, Gen. Rel. Grav. 36 (2004) 1883 [gr-qc/0304035] [INSPIRE].
P. Sharma, A. Tziolas, A. Wang and Z.-C. Wu, Spacetime singularities in string and its low dimensional effective theory, Int. J. Mod. Phys. A 26 (2011) 273 [arXiv:0901.2676] [INSPIRE].
J.W. Elliott, G.D. Moore and H. Stoica, Constraining the new Aether: Gravitational Cerenkov radiation, JHEP 08 (2005) 066 [hep-ph/0505211] [INSPIRE].
G.D. Moore and A.E. Nelson, Lower bound on the propagation speed of gravity from gravitational Cherenkov radiation, JHEP 09 (2001) 023 [hep-ph/0106220] [INSPIRE].
M. Pospelov and Y. Shang, On Lorentz violation in Hořava-Lifshitz type theories, Phys. Rev. D 85 (2012) 105001 [arXiv:1010.5249] [INSPIRE].
M. Pospelov and C. Tamarit, Lifshitz-sector mediated SUSY breaking, JHEP 01 (2014) 048 [arXiv:1309.5569] [INSPIRE].
D. Blas and S. Sibiryakov, Hořava gravity versus thermodynamics: The Black hole case, Phys. Rev. D 84 (2011) 124043 [arXiv:1110.2195] [INSPIRE].
P. Berglund, J. Bhattacharyya and D. Mattingly, Mechanics of universal horizons, Phys. Rev. D 85 (2012) 124019 [arXiv:1202.4497] [INSPIRE].
P. Berglund, J. Bhattacharyya and D. Mattingly, Thermodynamics of universal horizons in Einstein-aether theory, Phys. Rev. Lett. 110 (2013) 071301 [arXiv:1210.4940] [INSPIRE].
B. Cropp, S. Liberati and M. Visser, Surface gravities for non-Killing horizons, Class. Quant. Grav. 30 (2013) 125001 [arXiv:1302.2383] [INSPIRE].
M. Saravani, N. Afshordi and R.B. Mann, Dynamical emergence of universal horizons during the formation of black holes, arXiv:1310.4143 [INSPIRE].
B. Cropp, S. Liberati, A. Mohd and M. Visser, Ray tracing Einstein-aether black holes: universal versus Killing horizons, Phys. Rev. D 89 (2014) 064061 [arXiv:1312.0405] [INSPIRE].
K. Lin, F.-W. Shu, A. Wang and Q. Wu, in preparation.
R. Arnowitt, S. Deser and C.W. Misner, Republication of: the dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997.
C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman and Company, San Francisco, U.S.A. (1973).
S. Carlip, Quantum gravity in 2 + 1 dimensions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2003).
T. Zhu, Q. Wu, A. Wang and F.-W. Shu, U(1) symmetry and elimination of spin-0 gravitons in Hořava-Lifshitz gravity without the projectability condition, Phys. Rev. D 84 (2011) 101502 [arXiv:1108.1237] [INSPIRE].
T. Zhu, F.-W. Shu, Q. Wu and A. Wang, General covariant Hořava-Lifshitz gravity without projectability condition and its applications to cosmology, Phys. Rev. D 85 (2012) 044053 [arXiv:1110.5106] [INSPIRE].
K. Lin, S. Mukohyama, A. Wang and T. Zhu, Post-Newtonian approximations in the Hořava-Lifshitz gravity with extra U(1) symmetry, arXiv:1310.6666 [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
R.-G. Cai and A. Wang, Singularities in Hořava-Lifshitz theory, Phys. Lett. B 686 (2010) 166 [arXiv:1001.0155] [INSPIRE].
P. Painleve, La mécanique classique et la théorie de la relativité, C. R. Acad. Sci. (Paris) 173 (1921) 677.
A. Gullstrand, Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie, Arkiv. Mat. Astron. Fys. 16 (1922) 1
E.M. Lifshitz, On the theory of second-order phase transitions I, Zh. Eksp. Teor. Fiz. 11 (1941) 255.
E.M. Lifshitz, On the theory of second-order phase transitions II, Zh. Eksp. Teor. Fiz. 11 (1941) 269.
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: 1403.0946
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.
About this article
Cite this article
Shu, FW., Lin, K., Wang, A. et al. Lifshitz spacetimes, solitons, and generalized BTZ black holes in quantum gravity at a Lifshitz point. J. High Energ. Phys. 2014, 56 (2014). https://doi.org/10.1007/JHEP04(2014)056
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2014)056