Abstract
Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these “maximally symmetric” spacetimes by investigating their local geometry. For each such spacetime and relative to exponential coordinates, we calculate the (infinitesimal) action of the kinematical symmetries, paying particular attention to the action of the boosts, showing in almost all cases that they act with generic non-compact orbits. We also calculate the soldering form, the associated vielbein and any invariant aristotelian, galilean or carrollian structures. The (conformal) symmetries of the galilean and carrollian structures we determine are typically infinite-dimensional and reminiscent of BMS Lie algebras. We also determine the space of invariant affine connections on each homogeneous spacetime and work out their torsion and curvature.
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References
H. Bacry and J. Levy-Leblond, Possible kinematics, J. Math. Phys.9 (1968) 1605 [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [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].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
A. Ashtekar, M. Campiglia and A. Laddha, Null infinity, the BMS group and infrared issues, Gen. Rel. Grav. 50 (2018) 140 [arXiv:1808.07093] [INSPIRE].
J. Figueroa-O’Farrill and S. Prohazka, Spatially isotropic homogeneous spacetimes, JHEP01 (2019) 229 [arXiv:1809.01224] [INSPIRE].
K. Morand, Embedding Galilean and Carrollian geometries I. Gravitational waves, arXiv:1811.12681 [INSPIRE].
A. Farahmand Parsa, H.R. Safari and M.M. Sheikh-Jabbari, On rigidity of 3d asymptotic symmetry algebras, JHEP03 (2019) 143 [arXiv:1809.08209] [INSPIRE].
T. Harmark et al., Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence, JHEP11 (2018) 190 [arXiv:1810.05560] [INSPIRE].
C. Batlle, J. Gomis, S. Ray and J. Zanelli, Lie symmetries of nonrelativistic and relativistic motions, Phys. Rev.D 99 (2019) 064015 [arXiv:1812.05837] [INSPIRE].
A. Campoleoni et al., Two-dimensional fluids and their holographic duals, Nucl. Phys.B (2019) 114692 [arXiv:1812.04019] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field theories with conformal Carrollian symmetry, JHEP05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
H.R. Safari and M.M. Sheikh-Jabbari, BMS 4algebra, its stability and deformations, JHEP04 (2019) 068 [arXiv:1902.03260] [INSPIRE].
G.W. Gibbons, The Ashtekar-Hansen universal structure at spatial infinity is weakly pseudo-Carrollian, arXiv:1902.09170 [INSPIRE].
N. Ozdemir, M. Ozkan, O. Tunca and U. Zorba, Three-dimensional extended newtonian (super)gravity, JHEP05 (2019) 130 [arXiv:1903.09377] [INSPIRE].
D. Hansen, J. Hartong and N.A. Obers, Gravity between Newton and Einstein, arXiv:1904.05706 [INSPIRE].
E. Bergshoeff, J.M. Izquierdo, T. Ortín and L. Romano, Lie algebra expansions and actions for non-relativistic gravity, JHEP08 (2019) 048 [arXiv:1904.08304] [INSPIRE].
H. Bacry and J. Nuyts, Classification of ten-dimensional kinematical groups with space isotropy, J. Math. Phys.27 (1986) 2455.
J.M. Figueroa-O’Farrill, Kinematical Lie algebras via deformation theory, J. Math. Phys.59 (2018) 061701 [arXiv:1711.06111] [INSPIRE].
J.M. Figueroa-O’Farrill, Higher-dimensional kinematical Lie algebras via deformation theory, J. Math. Phys.59 (2018) 061702 [arXiv:1711.07363] [INSPIRE].
T. Andrzejewski and J.M. Figueroa-O’Farrill, Kinematical Lie algebras in 2 + 1 dimensions, J. Math. Phys.59 (2018) 061703 [arXiv:1802.04048] [INSPIRE].
S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, Coset construction for duals of non-relativistic CFTs, JHEP05 (2009) 038 [arXiv:0903.4245] [INSPIRE].
J.I. Jottar, R.G. Leigh, D. Minic and L.A. Pando Zayas, Aging and holography, JHEP11 (2010) 034 [arXiv:1004.3752] [INSPIRE].
A. Bagchi and A. Kundu, Metrics with Galilean conformal isometry, Phys. Rev.D 83 (2011) 066018 [arXiv:1011.4999] [INSPIRE].
C. Duval and S. Lazzarini, Schrödinger manifolds, J. Phys.A 45 (2012) 395203 [arXiv:1201.0683] [INSPIRE].
E. Bergshoeff, J. Gomis and L. Parra, The symmetries of the Carroll superparticle, J. Phys. A49 (2016) 185402 [arXiv:1503.06083] [INSPIRE].
X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view, J. Math. Phys.57 (2016) 022507 [arXiv:1412.8212] [INSPIRE].
K.T. Grosvenor, J. Hartong, C. Keeler and N.A. Obers, Homogeneous nonrelativistic geometries as coset spaces, Class. Quant. Grav.35 (2018) 175007 [arXiv:1712.03980] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].
Y. Voglaire, Strongly exponential symmetric spaces, Int. Math. Res. Not. IMRN21 (2014) 5974 [arXiv:1303.5925].
P.K. Rozanov, On the exponentiality of affine symmetric spaces, Funkt. Anal. i Prilozhen.43 (2009)68.
W. Rossmann, Lie groups. An introduction through linear groups, Oxford Graduate Texts in Mathematics volume 5, Oxford University Press, Oxford U.K. (2002).
K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math.76 (1954) 33.
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].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A269 (1962) 21 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev.128 (1962) 2851 [INSPIRE].
C. Duval, On Galileian isometries, Class. Quant. Grav.10 (1993) 2217 [arXiv:0903.1641] [INSPIRE].
G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav.24 (2007) F15 [gr-qc/0610130] [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett.105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
T. Banks, A critique of pure string theory: Heterodox opinions of diverse dimensions, hep-th/0306074 [INSPIRE].
A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev.D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys.A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
J. Matulich, S. Prohazka and J. Salzer, Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension, JHEP07 (2019) 118 [arXiv:1903.09165] [INSPIRE].
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ArXiv ePrint: 1905.00034
Amelie Prohazka gewidmet.
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Figueroa-O’Farrill, J., Grassie, R. & Prohazka, S. Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes. J. High Energ. Phys. 2019, 119 (2019). https://doi.org/10.1007/JHEP08(2019)119
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DOI: https://doi.org/10.1007/JHEP08(2019)119