Abstract
We continue our study of the global properties of the z = 2 Schrödinger space-time. In particular, we provide a codimension 2 isometric embedding which naturally gives rise to the previously introduced global coordinates. Furthermore, we study the causal structure by probing the space-time with point particles as well as with scalar fields. We show that, even though there is no global time function in the technical sense (Schrödinger space-time being non-distinguishing), the time coordinate of the global Schrödinger coordinate system is, in a precise way, the closest one can get to having such a time function. In spite of this and the corresponding strongly Galilean and almost pathological causal structure of this space-time, it is nevertheless possible to define a Hilbert space of normalisable scalar modes with a well-defined time-evolution. We also discuss how the Galilean causal structure is reflected and encoded in the scalar Wightman functions and the bulk-to-bulk propagator.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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] [SPIRES].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [SPIRES].
J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [SPIRES].
A. Adams, K. Balasubramanian and J. McGreevy, Hot spacetimes for cold atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [SPIRES].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].
J. McGreevy, Holographic duality with a view toward many-body physics, arXiv:0909.0518 [SPIRES].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [SPIRES].
S.F. Ross and O. Saremi, Holographic stress tensor for non-relativistic theories, JHEP 09 (2009) 009 [arXiv:0907.1846] [SPIRES].
M. Blau, J. Hartong and B. Rollier, Geometry of Schrödinger space-times, global coordinates and harmonic trapping, JHEP 07 (2009) 027 [arXiv:0904.3304] [SPIRES].
G.T. Horowitz and D. Marolf, Quantum probes of space-time singularities, Phys. Rev. D 52 (1995) 5670 [gr-qc/9504028] [SPIRES].
M. Blau, D. Frank and S. Weiss, Scalar field probes of power-law space-time singularities, JHEP 08 (2006) 011 [hep-th/0602207] [SPIRES].
V.E. Hubeny, M. Rangamani and S.F. Ross, Causal structures and holography, JHEP 07 (2005) 037 [hep-th/0504034] [SPIRES].
S. Hawking and G. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973) [SPIRES].
E. Minguzzi and M. Sanchez, The causal hierarchy of spacetimes, gr-qc/0609119 [SPIRES].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [SPIRES].
D. Brecher, A. Chamblin and H.S. Reall, AdS/CFT in the infinite momentum frame, Nucl. Phys. B 607 (2001) 155 [hep-th/0012076] [SPIRES].
J. Rosen, Embedding of various relativistic Riemannian spaces in pseudo-Euclidean spaces, Rev. Mod. Phys. 37 (1965) 204.
C. Collinson, Embeddings of the plane-fronted waves and other space-times, J. Math. Phys. 9 (1968) 403.
M. Blau, J.M. Figueroa-O’Farrill and G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Class. Quant. Grav. 19 (2002) 4753 [hep-th/0202111] [SPIRES].
M. Blau and M. O’Loughlin, Homogeneous plane waves, Nucl. Phys. B 654 (2003) 135 [hep-th/0212135] [SPIRES].
E. Minguzzi, Chronological spacetimes without lightlike lines are stably causal, Commun. Math. Phys. 288 (2009) 801 [arXiv:0806.0153] [SPIRES].
V.E. Hubeny, M. Rangamani and S.F. Ross, Causal inheritance in plane wave quotients, Phys. Rev. D 69 (2004) 024007 [hep-th/0307257] [SPIRES].
P. Hořava and C.M. Melby-Thompson, Anisotropic conformal infinity, arXiv:0909.3841 [SPIRES].
T. Heinzl, Light-cone quantization: Foundations and applications, Lect. Notes Phys. 572 (2001) 55 [hep-th/0008096] [SPIRES].
S. Hellerman and J. Polchinski, Compactification in the lightlike limit, Phys. Rev. D 59 (1999) 125002 [hep-th/9711037] [SPIRES].
P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Ann. Phys. 144 (1982) 249 [SPIRES].
S. Moroz, Below the Breitenlohner-Freedman bound in the nonrelativistic AdS/CFT correspondence, Phys. Rev. D 81 (2010) 066002 [arXiv:0911.4060] [SPIRES].
P. Minces and V.O. Rivelles, Energy and the AdS/CFT correspondence, JHEP 12 (2001) 010 [hep-th/0110189] [SPIRES].
E.D. Rainville, Special functions, Chelsea Publ. Co., Bronx New York U.S.A. (1971).
W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Third Edition, Springer-Verlag, Berlin Germany (1966).
A. Bezubik and A. Strasburger, A new form of the spherical expansion of zonal functions and Fourier transforms of SO(d)-finite functions, SIGMA 2 (2006) 033 [math-ph/0603011].
A. Volovich and C. Wen, Correlation functions in non-relativistic holography, JHEP 05 (2009) 087 [arXiv:0903.2455] [SPIRES].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [SPIRES].
R.G. Leigh and N.N. Hoang, Real-time correlators and non-relativistic holography, JHEP 11 (2009) 010 [arXiv:0904.4270] [SPIRES].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [SPIRES].
D. Marolf and S.F. Ross, Boundary conditions and new dualities: vector fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [SPIRES].
J. Moore and R. Schlafly, On equivariant isometric embeddings, Math. Z. 173 (1980) 119.
S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, Coset construction for duals of non-relativistic CFTs, JHEP 05 (2009) 038 [arXiv:0903.4245] [SPIRES].
I.S. Gradshteyn and I.H. Ryzhik, Tables of integrals, series and products, Academic, New York U.S.A. (1980).
U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Vacua, propagators and holographic probes in AdS/CFT, JHEP 01 (1999) 002 [hep-th/9812007] [SPIRES].
G. Compere, S. de Buyl, S. Detournay and K. Yoshida, Asymptotic symmetries of Schrödinger spacetimes, JHEP 10 (2009) 032 [arXiv:0908.1402] [SPIRES].
J. Hartong, E. Imeroni and B. Rollier, Thermodynamic properties of TsT transformed AdS black holes, work in progress.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1005.0760
Rights and permissions
About this article
Cite this article
Blau, M., Hartong, J. & Rollier, B. Geometry of Schrödinger space-times II: particle and field probes of the causal structure. J. High Energ. Phys. 2010, 69 (2010). https://doi.org/10.1007/JHEP07(2010)069
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2010)069