{"hits":{"hits":[{"metadata":{"citation_count_without_self_citations":31,"citation_count":31,"authors":[{"full_name_unicode_normalized":"philbin, thomas g.","full_name":"Philbin, Thomas G.","record":{"$ref":"https://inspirehep.net/api/authors/2234541"},"last_name":"Philbin","ids":[{"schema":"INSPIRE BAI","value":"T.G.Philbin.2"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903288"},"value":"Trinity Coll., Dublin"}],"signature_block":"FALBANt","first_name":"Thomas G.","uuid":"ce53c1ee-9398-4e70-a517-b8d661762cc7","recid":2234541}],"publication_info":[{"journal_volume":"13","page_end":"1232","year":1996,"journal_record":{"$ref":"https://inspirehep.net/api/journals/1214779"},"page_start":"1217","journal_title":"Class.Quant.Grav."}],"citeable":true,"$schema":"https://inspirehep.net/schemas/records/hep.json","references":[{"reference":{"publication_info":{"journal_volume":"24","artid":"551","page_start":"551","journal_title":"Gen.Rel.Grav."}}},{"reference":{"publication_info":{"journal_volume":"8","artid":"727","page_start":"727","journal_title":"Class.Quant.Grav."}}},{"reference":{"publication_info":{"journal_volume":"28","artid":"101","page_start":"101","journal_title":"Rend.Acc.Naz.Lincei"}}},{"reference":{"publication_info":{"journal_volume":"12","artid":"847","page_start":"847","journal_title":"J.Phys.A"}}},{"reference":{"publication_info":{"journal_volume":"21","artid":"152","page_start":"152","journal_title":"J.Math.Phys."}}},{"reference":{"publication_info":{"journal_volume":"9","artid":"2065","page_start":"2065","journal_title":"Class.Quant.Grav."}}},{"reference":{"arxiv_eprint":"gr-qc/9607058","publication_info":{"journal_volume":"36","artid":"3625","page_start":"3625","journal_title":"J.Math.Phys."}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/421106"}},{"reference":{"publication_info":{"journal_volume":"244","artid":"524","page_start":"524","journal_title":"Proc.Roy.Soc.Lond.A"}}},{"reference":{"publication_info":{"journal_volume":"53","artid":"472","page_start":"472","journal_title":"Annals Math."}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/9376"}},{"reference":{"publication_info":{"journal_volume":"15","artid":"571","page_start":"571","journal_title":"Found.Phys."}}},{"reference":{"publication_info":{"journal_volume":"19","artid":"1195","page_start":"1195","journal_title":"Gen.Rel.Grav."}}},{"reference":{"publication_info":{"journal_volume":"194","artid":"348","page_start":"348","journal_title":"Phys.Lett.A"}}},{"reference":{"publication_info":{"journal_volume":"11","artid":"1330","page_start":"1330","journal_title":"J.Math.Phys."}}},{"reference":{"publication_info":{"journal_volume":"45","artid":"1995","page_start":"1995","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/347897"}},{"reference":{"publication_info":{"journal_volume":"9","artid":"3605","page_start":"3605","journal_title":"Mod.Phys.Lett.A"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/385458"}},{"reference":{"publication_info":{"journal_volume":"17","artid":"1001","page_start":"1001","journal_title":"Gen.Rel.Grav."}}},{"reference":{"publication_info":{"journal_volume":"18","artid":"887","page_start":"887","journal_title":"Gen.Rel.Grav."}}},{"reference":{"publication_info":{"journal_volume":"26","artid":"917","page_start":"917","journal_title":"Gen.Rel.Grav."}}},{"reference":{"publication_info":{"journal_volume":"32","artid":"3135","page_start":"3135","journal_title":"J.Math.Phys."}},"record":{"$ref":"https://inspirehep.net/api/literature/2693036"}},{"reference":{"publication_info":{"journal_volume":"10","artid":"1303","page_start":"1303","journal_title":"J.Phys.A"}}},{"reference":{"publication_info":{"journal_volume":"5","artid":"393","page_start":"393","journal_title":"Class.Quant.Grav."}}},{"reference":{"publication_info":{"journal_volume":"22","artid":"553","page_start":"553","journal_title":"Gen.Rel.Grav."}}},{"reference":{"publication_info":{"journal_volume":"31","artid":"1972","page_start":"1972","journal_title":"J.Math.Phys."}}},{"reference":{"publication_info":{"journal_volume":"44S10","artid":"1","page_start":"1","journal_title":"Nuovo Cim.B"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/50008"}},{"reference":{"publication_info":{"journal_volume":"27","artid":"1731","page_start":"1731","journal_title":"Phys.Rev.D"}},"curated_relation":false,"record":{"$ref":"https://inspirehep.net/api/literature/192609"}}],"number_of_pages":22,"legacy_creation_date":"1995-12-15","preprint_date":"1995-12","author_count":1,"earliest_date":"1995-12","refereed":true,"external_system_identifiers":[{"schema":"MSNET","value":"1390111"},{"schema":"ADS","value":"1996CQGra..13.1217P"},{"schema":"SPIRES","value":"SPIRES-3273849"}],"facet_author_name":["2234541_Thomas G. Philbin"],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:403634","updated":"2023-09-04T14:38:12.013293"},"journal_title_variants":["Class. Quant. Grav.","Class.Quant.Grav."],"arxiv_eprints":[{"categories":["gr-qc"],"value":"gr-qc/9512029"}],"referenced_authors_bais":["M.F.A.da.Silva.1","L.Herrera.1","A.H.Taub.1","F.M.Paiva.1","E.Ruiz.2","O.Gron.1","N.O.Santos.3","R.G.Mclenaghan.2","W.Israel.1","J.M.M.Senovilla.1","An.Zhong.Wang.1","P.A.Amundsen.1","J.Carminati.1"],"legacy_version":"20180614221631.0","inspire_categories":[{"term":"Gravitation and Cosmology"}],"first_author":{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/02tyrky19"}],"full_name":"Philbin, Thomas G.","last_name":"Philbin","first_name":"Thomas G.","recid":2234541},"control_number":403634,"dois":[{"value":"10.1088/0264-9381/13/5/032"}],"document_type":["article"],"texkeys":["Philbin:1995iz"],"abstracts":[{"source":"arXiv","value":"The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (Levi--Civita) space--time. A match to a cylinder source is achieved for $-\\frac{1}{2}<\\si<\\frac{1}{2}$, where $\\si$ is the mass per unit length in the Newtonian limit $\\si\\to 0$, and a match to a wall source is possible for $|\\si|>\\frac{1}{2}$, this case being without a Newtonian limit; the positive (negative) values of $\\si$ correspond to a positive (negative) fluid density. The range of $\\si$ for which a source has previously been matched to the Levi--Civita metric is $0\\leq\\si<\\frac{1}{2}$ for a cylinder source.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["gr-qc"],"titles":[{"title":"Perfect fluid cylinders and walls: Sources for the Levi-Civita space-time"},{"source":"arXiv","title":"Perfect-fluid cylinders and walls - sources for the Levi-Civita space--time"}],"curated":true},"updated":"2023-09-04T14:38:12.013293+00:00","created":"1995-12-15T00:00:00+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/403634?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/403634?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/403634?format=latex-us","json":"https://inspirehep.net/api/literature/403634?format=json","json-expanded":"https://inspirehep.net/api/literature/403634?format=json-expanded","cv":"https://inspirehep.net/api/literature/403634?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A403634"},"id":"403634"},{"metadata":{"citation_count_without_self_citations":212,"documents":[{"filename":"njp8_5_053015.pdf","attachment":{"content":"T h e  o p e n – a c c e s s  j o u r n a l  f o r  p h y s i c s\n\nNew Journal of Physics\n\nObservation of negative-frequency waves in a water\ntank: a classical analogue to the Hawking effect?\n\nGermain Rousseaux 1,2, Christian Mathis 2, Philippe Maïssa 2,\nThomas G Philbin 3,4 and Ulf Leonhardt 3,5\n\n1 ACRI, Laboratoire Génimar, 260 route du Pin Montard, BP 234,\n06904 Sophia-Antipolis Cedex, France\n2 Université de Nice-Sophia Antipolis, Laboratoire J-A Dieudonné,\nUMR CNRS-UNSA 6621, Parc Valrose, 06108 Nice Cedex 02, France\n3 School of Physics and Astronomy, University of St Andrews, North Haugh,\nSt Andrews KY16 9SS, Scotland, UK\n4 Max Planck Research Group of Optics, Information and Photonics,\nGünther-Scharowsky-Strasse 1, Bau 24, D-91058 Erlangen, Germany\nE-mail: ulf@st-andrews.ac.uk\n\nNew Journal of Physics10 (2008) 053015 (12pp)\nReceived 24 December 2007\nPublished 13 May 2008\nOnline athttp://www.njp.org/\ndoi:10.1088/1367-2630/10/5/053015\n\nAbstract. The conversion of positive-frequency waves into negative-frequency\nwaves at the event horizon is the mechanism at the heart of the Hawking\nradiation of black holes. In black-hole analogues, horizons are formed for waves\npropagating in a medium against the current when and where the flow exceeds\nthe wave velocity. We report on the first direct observation of negative-frequency\nwaves converted from positive-frequency waves in a moving medium. The\nmeasured degree of mode conversion is significantly higher than that expected\nfrom the theory.\n\n5 Author to whom any correspondence should be addressed.\n\nNew Journal of Physics10 (2008) 053015\n1367-2630/08/053015+12$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft\n\nmailto:ulf@st-andrews.ac.uk\nhttp://www.njp.org/\n\n\n2\n\nContents\n\n1. Introduction 2\n2. Negative frequencies 5\n3. Water waves 7\n4. Experiment 8\n5. Numerical simulations 9\n6. Conclusions 11\nAcknowledgments 11\nReferences 12\n\n1. Introduction\n\nThe theory of Hawking radiation of black holes [1] connects three separate disciplines\nof physics—quantum mechanics, general relativity and thermodynamics [2]—and has been\napplied to test potential quantum theories of gravity [3, 4]. The radiation of astrophysical black\nholes is too feeble to be detectable, but laboratory analogues [5]–[8] of the event horizon may\ndemonstrate the physics behind Hawking radiation. Most candidates of artificial black holes rely\non quantum fluids [8]–[12], but here we report an experiment with a classical fluid: water [13]. A\nhorizon is formed when flowing water exceeds the wave velocity. We observed a key ingredient\nof the classical mechanism behind Hawking radiation, the generation of waves with negative\nfrequencies [1, 14, 15]. However, the measured conversion of positive- into negative-frequency\nwaves is significantly higher than that expected from the theory [13] for reasons we have not\nyet understood.\n\nIn 1974, Hawking [1] predicted that black holes are not black: they radiate. The event\nhorizon generates pairs of quanta; one particle of each pair emerges into space whereas its\npartner falls into the singularity. The quantum physics of pair creation at horizons is based\non the features of classical wave-packet propagation [14, 15, 21] as follows: figure1 shows\na wave packet escaping from the horizon. In a thought experiment, Hawking [21] traced such\nwave packets backwards in time and realized that they originate from two distinct waves: one\noscillating with positive frequencies and another with negative frequencies. Note that one can\nvisualize negative frequencies in the way waves propagate in space and time, i.e. in space–\ntime diagrams or videos, but negative frequencies do not directly appear in snapshots of wave\npackets. Figure2 compares the space–time diagrams of ordinary positive-frequency waves with\nthe behavior of negative-frequency waves. The figure shows that the lines of equal phase in\nspace–time have negative slopes for negative frequencies, as we discuss in section2.\n\nThe distinction between positive and negative frequencies is important for quantum\nfields [14, 15, 21]: the positive frequencies distinguish the annihilation and the negative\nfrequencies the creation operators. A process that mixes positive and negative frequencies thus\ncreates particles; the horizon spontaneously emits radiation. Figure1 illustrates the wave packets\nof the particles that escape into space; the particles that fall into the black hole are shown in\nfigure3. They originate from mixtures of the two initial wave packets of figure1. Therefore, the\ncreated quanta appear in entangled pairs, one escaping, other one falling into singularity.\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n3\n\nt ω ' > 0\n\nω ' > 0\n\nω' < 0\n\nFigure 1. Tracing wave packets backwards in time at the horizon of a black\nhole. Schematic space–time diagram showing a wave packet escaping into space\n(top), potentially reaching an observer. This wave packet oscillates at positive\nfrequencies, but it originates from two distinct waves, one with positive and\nanother one with negative frequencies, shown below the escaping wave packet\nin the space–time diagram (for times in the past). This mixing of positive and\nnegative frequencies is the classical root of the quantum Hawking radiation [1].\nNote that the deflection of the incident waves at the horizon depends on the\ndispersion properties of the ‘space–time medium’ [16]–[20]. In astrophysics,\nthese properties are unknown, in contrast to laboratory analogues.\n\nSeen from outside, the black hole turns out [14, 15, 21] to emit black-body radiation with\na temperature [1] that is proportional to the surface gravity at the horizon, or, equivalently,\ninversely proportional to the size of the black hole, the Schwarzschild radius. Since Hawking’s\nprediction, the radiation of horizons has been regarded as a confirmation for black-hole\nthermodynamics [2] and as a crucial test case for quantum theories of gravity such as superstring\ntheory [3] and loop quantum gravity [4].\n\nHowever, near the event horizon, fields are subject to frequency shifts beyond the Planck\nscale [16]–[20], as figure1 schematically illustrates: the incident wave packets oscillate at\nsignificantly higher frequencies than the outgoing waves. The mechanism that could limit the\nfrequency shifting at the horizon of the astrophysical black hole is unknown. Hawking radiation\nmay thus depend on as yet unknown physics or may not exist at all. There is no observational\nevidence for Hawking radiation in astrophysics yet; and it seems unlikely that there ever will\nbe for practical reasons—radiation with characteristic thermal wavelengths in the order of the\nSchwarzschild radius, a few km for solar-mass black holes, is obscured by the cm-waves of the\nCosmic Microwave Background.\n\nAstrophysical black holes are too large for noticeable Hawking radiation, but laboratory\nanalogues [5]–[8] of black holes offer valuable insights into the mechanism of radiating\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n4\n\nx\n\nt\n\nx\n\nt\n\nFigure 2. Positive- versus negative-frequency waves. The left diagram shows\nthe space–time diagram of a wave with positive frequency, whereas the right\ndiagram shows a negative-frequency wave. Section2 explains the physics of\nnegative-frequency waves in a moving medium. The pictures show space–time\ndiagrams of waves in a medium moving with uniform speed. The left diagram\ndisplays a wave with positive wavenumberk, whereas the right diagrams shows\na wave with negativek and negative frequencyω′ in the co-moving frame.\n\nt\n\nω' > 0\n\nω' > 0\n\nω' < 0\n\nFigure 3. Hawking partner. Schematic space–time diagram of a wave packet\npropagating against the ‘space–time flow’ on the other side of the horizon,\ndrifting toward the singularity of the black hole. Like the wave illustrated in\nfigure 1, this wave packet originates from waves with positive and negative\nfrequencies. These waves are mixtures of the escaping waves of figure1 traced\nbackwards in time; hence the escaping quanta and the in-falling quanta form\nentangled partners.\n\nhorizons. Most analogues are based on a simple idea [8]–[10]: black holes behave like moving\nfluids. Consider waves with phase velocityc′ in a medium of flow speedu. If the magnitude of\nu exceedsc′, waves can no longer propagate upstream; they are trapped beyond a horizon.\nThe horizon creates wave-quanta [5]–[8], the analogue of Hawking radiation [1], with an\neffective temperature that depends on the flow gradient at the horizon, the analogue [5]–[8] of\nthe surface gravity. The radiation is only noticeable if the temperature of the fluid lies below the\neffective Hawking temperature. Superfluids [8] like helium-3 or ultracold quantum gases\n[11, 12] may form radiating horizons for their elementary excitations and so would moving\noptical media for photons [7, 22].\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n5\n\nt\n\nω' > 0\n\nω' > 0\n\nω' < 0\n\nFigure 4. White-hole horizon. In order to demonstrate in a laboratory setting\nthe tracing of wave packets backwards in time at a black-hole horizon, one\nhas to time-reverse figure1. A time-reversed black hole is a white hole. The\narrow indicates the direction of the moving medium that establishes a horizon\nfor counter-propagating waves.\n\nOn the other hand, at the heart of the Hawking effect lies a classical process that can\nbe demonstrated with classical fluids such as water: the generation of waves with negative\nfrequencies. For this, one should reproduce the characteristic behavior of wave packets at\nhorizons traced backwards in time illustrated in figure1. This is possible with a time-reversed\nblack hole—a white-hole horizon—as shown in figure4. The horizon of the white hole\ncorresponds to the following analogy: imagine a fast river flowing out into the sea, getting\nslower. Waves cannot enter the river beyond the point where the flow speed exceeds the wave\nvelocity; beyond this point the river resembles an object that nothing can enter, the white\nhole. Such wave blocking has been comprehensively studied in the fluid-mechanics literature\n[23]–[28], but to our knowledge, the generation of negative-frequency waves has never been\nobserved before.\n\n2. Negative frequencies\n\nWhat are negative-frequency waves? Consider linear one-dimensional6 wave propagation in a\nmoving medium: a wave with phaseϕ propagates in thex-direction against the flowu. The\n\n6 The essential physics of horizons is contained in one-dimensional wave propagation, even in the case of the three-\ndimensional black hole, because near horizons the wavelength is dramatically reduced such that their curvatures\nare insignificant.\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n6\n\n−6 −4 −2 0 2 4 6\nk\n\n−2\n\n−1\n\n0\n\n1\n\n2\n\nω\n'\n\nki kb\n\nkh\n\nω −uk = ω'(k)\n\nFigure 5. Doppler formula (2) versus dispersion relation (4) for ω′ plotted in\narbitrary units. The wavenumberki describes the incident wave,kb the blue-\nshifted andkh the Hawking wave with negative wavenumberk and negative\nfrequencyω′.\n\nphase evolves in timet as\n\nϕ =\n\n∫\n(k dx − ω dt) , (1)\n\nwherek denotes the wavenumber andω the frequency in the laboratory frame. Imagine we\nconstruct at each pointx a frame that is co-moving with the fluid. In the locally co-moving\nframes7 dx = dx′ + u dt′ and dt = dt′, and so the phase evolves in terms of the co-moving\ncoordinates as the integral ofk dx′\n\n− ω′ dt′ with\n\nω′\n= ω − uk. (2)\n\nEquation (2) simply describes the Doppler effect—waves are frequency-shifted due to the\nmotion of the medium. In a locally co-moving frame,ω′ can only depend on the wavenumberk\nand the properties of the medium, but not explicitly on the position:ω′ is a functionω′(k) that\nis given by the dispersion relation. The phase velocityc′ is defined asω′/k, whereas the group\nvelocity is\n\nvg =\n∂ω\n\n∂k\n= v′\n\ng + u , v′\n\ng =\n∂ω′\n\n∂k\n. (3)\n\nWhat can we say about the dispersion relation in general? In isotropic media,ω′2 is an even\nfunction ofk, because waves should be able to propagate in positive and negative directions in\nthe same way. Without loss of generality, we assume that the medium moves in the negative\ndirection (from the right to the left). In this case, counter-propagating waves have positive phase\nvelocitiesc′. Therefore, we take the branch ofω′, whereω′/k is positive, i.e. wherec′ is an odd\nfunction ofk that is positive for positivek. We also assume that the counter-propagating waves\nmove with positive group-velocitiesv′\n\ng in the medium and that the group-velocity dispersion of\nthe medium is normal, i.e.v′\n\ng monotonically decreases for increasing|k|. Figure 5 shows our\nspecific case that satisfies these general requirements.\n\nFor a stationary flow the laboratory frequencyω is fixed. The wavenumberk is given by the\nDoppler formula (2) and the dispersion relationω′(k). In general, the solution of this equation is\n\n7 For simplicity, we ignore effects of relativistic velocities.\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n7\n\nFigure 6. Schematic diagram of the experiment.\n\nmulti-valued: each frequencyω corresponds to several wavenumbersk, i.e. to several physically\nallowed waves. As visualized in figure5, the physically allowed waves are determined by\nthe pointsk where the lineω − uk intersects the curveω′(k). One of these wavenumbersk\nis always negative, as figure5 illustrates. Sinceω′ is an odd function ofk, the co-moving\nfrequencyω′ must be negative for negativek, although the frequencyω in the laboratory frame is\nalways positive. We call waves with negative co-moving frequenciesnegative-frequency waves.\nImagine we display the wave propagation in a space–time diagram, see figure2. According to\nequation (1), the lines of constant phaseϕ have positive slopes dt/dx for positivek and negative\nslopes for negativek. We regard this behavior as the characteristic feature of negative-frequency\nwaves.\n\nFigure 5 shows that for negative-frequency waves the slope of the curveω′(k) is smaller\nthan the slope of the Doppler line, smaller than−u. As a consequence of equation (3), the\ngroup velocityvg in the laboratory frame must be negative. Therefore, negative-frequency waves\ncannot be launched directly, but they can be the result of a mode conversion from incident\npositive-frequency waves.\n\n3. Water waves\n\nFollowing a suggestion by Schützhold and Unruh [13], we studied water waves in the channel\nschematically shown in figure6. A ramp in the channel creates a gradient in flow speed. The\nflowing water forms a white-hole horizon, an object that waves cannot enter, when the flow\n|u| matches the group velocity∂ω′/∂k of the waves. Water waves—gravity waves—obey the\ndispersion relation [29]\n\nω′2\n= gk tanh(kh), (4)\n\nwhereg denotes the gravitational acceleration of the Earth at the water surface andh is the\nheight of the channel. In the limit of long wavelengths, i.e. small wavenumbersk, the dispersion\nrelation (4) reduces toω′2\n\n= gh k2; waves propagate withc′\n=\n\n√\ngh. We see from the Doppler\n\nformula (2) that, in this limit,ω′ is connected toω andk by a quadratic form, which defines\na space–time geometry [30]. A rigorous analysis [13] proves that the propagation of water\nwaves is exactly equivalent to wave propagation in space–time geometries, as long as|k| is\nmuch smaller than 1/h. So, in our case, the channel heighth serves as a simple analogue of the\nPlanck scale; waves with wavelengths shorter thanh do not experience the effective space–time\ngeometry anymore. Close to the horizon, the incident waves are compressed untilk reaches the\nscale of 1/h.\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n8\n\nTo characterize the waves, we use the graphical solution of the Doppler formula (2)\ncombined with the dispersion relation (4) shown in figure5. For a given positive frequency\nω, either one or three real solutions exist, one negative and possibly two positivek. Only in\nthe case of a positive solution will the wave-maker launch waves, because the group velocity\n(3) of the negative-frequency wave is negative. The slope ofω′ at the smallest positivek is\nhigher than the slope of the Doppler lineω − uk. For this wavenumber the group velocity is\npositive: thisk describes the incident wave. When the incident wave propagates against the\nrising current, the slope of the Doppler line rises until the two positivek merge. At this point,\nthe flow matches the group velocity of the wave. The incident wave is converted into a short-\nwavelength wave; it is blue-shifted below the effective Planck scaleh. For the blue-shifted wave,\n∂ω′/∂k lies below the flow speed|u|: the blue-shifted wave drifts back with negative group\nvelocity (3), butk is positive and so is the frequencyω′. Figure5 shows that such wave blocking\n[23]–[28] cannot occur below a critical flow speed. In order to estimate [26] the criticalu, we\nreplace tanh(kh) in the dispersion relation (4) by the asymptotic value of 1. A realk ceases\nto exist when the discriminant of the resulting quadratic equation vanishes, for|u| = u∗\n\n=\n\ng/(4ω). Since the dispersion curve (4) lies below the asymptotics, this procedure [26] gives\nan overestimation of the critical flow speed.\n\nThe horizon also converts [19]–[21] by tunneling a part of the incident wave into the\nnegative-k branch of figure5, generating a wave with negative co-moving frequency, the\nclassical analogue of Hawking radiation. In fluid dynamics, the blue-shifted waves have been\ndiscussed and observed in connection with wave-blocking [23]–[28] but to our knowledge,\nthe negative-frequency waves have neither been theoretically analyzed in the fluid-dynamics\nliterature, nor experimentally observed.\n\n4. Experiment\n\nWe performed our experiment at ACRI, a private research company working on environmental\nfluid mechanics problems such as coastal engineering. The Génimar Laboratory, a department of\nACRI, features a wave-tank 30 m long, 1.8 m wide and 1.8 m deep. The wave-maker is of piston-\ntype and can generate waves with periods ranging from 0.6 to 2.5 s with typical amplitudes\naround 5–30 cm. A current can be superimposed in the same direction as the wave propagation\nor in the opposite one, with a maximum flow rate around 1.2 m3 s−1. To generate a water-wave\nhorizon, we insert a ramp immersed in water, with positive and negative slopes separated by\na flat section; and send on it a train of waves against the reverse fluid flow produced by the\npump. At the place where the flow speed equals the group velocity of the waves a horizon is\ncreated. The geometrical parameters are: maximum water height 1.4 or 1.6 m; positive slope\n15.5◦; length of the flat part 6 m; minimum water height 30 or 50 cm; negative slope 18.5◦.\nWe fix the physical characteristics of the waves, period and amplitude, and only vary the\nbackground flow. We record the waves with the three video cameras indicated in figure6. As the\nbackground velocity is turbulent (the Reynolds number based on the water height is very large)\nand varies with depth, the horizon should be deduced from the mean velocity〈u(h, t)〉 measured\nat the interface between air and water; the brackets denote time averaging. Due to experimental\nconstraints, we measured the background flow with an MHD sensor averaged during 10 s. The\nvelocity profile on the flat part of the background flow is plug-like. Our first control parameter is\numax, the maximum of the counter-current plug velocity over the flat part of the geometric profile\nwithout water waves. We have checked that the velocity profiles are similar along a cross-section\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n9\n\nFigure 7. Phase diagram of our experiment. Each circle corresponds to a run with\nwave periodT = 2π/ω and maximal flow speedumax. The dots indicate runs\nwhere we observed negative-frequency waves, the squares run with horizons. In\nregimes without horizons, we saw a transition to mode conversion into purely\npositive frequencies below the lower green line in the diagram. The points\n(a) and (b) indicate the parameters used in figure8.\n\nof the tank. The second control parameter is the period of oscillations of the wave-maker. Both\nparameters are displayed in the phase diagram of figure7.\n\nIn our experiments, we observed indications of wave conversion in the presence of\nhorizons, but the cleanest data we obtained were for flow speeds just below the horizon\ncondition. In this case, the wave conversion still occurs [31], although it is reduced in magnitude.\nWithout a group-velocity horizon, the flow is much quieter, wave breaking and turbulence\nare significantly reduced. Figure8 shows the space–time diagrams of two typical cases, one\nillustrating the conversion into short waves with positive phase velocity, and the other showing\nwaves with negative frequency superposed on the incident waves.\n\n5. Numerical simulations\n\nIn order to test whether conversion into negative-frequency modes occurs even in the absence\nof a horizon, we applied Unruh’s method [19] for numerically simulating waves in a moving\nmedium. We consider wave packets propagating against the current in a simple one-dimensional\nmodel for the flow, using periodic boundary conditions, and analyze the mode conversion. This\nsimulation does not describe the influence of turbulence, nonlinearity, the three-dimensional\naspects of our experiment, nor the variation of the flow with water depth, but it captures the\nqualitative aspects of the Hawking effect and proves that the mode conversion can occur without\na horizon, a regime where the experiment is least affected by wave breaking and turbulence.\nA related example of Hawking radiation without horizon has been studied before [31] that\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n10\n\nFigure 8. Space–time diagrams, dimensions 1 m by 16 s, showing water waves\npropagating from the left to the right with the parameters (a) and (b) of figure7,\ninitial amplitude 5 cm and water height 1.4 m. No horizon is formed, but mode\nconversion still occurs. (a) Conversion into the positive-frequency waveskb of\nfigure 5; (b) waves with negative frequency (negative phase slope as shown\nin figure 2). The images were extracted from the video data recorded with\ncamera 1 of figure6. The right pictures display time traces along the lines\nindicated in the space–time diagrams. The traces show that the additional waves\nare periodic inT , indicating that they are converted incident waves.\n\nqualitatively agrees with our findings, although our case is significantly more extreme. Figure9\nshows the result of a wave packet interacting with the spatially dependent flow given by\n\nu(x) = −u0 − u1[tanh(ax) − tanh(a(x − x0))] ; (5)\n\nthe fluid moves left at velocity−u0 atx < 0, decreasing to−u0 − u1 betweenx = 0 andx = x0\n\nand returning to−u0 at x > x0. Gravity waves with the perturbationw(t, x) of the velocity\npotential obey the equation [13]\n\n(∂t + ∂xu)(∂t + u∂x)w = ig∂x tanh(−ih∂x)w , (6)\n\ngiving the dispersion relation (4). The wave packet propagates to the right; the flow speed\nnowhere reaches a value great enough to block the packet and create a white-hole horizon. When\nthe packet travels into the faster-flow regionx > 0 some of it tunnels into the blue-shifted root\nof the dispersion relation and this part propagates back to the left. There is also some tunneling\ninto the negativek root; this portion has shorter wavelength than the blue-shifted waves and\ntravels more quickly to the left. The simulation shows that negative-frequency waves can be\ngenerated without the presence of a horizon. The slope in the simulation is not realistic for\nour experiment, however, otherwise, there would be no visiblekh in the simulation. But in the\nexperiment negative-frequency waves were clearly observed. Apparently, the simple model [13]\nwe used does not capture all the complexity of our system.\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n11\n\nx\n\nw\n\nx\n\nw\n\nFigure 9. Wave–packet simulations. The left figure shows the incident\nwave packet traveling in the positivex-direction, the right figure its partial\nconversion into two wavelength components traveling in negativex-direction.\nThe components separate because of their different group velocities; the\nHawking component is visible in the center of the figure. The wrap-around is\ncaused by periodic boundary conditions and most of the packet that travels to\nthe right beyond the conversion region is not shown. We used the parameters\nu0 = 0.7 m s−1, u1 = 0.122 m s−1, a = 12 m−1, h = 0.6 m andT = 2.5 s.\n\n6. Conclusions\n\nWe believe we have made the first direct observation of the conversion of incident waves with\npositive- into negative-frequency waves in a moving medium. In astrophysics, such a mode\nconversion occurs at the event horizon of black holes. It represents the classical mechanism at\nthe heart of Hawking radiation [1]. However, we were surprised how strong the experimentally\nobserved mode conversion is, because in numerical simulations of a simple model [13], we saw a\nsignificantly lower conversion. This model takes into account the correct dispersion relation (4),\nbut it does not describe turbulence, nonlinearity, nor the three-dimensional nature of our\nexperiment. It would be highly desirable to find out exactly what happens to water waves at\nhorizons. Unfortunately, with the current set-up, we do not have sufficient data to characterize\nthe actual process of mode conversion in detail. It is conceivable that we have seen a new fluid-\nmechanics phenomenon that significantly enhances the Hawking effect. Could it be a nonlinear\nmode conversion, a nonlinear process generating harmonics with negative frequencies? We\nobserved that the incident waves become steeper as they propagate against the current. Hence,\nlocally, waves can be generated close to the crest, possibly with additional vorticity creation,\nwhere geometric cusps could develop through nonlinear effects. These crests waves are then\nswept away by the flow8. Moreover, it remains to be checked in future experiments whether a\ntransverse curvature of the wave crest could also be responsible for the creation of negative-\nfrequency waves. In any case, despite the limitations of our present experiment, we have found\nclear evidence for negative-frequency waves. In this way, we have demonstrated a key ingredient\nof the quantum radiation of black holes using a relatively simple classical laboratory analogue,\nwaves in a water tank.\n\nAcknowledgments\n\nWe thank Philippe Bardey, Jean Bougis, Mario Novello, Renaud Parentani, Viktor Ruban and\nMatt Visser for their discussions and encouragement, and Aurore de Gouvenain, Guillaume\n\n8 We are indebted to Viktor Ruban for pointing out this mechanism.\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n12\n\nBonnafoux, Jean-Francois Desté and Christian Perez for their technical support. This work has\nbeen financially supported by the Leverhulme Trust, a Royal Society Wolfson Research Merit\nAward and the University of St Andrews.\n\nReferences\n\n[1] Hawking S W 1974Nature24830\n[2] Bekenstein J D 1973Phys. Rev.D 7 2333\n[3] Green M B, Schwarz J H and Witten E 1987Superstring Theory(Cambridge: Cambridge University Press)\n[4] Rovelli C 1998Living Rev. 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(Sydney, 2000)vol 1 pp 227\n[28] Suastika I K 2004 Wave blockingPhD ThesisTechnische Universiteit Delft, The Netherlands Online at\n\nhttp://repository.tudelft.nl/file/275166/201607\n[29] Landau L D and Lifshitz E M 2004Fluid Mechanics(Amsterdam: Elsevier)\n[30] Landau L D and Lifshitz E M 1995The Classical Theory of Fields(Oxford: Butterworth-Heinemann)\n[31] Barceló C, Liberati S, Sonego S and Visser M 2006Phys. Rev. Lett.97 171301\n\nNew Journal of Physics10 (2008) 053015 (http://www.njp.org/)\n\nhttp://dx.doi.org/10.1038/248030a0\nhttp://dx.doi.org/10.1103/PhysRevD.7.2333\nhttp://dx.doi.org/10.1126/science.1153625\nhttp://dx.doi.org/10.1103/PhysRevLett.46.1351\nhttp://dx.doi.org/10.1088/0264-9381/15/6/024\nhttp://dx.doi.org/10.1103/PhysRevLett.85.4643\nhttp://dx.doi.org/10.1103/PhysRevLett.94.061302\nhttp://dx.doi.org/10.1103/PhysRevD.66.044019\nhttp://dx.doi.org/10.1016/0370-1573(95)00008-5\nhttp://dx.doi.org/10.1016/0550-3213(85)90418-3\nhttp://dx.doi.org/10.1103/PhysRevD.44.1731\nhttp://dx.doi.org/10.1103/PhysRevD.51.2827\nhttp://dx.doi.org/10.1103/PhysRevD.52.4559\nhttp://dx.doi.org/10.1103/PhysRevD.54.1568\nhttp://dx.doi.org/10.1007/BF02345020\nhttp://dx.doi.org/10.1088/0034-4885/66/7/203\nhttp://dx.doi.org/10.1029/2001JC001042\nhttp://repository.tudelft.nl/file/275166/201607\nhttp://dx.doi.org/10.1103/PhysRevLett.97.171301\nhttp://www.njp.org/\n\n\t1. Introduction\n\t2. Negative frequencies\n\t3. Water waves\n\t4. Experiment\n\t5. Numerical simulations\n\t6. Conclusions\n\tAcknowledgments\n\tReferences"},"fulltext":true,"key":"58a146616b412f8fca1185f2dfffe794","url":"https://inspirehep.net/files/58a146616b412f8fca1185f2dfffe794"}],"citation_count":240,"authors":[{"full_name_unicode_normalized":"rousseaux, germain","full_name":"Rousseaux, Germain","record":{"$ref":"https://inspirehep.net/api/authors/1941643"},"last_name":"Rousseaux","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903069"},"value":"Nice U."}],"ids":[{"schema":"INSPIRE BAI","value":"G.Rousseaux.2"}],"signature_block":"RASAXg","first_name":"Germain","uuid":"91bb9be6-b73e-40b9-9a28-00169dd42941","recid":1941643},{"full_name_unicode_normalized":"mathis, christian","full_name":"Mathis, Christian","record":{"$ref":"https://inspirehep.net/api/authors/2560843"},"last_name":"Mathis","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903069"},"value":"Nice U."}],"ids":[{"schema":"INSPIRE BAI","value":"C.Mathis.2"}],"signature_block":"MATc","first_name":"Christian","uuid":"9df8ea0b-6c56-426f-aa43-c7d205f4222b","recid":2560843},{"full_name_unicode_normalized":"maissa, philippe","full_name":"Maissa, Philippe","record":{"$ref":"https://inspirehep.net/api/authors/2559792"},"last_name":"Maissa","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/903069"},"value":"Nice U."}],"ids":[{"schema":"INSPIRE BAI","value":"P.Maissa.3"}],"signature_block":"MASp","first_name":"Philippe","uuid":"c7f6ca20-d3ac-4294-be0d-d2b31d47d4dc","recid":2559792},{"full_name_unicode_normalized":"philbin, thomas g.","full_name":"Philbin, Thomas G.","record":{"$ref":"https://inspirehep.net/api/authors/2234541"},"last_name":"Philbin","affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/909091"},"value":"St. Andrews U., Phys. 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In black-hole analogues, horizons are formed for waves propagating in a medium against the current when and where the flow exceeds the phase velocity. We report on the first direct observation of negative-frequency waves converted from positive-frequency waves in a moving medium.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["gr-qc"],"titles":[{"title":"Observation of negative phase velocity waves in a water tank: A classical analogue to the Hawking effect?"},{"source":"arXiv","title":"Observation of negative phase velocity waves in a water tank: A classical analogue to the Hawking effect?"}],"curated":true},"updated":"2023-03-07T03:46:21.976504+00:00","created":"2007-11-29T00:00:00+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/769034?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/769034?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/769034?format=latex-us","json":"https://inspirehep.net/api/literature/769034?format=json","json-expanded":"https://inspirehep.net/api/literature/769034?format=json-expanded","cv":"https://inspirehep.net/api/literature/769034?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A769034"},"id":"769034"},{"metadata":{"citation_count_without_self_citations":8,"citation_count":9,"authors":[{"full_name_unicode_normalized":"philbin, t.g.","full_name":"Philbin, T.G.","record":{"$ref":"https://inspirehep.net/api/authors/2234541"},"ids":[{"schema":"INSPIRE BAI","value":"T.G.Philbin.2"}],"last_name":"Philbin","first_name":"T.G.","uuid":"01182dc7-a4c8-449d-b6e3-a3b080bab3ef","recid":2234541},{"full_name_unicode_normalized":"allanson, o.","full_name":"Allanson, O.","record":{"$ref":"https://inspirehep.net/api/authors/3021569"},"ids":[{"schema":"INSPIRE BAI","value":"O.Allanson.1"}],"last_name":"Allanson","first_name":"O.","uuid":"bfae8f96-9eae-41ec-8ba0-5b637a500e11","recid":3021569}],"publication_info":[{"journal_volume":"86","pubinfo_freetext":"Phys. 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The block extends infinitely in the $y$ and $z$ directions and has boundaries at $x=0$ and $x=L$. The surrounding region is vacuum, written as the vacuum limit $\\varepsilon_2(\\omega)\\to 1$ of another material. We consider light linearly polarized in the $y$-direction and propagating in the $x$-direction perpendicular to the block faces at $x=0$ and $x=L$.","source":"arxiv","label":"fig:block","key":"4af7a0091aa46ece03bd665fb26eec37","url":"https://inspirehep.net/files/4af7a0091aa46ece03bd665fb26eec37"},{"filename":"figure2a.png","material":"preprint","caption":"Electromagnetic zero-point energy per unit frequency (\\ref{W}) inside a metal of length $L$ (blue curves). The material has permittivity (\\ref{osc}) with $\\omega_0=0$, $\\Omega=8.45\\,\\mathrm{eV}$ and $\\gamma=0.047\\,\\mathrm{eV}$. The dashed red lines are $\\hbar \\omega L/(2\\pi c)$, the value of $W(\\omega)$ in the same spatial region  but without the block. In the top plot $L=1\\,\\mu\\mathrm{m}$, in the bottom plot $L=10\\,\\mu\\mathrm{m}$. The zero-point energy is less than the free-space value for frequencies $\\omega\\lesssim\\Omega$.","source":"arxiv","label":"fig:gold1","key":"a73046e930c79318b7d8b70fd9762bad","url":"https://inspirehep.net/files/a73046e930c79318b7d8b70fd9762bad"},{"filename":"figure2b.png","material":"preprint","caption":"Electromagnetic zero-point energy per unit frequency (\\ref{W}) inside a metal of length $L$ (blue curves). The material has permittivity (\\ref{osc}) with $\\omega_0=0$, $\\Omega=8.45\\,\\mathrm{eV}$ and $\\gamma=0.047\\,\\mathrm{eV}$. The dashed red lines are $\\hbar \\omega L/(2\\pi c)$, the value of $W(\\omega)$ in the same spatial region  but without the block. In the top plot $L=1\\,\\mu\\mathrm{m}$, in the bottom plot $L=10\\,\\mu\\mathrm{m}$. The zero-point energy is less than the free-space value for frequencies $\\omega\\lesssim\\Omega$.","source":"arxiv","label":"fig:gold1","key":"ba24de457684b4f0e0e64d59f854a246","url":"https://inspirehep.net/files/ba24de457684b4f0e0e64d59f854a246"},{"filename":"figure3a.png","material":"preprint","caption":"Casimir energy per unit frequency (\\ref{WC}) of the metal blocks described in Fig.~\\ref{fig:gold1}. The total Casimir energy (\\ref{encas}) is positive for all lengths $L$.","source":"arxiv","label":"fig:gold2","key":"3e03bea8ff92dde646fa040a0d2627ae","url":"https://inspirehep.net/files/3e03bea8ff92dde646fa040a0d2627ae"},{"filename":"figure3b.png","material":"preprint","caption":"Casimir energy per unit frequency (\\ref{WC}) of the metal blocks described in Fig.~\\ref{fig:gold1}. The total Casimir energy (\\ref{encas}) is positive for all lengths $L$.","source":"arxiv","label":"fig:gold2","key":"5302ba206698d5d731eaabbe262b6189","url":"https://inspirehep.net/files/5302ba206698d5d731eaabbe262b6189"},{"filename":"figure4a.png","material":"preprint","caption":"Electromagnetic zero-point energy per unit frequency (\\ref{W}) inside a non-metallic dielectric (blue curves) of length $L$. The material has permittivity (\\ref{osc}) with $\\omega_0=5\\,\\mathrm{eV}$, $\\Omega=8\\,\\mathrm{eV}$ and $\\gamma=0.5\\,\\mathrm{eV}$.  The dashed red lines are $\\hbar \\omega L/(2\\pi c)$, the value of $W(\\omega)$ in the same spatial region but with the block replaced by empty space. In the top plot $L=1\\,\\mu\\mathrm{m}$, in the bottom plot $L=10\\,\\mu\\mathrm{m}$.","source":"arxiv","label":"fig:diel1","key":"8df54c0d48e1e616a1d7bb23a176e6be","url":"https://inspirehep.net/files/8df54c0d48e1e616a1d7bb23a176e6be"},{"filename":"figure4b.png","material":"preprint","caption":"Electromagnetic zero-point energy per unit frequency (\\ref{W}) inside a non-metallic dielectric (blue curves) of length $L$. The material has permittivity (\\ref{osc}) with $\\omega_0=5\\,\\mathrm{eV}$, $\\Omega=8\\,\\mathrm{eV}$ and $\\gamma=0.5\\,\\mathrm{eV}$.  The dashed red lines are $\\hbar \\omega L/(2\\pi c)$, the value of $W(\\omega)$ in the same spatial region but with the block replaced by empty space. In the top plot $L=1\\,\\mu\\mathrm{m}$, in the bottom plot $L=10\\,\\mu\\mathrm{m}$.","source":"arxiv","label":"fig:diel1","key":"fa8b5e790de01548ab1e92517ec14858","url":"https://inspirehep.net/files/fa8b5e790de01548ab1e92517ec14858"},{"filename":"figure5a.png","material":"preprint","caption":"Casimir energy per unit frequency (\\ref{WC}) of the dielectric blocks described in Fig.~\\ref{fig:diel1}. The total Casimir energy (\\ref{encas}) is positive for all lengths $L$ of the dielectric.","source":"arxiv","label":"fig:diel2","key":"42464aa6c9124a1d632f9111cbfbf412","url":"https://inspirehep.net/files/42464aa6c9124a1d632f9111cbfbf412"},{"filename":"figure5b.png","material":"preprint","caption":"Casimir energy per unit frequency (\\ref{WC}) of the dielectric blocks described in Fig.~\\ref{fig:diel1}. The total Casimir energy (\\ref{encas}) is positive for all lengths $L$ of the dielectric.","source":"arxiv","label":"fig:diel2","key":"0acd3efd2770557cc5fc1f8c2cc8b017","url":"https://inspirehep.net/files/0acd3efd2770557cc5fc1f8c2cc8b017"}],"first_author":{"emails":["t.g.philbin@exeter.ac.uk"],"full_name":"Philbin, T.G.","last_name":"Philbin","first_name":"T.G.","recid":2234541},"control_number":3031137,"dois":[{"material":"publication","source":"arXiv","value":"10.1088/2040-8978/18/9/095201"}],"document_type":["article"],"texkeys":["Philbin:2016vwn"],"abstracts":[{"source":"arXiv","value":"We consider one-dimensional propagation of quantum light in the presence of a block of material, with a full account of dispersion and absorption. The electromagnetic zero-point energy for some frequencies is damped (suppressed) by the block below the free-space value, while for other frequencies it is increased. We also calculate the regularized (Casimir) zero-point energy at each frequency and find that it too is damped below the free-space value (zero) for some frequencies. The total Casimir energy is positive.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"source":"arXiv","title":"Damped vacuum states of light"}],"curated":false},"updated":"2026-02-09T15:20:59.524790+00:00","created":"2025-10-02T16:21:43.140855+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/3031137?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/3031137?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/3031137?format=latex-us","json":"https://inspirehep.net/api/literature/3031137?format=json","json-expanded":"https://inspirehep.net/api/literature/3031137?format=json-expanded","cv":"https://inspirehep.net/api/literature/3031137?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A3031137"},"id":"3031137"},{"metadata":{"citation_count_without_self_citations":29,"citation_count":35,"authors":[{"raw_affiliations":[{"value":"Université François Rabelais de Tours , 60 Rue du Plat d’Etain, 37000 Tours, France"}],"full_name_unicode_normalized":"peloquin, cédric","full_name":"Peloquin, Cédric","record":{"$ref":"https://inspirehep.net/api/authors/2589508"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/1289918"},"value":"U. 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The positive relative frequency $\\omega'=\\omega -Uk=\\pm ck'=\\pm ck$ corresponds to the green color (the ' corresponds to the flow current frame of reference); the negative relative frequency is in blue. The conserved frequency of an incident wave as generated by a wave-maker is the horizontal line in dotted black.","source":"arxiv","key":"fe32bef5951ea31cc552c46181142f29","url":"https://inspirehep.net/files/fe32bef5951ea31cc552c46181142f29"},{"filename":"relativist_bump.png","caption":"Left bottom: velocity field for a bump geometry. Right bottom: velocity field for a trough geometry. Left and right top: scheme of the associated space-time diagrams. For both cases, we set $U_{min}=0.5\\cdot c$ and $U_{max}=1.5\\cdot c$.","source":"arxiv","key":"3f8b02cb3160974c2eb1941ef6170616","url":"https://inspirehep.net/files/3f8b02cb3160974c2eb1941ef6170616"},{"filename":"relativist_dip.png","caption":"Left bottom: velocity field for a bump geometry. Right bottom: velocity field for a trough geometry. Left and right top: scheme of the associated space-time diagrams. For both cases, we set $U_{min}=0.5\\cdot c$ and $U_{max}=1.5\\cdot c$.","source":"arxiv","key":"159c5881dd56d87fbe382593a99bc37f","url":"https://inspirehep.net/files/159c5881dd56d87fbe382593a99bc37f"},{"filename":"wavepacket.png","caption":"The wave-packet as described by the equation (\\ref{Gaussian_Wave_Packet}) used in all simulations with $\\sigma=\\lambda_0$.","source":"arxiv","key":"63ee6e610096ea34c46535c553d45d87","url":"https://inspirehep.net/files/63ee6e610096ea34c46535c553d45d87"},{"filename":"velocityfieldBW.png","caption":"Left: velocity field for the travel - Black$\\rightarrow$White - (co-current $U>0$). Right: velocity field for the travel - White$\\rightarrow$Black - (counter-current $U&lt;0$).","source":"arxiv","key":"ab8cafce7ef9f809aae2052fcd9f510a","url":"https://inspirehep.net/files/ab8cafce7ef9f809aae2052fcd9f510a"},{"filename":"velocityfieldWB.png","caption":"Left: velocity field for the travel - Black$\\rightarrow$White - (co-current $U>0$). Right: velocity field for the travel - White$\\rightarrow$Black - (counter-current $U&lt;0$).","source":"arxiv","key":"b8617e0973fd4c7e182d6a93b28b2615","url":"https://inspirehep.net/files/b8617e0973fd4c7e182d6a93b28b2615"},{"filename":"horizon.png","caption":"Left: the horizons in the Fourier space (for the dispersion relation of capillary-gravity waves with $U&lt;0$): white/black horizon (black diamond), blue/red horizon (white diamond), negative-white/negative-black horizon (gray diamond); the branches with a positive relative frequency are in green and the ones with a negative relative frequency are in blue. Rightthe six solutions for a given frequency for a white hole configuration: incident wave (I), retrograde (R), blue-shifted (B), capillary positive (cB), negative (N), capillary negative (cN) modes.","source":"arxiv","key":"39df1a4f9e1729cbfd9c938270932f54","url":"https://inspirehep.net/files/39df1a4f9e1729cbfd9c938270932f54"},{"filename":"solution.png","caption":"Left: the horizons in the Fourier space (for the dispersion relation of capillary-gravity waves with $U&lt;0$): white/black horizon (black diamond), blue/red horizon (white diamond), negative-white/negative-black horizon (gray diamond); the branches with a positive relative frequency are in green and the ones with a negative relative frequency are in blue. Rightthe six solutions for a given frequency for a white hole configuration: incident wave (I), retrograde (R), blue-shifted (B), capillary positive (cB), negative (N), capillary negative (cN) modes.","source":"arxiv","key":"3b8edb820b1306f618238a8d65c58490","url":"https://inspirehep.net/files/3b8edb820b1306f618238a8d65c58490"},{"filename":"Tc.png","caption":"Left: the dispersion relation with $U_c=-0.178m.s^{-1}$, $h=0.05m$ and $\\omega_c=\\frac{2\\pi}{T_c}$ (black dot), $T_c=0.425s$. Right: the dispersion relation with $U_b=-0.255m.s^{-1}$, $h=0.05m$ and $\\omega_c=\\frac{2\\pi}{T_c}$ (black dot), $T_c=0.647s$).","source":"arxiv","key":"621f1cfe6b0a61543d00651b0bc8c363","url":"https://inspirehep.net/files/621f1cfe6b0a61543d00651b0bc8c363"},{"filename":"Tb.png","caption":"Left: the dispersion relation with $U_c=-0.178m.s^{-1}$, $h=0.05m$ and $\\omega_c=\\frac{2\\pi}{T_c}$ (black dot), $T_c=0.425s$. Right: the dispersion relation with $U_b=-0.255m.s^{-1}$, $h=0.05m$ and $\\omega_c=\\frac{2\\pi}{T_c}$ (black dot), $T_c=0.647s$).","source":"arxiv","key":"0c6e4c24a94b4d8e4507c60cbf125d2d","url":"https://inspirehep.net/files/0c6e4c24a94b4d8e4507c60cbf125d2d"},{"filename":"realhorizon.png","caption":"Left: scheme of the main conversions in the travel - White$\\rightarrow$Black -. The vertical lines represent the horizons positions, white(right)/black(left) horizons(continuous lines) where $U=-0.337m^2/s$, blue(right)/red(left)  horizons(dashed lines) where $U=-0.213m^2/s$, negative(right)/negative-black(left) horizons(dot-dashed lines) where $U=-0.248m^2/s$, $h=0.05m$, $T=1s$. Right: the corresponding dispersion relations at the different horizons for the corresponding velocities.","source":"arxiv","key":"c521d3136ae8806c4be6b29e19e392cd","url":"https://inspirehep.net/files/c521d3136ae8806c4be6b29e19e392cd"},{"filename":"rdhorizon.png","caption":"Left: scheme of the main conversions in the travel - White$\\rightarrow$Black -. The vertical lines represent the horizons positions, white(right)/black(left) horizons(continuous lines) where $U=-0.337m^2/s$, blue(right)/red(left)  horizons(dashed lines) where $U=-0.213m^2/s$, negative(right)/negative-black(left) horizons(dot-dashed lines) where $U=-0.248m^2/s$, $h=0.05m$, $T=1s$. Right: the corresponding dispersion relations at the different horizons for the corresponding velocities.","source":"arxiv","key":"dc793f217a19d5d9f65be3f29a0943f8","url":"https://inspirehep.net/files/dc793f217a19d5d9f65be3f29a0943f8"},{"filename":"w_to_b1_colorscale.png","caption":"Space-time diagram of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (directionWhite$\\rightarrow$Black). The horizons are located roughly at $x_1=0m$ and $x_2=10m$. $h=0.05m$, $T=1s$, $U_{min}=-0.1m.s^{-1}$, $U_{max}=-0.9m.s^{-1}$. Left$a_1=a_2=1.5m^{-1}$. Right$a_1=a_2=20m^{-1}$","source":"arxiv","key":"018806cc9491c19e5b82522a54ed2c92","url":"https://inspirehep.net/files/018806cc9491c19e5b82522a54ed2c92"},{"filename":"w_to_b2_colorscale.png","caption":"Space-time diagram of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (directionWhite$\\rightarrow$Black). The horizons are located roughly at $x_1=0m$ and $x_2=10m$. $h=0.05m$, $T=1s$, $U_{min}=-0.1m.s^{-1}$, $U_{max}=-0.9m.s^{-1}$. Left$a_1=a_2=1.5m^{-1}$. Right$a_1=a_2=20m^{-1}$","source":"arxiv","key":"74cf3789999089fbb89707f402e7a137","url":"https://inspirehep.net/files/74cf3789999089fbb89707f402e7a137"},{"filename":"snapshot_w_to_b1.png","caption":"Left: successive snapshots of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (direction: White$\\rightarrow$Black) corresponding to the space-time diagram of the Figure \\ref{WtoB} left (amplitude scale [-1:1]). Right: corresponding to the space-time diagram of the Figure \\ref{WtoB} right (amplitude scale [-0.2:0.2]).","source":"arxiv","key":"de9ec3b51e03d5f61e08e00b925927a4","url":"https://inspirehep.net/files/de9ec3b51e03d5f61e08e00b925927a4"},{"filename":"snapshot_w_to_b2.png","caption":"Left: successive snapshots of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (direction: White$\\rightarrow$Black) corresponding to the space-time diagram of the Figure \\ref{WtoB} left (amplitude scale [-1:1]). Right: corresponding to the space-time diagram of the Figure \\ref{WtoB} right (amplitude scale [-0.2:0.2]).","source":"arxiv","key":"70381f0c0445d481cb0afd9d5ab3fd01","url":"https://inspirehep.net/files/70381f0c0445d481cb0afd9d5ab3fd01"},{"filename":"b_to_w1_colorscale.png","caption":"Space-time diagram of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (direction: Black$\\rightarrow$White). The horizons are located roughly at $x_1=0m$ and $x_2=10m$. $h=0.05m$, $T=1s$, $U_{min}=0.1m.s^{-1}$, $U_{max}=0.9m.s^{-1}$. Left: $a_1=a_2=1.5m^{-1}$. Right: $a_1=a_2=20m^{-1}$","source":"arxiv","key":"519224c25a2aa4532d90429715affb15","url":"https://inspirehep.net/files/519224c25a2aa4532d90429715affb15"},{"filename":"b_to_w2_colorscale.png","caption":"Space-time diagram of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (direction: Black$\\rightarrow$White). The horizons are located roughly at $x_1=0m$ and $x_2=10m$. $h=0.05m$, $T=1s$, $U_{min}=0.1m.s^{-1}$, $U_{max}=0.9m.s^{-1}$. Left: $a_1=a_2=1.5m^{-1}$. Right: $a_1=a_2=20m^{-1}$","source":"arxiv","key":"c08f10364407c1cf114f2c58e864aa96","url":"https://inspirehep.net/files/c08f10364407c1cf114f2c58e864aa96"},{"filename":"snapshot_b_to_w1.png","caption":"Left: successive snapshots of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (direction: Black$\\rightarrow$White) corresponding to the space-time diagram of the Figure \\ref{BtoW} left (amplitude scale [-1:1]). Right: corresponding to the space-time diagram of the Figure \\ref{BtoW} right (amplitude scale [-0.05:0.05]).","source":"arxiv","key":"33347063a4705e26fc4ad01dfd0d3909","url":"https://inspirehep.net/files/33347063a4705e26fc4ad01dfd0d3909"},{"filename":"snapshot_b_to_w2.png","caption":"Left: successive snapshots of the propagation of a Gaussian wave-packet in an analogue dispersive bi-directional wormhole (direction: Black$\\rightarrow$White) corresponding to the space-time diagram of the Figure \\ref{BtoW} left (amplitude scale [-1:1]). Right: corresponding to the space-time diagram of the Figure \\ref{BtoW} right (amplitude scale [-0.05:0.05]).","source":"arxiv","key":"11daca0638d99453cf7d3648ff29e38e","url":"https://inspirehep.net/files/11daca0638d99453cf7d3648ff29e38e"}],"legacy_version":"20170316203458.0","inspire_categories":[{"term":"General Physics"},{"term":"Gravitation and Cosmology"}],"first_author":{"full_name":"Peloquin, Cédric","last_name":"Peloquin","first_name":"Cédric","recid":2589508},"control_number":1409102,"dois":[{"source":"bibmatch","value":"10.1103/PhysRevD.93.084032"}],"document_type":["article"],"texkeys":["Peloquin:2015rnl"],"abstracts":[{"source":"APS","value":"We numerically study water wave packets on a spatially varying countercurrent in the presence of surface tension. Depending on the details of the velocity profile, we show that traversable and bidirectional analogue wormholes exist in fluid mechanics. The limitations on traversability of wormholes in general relativity are absent here because of the dispersion of water waves and the ability to form flow profiles that are not solutions of Einstein’s equations. We observe that negative energy can be trapped between analogue horizons forming a laserlike cavity. Six horizons are involved in the trapping cavity because of the existence of two dispersive scales, in contrast to previous treatments which considered two horizons and one dispersive scale.","abstract_source_suggest":{"input":"APS"}},{"source":"arXiv","value":"We numerically study water wave packets on a spatially varying counter-current in the presence of surface tension. Depending on the details of the velocity profile, we show that traversable and bi-directional analogue wormholes exist in fluid mechanics. The limitations on traversability of wormholes in general relativity are absent here because of the dispersion of water waves and the ability to form flow profiles that are not solutions of Einstein's equations. We observe that negative energy can be trapped between analogue horizons forming a LASER-like cavity. Six horizons are involved in the trapping cavity because of the existence of two dispersive scales, in contrast to previous treatments which considered two horizons and one dispersive scale.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["physics.flu-dyn"],"titles":[{"title":"Analog wormholes and black hole laser effects in hydrodynamics"},{"source":"arXiv","title":"Analogue Wormholes and Black Hole LASER Effect in Hydrodynamics"}],"imprints":[{"date":"2016-04-18"}],"curated":true},"updated":"2023-03-06T19:57:29.845239+00:00","created":"2015-12-11T00:00:00+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/1409102?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/1409102?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/1409102?format=latex-us","json":"https://inspirehep.net/api/literature/1409102?format=json","json-expanded":"https://inspirehep.net/api/literature/1409102?format=json-expanded","cv":"https://inspirehep.net/api/literature/1409102?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A1409102"},"id":"1409102"},{"metadata":{"citation_count_without_self_citations":17,"citation_count":18,"authors":[{"raw_affiliations":[{"value":"Physics and Astronomy Department, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom"}],"full_name_unicode_normalized":"philbin, t.g.","full_name":"Philbin, T.G.","record":{"$ref":"https://inspirehep.net/api/authors/2234541"},"ids":[{"schema":"INSPIRE BAI","value":"T.G.Philbin.2"}],"last_name":"Philbin","uuid":"0c2a9740-c1bd-4b31-9986-86bd99c8e76b","first_name":"T.G.","recid":2234541}],"publication_info":[{"journal_volume":"87","pubinfo_freetext":"Phys. Rev. A 87, 043843 (2013)","artid":"043843","material":"publication","year":2013,"journal_title":"Phys.Rev.A"}],"citeable":true,"$schema":"https://inspirehep.net/schemas/records/hep.json","references":[{"reference":{"label":"1","publication_info":{"journal_volume":"5","artid":"696","year":1964,"page_start":"696","journal_title":"J.Math.Phys."},"authors":[{"full_name":"Lipkin, D.M."}]},"raw_refs":[{"schema":"text","source":"arXiv","value":"[1] D. M. Lipkin, J. Math. Phys. 5, 696 (1964)."}]},{"reference":{"label":"2","publication_info":{"journal_volume":"5","artid":"1659","year":1964,"page_start":"1659","journal_title":"J.Math.Phys."},"authors":[{"full_name":"Morgan, T.A."}]},"raw_refs":[{"schema":"text","source":"arXiv","value":"[2] T. A. Morgan, J. Math. Phys. 5, 1659 (1964)."}]},{"reference":{"label":"3","publication_info":{"journal_volume":"6","artid":"1952","year":1965,"page_start":"1952","journal_title":"J.Math.Phys."},"authors":[{"full_name":"O'Connell, R.F."},{"full_name":"Tompkins, D.R."}]},"raw_refs":[{"schema":"text","source":"arXiv","value":"[3] R. F. O’Connell and D. R. Tompkins, J. Math. Phys. 6, 1952 (1965)."}]},{"reference":{"label":"4","publication_info":{"journal_volume":"6","artid":"1022","year":1965,"page_start":"1022","journal_title":"J.Math.Phys."},"authors":[{"full_name":"Kibble, T.W.B."}]},"raw_refs":[{"schema":"text","source":"arXiv","value":"[4] T. W. B. Kibble, J. Math. Phys. 6, 1022 (1965)."}]},{"reference":{"label":"5","publication_info":{"journal_volume":"37","artid":"1390","year":1965,"page_start":"1390","journal_title":"Nuovo Cim."},"authors":[{"full_name":"Candlin, D.J."}]},"raw_refs":[{"schema":"text","source":"arXiv","value":"[5] D. J. 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Ohmic damping of the oscillator can be exactly treated with a 1D scalar field reservoir, whereas general non-Ohmic damping is conveniently treated with a continuum reservoir of harmonic oscillators. Using the diagonalized Hamiltonian of the total system, we calculate a number of thermodynamic quantities for the damped oscillator: the mean force internal energy, mean force free energy, and another internal energy based on the free-oscillator Hamiltonian. The classical mean force energy is equal to that of a free oscillator, for both Ohmic and non-Ohmic damping and no matter how strong the coupling to the reservoir. In contrast, the quantum mean force energy depends on the details of the damping and diverges for strictly Ohmic damping. 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The squares of the uncertainties are scaled with an appropriate power of $\\omega_0$ to have the same units as the product $\\Delta q\\,\\Delta p$. The uncertainty relation is satisfied for all parameters obeying (\\ref{ineq}).","source":"arxiv","label":"fig:up","key":"8c461849ad49ae9c2f8e883b0277283f","url":"https://inspirehep.net/files/8c461849ad49ae9c2f8e883b0277283f"},{"filename":"figure2.png","material":"preprint","caption":"Plots of the energy of the harmonic oscillator versus damping $\\gamma_2$, with $\\omega_0=10^{10}\\,\\mathrm{s}^{-1}$ and $\\gamma_1=\\omega_0/4$, for $T=0$ and $T=\\hbar\\omega_0/k_B$. The zero-point energy ($T=0$) is damped below the free-oscillator value $\\hbar\\omega_0/2$. The energy for $T>0$ is also damped below the free-oscillator value, though this damping is not very apparent except for very low $T$. The energy that can be extracted from the oscillator at $T>0$ (i.e.\\ the $T>0$ energy minus the zero-point energy) increases with damping.","source":"arxiv","label":"fig:zpe","key":"2a67553c7bdb95ba93afa59ddf5bd0c3","url":"https://inspirehep.net/files/2a67553c7bdb95ba93afa59ddf5bd0c3"}],"preprint_date":"2013-07-31","author_count":2,"first_author":{"emails":["t.g.philbin@exeter.ac.uk"],"full_name":"Philbin, T.G.","last_name":"Philbin","first_name":"T.G.","recid":2234541},"public_notes":[{"source":"arXiv","value":"10 pages"}],"control_number":3009396,"earliest_date":"2013-07-31","document_type":["article"],"texkeys":["Philbin:2013stu"],"abstracts":[{"source":"arXiv","value":"The physics of quantum electromagnetism in an absorbing medium is that of a field of damped harmonic oscillators. Yet until recently the damped harmonic oscillator was not treated with the same kind of formalism used to describe quantum electrodynamics in a arbitrary medium. Here we use the techniques of macroscopic QED, based on the Huttner--Barnett reservoir, to describe the quantum mechanics of a damped oscillator. We calculate the thermal and zero-point energy of the oscillator for a range of damping values from zero to infinity. While both the thermal and zero-point energies decrease with damping, the energy stored in the oscillator at fixed temperature increases with damping, an effect that may be experimentally observable. As the results follow from canonical quantization, the uncertainty principle is valid for all damping levels.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"source":"arXiv","title":"Damping the zero-point energy of a harmonic oscillator"}],"facet_author_name":["2034487_S.A.R. Horsley","2234541_Thomas G. 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