Abstract
We present a method of contraction that can be applied to re-construct the recent extended non-relativistic and ultra-relativistic algebras as well as corresponding action principles. The methodology involves the use of multiple copies of Poincaré algebra. Consequently, the contraction defines non-relativistic or ultra-relativistic limits of multimetric theories of gravity. In particular, we show that the non-relativistic scaling limit of bi-metric gravity corresponds to the recent formulation of an action principle for Newtonian gravity with a constant background mass density.
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M. Hatsuda and M. Sakaguchi, Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction, Prog. Theor. Phys. 109 (2003) 853 [hep-th/0106114] [INSPIRE].
J.A. de Azcarraga, J.M. Izquierdo, M. Picón and O. Varela, Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity, Nucl. Phys. B 662 (2003) 185 [hep-th/0212347] [INSPIRE].
J.A. de Azcarraga, J.M. Izquierdo, M. Picón and O. Varela, Expansions of algebras and superalgebras and some applications, Int. J. Theor. Phys. 46 (2007) 2738 [hep-th/0703017] [INSPIRE].
E. Bergshoeff, J.M. Izquierdo, T. Ortín and L. Romano, Lie Algebra Expansions and Actions for Non-Relativistic Gravity, JHEP 08 (2019) 048 [arXiv:1904.08304] [INSPIRE].
E.A. Bergshoeff, M. Ozkan and M.S. Zog, The holographic c-theorem and infinite-dimensional Lie algebras, JHEP 01 (2022) 010 [arXiv:2110.09542] [INSPIRE].
D. Van den Bleeken, Torsional Newton-Cartan gravity from the large c expansion of general relativity, Class. Quant. Grav. 34 (2017) 185004 [arXiv:1703.03459] [INSPIRE].
D. Hansen, J. Hartong and N.A. Obers, Action Principle for Newtonian Gravity, Phys. Rev. Lett. 122 (2019) 061106 [arXiv:1807.04765] [INSPIRE].
G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions, JHEP 11 (2009) 009 [arXiv:0907.2880] [INSPIRE].
E.A. Bergshoeff and J. Rosseel, Three-Dimensional Extended Bargmann Supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].
J. Hartong, Y. Lei and N.A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hořava-Lifshitz gravity, Phys. Rev. D 94 (2016) 065027 [arXiv:1604.08054] [INSPIRE].
L. Avilés, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli, Non-Relativistic Maxwell Chern-Simons Gravity, JHEP 05 (2018) 047 [arXiv:1802.08453] [INSPIRE].
N. Ozdemir, M. Ozkan, O. Tunca and U. Zorba, Three-Dimensional Extended Newtonian (Super)Gravity, JHEP 05 (2019) 130 [arXiv:1903.09377] [INSPIRE].
J.A. de Azcárraga, D. Gútiez and J.M. Izquierdo, Extended D = 3 Bargmann supergravity from a Lie algebra expansion, Nucl. Phys. B 946 (2019) 114706 [arXiv:1904.12786] [INSPIRE].
P. Concha and E. Rodríguez, Non-Relativistic Gravity Theory based on an Enlargement of the Extended Bargmann Algebra, JHEP 07 (2019) 085 [arXiv:1906.00086] [INSPIRE].
D.M. Peñafiel and P. Salgado-ReboLledó, Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra, Phys. Lett. B 798 (2019) 135005 [arXiv:1906.02161] [INSPIRE].
J. Gomis, A. Kleinschmidt and J. Palmkvist, Galilean free Lie algebras, JHEP 09 (2019) 109 [arXiv:1907.00410] [INSPIRE].
N. Ozdemir, M. Ozkan and U. Zorba, Three-dimensional extended Lifshitz, Schrödinger and Newton-Hooke supergravity, JHEP 11 (2019) 052 [arXiv:1909.10745] [INSPIRE].
J. Gomis, A. Kleinschmidt, J. Palmkvist and P. Salgado-ReboLledó, Symmetries of post-Galilean expansions, Phys. Rev. Lett. 124 (2020) 081602 [arXiv:1910.13560] [INSPIRE].
J. Gomis, A. Kleinschmidt, J. Palmkvist and P. Salgado-ReboLledó, Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity, JHEP 02 (2020) 009 [arXiv:1912.07564] [INSPIRE].
O. Kasikci, N. Ozdemir, M. Ozkan and U. Zorba, Three-dimensional higher-order Schrödinger algebras and Lie algebra expansions, JHEP 04 (2020) 067 [arXiv:2002.03558] [INSPIRE].
P. Concha, M. Ipinza and E. Rodríguez, Generalized Maxwellian exotic Bargmann gravity theory in three spacetime dimensions, Phys. Lett. B 807 (2020) 135593 [arXiv:2004.01203] [INSPIRE].
P. Concha, L. Ravera, E. Rodríguez and G. Rubio, Three-dimensional Maxwellian Extended Newtonian gravity and flat limit, JHEP 10 (2020) 181 [arXiv:2006.13128] [INSPIRE].
P. Concha, M. Ipinza, L. Ravera and E. Rodríguez, Non-relativistic three-dimensional supergravity theories and semigroup expansion method, JHEP 02 (2021) 094 [arXiv:2010.01216] [INSPIRE].
P. Concha, L. Ravera and E. Rodríguez, Three-dimensional non-relativistic extended supergravity with cosmological constant, Eur. Phys. J. C 80 (2020) 1105 [arXiv:2008.08655] [INSPIRE].
P. Concha, L. Ravera and E. Rodríguez, Three-dimensional exotic Newtonian supergravity theory with cosmological constant, Eur. Phys. J. C 81 (2021) 646 [arXiv:2104.12908] [INSPIRE].
J. Gomis and A. Kleinschmidt, Infinite-Dimensional Algebras as Extensions of Kinematic Algebras, Front. in Phys. 10 (2022) 892812 [arXiv:2202.05026] [INSPIRE].
D. Grumiller, J. Hartong, S. Prohazka and J. Salzer, Limits of JT gravity, JHEP 02 (2021) 134 [arXiv:2011.13870] [INSPIRE].
J. Gomis, D. Hidalgo and P. Salgado-ReboLledó, Non-relativistic and Carrollian limits of Jackiw-Teitelboim gravity, JHEP 05 (2021) 162 [arXiv:2011.15053] [INSPIRE].
L. Ravera and U. Zorba, Carrollian and Non-relativistic Jackiw-Teitelboim Supergravity, arXiv:2204.09643 [INSPIRE].
P. Concha, E. Rodríguez, G. Rubio and P. Yañez, Three-dimensional Newtonian gravity with cosmological constant and torsion, arXiv:2204.11763 [INSPIRE].
P. Concha, L. Ravera and E. Rodríguez, Three-dimensional non-relativistic supergravity and torsion, Eur. Phys. J. C 82 (2022) 220 [arXiv:2112.05902] [INSPIRE].
L. Ravera, AdS Carroll Chern-Simons supergravity in 2 + 1 dimensions and its flat limit, Phys. Lett. B 795 (2019) 331 [arXiv:1905.00766] [INSPIRE].
F. Ali and L. Ravera, \( \mathcal{N} \)-extended Chern-Simons Carrollian supergravities in 2 + 1 spacetime dimensions, JHEP 02 (2020) 128 [arXiv:1912.04172] [INSPIRE].
P. Concha, D. Peñafiel, L. Ravera and E. Rodríguez, Three-dimensional Maxwellian Carroll gravity theory and the cosmological constant, Phys. Lett. B 823 (2021) 136735 [arXiv:2107.05716] [INSPIRE].
E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional Spin-3 Theories Based on General Kinematical Algebras, JHEP 01 (2017) 114 [arXiv:1612.02277] [INSPIRE].
J. Matulich, S. Prohazka and J. Salzer, Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension, JHEP 07 (2019) 118 [arXiv:1903.09165] [INSPIRE].
M.F. Paulos and A.J. Tolley, Massive Gravity Theories and limits of Ghost-free Bigravity models, JHEP 09 (2012) 002 [arXiv:1203.4268] [INSPIRE].
H.R. Afshar, E.A. Bergshoeff and W. Merbis, Interacting spin-2 fields in three dimensions, JHEP 01 (2015) 040 [arXiv:1410.6164] [INSPIRE].
E.A. Bergshoeff, S. de Haan, O. Hohm, W. Merbis and P.K. Townsend, Zwei-Dreibein Gravity: A Two-Frame-Field Model of 3D Massive Gravity, Phys. Rev. Lett. 111 (2013) 111102 [Erratum ibid. 111 (2013) 259902] [arXiv:1307.2774] [INSPIRE].
M. Ozkan, Y. Pang and U. Zorba, Unitary Extension of Exotic Massive 3D Gravity from Bigravity, Phys. Rev. Lett. 123 (2019) 031303 [arXiv:1905.00438] [INSPIRE].
S. Sevim and M.S. Zöğ, Unitarity flow in 2+1 dimensional massive gravity, Phys. Rev. D 102 (2020) 064050 [arXiv:1912.04710] [INSPIRE].
D. Van den Bleeken, Torsional Newton-Cartan gravity and strong gravitational fields, in 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, (2019) [arXiv:1903.10682] [INSPIRE].
D. Hansen, J. Hartong and N.A. Obers, Gravity between Newton and Einstein, Int. J. Mod. Phys. D 28 (2019) 1944010 [arXiv:1904.05706] [INSPIRE].
M. Ergen, E. Hamamci and D. Van den Bleeken, Oddity in nonrelativistic, strong gravity, Eur. Phys. J. C 80 (2020) 563 [Erratum ibid. 80 (2020) 657] [arXiv:2002.02688] [INSPIRE].
D. Hansen, J. Hartong and N.A. Obers, Non-Relativistic Gravity and its Coupling to Matter, JHEP 06 (2020) 145 [arXiv:2001.10277] [INSPIRE].
O. Kasikci and M. Ozkan, Lie algebra expansions, non-relativistic matter multiplets and actions, JHEP 01 (2022) 081 [arXiv:2111.14568] [INSPIRE].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive Gravity in Three Dimensions, Phys. Rev. Lett. 102 (2009) 201301 [arXiv:0901.1766] [INSPIRE].
A. Sinha, On the new massive gravity and AdS/CFT, JHEP 06 (2010) 061 [arXiv:1003.0683] [INSPIRE].
M.F. Paulos, New massive gravity extended with an arbitrary number of curvature corrections, Phys. Rev. D 82 (2010) 084042 [arXiv:1005.1646] [INSPIRE].
E. Bergshoeff, J. Gomis and G. Longhi, Dynamics of Carroll Particles, Class. Quant. Grav. 31 (2014) 205009 [arXiv:1405.2264] [INSPIRE].
C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A 47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].
J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].
E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [arXiv:1701.06156] [INSPIRE].
A. Guerrieri and R.F. Sobreiro, Carroll limit of four-dimensional gravity theories in the first order formalism, Class. Quant. Grav. 38 (2021) 245003 [arXiv:2107.10129] [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].
D. Hansen, N.A. Obers, G. Oling and B.T. Søgaard, Carroll Expansion of General Relativity, SciPost Phys. 13 (2022) 055 [arXiv:2112.12684] [INSPIRE].
E.A. Bergshoeff, K.T. Grosvenor, C. Simsek and Z. Yan, An Action for Extended String Newton-Cartan Gravity, JHEP 01 (2019) 178 [arXiv:1810.09387] [INSPIRE].
E. Bergshoeff, J. Gomis and P. Salgado-Rebolledo, Non-relativistic limits and three-dimensional coadjoint Poincaré gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200106.
S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow Valley of Colored (Anti) de Sitter Gravity in Three Dimensions, JHEP 04 (2016) 055 [arXiv:1511.05220] [INSPIRE].
S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, Rainbow vacua of colored higher-spin (A)dS3 gravity, JHEP 05 (2016) 150 [arXiv:1511.05975] [INSPIRE].
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Ekiz, E., Kasikci, O., Ozkan, M. et al. Non-relativistic and ultra-relativistic scaling limits of multimetric gravity. J. High Energ. Phys. 2022, 151 (2022). https://doi.org/10.1007/JHEP10(2022)151
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DOI: https://doi.org/10.1007/JHEP10(2022)151