Abstract
We define a ‘non-relativistic conformal method’, based on a Schrödinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Hořava-Lifshitz gravity by classifying all possible Schrödinger invariant scalar field theories of a complex scalar up to second order in time derivatives.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Properties of Conformal Supergravity, Phys. Rev. D 17 (1978) 3179 [INSPIRE].
M. Kaku and P.K. Townsend, Poincaré Supergravity as Broken Superconformal Gravity, Phys. Lett. B 76 (1978) 54 [INSPIRE].
S. Ferrara, M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauging the Graded Conformal Group with Unitary Internal Symmetries, Nucl. Phys. B 129 (1977) 125 [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales Sci. Ecole Norm. Sup. 40 (1923) 325.
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée. (première partie) (Suite), Annales Sci. Ecole Norm. Sup. 41 (1924) 1.
P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
P. Hořava, Membranes at Quantum Criticality, JHEP 03 (2009) 020 [arXiv:0812.4287] [INSPIRE].
G. Dautcourt, On the newtonian limit of general relativity, Acta Phys. Polon. B21 (1990) 755.
S. Mukohyama, Hořava-Lifshitz Cosmology: A Review, Class. Quant. Grav. 27 (2010) 223101 [arXiv:1007.5199] [INSPIRE].
T.P. Sotiriou, Hořava-Lifshitz gravity: a status report, J. Phys. Conf. Ser. 283 (2011) 012034 [arXiv:1010.3218] [INSPIRE].
J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].
A. Trautman, Sur la theorie newtonienne de la gravitation, Compt. Rend. Acad. Sci. 257 (1963) 617.
H.P. Kuenzle, Galilei and Lorentz structures on space-time — comparison of the corresponding geometry and physics, Annales Poincaré Phys. Theor. 17 (1972) 337 [INSPIRE].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
C. Leiva and M.S. Plyushchay, Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence, Annals Phys. 307 (2003) 372 [hep-th/0301244] [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [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].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].
C. Duval, M. Hassaine and P.A. Horvathy, The Geometry of Schrödinger symmetry in gravity background/non-relativistic CFT, Annals Phys. 324 (2009) 1158 [arXiv:0809.3128] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
S. Janiszewski and A. Karch, Non-relativistic holography from Hořava gravity, JHEP 02 (2013) 123 [arXiv:1211.0005] [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].
D.M. Hofman and B. Rollier, Warped Conformal Field Theory as Lower Spin Gravity, Nucl. Phys. B 897 (2015) 1 [arXiv:1411.0672] [INSPIRE].
A. Castro, D.M. Hofman and N. Iqbal, Entanglement Entropy in Warped Conformal Field Theories, JHEP 02 (2016) 033 [arXiv:1511.00707] [INSPIRE].
K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger Invariance from Lifshitz Isometries in Holography and Field Theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Field Theory on Newton-Cartan Backgrounds and Symmetries of the Lifshitz Vacuum, JHEP 08 (2015) 006 [arXiv:1502.00228] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Curved non-relativistic spacetimes, Newtonian gravitation and massive matter, J. Math. Phys. 56 (2015) 103505 [arXiv:1503.02682] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].
D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].
A. Gromov and A.G. Abanov, Thermal Hall Effect and Geometry with Torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [INSPIRE].
M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime Symmetries of the Quantum Hall Effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].
S. Moroz and C. Hoyos, Effective theory of two-dimensional chiral superfluids: gauge duality and Newton-Cartan formulation, Phys. Rev. B 91 (2015) 064508 [arXiv:1408.5911] [INSPIRE].
A.J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43 (1994) 357 [INSPIRE].
M. Henkel, M. Pleimling, C. Godreche and J.-M. Luck, Aging and conformal invariance, Phys. Rev. Lett. 87 (2001) 265701 [hep-th/0107122] [INSPIRE].
M. Henkel, Phenomenology of local scale invariance: From conformal invariance to dynamical scaling, Nucl. Phys. B 641 (2002) 405 [hep-th/0205256] [INSPIRE].
M. Henkel and F. Baumann, Autocorrelation functions in phase-ordering kinetics from local scale-invariance, J. Stat. Mech. 0707 (2007) P07015 [cond-mat/0703226] [INSPIRE].
M. Henkel and M. Pleimling, Non-Equilibrium Phase Transitions. Volume 2: Ageing and Dynamical Scaling Far from Equilibrium, Springer, Dordrecht Netherlands (2010).
D. Minic and M. Pleimling, Non-relativistic AdS/CFT and Aging/Gravity Duality, Phys. Rev. E 78 (2008) 061108 [arXiv:0807.3665] [INSPIRE].
J.I. Jottar, R.G. Leigh, D. Minic and L.A. Pando Zayas, Aging and Holography, JHEP 11 (2010) 034 [arXiv:1004.3752] [INSPIRE].
D. Minic, D. Vaman and C. Wu, On the 3-point functions of Aging Dynamics and the AdS/CFT Correspondence, Phys. Rev. Lett. 109 (2012) 131601 [arXiv:1207.0243] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan Geometry and Lifshitz Holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].
M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].
R. Andringa, E. Bergshoeff, S. Panda and M. de Roo, Newtonian Gravity and the Bargmann Algebra, Class. Quant. Grav. 28 (2011) 105011 [arXiv:1011.1145] [INSPIRE].
E.A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton-Cartan geometry and the Schrödinger algebra, Class. Quant. Grav. 32 (2015) 135017 [arXiv:1409.5555] [INSPIRE].
C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [INSPIRE].
A.O. Barut, Conformal Group → Schrödinger Group → Dynamical Group — The Maximal Kinematical Group of the Massive Schrödinger Particle, Helv. Phys. Acta 46 (1973) 496.
P. Havas and J. Plebánski, Conformal extensions of the Galilei group and their relation to the Schrödinger group., J. Math. Phys 19 (1978) 482.
R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D 42 (1990) 3500 [Erratum ibid. D 48 (1993) 3929] [INSPIRE].
M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Statist. Phys. 75 (1994) 1023 [hep-th/9310081] [INSPIRE].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].
R. Banerjee, A. Mitra and P. Mukherjee, Localization of the Galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 045010 [arXiv:1407.3617] [INSPIRE].
R. Banerjee and P. Mukherjee, New approach to nonrelativistic diffeomorphism invariance and its applications, arXiv:1509.05622 [INSPIRE].
R. De Pietri, L. Lusanna and M. Pauri, Standard and generalized Newtonian gravities as ‘gauge’ theories of the extended Galilei group. I. The standard theory, Class. Quant. Grav. 12 (1995) 219 [gr-qc/9405046] [INSPIRE].
C. Duval and H.P. Kunzle, Minimal Gravitational Coupling in the Newtonian Theory and the Covariant Schrödinger Equation, Gen. Rel. Grav. 16 (1984) 333 [INSPIRE].
J. Ehlers, Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie, in Grundlagenprobleme der modernen Physik, J. Nitsch, J. Pfarr and E.W. Stachow eds., Bibliographisches Institut, Mannheim Germany (1981), pp. 65-84.
E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav. 32 (2015) 205003 [arXiv:1505.02095] [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].
D. Blas, O. Pujolàs and S. Sibiryakov, Consistent Extension of Hořava Gravity, Phys. Rev. Lett. 104 (2010) 181302 [arXiv:0909.3525] [INSPIRE].
T.P. Sotiriou, M. Visser and S. Weinfurtner, Phenomenologically viable Lorentz-violating quantum gravity, Phys. Rev. Lett. 102 (2009) 251601 [arXiv:0904.4464] [INSPIRE].
E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan supergravity with torsion and Schrödinger supergravity, JHEP 11 (2015) 180 [arXiv:1509.04527] [INSPIRE].
R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].
K. Peeters, A Field-theory motivated approach to symbolic computer algebra, Comput. Phys. Commun. 176 (2007) 550 [cs/0608005] [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: 1512.06277
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
Afshar, H.R., Bergshoeff, E.A., Mehra, A. et al. A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities. J. High Energ. Phys. 2016, 145 (2016). https://doi.org/10.1007/JHEP04(2016)145
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2016)145