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Jürgen Ehlers

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Jürgen Ehlers
At the award ceremony for the Charles University Medal in Potsdam, September 2007
Born(1929-12-29)29 December 1929
Hamburg,Germany
Died20 May 2008(2008-05-20)(aged 78)
Potsdam,Brandenburg,Germany
NationalityGerman
Alma materUniversity of Hamburg
Known forGeneral relativity
Mathematical physics
AwardsMax Planck Medal(2002)
Scientific career
FieldsPhysics
InstitutionsUniversity of Hamburg
Max Planck Institute for Astrophysics
Max Planck Institute for Gravitational Physics
Doctoral advisorPascual Jordan
Part of a series on
Physical cosmology
Full-sky image derived from nine years' WMAP data
Early universe
Backgrounds
Expansion·Future
Components·Structure
Components
Structure
Experiments
Scientists
Subject history

Jürgen Ehlers(German:[ˈjʏʁɡŋ̩ˈeːlɐs];29 December 1929 – 20 May 2008) was a Germanphysicistwho contributed to the understanding ofAlbert Einstein's theory ofgeneral relativity.From graduate and postgraduate work inPascual Jordan's relativity research group atHamburg University,he held various posts as a lecturer and, later, as a professor before joining theMax Planck Institute for AstrophysicsinMunichas a director. In 1995, he became the founding director of the newly createdMax Planck Institute for Gravitational PhysicsinPotsdam,Germany.

Ehlers' research focused on the foundations of general relativity as well as on the theory's applications toastrophysics.He formulated a suitable classification ofexact solutionstoEinstein's field equationsand proved theEhlers–Geren–Sachs theoremthat justifies the application of simple, general-relativistic model universes to moderncosmology.He created aspacetime-oriented description ofgravitational lensingand clarified the relationship between models formulated within the framework of general relativity and those ofNewtonian gravity.In addition, Ehlers had a keen interest in both the history andphilosophy of physicsand was an ardent populariser of science.

Biography[edit]

Early life[edit]

Jürgen Ehlers was born in Hamburg on 29 December 1929.[1]He attended public schools from 1936 to 1949, and then went on to study physics, mathematics and philosophy atHamburg Universityfrom 1949 to 1955. In the winter term of 1955–56, he passed the high school teacher's examination (Staatsexamen), but instead of becoming a teacher undertook graduate research withPascual Jordan,who acted as his thesis advisor. Ehlers' doctoral work was on the construction and characterization of solutions of theEinstein field equations.He earned his doctorate in physics from Hamburg University in 1958.[2]

Prior to Ehlers' arrival, the main research of Jordan's group had been dedicated to ascalar-tensormodification of general relativity that later became known asJordan–Brans–Dicke theory.This theory differs from general relativity in that thegravitational constantis replaced by a variablefield.Ehlers was instrumental in changing the group's focus to the structure and interpretation of Einstein's original theory.[3]Other members of the group included Wolfgang Kundt,Rainer K. Sachsand Manfred Trümper.[4] The group had a close working relationship withOtto Heckmannand his studentEngelbert SchückingatHamburger Sternwarte,the city's observatory. Guests at the group's colloquium includedWolfgang Pauli,Joshua Goldberg andPeter Bergmann.[5]

In 1961, as Jordan's assistant, Ehlers earned hishabilitation,qualifying him for a German professorship. He then held teaching and research positions in Germany and in the US, namely at theUniversity of Kiel,Syracuse Universityand Hamburg University. From 1964 to 1965, he was at theGraduate Research Center of the SouthwestinDallas.From 1965 to 1971, he held various positions inAlfred Schild's group at theUniversity of Texas at Austin,starting as anassociate professorand, in 1967, obtaining a position as full professor. During that time, he held visiting professorships at the universities ofWürzburgandBonn.[6]

Munich[edit]

In 1970, Ehlers received an offer to join theMax Planck Institute for Physics and AstrophysicsinMunichas the director of its gravitational theory department.[7]Ehlers had been suggested byLudwig Biermann,the institute's director at the time. When Ehlers joined the institute in 1971, he also became an adjunct professor at Munich'sLudwig Maximilian University.In March 1991, the institute split into theMax Planck Institute for Physicsand theMax Planck Institute for Astrophysics,where Ehlers' department found a home.[8]Over the 24 years of his tenure, his research group was home to, among others,Gary Gibbons,John Stewart and Bernd Schmidt, as well as visiting scientists includingAbhay Ashtekar,Demetrios ChristodoulouandBrandon Carter.[9]

One of Ehlers'postdoctoral studentsin Munich was Reinhard Breuer, who later became editor-in-chief ofSpektrum der Wissenschaft,the German edition of the popular-science journalScientific American.[10]

Potsdam[edit]

When German science institutions reorganized afterGerman reunificationin 1990, Ehlers lobbied for the establishment of an institute of the Max Planck Society dedicated to research on gravitational theory. On 9 June 1994, the Society decided to open theMax Planck Institute for Gravitational PhysicsinPotsdam.The institute started operations on 1 April 1995, with Ehlers as its founding director and as the leader of its department for the foundations and mathematics of general relativity.[11]Ehlers then oversaw the founding of a second institute department devoted togravitational waveresearch and headed byBernard F. Schutz.On 31 December 1998, Ehlers retired to become founding directoremeritus.[12]

Ehlers continued to work at the institute until his death on 20 May 2008.[13]He left behind his wife Anita Ehlers, his four children, Martin, Kathrin, David, and Max, as well as five grandchildren.[14]

Research[edit]

Ehlers' research was in the field of general relativity. In particular, he made contributions tocosmology,the theory ofgravitational lensesandgravitational waves.His principal concern was to clarify general relativity's mathematical structure and its consequences, separating rigorous proofs fromheuristicconjectures.[15]

Exact solutions[edit]

For his doctoral thesis, Ehlers turned to a question that was to shape his lifetime research. He sought exact solutions ofEinstein's equations:model universesconsistent with the laws of general relativity that are simple enough to allow for an explicit description in terms of basic mathematical expressions. These exact solutions play a key role when it comes to building general-relativistic models of physical situations. However, general relativity is a fullycovarianttheory – its laws are the same, independent of whichcoordinatesare chosen to describe a given situation. One direct consequence is that two apparently different exact solutions could correspond to the same model universe, and differ only in their coordinates. Ehlers began to look for serviceable ways of characterizing exact solutionsinvariantly,that is, in ways that do not depend on coordinate choice. In order to do so, he examined ways of describing the intrinsic geometric properties of the known exact solutions.[16]

During the 1960s, following up on his doctoral thesis, Ehlers published a series of papers, all but one in collaboration with colleagues from the Hamburg group, which later became known as the "Hamburg Bible".[17] The first paper, written with Jordan and Kundt, is a treatise on how to characterize exact solutions to Einstein's field equations in a systematic way. The analysis presented there uses tools fromdifferential geometrysuch as thePetrov classificationofWeyl tensors(that is, those parts of theRiemann tensordescribing thecurvatureofspace-timethat are not constrained by Einstein's equations),isometry groupsandconformaltransformations. This work also includes the first definition and classification ofpp-waves,a class of simple gravitational waves.[18]

The following papers in the series were treatises ongravitational radiation(one with Sachs, one with Trümper). The work with Sachs studies, among other things,vacuum solutionswith specialalgebraicproperties, using the 2-componentspinorformalism. It also gives a systematic exposition of the geometric properties of bundles (in mathematical terms: congruences) of light beams. Spacetime geometry can influence the propagation of light, making them converge on or diverge from each other, or deforming the bundle's cross section without changing its area. The paper formalizes these possible changes in the bundle in terms of the bundle's expansion (convergence/divergence), and twist and shear (cross-section area-conserving deformation), linking those properties to spacetime geometry. One result is theEhlers-Sachs theoremdescribing the properties of the shadow produced by a narrow beam of light encountering an opaque object. The tools developed in that work would prove essential for the discovery byRoy Kerrof hisKerr solution,describing a rotatingblack hole– one of the most important exact solutions.[19]

The last of these seminal papers addressed the general-relativistic treatment of the mechanics of continuous media. However, useful the notion of a point mass may be in classical physics; in general relativity, such an idealized mass concentration into a single point of space is not even well-defined. That is why relativistichydrodynamics,that is, the study of continuous media, is an essential part of model-building in general relativity. The paper systematically describes the basic concepts and models in what the editor of the journalGeneral Relativity and Gravitation,on the occasion of publishing an English translation 32 years after the original publication date, called "one of the best reviews in this area".[20]

Another part of Ehlers' exploration of exact solutions in his thesis led to a result that proved important later. At the time he started his research on his doctoral thesis, theGolden age of general relativityhad not yet begun and the basic properties and concepts of black holes were not yet understood. In the work that led to his doctoral thesis, Ehlers proved important properties of the surface around a black hole that would later be identified as itshorizon,in particular that thegravitational fieldinside cannot be static, but must change over time. The simplest example of this is the "Einstein-Rosen bridge", orSchwarzschild wormholethat is part of the Schwarzschild solution describing an idealized, spherically symmetric black hole: the interior of the horizon houses a bridge-like connection that changes over time, collapsing sufficiently quickly to keep any space-traveler from traveling through the wormhole.[21]

Ehlers group[edit]

Main article:Ehlers group

In physics,dualitymeans that two equivalent descriptions of a particular physical situation exist, using different physical concepts. This is a special case of a physicalsymmetry,that is, a change that preserves key features of a physical system. A simple example for a duality is that between theelectric fieldEand themagnetic fieldBelectrodynamics:In the complete absence of electrical charges, the replacementE–B,BEleavesMaxwell's equationsinvariant. Whenever a particular pair of expressions forBandEconform to the laws of electrodynamics, switching the two expressions around and adding a minus sign to the newBis also valid.[22]

In his doctoral thesis, Ehlers pointed out a duality symmetry between different components of themetricof a stationaryvacuumspacetime,which maps solutions of Einstein's field equations to other solutions. This symmetry between the tt-component of the metric, which describes time as measured by clocks whose spatial coordinates do not change, and a term known as thetwist potentialis analogous to the aforementioned duality betweenEandB.[23]

The duality discovered by Ehlers was later expanded to a larger symmetry corresponding to thespecial linear group.This largersymmetry grouphas since become known as theEhlers group.Its discovery led to further generalizations, notably the infinite-dimensionalGeroch group(the Geroch group is generated by twonon-commutingsubgroups,one of which is the Ehlers group). These so-calledhidden symmetriesplay an important role in theKaluza–Klein reductionof both general relativity and its generalizations, such aseleven-dimensional supergravity.Other applications include their use as a tool in the discovery of previously unknown solutions and their role in a proof that solutions in the stationaryaxi-symmetriccase form anintegrable system.[24]

Cosmology: Ehlers–Geren–Sachs theorem[edit]

Main article:Ehlers–Geren–Sachs theorem
The inhomogeneities in the temperature of the cosmic background radiation recorded in this image from the satellite probeWMAPamount to no more than 10−4kelvins.

The Ehlers–Geren–Sachs theorem, published in 1968, shows that in a given universe, if all freely falling observers measure thecosmic background radiationto have exactly the same properties in all directions (that is, they measure the background radiation to beisotropic), then that universe is an isotropic and homogeneousFriedmann–Lemaîtrespacetime.[25]Cosmic isotropy and homogeneity are important as they are the basis of the modern standard model of cosmology.[26]

Fundamental concepts in general relativity[edit]

In the 1960s, Ehlers collaborated withFelix PiraniandAlfred Schildon a constructive-axiomatic approach to general relativity: a way of deriving the theory from a minimal set of elementary objects and a set of axioms specifying these objects' properties. The basic ingredients of their approach are primitive concepts such asevent,lightray,particleandfreely falling particle.At the outset, spacetime is a mere set of events, without any further structure. They postulated the basic properties of light and freely falling particles as axioms, and with their help constructed thedifferential topology,conformal structureand, finally, themetricstructure of spacetime, that is: the notion of when two events are close to each other, the role of light rays in linking up events, and a notion of distance between events. Key steps of the construction correspond to idealized measurements, such the standard range finding used inradar.The final step derived Einstein's equations from the weakest possible set of additional axioms. The result is a formulation that clearly identifies the assumptions underlying general relativity.[27]

In the 1970s, in collaboration with Ekkart Rudolph, Ehlers addressed the problem of rigid bodies in general relativity. Rigid bodies are a fundamental concept in classical physics. However, the fact that by definition their different parts move simultaneously is incompatible with the relativistic concept of thespeed of lightas a limiting speed for the propagation of signals and other influences. While, as early as 1909,Max Bornhad given a definition of rigidity that was compatible with relativistic physics, his definition depends on assumptions that are not satisfied in a general space-time, and are thus overly restrictive. Ehlers and Rudolph generalized Born's definition to a more readily applicable definition they called "pseudo-rigidity", which represents a more satisfactory approximation to the rigidity of classical physics.[28]

Gravitational lensing[edit]

Most astrophysical modeling of gravitational lens systems makes use of the quasi-Newtonian approximation

With Peter Schneider, Ehlers embarked on an in-depth study of the foundations ofgravitational lensing.One result of this work was a 1992 monograph co-authored with Schneider and Emilio Falco. It was the first systematic exposition of the topic that included both the theoretical foundations and the observational results. From the viewpoint of astronomy, gravitational lensing is often described using a quasi-Newtonian approximation—assuming thegravitational fieldto be small and the deflection angles to be minute—which is perfectly sufficient for most situations of astrophysical relevance. In contrast, the monograph developed a thorough and complete description of gravitational lensing from a fully relativistic space-time perspective. This feature of the book played a major part in its long-term positive reception.[29]In the following years, Ehlers continued his research on the propagation of bundles of light in arbitrary spacetimes.[30]

Frame theory and Newtonian gravity[edit]

A basic derivation of the Newtonian limit of general relativity is as old as the theory itself. Einstein used it to derive predictions such as theanomalous perihelion precessionof the planetMercury.Later work byÉlie Cartan,Kurt Friedrichsand others showed more concretely how a geometrical generalization ofNewton's theory of gravityknown asNewton–Cartan theorycould be understood as a (degenerate) limit ofgeneral relativity.This required letting a specific parametergo to zero. Ehlers extended this work by developing aframe theorythat allowed for constructing the Newton–Cartan limit, and in a mathematically precise way, not only for the physical laws, but for any spacetime obeying those laws (that is, solutions of Einstein's equations). This allowed physicists to explore what the Newtonian limit meant in specific physical situations. For example, the frame theory can be used to show that the Newtonian limit of aSchwarzschild black holeis a simplepoint particle.Also, it allows Newtonian versions of exact solutions such as theFriedmann–Lemaître modelsor theGödel universeto be constructed.[31]Since its inception, ideas Ehlers introduced in the context of his frame theory have found important applications in the study of both the Newtonian limit of general relativity and of thePost-Newtonian expansion,where Newtonian gravity is complemented by terms of ever higher order inin order to accommodate relativistic effects.[32]

General relativity isnon-linear:the gravitational influence of two masses is not simply the sum of those masses' individual gravitational influences, as had been the case in Newtonian gravity. Ehlers participated in the discussion of how theback-reactionfrom gravitational radiation onto a radiating system could be systematically described in a non-linear theory such as general relativity, pointing out that the standardquadrupoleformula for the energy flux for systems like thebinary pulsarhad not (yet) been rigorously derived: a priori, a derivation demanded the inclusion of higher-order terms than was commonly assumed, higher than were computed until then.[33]

His work on the Newtonian limit, particularly in relation tocosmologicalsolutions, led Ehlers, together with his former doctoral student Thomas Buchert, to a systematic study ofperturbationsand inhomogeneities in a Newtonian cosmos. This laid the groundwork for Buchert's later generalization of this treatment of inhomogeneities. This generalization was the basis of his attempt to explain what is currently seen as the cosmic effects of acosmological constantor, in modern parlance,dark energy,as a non-linear consequence of inhomogeneities in general-relativistic cosmology.[34]

History and philosophy of physics[edit]

Complementing his interest in the foundations of general relativity and, more generally, of physics, Ehlers researched the history of physics. Up until his death, he collaborated in a project on the history of quantum theory at theMax Planck Institute for the History of Sciencein Berlin.[35]In particular, he explored Pascual Jordan's seminal contributions to the development ofquantum field theorybetween 1925 and 1928.[36]Throughout his career, Ehlers had an interest in the philosophical foundations and implications of physics and contributed to research on this topic by addressing questions such as the basic status of scientific knowledge in physics.[37]

Science popularization[edit]

Ehlers showed a keen interest in reaching a general audience. He was a frequent public lecturer, at universities as well as at venues such as theUraniainBerlin.He authored popular-science articles, including contributions to general-audience journals such asBild der Wissenschaft.He edited a compilation of articles on gravity for the German edition ofScientific American.[38] Ehlers directly addressed physics teachers, in talks and journal articles on the teaching of relativity and related basic ideas, such asmathematicsas the language of physics.[39]

Honours and awards[edit]

Ehlers became a member of theBerlin-Brandenburg Academy of Sciences and Humanities(1993), theAkademie der Wissenschaften und der Literatur,Mainz(1972), theLeopoldinainHalle(1975) and theBavarian Academy of Sciences and Humanitiesin Munich (1979).[40]From 1995 to 1998, he served as president of theInternational Society on General Relativity and Gravitation.[41]He also received the 2002Max Planck Medalof theGerman Physical Society,theVoltaGold Medal ofPavia University(2005) and the medal of the Faculty of Natural Sciences ofCharles University,Prague(2007).[42]

In 2008, the International Society on General Relativity and Gravitation instituted the "Jürgen Ehlers Thesis Prize" in commemoration of Ehlers. It is sponsored by the scientific publishing houseSpringerand is awarded triennially, at the society's international conference, to the best doctoral thesis in the areas of mathematical and numerical general relativity.[43]Issue 9 of volume 41 of the journalGeneral Relativity and Gravitationwas dedicated to Ehlers, in memoriam.[44]

Selected publications[edit]

  • Börner, G.; Ehlers, J., eds. (1996),Gravitation,Spektrum Akademischer Verlag,ISBN3-86025-362-X
  • Ehlers, Jürgen (1973), "Survey of general relativity theory", in Israel, Werner (ed.),Relativity, Astrophysics and Cosmology,D. Reidel, pp. 1–125,ISBN90-277-0369-8
  • Schneider, P.; Ehlers, J.; Falco, E. E. (1992),Gravitational lenses,Springer,ISBN3-540-66506-4

Notes[edit]

  1. ^Arimondo, E.; Ertmer, W.; Schleich, W.; Rasel, E.M. (2009).Atom Optics and Space Physics: Proceedings of the International School of Physics "Enrico Fermi", Course CLXVIII, Varenna on Lake Como, Villa Monastero, 3-13 July 2007.International School of Physics Enrico Fermi. IOS Press. p. 9.ISBN978-1-58603-990-5.Retrieved28 December2022.
  2. ^The dissertation isEhlers 1957;cf.Ellis 2009.
  3. ^Schücking, Engelbert (2006), "Jürgen Ehlers", in Schmidt, Bernd G. (ed.),Einstein's Field Equations and Their Physical Implications,Springer, pp. V–VI,ISBN3-540-67073-4
  4. ^As described inEllis & Krasiński 2007andSachs 2009.
  5. ^Ellis 2009
  6. ^Huisken, Nicolai & Schutz 2009,cf. the English version online asHuisken, Nicolai & Schutz 2008,and the associated CV,Lebenslauf von Prof. Dr. Jürgen Ehlers(PDF),Max Planck Institute for Gravitational Physics, 27 May 2008, archived fromthe original(PDF)on 19 May 2009,retrieved27 May2008(in German, English translation of title: "CV for Prof. Dr. Jürgen Ehlers" ). Dates and positions also summarized inWeber & Borissoff 1998.
  7. ^Henning & Kazemi 2011,p. 472
  8. ^Henning & Kazemi 2011,p. 634
  9. ^As described inBreuer 2008
  10. ^Breuer 2008
  11. ^Henning & Kazemi 2011,p. 676
  12. ^Henning & Kazemi 2011,p. 737
  13. ^See p. 520 in the Max Planck Society's annual report for 2000,Jahrbuch 2000,Max-Planck-Gesellschaft, 2000.Time as emeritus and death cf.Braun 2008.
  14. ^Huisken, Nicolai & Schutz 2009;English version online asHuisken, Nicolai & Schutz 2008
  15. ^Schücking 2000
  16. ^B. Schmidt, Preface toSchmidt 2000
  17. ^Ellis 2009,p. 2180
  18. ^A later version of this paper isEhlers & Kundt 1962. For an assessment, see J. Bicak, p. 14f. inSchmidt 2000
  19. ^Ehlers-Sachs theorem see sec. 5.3 inFrolov & Novikov 1998.An assessment of the work and its connection with Kerr solution is given by J. Bicak on p. 14f. ofSchmidt 2000.The original work with Sachs isJordan, Ehlers & Sachs 1961.
  20. ^The English translation, byG. F. R. Ellis,isEhlers 1993.The quotation can be found on p. 1225 in the editor's comments section.
  21. ^The changing views of what eventually be regarded as black holes can be found inIsrael 1987.Ehlers' thesis isEhlers 1957.
  22. ^Olive 1996
  23. ^Cf. Dieter Maison's contribution "Duality and Hidden Symmetries in Gravitational Theories", pp. 273–323 inSchmidt 2000.
  24. ^ Maison op. cit.,Geroch 1971,and, for various applications,Mars 2001.
  25. ^Hawking & Ellis 1973,p. 351ff. The original work isEhlers, Geren & Sachs 1968.
  26. ^E.g.Liddle 2003,p.2
  27. ^SeeEhlers, Pirani & Schild 1972;a summary can be found inEhlers 1973.
  28. ^SeeKöhler & Schattner 1979.The original publication isEhlers & Rudolph 1977.
  29. ^A review of the book itself isBleyer 1993.The long-term impact can be judged by the way it is held up as a reference in the reviews of later books on the same topic, e.g.Perlick 2005andBozza 2005;see also the assessment ofTrümper 2009,p. 154.
  30. ^Seitz, Schneider & Ehlers 1994,cf. section 3.5 ofAnnual Report 1994,Max Planck Institute for Astrophysics, 1995, archived fromthe originalon 19 May 2009
  31. ^Ehlers 1997;a description can be found on p. 216f. in Luc Blanchet's contribution "Post-Newtonian Gravitational Radiation", pp. 225–271 inSchmidt 2000.
  32. ^ Oliynyk & Schmidt 2009
  33. ^A description that includes the historical context can be found inSchutz 1996.The original work isEhlers et al. 1976.
  34. ^SeeBuchert & Ehlers 1993,Buchert & Ehlers 1997aandBuchert & Ehlers 1997b.The current status of Buchert's further work is summarized inBuchert 2008.
  35. ^Cf.Braun 2008.Details about the project can be found on itswebsite.
  36. ^ Ehlers 2007
  37. ^SeeEhlers 2006aandBreuer & Springer 2001as well as its later English translationBreuer & Springer 2009,as well asEhlers 2005.
  38. ^Public lectures:Biennial Report 2004/2005(PDF),Max Planck Institute for Gravitational Physics, 2006, archived fromthe original(PDF)on 11 June 2007,lists 25 popular talks (p. 158f.) for that time-frame alone. The compilation of articles is Börner & Ehlers 1996, listed underSelected Publications.An example for a popular article isEhlers & Fahr 1994.
  39. ^Biennial Report 2004/2005(PDF),Max Planck Institute for Gravitational Physics, 2006, archived fromthe original(PDF)on 11 June 2007lists 11 talks to teachers or in an interdisciplinary setting (p. 147f., p. 154f.). Mathematics and physicsEhlers 2006b
  40. ^Berlin:Huisken, Nicolai & Schutz 2009;initial membership date in brief note on p.35 of the same publication. Mainz: p. 13 ofLütjen-Drecoll 2008.Leopoldina: listed as member onMitgliederverzeichnis,Deutsche Akademie der Naturforscher Leopoldina,retrieved28 May2012(in German, English translation of title:Members list). Bavarian Academy:Trümper 2009.
  41. ^GRG Society History,International Society on General Relativity and Gravitation,retrieved28 May2013.
  42. ^Max Planck Medal: Press release about the 2002 awards,Physikalische Spitzenleistung,Deutsche Physikalische Gesellschaft, 17 December 2001, archived fromthe originalon 13 February 2007,retrieved27 May2008(in German, English translation of title:Top achievement in physics), andRogalla 2001.Volta Medal:"Namen: Prof. Dr. Jürgen Ehlers",Berliner Zeitung,18 May 2005,retrieved27 May2008(in German) and"Medaille für Golmer Forscher",Märkische Allgemeine Zeitung,19 May 2005(in German, English translation of title:Medal for researcher from Golm). Charles University Medal:Trümper 2009,p. 154.
  43. ^The Jürgen Ehlers Thesis Prize,Website of the International Society on General Relativity and Gravitation,retrieved28 May2013
  44. ^Nicolai, Ellis & Schmidt 2009

References[edit]

External links[edit]

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