{"id":2845,"date":"2020-08-29T15:19:32","date_gmt":"2020-08-29T19:19:32","guid":{"rendered":"http:\/\/blogs.ams.org\/beyondreviews\/?p=2845"},"modified":"2020-10-06T15:50:19","modified_gmt":"2020-10-06T19:50:19","slug":"roger-penrose-on-numberphile","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/beyondreviews\/2020\/08\/29\/roger-penrose-on-numberphile\/","title":{"rendered":"Roger Penrose on Numberphile"},"content":{"rendered":"<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" title=\"Why Did The Mathematician Cross The Road? - with Roger Penrose\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/JiDWGbsVEno?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<hr \/>\n<p><strong>UPDATE <\/strong>[6 October 2020]:\u00a0 Roger Penrose is one of the winners of the <a href=\"https:\/\/www.nobelprize.org\/prizes\/physics\/2020\/summary\/\">2020 Nobel Prize in Physics<\/a>.\u00a0 Congratulations!<\/p>\n<hr \/>\n<p><a href=\"https:\/\/www.numberphile.com\/\">Numberphile<\/a> has posted an audio interview with <a href=\"https:\/\/en.wikipedia.org\/wiki\/Roger_Penrose\">Roger Penrose<\/a> on their YouTube channel <a href=\"https:\/\/www.youtube.com\/channel\/UCyp1gCHZJU_fGWFf2rtMkCg\">Numberphile2<\/a>.\u00a0 You can also access it as audio-only from their <a href=\"https:\/\/www.numberphile.com\/videos\/podcast-roger-penrose\">website<\/a>.\u00a0 <!--more--><\/p>\n<p>The podcast is quite good.\u00a0\u00a0Penrose is very relaxed and thoughtfully answers all of the questions from the host, <a href=\"https:\/\/www.bradyharanblog.com\/about\/\">Brady Haran<\/a>.\u00a0 The title refers to a story Penrose tells about coming up with the notion of trapped surfaces, the key idea to his singularity theorems on black holes.\u00a0 Penrose was my supervisor during my time as a post-doc at Oxford, and he sounds exactly the same now as he did in the late 1980s.<\/p>\n<p>At the beginning of the podcast, we learn about Penrose&#8217;s childhood and his family &#8211; they are quite famous.\u00a0 \u00a0His father, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Lionel_Penrose\">Lionel Penrose<\/a>, was a famous biologist, physician, and geneticist.\u00a0 His brother <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jonathan_Penrose\">Jonathan<\/a> is a chess grandmaster, who was the British chess champion ten times.\u00a0 His brother <a href=\"https:\/\/en.wikipedia.org\/wiki\/Oliver_Penrose\">Oliver<\/a> is a physicist, with <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/137775\">55 entries in MathSciNet<\/a>.\u00a0 His sister <a href=\"https:\/\/en.wikipedia.org\/wiki\/Shirley_Hodgson\">Shirley Hodgson<\/a> is a geneticist.\u00a0 Penrose readily admits that he was not &#8220;quick&#8221; as a student, something that sometimes influenced his schooling and his career.\u00a0 Moreover, he says that he never became quick.\u00a0 He needs time to think about a problem.\u00a0 In my experience, if someone asks Penrose a question and he answers right away, it is something that he has already thought about.\u00a0 In this case, you will get a really good answer on the spot.\u00a0 Fortunately, he has thought about a lot of interesting subjects.\u00a0 In other cases, Penrose will get back to you days later with a really good answer.<\/p>\n<p>The host asks Penrose about his affiliation with Stephen Hawking.\u00a0 In the podcast, Penrose corrects some little details from the film, <em>The Theory of Everything<\/em>.\u00a0 A couple of times, Brady Haran asks whether Penrose and Hawking were friends.\u00a0 My understanding is that the answer to that question would have varied over the years.\u00a0 I remember being at a conference at Durham University where both Penrose and Hawking were present.\u00a0 One evening in the middle of the conference, a bunch of us found ourselves at a pub with Hawking and Penrose.\u00a0 Hawking had great fun ordering round after round for the table, all the while asking us math questions.\u00a0 \u00a0The one that sticks with me is Hawking chiding us that we couldn&#8217;t figure out for which real values of $x$, the function $x$^$x$^$x$^$x$^$\\dots$ converged. It&#8217;s an interesting problem, but tricky after a few rounds of beer with brandy chasers. (As I hazily recall, the brandies were Hawking&#8217;s idea.)\u00a0 After you have spent some time with the problem, check out <a href=\"https:\/\/www.degruyter.com\/view\/journals\/crll\/1844\/28\/article-p49.xml\">one<\/a> of Eisenstein&#8217;s ten papers in <a href=\"https:\/\/www.degruyter.com\/view\/journals\/crll\/1844\/28\/crll.1844.issue-28.xml\">volume 28<\/a> of <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/journal\/profile?groupId=2764\">J. Reine Angew. Math. [Crelle&#8217;s Journal]<\/a>.<\/p>\n<p>Here are a couple more stories from my time in Penrose&#8217;s research group.<\/p>\n<p>Every Friday, everyone in the group met in Penrose&#8217;s office for the so-called &#8220;Friday meeting&#8221;.\u00a0 The office was large enough, though we had to bring chairs from other rooms.\u00a0 \u00a0There was a loose agenda for each meeting, something like:\u00a0 short announcements (just published a paper, upcoming conferences, etc.); brief reports &#8211; usually from someone who had been to a conference; questions from grad students; questions \/ requests for help on a research problem from others; general discussion.\u00a0 The meeting would start around noon, maybe 1:00pm.\u00a0 (I forget.)\u00a0 People would bring lunches, either something from home or carry-out.\u00a0 A favorite for carry-out was the &#8220;pink bag shop&#8221;, which was actually Maison Blanc (&#8220;Blanc&#8221; after the proprietor, Raymond Blanc).\u00a0 \u00a0The meetings would sometimes go long, in which case we would break to join the Institute&#8217;s regular tea downstairs.\u00a0 One of the rules of the meetings concerned the graduate students&#8217; questions.\u00a0 A student was allowed to ask any question, the others were not allowed to criticize or to make fun of the question, and it had to be answered.\u00a0 Well, one Friday, a new-ish graduate student asked, &#8220;What&#8217;s renormalization?&#8221;\u00a0 At first, the room went quiet.\u00a0 Then various people took a turn at trying to explain it rigorously.\u00a0 I don&#8217;t remember exactly, but that Friday meeting probably went long.<\/p>\n<p>While in Oxford, I happened to attend the British Mathematics Colloquium with Penrose.\u00a0 Besides the talks, a small number of publishers and vendors had booths or tables set up.\u00a0 One vendor had various puzzles, many of them were of the string and rings on a stick variety.\u00a0 Penrose looked at the table.\u00a0 After a minute or two, he pointed at one and said, &#8220;Well that one&#8217;s different &#8211; isn&#8217;t it?&#8221;\u00a0 I asked how it was different.\u00a0 Penrose replied, &#8220;It&#8217;s the only one that can&#8217;t be done.&#8221;\u00a0 The vendor confirmed it, and proceeded to have a hearty conversation with Penrose.<\/p>\n<p>If you look up <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/137780\">Roger Penrose in MathSciNet<\/a>, you will see that we have papers by him in various subjects, but mostly in <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscbrowse.html?pc=83\">relativity and gravitational theory<\/a>, followed by <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscbrowse.html?pc=81\">quantum theory<\/a>.\u00a0\u00a0Penrose is famous for many things in both mathematics and physics.\u00a0 By far his most cited paper, however, is his paper on generalized inverses:<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=0069793\"><strong>MR0069793<\/strong><\/a>\u00a0<span style=\"color: #000000\"><b>\u00a0<\/b><\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, R.<\/a><br \/>\n<span class=\"title\">A generalized inverse for matrices.<\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/journaldoc.html?id=9202\"><em>Proc. Cambridge Philos. Soc.<\/em><\/a>\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publications.html?pg1=ISSI&amp;s1=374083\">51\u00a0<\/a><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publications.html?pg1=ISSI&amp;s1=374083\">(1955),\u00a0<\/a>406\u2013413.<br \/>\n(Reviewer: <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/171090\">O. Taussky-Todd<\/a>)<\/p>\n<p>This is also know as the Moore-Penrose pseudoinverse.\u00a0 The generalized inverse is defined as the unique solution of a certain set of equations involving the matrix and its conjugate transpose.\u00a0\u00a0There is a function hard-coded into <a href=\"https:\/\/www.mathworks.com\/products\/matlab.html\">MATLAB<\/a>, called <a href=\"https:\/\/www.mathworks.com\/help\/matlab\/ref\/pinv.html\"><code>pinv<\/code><\/a>, that computes it (using SVD).\u00a0 We have 421 citations for it.\u00a0 Google Scholar has 4766.\u00a0 I was unable to find the paper in Web of Science.\u00a0 It seems to be from before their coverage starts.<\/p>\n<p>It is a good idea to brush up your knowledge of spinors when talking to Penrose, as he prefers that formalism to the traditional tensors.\u00a0 The canonical place is his two-volume book with Wolfgang Rindler:<\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=0776784\"><strong>MR0776784<\/strong><\/a>\u00a0<span style=\"color: #000000\"><b>\u00a0<\/b><\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, Roger<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=4-OX\">(4-OX)<\/a><\/span>;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=148430\">Rindler, Wolfgang<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=1-TX-P\">(1-TX-P)<\/a><\/span><br \/>\n<span class=\"title\">Spinors and space-time. Vol. 1.<\/span><br \/>\nTwo-spinor calculus and relativistic fields.\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/series.html?id=27\">Cambridge Monographs on Mathematical Physics.<\/a>\u00a0<em>Cambridge University Press, Cambridge,<\/em>\u00a01984.\u00a0<span class=\"rm\">x<\/span>+458 pp. ISBN: 0-521-24527-3<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=83-02,(53A50,53B50,53C80,83Cxx)\">83-02 (53A50 53B50 53C80 83Cxx)<\/a><br \/>\n<span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=83855\">Adam Helfer<\/a><\/span><\/p>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=0838301\"><strong>MR0838301<\/strong><\/a>\u00a0<span style=\"color: #000000\"><b>\u00a0<\/b><\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, Roger<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=4-OX\">(4-OX)<\/a><\/span>;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=148430\">Rindler, Wolfgang<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=1-TXD\">(1-TXD)<\/a><\/span><br \/>\n<span class=\"title\">Spinors and space-time. Vol. 2.<\/span><br \/>\nSpinor and twistor methods in space-time geometry.\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/series.html?id=27\">Cambridge Monographs on Mathematical Physics.<\/a>\u00a0<em>Cambridge University Press, Cambridge,<\/em>\u00a01986.\u00a0<span class=\"rm\">x<\/span>+501 pp. ISBN: 0-521-25267-9<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=83Cxx,(32L25,53A50,53C80,81D25,81D27,83-02,83C60)\">83Cxx (32L25 53A50 53C80 81D25 81D27 83-02 83C60)<\/a><br \/>\n<span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=83855\">Adam Helfer<\/a><\/span><\/p>\n<p>Penrose has written some other very good books.\u00a0 His books on consciousness are well-known, and created a stir.\u00a0 He also wrote a general physics book, The Road to Reality, which begins with about 300 pages of mathematical preparation, has amazing accounts of classical physics (especially Lagrangian and Hamiltonian dynamics), quantum mechanics, and relativity, and finishes with Penrose&#8217;s take on some of the hot topics in modern physics.\u00a0 I would be hard-pressed to name better treatments for the first and second parts of the book.<\/p>\n<p><strong><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=2116746\">MR2116746<\/a><\/strong><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, Roger<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=4-OX\">(4-OX)<\/a><\/span><br \/>\n<span class=\"title\">The road to reality.<\/span><br \/>\nA complete guide to the laws of the universe.\u00a0<em>Alfred A. Knopf, Inc., New York,<\/em>\u00a02005. xxviii+1099 pp. ISBN: 0-679-45443-8<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=83-02,(00A05,00A79,81-02)\">83-02 (00A05 00A79 81-02)<\/a><br \/>\n<span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=111020\">Peter R. Law<\/a><\/span><\/p>\n<p>The publication of the podcast was timed to coincide with Roger Penrose&#8217;s birthday (August 8).\u00a0 I didn&#8217;t get around to listening to it until August 27.\u00a0 So, belated happy birthday, Roger Penrose!<\/p>\n<p>Below are the texts of some reviews of Penrose&#8217;s work from <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\">MathSciNet<\/a>.<\/p>\n<hr \/>\n<h2>Some reviews of Penrose&#8217;s works<\/h2>\n<p class=\"headline\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=0069793\"><strong>MR0069793<\/strong><\/a>\u00a0<span style=\"color: #000000\"><b>\u00a0<\/b><\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, R.<\/a><br \/>\n<span class=\"title\">A generalized inverse for matrices.<\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/journaldoc.html?id=9202\"><em>Proc. Cambridge Philos. Soc.<\/em><\/a>\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publications.html?pg1=ISSI&amp;s1=374083\">51\u00a0<\/a><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publications.html?pg1=ISSI&amp;s1=374083\">(1955),\u00a0<\/a>406\u2013413.<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=09.0X\">09.0X<\/a><\/p>\n<p class=\"headlineMenu\">A generalized inverse (g.i.) is introduced for arbitrary (possibly rectangular) matrices <span class=\"MathTeX\">$A$<\/span>\u00a0with complex elements. It is the unique solution\u00a0<span class=\"MathTeX\">$X$<\/span>\u00a0of the four equations\u00a0<span class=\"MathTeX\">$AXA=X$<\/span>,\u00a0<span class=\"MathTeX\">$XAX=X$<\/span>,\u00a0<span class=\"MathTeX\">$(AX)^\\ast=AX$<\/span>,\u00a0<span class=\"MathTeX\">$(XA)^\\ast=XA$<\/span>. [Abstract rings with inverses satisfying\u00a0<span class=\"MathTeX\">$axa=a$<\/span>\u00a0had been studied by v. Neumann, Proc. Nat. Acad. Sci. U. S. A.\u00a0<span class=\"bf\">22<\/span>, 707\u2013713 (1936).] The g.i. is used to formulate a necessary and sufficient condition for the solvability of the matrix equation\u00a0<span class=\"MathTeX\">$AXB=C$<\/span>\u00a0and of the system\u00a0<span class=\"MathTeX\">$AX=C$<\/span>,\u00a0<span class=\"MathTeX\">$XB=D$<\/span>\u00a0and to give an explicit solution. They are further used to give explicit expressions for the principal idempotents of a matrix and for a new type of spectral decomposition which allows the g.i. of\u00a0<span class=\"MathTeX\">$A$<\/span>\u00a0to be expressed in a simple manner in terms of the principal idempotents even for non-normal matrices.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=171090\">O. Taussky-Todd<\/a><\/span><\/p>\n<hr \/>\n<p class=\"headline\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=172678\"><strong>MR0172678<\/strong><\/a>\u00a0<span style=\"color: #000000\"><b>\u00a0<\/b><\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, Roger<\/a><br \/>\n<span class=\"title\">Gravitational collapse and space-time singularities.<\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/journaldoc.html?id=341\"><em>Phys. Rev. Lett.<\/em><\/a>\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publications.html?pg1=ISSI&amp;s1=394974\">14\u00a0<\/a><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publications.html?pg1=ISSI&amp;s1=394974\">(1965),\u00a0<\/a>57\u201359.<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=83.53\">83.53<\/a><\/p>\n<p class=\"review\">It is shown that even in the absence of spherical symmetry the occurrence and persistence of &#8220;trapped surfaces&#8221; will, within a finite time, lead to true singularities of the metric field. A trapped surface is a space-like closed 2-surface (topologically equivalent to a spherical surface) which shrinks if it is mapped on the two one-parametric sets of similar 2-surfaces by means of the two pencils of null geodesics passing through the original 2-surface and perpendicular to it, both times proceeding in the future direction. The very notion of the trapped surface appears to be a major conceptual contribution to the clarification of what usually is referred to as Schwarzschild &#8220;singularities&#8221;, the quotes being used because locally the Schwarzschild radius is in no sense singular. Trapped surfaces will occur inside, but not outside, the Schwarzschild radius; this definition is independent of spherical symmetry, of the presence of isometries, and of the choice of coordinate system.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=35300\">P. G. Bergmann<\/a><\/span><\/p>\n<hr \/>\n<p class=\"headline\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=0776784\"><strong>MR0776784<\/strong><\/a>\u00a0<span style=\"color: #000000\"><b>\u00a0<\/b><\/span><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, Roger<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=4-OX\">(4-OX)<\/a><\/span>;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=148430\">Rindler, Wolfgang<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=1-TX-P\">(1-TX-P)<\/a><\/span><br \/>\n<span class=\"title\">Spinors and space-time. Vol. 1.<\/span><br \/>\nTwo-spinor calculus and relativistic fields.\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/series.html?id=27\">Cambridge Monographs on Mathematical Physics.<\/a>\u00a0<em>Cambridge University Press, Cambridge,<\/em>\u00a01984.\u00a0<span class=\"rm\">x<\/span>+458 pp. ISBN: 0-521-24527-3<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=83-02,(53A50,53B50,53C80,83Cxx)\">83-02 (53A50 53B50 53C80 83Cxx)<\/a><\/p>\n<p class=\"review\">The space-times considered in general relativity are 4-dimensional manifolds with metrics of signature\u00a0<span class=\"MathTeX\">$+\\,-\\,-\\,-$<\/span>. On such a manifold there is, at least locally, a 2-complex-dimensional vector bundle\u00a0<span class=\"MathTeX\">${\\scr S}_A$<\/span>, called the bundle of (2-component) spinors. Such spinors have been used extensively in modern general relativity and relativistic field theory for two reasons, one algebraic and one geometric. The tensor algebra of spinors is particularly simple and elegant, due to the two-dimensionality of the spin space. In fact, this algebra subsumes the usual world-tensor and 4-spinor algebras of the manifold, and many computations which are opaque in these terms become transparent with 2-component spinors. Geometrically, the spinor structure is tied in a deep way to the causal structure of space-time, since each spinor represents a future-pointing null vector. Arguably, the existence of\u00a0<span class=\"MathTeX\">${\\scr S}_A$<\/span> is the most primitive distinction between space-times and general Lorentzian manifolds.<\/p>\n<p class=\"review\">The volume under review is an exposition of the calculus of spinors and of their use in the theory of relativistic fields. (Volume 2 is to concentrate on the applications of spinors and twistors to space-time geometry.) It is both lucid and rigorous enough to satisfy mathematicians. It will be a useful and very likely a standard reference book, and even experts will find something to learn from it.<\/p>\n<p class=\"review\">Chapter 1 introduces spinors and vectors geometrically. Spinors are defined as null flags (null vectors together with null half-planes). The spinor operations of addition, scalar multiplication and inner product are interpreted in terms of these null flags. There is also a discussion of the topological properties of the space-time required for the existence of <span class=\"MathTeX\">${\\scr S}_A$<\/span>. This is a clear account of some rather sophisticated geometry.<\/p>\n<p class=\"review\">Chapter 2 describes the abstract-index formalism. This is a general formalism for multilinear algebra. It combines the most attractive features of the usual modern notation and the old-fashioned component notation: it is an invariant notation in which indices are retained. This makes many computations easy, and also enables one to see immediately to what space a tensorial object belongs.<\/p>\n<p class=\"review\">Chapter 3 establishes the relation between the spinor and world-tensor algebras. Every tensor equation has a straightforward interpretation as a spinor equation. To convert spinor equations to tensors is in general much more complicated, and there is a useful section which gives a general procedure for doing so.<\/p>\n<p class=\"review\">In Chapter 4, differentiation of spinor and tensor fields is introduced. First the covariant derivative is discussed, and then there is a section on connection-independent derivatives. The spinor expressions for the various curvature quantities are given both for the usual torsion-free case, and for connections which allow torsion (which have been of occasional interest in relativity theory). Formulae for the differentiation of spinor components are given, and there is an account of the spin coefficient formalism with useful equations for their changes with conformal transformations. There is also a discussion of the compacted spin coefficient formalism [R. Geroch, A. Held and Penrose, J. Math. Phys.\u00a0<span class=\"bf\">14<\/span>\u00a0(1973), 874\u2013881;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=323287&amp;loc=fromrevtext\">MR0323287<\/a>]. This is especially useful for the study of 2-surfaces in space-time, and there is a section on spin-weighted spherical harmonics, which are of interest in spin-coefficient problems. Additionally, the Cartan description of connections and curvature by means of differential forms is presented.<\/p>\n<p class=\"review\">Chapter 5 concerns relativistic fields. The Maxwell and Einstein-Maxwell equations are given in spinor form. There is a section explaining vector bundles, and the spinor Yang-Mills equations are given and interpreted as equations on the curvatures of connections for vector bundles. The equations for (possibly charged) massless test fields (i.e., fields which do not produce a back-reaction on the gravitational field) of arbitrary helicity are given. The consistency conditions for such fields [M. Fierz and W. Pauli, Proc. Roy. Soc. London Ser. A\u00a0<span class=\"bf\">173<\/span>\u00a0(1939), 211\u2013232;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1173&amp;loc=fromrevtext\">MR0001173<\/a>; H. A. Buchdahl, Nuovo Cimento (10)\u00a0<span class=\"bf\">25<\/span>\u00a0(1962), 486\u2013496;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=145855&amp;loc=fromrevtext\">MR0145855<\/a>] are discussed. The conformal invariance of these various fields is established, and a useful conformally-invariant spin-coefficient formalism is introduced. There is a section on exact sets of fields, which are systems of field equations for which the data may be specified completely and without constraints by a set of functions on a null hypersurface. Such systems allow natural formulations as characteristic initial-value problems, and the specification of initial data for various fields, including gravity, on a light-cone is discussed. The last section gives some explicit integrals for the fields in certain exact sets from initial data on a light-cone in Minkowski space.<\/p>\n<p class=\"review\">Lastly, there is an appendix describing Penrose&#8217;s &#8220;musical&#8221; notation for index manipulation, which is useful for calculations involving lots of index permutation.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=83855\">Adam Helfer<\/a><\/span><\/p>\n<hr \/>\n<p class=\"headline\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=0838301\"><strong>MR0838301<\/strong><\/a><br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=137780\">Penrose, Roger<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=4-OX\">(4-OX)<\/a><\/span>;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=148430\">Rindler, Wolfgang<\/a><span class=\"instInfo\"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/institution.html?code=1-TXD\">(1-TXD)<\/a><\/span><br \/>\n<span class=\"title\">Spinors and space-time. Vol. 2.<\/span><br \/>\nSpinor and twistor methods in space-time geometry.\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/series.html?id=27\">Cambridge Monographs on Mathematical Physics.<\/a>\u00a0<em>Cambridge University Press, Cambridge,<\/em>\u00a01986.\u00a0<span class=\"rm\">x<\/span>+501 pp. ISBN: 0-521-25267-9<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/mscdoc.html?code=83Cxx,(32L25,53A50,53C80,81D25,81D27,83-02,83C60)\">83Cxx (32L25 53A50 53C80 81D25 81D27 83-02 83C60)<\/a><\/p>\n<p class=\"headlineMenu\">A space-time is a four-dimensional manifold with a Lorentzian metric. If no gravitational forces are present, space-time is at least locally isometric to Minkowski space <span class=\"MathTeX\">${\\bf M},\\ {\\bf R}^4$<\/span>\u00a0with a flat metric. The principle of special relativity is that, in the absence of gravity, the laws of physics are invariant under the group of isometries of\u00a0<span class=\"MathTeX\">${\\bf M}$<\/span>\u00a0(called the Poincar\u00e9 group,\u00a0<span class=\"MathTeX\">$P(1,3)$<\/span>).<\/p>\n<p class=\"headlineMenu\">The Poincar\u00e9 group is a subgroup of the conformal group\u00a0<span class=\"MathTeX\">$C(1,3)$<\/span>\u00a0of diffeomorphisms preserving the metric up to scale. (More precisely,\u00a0<span class=\"MathTeX\">$C(1,3)$<\/span>\u00a0acts on the conformal completion\u00a0<span class=\"MathTeX\">$\\widetilde{{\\bf M}}$<\/span>\u00a0of\u00a0<span class=\"MathTeX\">${\\bf M}$<\/span>.) The conformal group is of interest for two reasons. First, it is a symmetry group of certain physical theories without length scales, most importantly those of the classical Yang-Mills, Maxwell and free massless fields. Second,\u00a0<span class=\"MathTeX\">$C(1,3)$<\/span>\u00a0is a simple Lie group and so has nicer mathematical properties than\u00a0<span class=\"MathTeX\">$P(1,3)$<\/span>.<\/p>\n<p class=\"headlineMenu\">The identity-connected component of\u00a0<span class=\"MathTeX\">$C(1,3)$<\/span>\u00a0is 4-1 covered by\u00a0<span class=\"MathTeX\">$\\textrm{SU}(2,2)$<\/span>. The defining representation of this group is twistor space\u00a0<span class=\"MathTeX\">${\\bf T}\\simeq {\\bf C}^4$<\/span>, and all other finite-dimensional representations can be obtained from tensor powers of\u00a0<span class=\"MathTeX\">${\\bf T}$<\/span>. In particular, the conformal geometry of\u00a0<span class=\"MathTeX\">${\\bf M}$<\/span>\u00a0can be studied by means of twistors, and its metrical geometry by twistors and reducing\u00a0<span class=\"MathTeX\">$\\textrm{SU}(2,2)$<\/span>\u00a0to (a covering group of the identity-connected component of)\u00a0<span class=\"MathTeX\">$P(1,3)$<\/span>. Although by their definition twistors seem creatures of Minkowski space, it has been possible to adapt them with some, and occasionally remarkable, success to general space-times.<\/p>\n<p class=\"headlineMenu\">The most important mathematical results in twistor theory so far are the Penrose transform, which establishes isomorphisms between the space of massless free fields of fixed helicity and analytic cohomology groups on projective twistor space\u00a0<span class=\"MathTeX\">$\\bf P$<\/span>, the Ward construction, which gives a one-to-one correspondence between self-dual Yang-Mills fields and certain bundles over\u00a0<span class=\"MathTeX\">$\\bf P$<\/span>, Penrose&#8217;s nonlinear graviton, which gives a one-to-one correspondence between space-times with self-dual curvature tensors and certain deformed twistor spaces, the hypersurface twistor construction, which reformulates the initial value problem for Einstein&#8217;s equations as a problem in CR structures, and the 2-surface twistors introduced by Penrose to give a definition of the mass enclosed within a sphere in a general space-time.<\/p>\n<p class=\"headlineMenu\">The volume under review is mostly about the applications of twistors to general relativity. Such a work is necessarily selective, and the authors have chosen to concentrate on those facets of twistor theory most closely allied to classical relativity. The exposition is lucid, the level is rigorous but not pedantic. There are two marked contrasts with the first volume: there is a great deal of emphasis on geometry, and the book is generally an account of the state of the art rather than a definitive treatise.<\/p>\n<p class=\"headlineMenu\">In Chapter 6, twistors are introduced as spinor fields satisfying the &#8220;twistor equation&#8221;. Such fields are &#8220;square roots&#8221; of null conformal Killing vectors, and a full 4-dimensional space of twistors can exist only if the space-time is conformally flat. Other elements of the tensor algebra of\u00a0<span class=\"MathTeX\">${\\bf T}$<\/span> are shown to be solutions of the general Killing spinor equations, and some of the consequences of the existence of such fields are discussed. The role of Killing vectors in obtaining conserved quantities from the stress-energy tensor is explained and, for linearized gravity, interpreted in twistor terms. Twistor fields are also used to give relations between various conformally invariant first-order differential operators on spinor fields. A noncohomological discussion of the Penrose transform is given, and a sketch of the cohomological result. The Ward construction is described, but not the nonlinear graviton.<\/p>\n<p class=\"headlineMenu\">Chapter 7 is about the geometry of congruences of null geodesics; it complements the more algebraic treatment of spin coefficients given in Volume 1. Also, there is a clear account of the CR twistor space associated with a hypersurface. (Since this book was written, progress has been made on CR twistor spaces; see articles by L. J. Mason [Twistor Newsletter No. 20 (1985), 75\u201379; ibid. No. 21 (1986), 49\u201355; ibid. No. 21 (1986), 56\u201357; addendum, ibid. No. 22 (1986), 43; ibid. No. 21 (1986), 57\u201358; ibid. No. 22 (1986), 26\u201327; ibid. No. 22 (1986), 28\u201331; ibid. No. 22 (1986), 41\u201342] and his dissertation [&#8220;Twistors in curved space-time&#8221;, Ph.D. Thesis, Oxford Univ., Oxford, 1985].<\/p>\n<p class=\"headlineMenu\">Chapter 8 is an algebraic classification of curvature spinors and a detailed analysis of the associated homogeneous functions on spin space.<\/p>\n<p class=\"headlineMenu\">Chapter 9 discusses the conformal compactification of space-time and some of its applications. The two spin structures on\u00a0<span class=\"MathTeX\">$\\widetilde{{\\bf M}}$<\/span>\u00a0are described, and their relation to twistor geometry and the &#8220;Grgin phenomenon&#8221;. There is a section on twistors and (conformally flat) cosmological models. The structure of conformal infinity for more general space-times is analyzed in detail. There is then a section giving Penrose&#8217;s proposed definitions for the momentum\u00a0<span class=\"MathTeX\">$P_a$<\/span>\u00a0and angular momentum\u00a0<span class=\"MathTeX\">$M_{ab}$<\/span>\u00a0enclosed by a space-like 2-surface of spherical topology. This is the only one of many definitions which have been put forward which has given reasonable results in all cases investigated. However, a proof that\u00a0<span class=\"MathTeX\">$M_{ab}$<\/span>\u00a0is real and\u00a0<span class=\"MathTeX\">$P_a$<\/span>\u00a0is real and future-pointing is lacking, and there are still ambiguities in the prescription. These problems are discussed. The Bondi-Sachs mass is introduced as the limit of this quasilocal mass as the 2-surface approaches a cut of\u00a0<span class=\"MathTeX\">$\\scr I^+$<\/span>. The various proofs that the Bondi-Sachs mass is positive are outlined, and a connection between them and a spinor reformulation of the vacuum equations due to Sparling is noted.<\/p>\n<p class=\"headlineMenu\">Lastly, there is an appendix on spinors in arbitrary (finite) dimensions.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=83855\">Adam Helfer<\/a><\/span><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>&nbsp; UPDATE [6 October 2020]:\u00a0 Roger Penrose is one of the winners of the 2020 Nobel Prize in Physics.\u00a0 Congratulations! Numberphile has posted an audio interview with Roger Penrose on their YouTube channel Numberphile2.\u00a0 You can also access it as &hellip; <a href=\"https:\/\/blogs.ams.org\/beyondreviews\/2020\/08\/29\/roger-penrose-on-numberphile\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/beyondreviews\/2020\/08\/29\/roger-penrose-on-numberphile\/><\/div>\n","protected":false},"author":86,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[8,44],"tags":[],"class_list":["post-2845","post","type-post","status-publish","format-standard","hentry","category-mathematicians","category-prizes-and-awards"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6C2KK-JT","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/2845","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/users\/86"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/comments?post=2845"}],"version-history":[{"count":31,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/2845\/revisions"}],"predecessor-version":[{"id":2917,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/2845\/revisions\/2917"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/media?parent=2845"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/categories?post=2845"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/tags?post=2845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}