Noncommutative Schwarzschild geometry and generalized uncertainty principle
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Abstract
We discuss a possible link between the deformation parameter \(\Theta ^{\mu \nu }\) arising in the framework of noncommutative geometry and the parameter \(\beta \) of the generalized uncertainty principle (GUP). We compute the shift of the Hawking temperature induced by the \(\Theta ^{\mu \nu }\)deformed Schwarzschild geometry, and then we relate it to one obtained by GUP. Results suggest a granular structure of specetime at the Planck scales. The current bounds on \(\beta \) allow to constraint the noncommutative parameter \(\Theta ^{\mu \nu }\).
1 Introduction
The possibility to describe spacetime in noncommutive frameworks was noted long time ago [1], and its interest renewed recently owing to the discovery of SeibergWitten map [2], which relates noncommutative to commutative gauge theories. Since then there has been a more and more interest to understand the impact of noncommutativity on fundamental issues. From a side by studying the the spacetime symmetry^{1} and unitary properties of these theories [3, 4, 5, 6, 7, 8, 9, 10, 11], from the other side, investigate on possible experimental evidences [12, 13, 14] (see the review [19, 20] and references therein). Moreover, the interest increased also thanks to the fact that the lowenergy limit of string theory with an antisymmetric Bfield background provides a quantized structure of the spacetimes [2, 19, 20, 21].
The idea of noncommutativity of spacetime might provide deep indications about the quantum nature of spacetime at very high energy scales, where (gravitational) singularities are inevitable. In fact, the noncommutativity of spacetime could be intrinsically connected with gravity [2, 6, 7], and several studies have been proposed in literature to conciliate General Relativity with noncommutative spacetime models. The general idea is to define the fields over phase space by replacing the ordinary product of fields with the GronewaldMoyal product and then map (via the SeibergWitten) this theory in the equivalent commutative theory with expansion of the fields in terms of the noncommutative parameter. This approach has been extensively used to study many gauge theories [22, 23, 24, 25, 26, 27, 28] (see also [29, 30, 31, 32]), and since gravity can be considered as a gauge theory, the commutative equivalent approach appears to be a promising formulation^{2} [33, 48, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72].
2 GUP and Hawking temperature
3 Temperature from a \(\Theta ^{\mu \nu }\)Schwarzschild metric
In this section, we shall derive the relation between the parameters \(\Theta ^{\mu \nu }\) and \(\beta \). To this aim, we first recall the modifications to the Schwarzschild metric induced by the noncommutative geometry. Then we compute the correction/shift to the Hawking temperature.
3.1 \(\Theta ^{\mu \nu }\)Schwarzschild metric
The noncommutative structure allows to derive \(\Theta ^{\mu \nu }\)corrections to a given geometry, in particular to the Schwarzschild geometry, to which we are interested in [71].
3.2 Temperature shift from the deformed Schwarzschild metric

Remarkably, the relation between \(\beta \) and \(\Upsilon \) does not depend on gravitational mass M. This is particularly important because, as Eq. (3.13) shows, it is related to the universal character of the deformation parameter \(\Theta \), suggesting its deep connection to Planck scale, then to quantum gravity.

The above point is corroborated by the fact that \(\beta \simeq (M_{Pl}\Upsilon )^2<0\). A negative value of the GUP parameter typically arises in nontrivial spacetime structures such as a (fundamental) discreteness of space, see for example [74, 75, 93]. Interestingly, a similar result has been also obtained in the framework of the crystal lattice [93], providing therefore a further hint that the physical spacetime could have a lattice or granular structure at the level of Planck scale.
 To infer bounds on the parameter \(\Upsilon \), we require that the \(\Upsilon \)correction is smaller or equal to the \(\beta \)term, hencewhere \(\beta _{exp}\) is the experimental upper bound on \(\beta \). For the sake of completeness, in Tables 1 and 2 are reported experimental bounds on \(\beta \) obtained in different frameworks. Using the the more stringent upper bound \(\beta _{exp}= 10^{21}\), obtained in the gravitational sector, it follows \(\Upsilon < 10^{10}\)GeV\(^{1}\). Such a bound improves one order of magnitude, \(\Upsilon <10^{11}\)GeV\(^{1}\), for \(\beta \) bounded from nongravitational experiments.$$\begin{aligned} \Upsilon \leqslant \frac{2}{7\pi ^2}\frac{\sqrt{\beta }}{M_{Pl}}< \frac{2}{7\pi ^2}\frac{\sqrt{\beta _{exp}}}{M_{Pl}}\,, \end{aligned}$$(3.14)

As pointed out in the Introduction, the deformation parameter \(\beta \) is not fixed by the theory, and it is generally assumed that \(\beta \sim \mathcal{O}(110)\), as suggested by some models of string theory [52, 53, 54, 55, 56]. It is hence interesting to observe that if the parameter \(\Upsilon \) is of the order of the Planck scale, \(\Upsilon \sim M_{Pl}^{1}\) (the quantum gravity scale), then the GUP deformation parameter \(\beta \) can be fixed to the value \(\beta \sim \frac{7\pi ^2}{2}\sim \mathcal{O}(110)\).
Upper bounds on \(\beta \) derived from gravitational experiments
\(\beta<\)  Physical framework  Refs. 

\(10^{21}\)  Violation of equivalence principle (on Earth)  [102] (2014) 
Law of reciprocal action  
\(10^{60}\)  GW 150914  [101] 
\(10^{69}\)  Perihelion precession  [94] (2015) 
(Solar system data)  
\(10^{71}\)  Perihelion precession  [94] (2015) 
(Pulsar PRS B 1913+16 data)  
\(10^{78}\)  Modified masstemperature relation  [94] (2015) 
Light deflection 
Upper bounds on \(\beta \) derived from nongravitational experiments
\(\beta<\)  Physical framework  Refs. 

\(10^{18}\)  Evolution of micro and nano mechanical oscillators (masses \(\sim m_\mathrm{p}\))  [103] (2015) 
\(10^{20}\)  Lamb shift  
\(10^{21}\)  Scanning tunneling microscope  
\(10^{33}\)  Gravitational bar detectors\({}^{\mathrm{a}}\)  [100] (2013) 
\(10^{34}\)  Electroweak measurement  
\(10^{34}\)  Charmonium levels  
Energy difference in Hydrogen  [113] (2010)  
levels \(1S2S\)  
\(10^{39} \)  \({}^{87}\hbox {Rb}\) coldatomrecoil experiment  [114] (2016) 
\(10^{46}\)  Landau levels 
4 Conclusions
In this paper we have derived an upper bound on the deformation parameter \(\Upsilon \) of the noncommutative geometry (referring in particular to the gravitational sector of noncommutative geometry), by relating \(\Upsilon \) to the coefficients \(\beta \) of GUP. The shift of the Hawking temperature, for which the GUP is relevant, is derived by means of pure quantum mechanics principles, and no specific representations of canonical commutator relation is postulated. On the other hand, the same temperature is derived geometrically for a deformed Schwarzschild metric, allowing to link the deformed uncertainty relation with the \(\Theta \)deformed metric. We have found that the \(\Theta ^2\)correction to the canonical commutation relations of Heisenberg algebra is negative, suggesting a discrete nature of spacetime at the Planck scales, and that the more stringent bound that the current experiments allow to obtain is \(\Upsilon <10^{11}10^{10}\) (here \(\Upsilon \equiv \Theta ^{12}=\Theta ^{r\theta }\)).
Here we focused on noncommutative geometry putting attention to the gravitational sector, but understanding whenever other algebras may affect GUP, or specific representations of canonical operators, is certainly a non trivial task, especially for the possible links with quantum gravity. There is indeed a wide discussion on the implications of various models yielding GUPs, and a common aspect of all these models is related to test the size of these modifications. These aspects appear particularly interesting in perspective of laboratoryscale imitation of the black hole horizon, with the subsequent possible emission of an analogue Hawking radiation [115, 116].
Footnotes
 1.
Spacetime properties of noncommutative field theories are essentially either spacetime symmetries are manifestly violated [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], or the full Lorentz invariance is imposed on some parameters characterizing the noncommutative model, yielding to a quantum spacetime with the same classical global symmetries [6, 7, 17, 18]).
 2.
More precisely, some formulations of General Relativity on noncommutative spacetimes have been studied in different frameworks: 1) By gauging the noncommutative SO(4, 1) de Sitter group and using the SeibergWitten map followed by a contraction to the Poincaré group ISO(3, 1) [62]; 2) By twisting the Poincaré algebra in such a way that the latter insures the invariance of the algebra (1.1) (canonical structure) defining the noncommutativity of the spacetime [67]; 3) By considering a restrictive class of coordinate transformations which preserve the canonical structure [69, 70](by gauging the Lorentz algebra so(3, 1) within the enveloping algebra approach one infers a noncommutative general relativity restricted to the volumepreserving transformations (unimodular theory of gravity)); 4) By twisting the gauge Poincaré algebra [48]; 5) By considering geometrical approach to noncommutative gravity [33].
 3.
There are other different approaches in which the noncommutativity of the coordinates could take place, such as the Liealgebraic and the coordinatedependent (qdeformed) formulations [57].
 4.
Notice that \(\Upsilon \equiv \Theta ^{12}=\Theta ^{r\theta }\). Consistently with results of Ref. [71], indeed, in spherical coordinates one has \(x^1=r\) and \(x^2=\theta \), therefore \([\Theta ]=L\).
 5.
 6.
It is worth to mention that in the case of modifications of the BTZ black hole in threedimensional antide Sitter (\(AdS_3\)) [124], the commutation relations are \([r, \phi ]=i{\hat{\theta }}\), with \({\hat{\theta }}\equiv \theta ^{r\phi }\). These differ from the Cartesian ones \([x^i, x^j]=i \theta ^{ij}\) with a constant \(\theta ^{ij}\) since it corresponds to a nonconstant \(\theta ^{ij} = r {\hat{\theta }} \varepsilon ^{ij}\) (\(\varepsilon ^{ij}=\varepsilon ^{ji}\)). However, as argued in [124], the Moyal product can be still consistently defined in the polar coordinate with a constant \(\theta ^{r\phi }\) (spherically symmetry case).
Notes
Acknowledgements
GL and GV thank Prof. A. Yoshioka and Dr. T. Kanazawa for the kind hospitality, and the Tokyo University of Science for support. GL thanks F. Scardigli for discussions and suggestions. The authors acknowledge the referee for very constructive comments.
References
 1.H. Snyder, Phys. Rev. 71, 38 (1947)ADSCrossRefGoogle Scholar
 2.N. Seiberg, E. Witten, J. High Energy Phys. 09, 32 (1999)ADSCrossRefGoogle Scholar
 3.L. AlvarezGaumé, M.A. VazquezMozo, Nucl. Phys. B 668, 293 (2003)ADSCrossRefGoogle Scholar
 4.S. Carroll, J. Harvey, V.A. Kostelecky, C. Lane, T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001)ADSMathSciNetCrossRefGoogle Scholar
 5.M. Chaichian, K. Nishijima, A. Tureanu, Phys. Lett. B 568, 146 (2003)ADSMathSciNetCrossRefGoogle Scholar
 6.S. Doplicher, in Proceedings of the 37th Karpacz Winter School of Theoretical Physics, 2001, p. 204, hepth/0105251;Google Scholar
 7.S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Phys. 172, 187 (1995)ADSCrossRefGoogle Scholar
 8.D. Bahns, S. Doplicher, K. Fredenhagen, G. Piacitelli, Phys. Lett. B 533, 178 (2002)ADSMathSciNetCrossRefGoogle Scholar
 9.A. Iorio, T. Sykora, Int. J. Mod. Phys. A 17, 2369 (2002)ADSCrossRefGoogle Scholar
 10.R. Jackiw, S.Y. Pi, Phys. Rev. Lett. 88, 111603 (2002)ADSMathSciNetCrossRefGoogle Scholar
 11.R. Jackiw, ibid. 41, 1635 (1978)CrossRefGoogle Scholar
 12.G. AmelinoCamelia, G. Mandanici, K. Yoshida, On the IR/UV mixing and experimental limits on the parameters of canonical noncommutative spacetimes, arxiv:hepth/0209254
 13.Z. Guralnik, R. Jackiw, S.Y. Pi, A.P. Polychronakos, Phys. Lett. B 517, 450 (2001)ADSCrossRefGoogle Scholar
 14.R.G. Cai, ibid. 517, 457 (2001)Google Scholar
 15.J.M. Grimstrup, B. Kloibock, L. Popp, V. Putz, M. Schweda, M. Wickenhauser, The energymomentum tensor in noncommutative gauge field models, arxiv:hepth/0210288
 16.A.A. Bichl, J.M. Grimstrup, H. Grosse, E. Kraus, L. Popp, M. Schweda, R. Wulkenhaar, Eur. Phys. J. C 24, 165 (2002)ADSCrossRefGoogle Scholar
 17.M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001)ADSCrossRefGoogle Scholar
 18.H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, Lett. Math. Phys. 82, 153 (2007)ADSMathSciNetCrossRefGoogle Scholar
 19.M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001)ADSCrossRefGoogle Scholar
 20.R.J. Szabo, Phys. Rep. 378, 207 (2003)ADSMathSciNetCrossRefGoogle Scholar
 21.F. Ardalan, H. Arfaei, M.M. SheikhJabbari, J. High Energy Phys. 02, 016 (1999)ADSCrossRefGoogle Scholar
 22.O.F. Dayi, B. Yapiskann, J. High Energy Phys. 10, 022 (2002)ADSCrossRefGoogle Scholar
 23.S. Ghosh, Nucl. Phys. B 670, 359 (2003)ADSCrossRefGoogle Scholar
 24.B. Chakraborty, S. Gangopadhyay, A. Saha, Phys. Rev. D 70, 107707 (2004)ADSMathSciNetCrossRefGoogle Scholar
 25.S. Ghosh, Phys. Rev. D 70, 085007 (2004)ADSMathSciNetCrossRefGoogle Scholar
 26.P. Mukherjee, A. Saha, Mod. Phys. Lett. A 21, 821 (2006)ADSCrossRefGoogle Scholar
 27.A. Saha, A. Rahaman, P. Mukherjee, Phys. Lett. B 638, 292 (2006)ADSMathSciNetCrossRefGoogle Scholar
 28.X. Calmet, A. Kobakhidze, Phys. Rev. D 72, 045010 (2005)ADSMathSciNetCrossRefGoogle Scholar
 29.M. Chaichian, P. Presnajder, M.M. SheikhJabbari, A. Tureanu, Eur. Phys. J. C 29, 413 (2003)ADSCrossRefGoogle Scholar
 30.M. Chaichian, A. Kobakhidze, A. Tureanu, Eur. Phys. J. C 47, 241 (2006)ADSCrossRefGoogle Scholar
 31.X. Calmet, B. Jurco, P. Schupp, J. Wess, M. Wohlgenannt, Eur. Phys. J. C 23, 363 (2002)ADSCrossRefGoogle Scholar
 32.P. Aschieri, B. Jurco, P. Schupp, J. Wess, Nucl. Phys. B 651, 45 (2003)ADSCrossRefGoogle Scholar
 33.M. Chaichian, A. Tureanu, R.B. Zhang, X. Zhang, J. Math. Phys. 49, 073511 (2008)ADSMathSciNetCrossRefGoogle Scholar
 34.I. Mocioiu, M. Pospelov, R. Roiban, Phys. Lett. B 489, 390 (2000)ADSMathSciNetCrossRefGoogle Scholar
 35.M. Chaichian, M. SheikhJabbari, A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001)ADSCrossRefGoogle Scholar
 36.M. Chaichian, M. SheikhJabbari, A. Tureanu, Eur. Phys. J. C 36, 251 (2004)ADSCrossRefGoogle Scholar
 37.P.K. Joby, P. Chingangbam, S. Das, Phys. Rev. D 91, 083503 (2015)ADSCrossRefGoogle Scholar
 38.S.M. Carroll, J.A. Harvey, V.A. Kostelecky, C.D. Lane, T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001)ADSMathSciNetCrossRefGoogle Scholar
 39.X. Calmet, Eur. Phys. J. C 41, 269 (2005)ADSMathSciNetCrossRefGoogle Scholar
 40.P. Joby, P. Chingangbam, S. Das, Phys. Rev. D 91, 083503 (2015)ADSCrossRefGoogle Scholar
 41.X. Calmet, C. Fritz, Phys. Lett. B 747, 406 (2015)ADSCrossRefGoogle Scholar
 42.G. Lambiase, G. Vilasi, A. Yoshioka, Class. Quantum Gravity 34, 025004 (2017)ADSCrossRefGoogle Scholar
 43.A. Kobakhidze, C. Lagger, A. Manning, Phys. Rev. D 94, 064033 (2016)ADSMathSciNetCrossRefGoogle Scholar
 44.A. Kobakhidze, Phys. Rev. D 79, 047701 (2009)ADSCrossRefGoogle Scholar
 45.P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess, Class. Quantum Gravity 22, 3511 (2005)ADSCrossRefGoogle Scholar
 46.X. Calmet, A. Kobakhidze, Phys. Rev. D 72, 045010 (2005)ADSMathSciNetCrossRefGoogle Scholar
 47.P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess, Class. Quantum Gravity 23, 1883 (2006)ADSCrossRefGoogle Scholar
 48.A. Kobakhidze, Int. J. Mod. Phys. A 23, 2541 (2008)ADSMathSciNetCrossRefGoogle Scholar
 49.R.J. Szabo, Class. Quantum Gravity 23, R199 (2006)ADSCrossRefGoogle Scholar
 50.X. Calmet, A. Kobakhidze, Phys. Rev. D 74, 047702 (2006)ADSCrossRefGoogle Scholar
 51.P. Mukherjee, A. Saha, Phys. Rev. D 74, 027702 (2006)ADSCrossRefGoogle Scholar
 52.D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 197, 81 (1987)ADSCrossRefGoogle Scholar
 53.D.J. Gross, P.F. Mende, Phys. Lett. B 197, 129 (1987)ADSMathSciNetCrossRefGoogle Scholar
 54.D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 216, 41 (1989)ADSCrossRefGoogle Scholar
 55.K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B 234, 276 (1990)ADSMathSciNetCrossRefGoogle Scholar
 56.S. Capozziello, G. Lambiase, G. Scarpetta, Int. J. Theor. Phys. 39, 15 (2000)CrossRefGoogle Scholar
 57.J. Madore, S. Schraml, P. Schupp, J. Wess, Eur. Phys. J. C 16, 161 (2000)ADSCrossRefGoogle Scholar
 58.H.S. Snyder, Phys. Rev. 71, 38 (1947)ADSCrossRefGoogle Scholar
 59.C.N. Yang, Phys. Rev. 72, 874 (1947)ADSCrossRefGoogle Scholar
 60.C.A. Mead, Phys. Rev. 135, B849 (1964)ADSCrossRefGoogle Scholar
 61.F. Karolyhazy, Nuovo Cim. A 42, 390 (1966)ADSCrossRefGoogle Scholar
 62.A.H. Chamseddine, Phys. Lett. B 504, 33 (2001)ADSMathSciNetCrossRefGoogle Scholar
 63.L. Bonora, M. Schnabl, M. SheikhJabbari, A. Tomasiello, Nucl. Phys. B 589, 461 (2000)ADSCrossRefGoogle Scholar
 64.B. Jurco, S. Schraml, P. Schupp, J. Wess, Eur. Phys. J. C 17, 521 (2000)ADSCrossRefGoogle Scholar
 65.M. Chaichian, P.P. Kulish, K. Nishijima, A. Tureanu, Phys. Lett. B 604, 98 (2004)ADSMathSciNetCrossRefGoogle Scholar
 66.M. Chaichian, P. Prenajder, A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005)ADSCrossRefGoogle Scholar
 67.P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess, Class. Quantum Gravity 22, 3511 (2005)ADSCrossRefGoogle Scholar
 68.L. lvarezGaum, F. Meyer, M.A. VazquezMozo, Nucl. Phys. B 75, 392 (2006)Google Scholar
 69.X. Calmet, A. Kobakhidze, Phys. Rev. D 72, 045010 (2005)ADSMathSciNetCrossRefGoogle Scholar
 70.X. Calmet, A. Kobakhidze, Phys. Rev. D 74, 047702 (2006)ADSCrossRefGoogle Scholar
 71.M. Chaichian, A. Tureanu, Phys. Lett. B 637, 199 (2006)ADSMathSciNetCrossRefGoogle Scholar
 72.M. Chaichian, A. Tureanu, G. Zet, Phys. Lett. B 660, 573 (2008)ADSMathSciNetCrossRefGoogle Scholar
 73.M. Maggiore, Phys. Lett. B 304, 65 (1993)ADSMathSciNetCrossRefGoogle Scholar
 74.A. Kempf, G. Mangano, R.B. Mann, Phys. Rev. D 52, 1108 (1995)ADSMathSciNetCrossRefGoogle Scholar
 75.M. Bojowald, A. Kempf, Phys. Rev. D 86, 085017 (2012)ADSCrossRefGoogle Scholar
 76.F. Scardigli, Phys. Lett. B 452, 39 (1999)ADSCrossRefGoogle Scholar
 77.R.J. Adler, D.I. Santiago, Mod. Phys. Lett. A 14, 1371 (1999)ADSCrossRefGoogle Scholar
 78.F. Scardigli, R. Casadio, Class. Quantum Gravity 20, 3915 (2003)ADSCrossRefGoogle Scholar
 79.W. Heisenberg, Zeitschrift für Physik 43, 172 (1927)ADSCrossRefGoogle Scholar
 80.K. Nozari, B. Fazlpour, Chaos Solitons Fractals 34, 224 (2007)ADSCrossRefGoogle Scholar
 81.K. Nozari, Phys. Lett. B 629, 41 (2005)ADSMathSciNetCrossRefGoogle Scholar
 82.K. Nozari, T. Azizi, Gen. Relativ. Gravit. 38, 735 (2006)ADSCrossRefGoogle Scholar
 83.F. Scardigli, Nuovo Cim. B 110, 1029 (1995)ADSCrossRefGoogle Scholar
 84.R.J. Adler, P. Chen, D.I. Santiago, Gen. Relativ. Gravit. 33, 2101 (2001)ADSCrossRefGoogle Scholar
 85.M. Cavaglia, S. Das, Class. Quantum Gravity 21, 4511 (2004)ADSCrossRefGoogle Scholar
 86.M. Cavaglia, S. Das, R. Maartens, Class. Quantum Gravity 20, L205 (2003)ADSCrossRefGoogle Scholar
 87.L. Susskind, J. Lindesay, An Introduction to Black Holes, Information, and the String Theory Revolution (World Scientific, Singapore, 2005). See chapter 10Google Scholar
 88.K. Nouicer, Class. Quantum Gravity 24, 5917 (2007)ADSMathSciNetCrossRefGoogle Scholar
 89.F. Scardigli, Glimpses on the micro black hole planck phase. arXiv:0809.1832
 90.F. Scardigli, G. Lambiase, E. Vagenas, Phys. Lett. B 767, 242 (2017)ADSMathSciNetCrossRefGoogle Scholar
 91.G. Dvali, C. Gomez, Black Hole’s Quantum NPortrait. arXiv:1112.3359
 92.S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)Google Scholar
 93.P. Jizba, H. Kleinert, F. Scardigli, Phys. Rev. D 81, 084030 (2010)ADSCrossRefGoogle Scholar
 94.F. Scardigli, R. Casadio, Eur. Phys. J. C 75, 425 (2015)ADSCrossRefGoogle Scholar
 95.G. Lambiase, F. Scardigli, Phys. Rev. D 97, 075003 (2018)ADSMathSciNetCrossRefGoogle Scholar
 96.F. Scardigli, G. Lambiase, E. Vagenas, Phys. Lett. B 767, 242 (2017)ADSMathSciNetCrossRefGoogle Scholar
 97.F. Scardigli, G. Lambiase, E. Vagenas, J. Phys. Conf. Ser. 880, 012044 (2017)CrossRefGoogle Scholar
 98.F. Scardigli, M. Blasone, G. Luciano, R. Casadio, Eur. Phys. J. C 78, 728 (2018)ADSCrossRefGoogle Scholar
 99.A.N. Tawfik, A.M. Diab, Rep. Prog. Phys. 78, N. 12 (2015)Google Scholar
 100.F. Marin, M. Cerdonio et al., Nat. Phys. 9, 71 (2013)CrossRefGoogle Scholar
 101.Z.W. Feng, S.Z. Yang, H.L. Li, X.T. Zu, Phys. Lett. B 768, 81 (2017)ADSCrossRefGoogle Scholar
 102.S. Ghosh, Class. Quantum Gravity 31, 025025 (2014)ADSCrossRefGoogle Scholar
 103.M. Bawaj, C. Biancofiore, F. Marin, Nat. Commun. 6, 7503 (2015). arXiv:1411.6410 CrossRefGoogle Scholar
 104.S. Das, E.C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008)ADSCrossRefGoogle Scholar
 105.A.F. Ali, S. Das, E.C. Vagenas, Phys. Rev. D 84, 044013 (2011)ADSCrossRefGoogle Scholar
 106.S. Das, E.C. Vagenas Can. J. Phys. 84, 233 (2009)Google Scholar
 107.E.C. Vagenas, L. Alasfar, S.M. Alsaleh, A.F. Ali, Nucl. Phys. B 931, 72 (2018)ADSCrossRefGoogle Scholar
 108.E.C. Vagenas, S.M. Alsaleh, A. Farag, EPL 120, 40001 (2017)ADSCrossRefGoogle Scholar
 109.M. Faizal, A.F. Ali, A. Nassar, Phys. Lett. B 765, 238 (2017)ADSCrossRefGoogle Scholar
 110.L. Perivolaropoulos, Phys. Rev. D 95, 103523 (2017)ADSMathSciNetCrossRefGoogle Scholar
 111.A. AlonsoSerrano, M.P. Dabrowski, H. Gohar, Phys. Rev. D 97, 044029 (2018)ADSCrossRefGoogle Scholar
 112.B. Khosropour, M. Eghbali, S. Ghorbanali, Gen. Relativ. Gravit. 50, 25 (2018)ADSCrossRefGoogle Scholar
 113.C. Quesne, V.M. Tkachuk, Phys. Rev. A 81, 012106 (2010)ADSCrossRefGoogle Scholar
 114.D. Gao, M. Zhan, Phys. Rev. A 94, 013607 (2016)ADSCrossRefGoogle Scholar
 115.J. Steinhauer, Nat. Phys. 10, 864 (2014)CrossRefGoogle Scholar
 116.R. Cowen, Nature News, 12 October 2014, Hawking radiation mimicked in the lab, https://doi.org/10.1038/nature.2014.16131
 117.M. Chaichian, A. Tureanu, R.B. Zhang, X. Zhang, J. Math. Phys. 49, 073511 (2008)ADSMathSciNetCrossRefGoogle Scholar
 118.D. Wang, R.B. Zhang, X. Zhang, Class. Quantum Gravity 26, 085014 (2009)ADSCrossRefGoogle Scholar
 119.D. Wang, R.B. Zhang, X. Zhang, Eur. Phys. J. C 64, 439 (2009)ADSCrossRefGoogle Scholar
 120.W. Sun, D. Wang, N. Xie, R.B. Zhang, X. Zhang, Eur. Phys. J. C 69, 271 (2010)ADSCrossRefGoogle Scholar
 121.M. Chaichian, A. Tureanu, M.R. Setare, G. Zet, JHEP04, 064 (2008)Google Scholar
 122.D.V. Singh, M.S. Ali, S.G. Ghosh, Int. J. Mod. Phys. D 27, 1850108 (2018)ADSCrossRefGoogle Scholar
 123.S.A. Alavi, S. Nodeh, Phys. Scr. 90, 035301 (2015)ADSCrossRefGoogle Scholar
 124.HC. Kim, MuIn Park, C. Rim, J.H. Yee, JHEP 0810 060 (2008)Google Scholar
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