In 1955, while a doctoral student, Penrose reintroduced theE. H. Moore generalised matrix inverse, also known as theMoore–Penrose inverse,[21] after it had been reinvented byArne Bjerhammar in 1951.[22] Having started research under the professor of geometry and astronomyW. V. D. Hodge, Penrose received hisPhD inalgebraic geometry atSt John's College, Cambridge, in 1957, with his thesis titled "Tensor Methods in Algebraic Geometry"[23] supervised by the algebraist and geometerJohn A. Todd.[24] He devised and popularised thePenrose triangle in the 1950s in collaboration with his father, describing it as "impossibility in its purest form", and exchanged material with the artistM. C. Escher, whose earlier depictions of impossible objects partly inspired it.[25][26] Escher'sWaterfall andAscending and Descending were in turn inspired by Penrose.[27]
As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design astaircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.[28]
In 1964, while areader atBirkbeck College, London, (and having had his attention drawn from pure mathematics to astrophysics by the cosmologistDennis Sciama, then at Cambridge)[16] in the words ofKip Thorne of theCalifornia Institute of Technology, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime".[30][31] Until then, work on the curved geometry of general relativity had been confined to configurations with sufficiently high symmetry for Einstein's equations to be solvable explicitly, and there was doubt about whether such cases were typical. One approach to this issue was by the use ofperturbation theory, as developed under the leadership ofJohn Archibald Wheeler at Princeton.[32] The other, and more radically innovative, approach initiated by Penrose was to overlook the detailed geometrical structure of spacetime and instead concentrate attention just on the topology of the space, or at most itsconformal structure, since it is the latter – as determined by the lay of the lightcones – that determines the trajectories of lightlike geodesics, and hence their causal relationships. The importance of Penrose's paper "Gravitational Collapse and Space-Time Singularities"[33] (summarised roughly as that if an object such as a dying star implodes beyond a certain point, then nothing can prevent the gravitational field getting so strong as to form some kind of singularity) was not its only result. It also showed a way to obtain similarly general conclusions in other contexts, notably that of the cosmologicalBig Bang, which he dealt with in collaboration with Sciama's studentStephen Hawking.[34][35][36]
It was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture,[37] to the effect that any ensuing singularities would be confined within a well-behavedevent horizon surrounding a hidden space-time region for which Wheeler coined the termblack hole, leaving a visible exterior region with strong but finite curvature, from which some of the gravitational energy may be extractable by what is known as thePenrose process, while accretion of surrounding matter may release further energy that can account for astrophysical phenomena such asquasars.[38][39][40]
Following up his "weakcosmic censorship hypothesis", Penrose went on, in 1979, to formulate a stronger version called the "strong censorship hypothesis". Together with theBelinski–Khalatnikov–Lifshitz conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems ingeneral relativity. Also from 1979 dates Penrose's influentialWeyl curvature hypothesis on the initial conditions of the observable part of the universe and the origin of thesecond law of thermodynamics.[41] Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation. This effect has come to be called theTerrell rotation or Penrose–Terrell rotation.[42][43]
In 1967 Penrose invented thetwistor theory, which maps geometric objects inMinkowski space into the 4-dimensional complex space with the metric signature (2,2).[44][45]
Penrose is well known for his 1974 discovery ofPenrose tilings, which are formed from two tiles that can onlytile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. In 1984 such patterns were observed in the arrangement of atoms inquasicrystals.[46] Another noteworthy contribution is his 1971 invention ofspin networks, which later came to form the geometry ofspacetime inloop quantum gravity.[47] He was influential in popularising what are commonly known asPenrose diagrams (causal diagrams).[48]
In 2010 Penrose reported possible evidence, based on concentric circles found inWilkinson Microwave Anisotropy Probe data of thecosmic microwave background sky, of an earlier universe existing before theBig Bang of the present universe.[55] He mentions this evidence in the epilogue of his 2010 bookCycles of Time,[56] a book in which he presents his reasons, to do withEinstein's field equations, theWeyl curvature C, and theWeyl curvature hypothesis (WCH), that the transition at the Big Bang could have been smooth enough for a previous universe to survive it.[57][58] He made several conjectures about C and the WCH, some of which were subsequently proved by others, and he also popularized hisconformal cyclic cosmology (CCC) theory.[59] In this theory, Penrose postulates that at the end of the universe all matter is eventually contained within black holes, which subsequently evaporate viaHawking radiation. At this point, everything contained within the universe consists ofphotons, which "experience" neither time nor space. There is essentially no difference between an infinitely large universe consisting only of photons and an infinitely small universe consisting only of photons. Therefore, a singularity for a Big Bang and an infinitely expanded universe are equivalent.[60]
In simple terms, Penrose believes that the singularity inEinstein's field equation at the Big Bang is only an apparent singularity, similar to the well-known apparent singularity at theevent horizon of ablack hole.[38] The latter singularity can be removed by a change ofcoordinate system, and Penrose proposes a different change of coordinate system that will remove the singularity at the big bang.[61] One implication of this is that the major events at the Big Bang can be understood without unifying general relativity and quantum mechanics, and therefore we are not necessarily constrained by theWheeler–DeWitt equation, which disrupts time.[62][63] Alternatively, one can use the Einstein–Maxwell–Dirac equations.[64]
Penrose has written books on the connection between fundamental physics and human (or animal) consciousness. InThe Emperor's New Mind (1989), he argues that known laws of physics are inadequate to explain the phenomenon of consciousness.[65] Penrose proposes the characteristics this new physics may have and specifies the requirements for a bridge between classical and quantum mechanics (what he callscorrect quantum gravity).[66] Penrose uses a variant ofTuring's halting theorem to demonstrate that a system can bedeterministic without beingalgorithmic. (For example, imagine a system with only two states, ON and OFF. If the system's state is ON when a givenTuring machine halts and OFF when the Turing machine does not halt, then the system's state is completely determined by the machine; nevertheless, there is no algorithmic way to determine whether the Turing machine stops.)[67][68]
Penrose believes that such deterministic yet non-algorithmic processes may come into play in the quantum mechanicalwave function reduction, and may be harnessed by the brain. He argues that computers today are unable to have intelligence because they are algorithmically deterministic systems. He argues against the viewpoint that the rational processes of the mind are completely algorithmic and can thus be duplicated by a sufficiently complex computer.[69] This contrasts with supporters ofstrong artificial intelligence, who contend that thought can be simulated algorithmically. He bases this on claims that consciousness transcendsformal logic because factors such as the insolubility of thehalting problem andGödel's incompleteness theorem prevent an algorithmically based system of logic from reproducing such traits of human intelligence as mathematical insight.[69] These claims were originally espoused by the philosopherJohn Lucas ofMerton College,Oxford.[70]
ThePenrose–Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been criticised by mathematicians, computer scientists and philosophers. Many experts in these fields assert that Penrose's argument fails, though different authors choose different aspects of the argument to attack.[71]Marvin Minsky, a leading proponent of artificial intelligence, was particularly critical, writing that Penrose "tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle." Minsky's position is exactly the opposite – he believed that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintained that "one can carry that quest [for scientific explanation] too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose's quest for a new basic principle of physics that will account for consciousness."[72]
Penrose and Hameroff have argued that consciousness is the result of quantum gravity effects inmicrotubules, which they dubbedOrch-OR (orchestrated objective reduction).Max Tegmark, in a paper inPhysical Review E,[74] calculated that the time scale of neuron firing and excitations in microtubules is slower than thedecoherence time by a factor of at least 10 billion. The paper's reception is summed up by this statement in Tegmark's support: "Physicists outside the fray, such as IBM'sJohn A. Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior'".[75] Tegmark's paper has been widely cited by critics of the Penrose–Hameroff position.
Phillip Tetlow, although himself supportive of Penrose's views, acknowledges that Penrose's ideas about the human thought process are a minority view in scientific circles, citing Minsky's criticisms and quoting the science journalistCharles Seife's description of Penrose as "one of a handful of scientists" who believe that the nature of consciousness suggests a quantum process.[75]
In January 2014 Hameroff and Penrose ventured that a discovery of quantum vibrations in microtubules byAnirban Bandyopadhyay of the National Institute for Materials Science in Japan[76] supports the hypothesis ofOrch-OR theory. A reviewed and updated version of the theory was published along with critical commentary and debate in the March 2014 issue ofPhysics of Life Reviews.[77]
Techniques of Differential Topology in Relativity (1972,ISBN0-89871-005-7)
Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (withWolfgang Rindler, 1987)ISBN0-521-33707-0 (paperback)
Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler, 1988) (reprint),ISBN0-521-34786-6 (paperback)
His deep work on General Relativity has been a major factor in our understanding of black holes. His development ofTwistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.[106]
In 2012 Penrose was awarded the Richard R. Ernst Medal byETH Zürich (Switzerland) for his contributions to science and strengthening the connection between science and society. In that year he was also awarded the honorary degree ofDoctor of Science (DSc) by theTrinity College Dublin (Ireland)[113] as well an honorary doctorate degree by theIgor Sikorsky Kyiv Polytechnic Institute (Ukraine).[114]
Penrose's first marriage was to Joan Isabel Penrose (born Wedge), an American he married in 1959. They had three sons.[121][122] Penrose is now married to Vanessa Thomas, director of Academic Development atCokethorpe School inWitney,Oxfordshire, and former head of mathematics atAbingdon School.[123][124] They have one son.[125][123]
During an interview withBBC Radio 4 on 25 September 2010, Penrose stated, "I'm not a believer myself. I don't believe in established religions of any kind."[126] He regards himself as an agnostic.[127] In the 1991 filmA Brief History of Time, he also said, "I think I would say that the universe has a purpose, it's not somehow just there by chance ... some people, I think, take the view that the universe is just there and it runs along—it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it."[128]
^Fernandez, Rodrigo L. (21 July 2020). "Cosmic Censorship Conjecture violation: A semiclassical approach".arXiv:2007.10601 [gr-qc].
^abCuriel, Erik (2020),"Singularities and Black Holes", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Summer 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved7 October 2020
^"Penrose process".Oxford Reference.Archived from the original on 2 February 2021. Retrieved7 October 2020.
^Penrose, Roger (1979). "Singularities and Time-Asymmetry". In Hawking, Stephen W.; Israel, W. (eds.).General Relativity: An Einstein Centenary Survey. Cambridge University Press. pp. 581–638.
^Gurzadyan, V. G.; Penrose, R. (2010). "Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity". volume "v1".arXiv:1011.3706 [astro-ph.CO].
^Roger Penrose,Cycles of Time, Vintage; Reprint edition (1 May 2012)
^Penrose, Roger (1979). "Singularities and Time-Asymmetry". In Hawking, S. W.; Israel, W. (eds.).General Relativity: An Einstein Centenary Survey.Cambridge University Press. pp. 581–638.
In an article at"King's College London – Department of Mathematics". Archived fromthe original on 25 January 2001. Retrieved22 October 2010. L. J. Landau at the Mathematics Department of King's College London writes that "Penrose's argument, its basis and implications, is rejected by experts in the fields which it touches."
Sources that also note that different sources attack different points of the argument:
Princeton Philosophy professor John Burgess writes inOn the Outside Looking In: A Caution about Conservativeness.Archived 19 October 2012 at theWayback Machine, (published in Kurt Gödel: Essays for his Centennial, with the following comments found onpp. 131–132.Archived 27 December 2016 at theWayback Machine) that "the consensus view of logicians today seems to be that the Lucas–Penrose argument is fallacious, though as I have said elsewhere, there is at least this much to be said for Lucas and Penrose, that logicians are not unanimously agreed as to where precisely the fallacy in their argument lies. There are at least three points at which the argument may be attacked."
^Marvin Minsky. "Conscious Machines." Machinery of Consciousness, Proceedings,National Research Council of Canada, 75th Anniversary Symposium on Science in Society, June 1991.
Awake in the Universe – Penrose debates how creativity, the most elusive of faculties, has helped us unlock the country of the mind and the mysteries of the cosmos withBonnie Greer.
Dangerous Knowledge onYouTube – Penrose was one of the principal interviewees in a BBC documentary about the mathematics of infinity directed byDavid Malone
Penrose's new theory "Aeons Before the Big Bang?":
"Toilet Paper Plagiarism" at theWayback Machine (archived 12 March 2005) – D. Trull about Penrose's lawsuit concerning the use of his Penrose tilings on toilet paper
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