# Roger Penrose on Numberphile

UPDATE [6 October 2020]:  Roger Penrose is one of the winners of the 2020 Nobel Prize in Physics.  Congratulations!

Numberphile has posted an audio interview with Roger Penrose on their YouTube channel Numberphile2.  You can also access it as audio-only from their website

The podcast is quite good.  Penrose is very relaxed and thoughtfully answers all of the questions from the host, Brady Haran.  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.  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.

The host asks Penrose about his affiliation with Stephen Hawking.  In the podcast, Penrose corrects some little details from the film, The Theory of Everything.  A couple of times, Brady Haran asks whether Penrose and Hawking were friends.  My understanding is that the answer to that question would have varied over the years.  I remember being at a conference at Durham University where both Penrose and Hawking were present.  One evening in the middle of the conference, a bunch of us found ourselves at a pub with Hawking and Penrose.  Hawking had great fun ordering round after round for the table, all the while asking us math questions.   The one that sticks with me is Hawking chiding us that we couldn’t figure out for which real values of $x$, the function $x$^$x$^$x$^$x$^$\dots$ converged. It’s an interesting problem, but tricky after a few rounds of beer with brandy chasers. (As I hazily recall, the brandies were Hawking’s idea.)  After you have spent some time with the problem, check out one of Eisenstein’s ten papers in volume 28 of J. Reine Angew. Math. [Crelle’s Journal].

Here are a couple more stories from my time in Penrose’s research group.

Every Friday, everyone in the group met in Penrose’s office for the so-called “Friday meeting”.  The office was large enough, though we had to bring chairs from other rooms.   There was a loose agenda for each meeting, something like:  short announcements (just published a paper, upcoming conferences, etc.); brief reports – 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.  The meeting would start around noon, maybe 1:00pm.  (I forget.)  People would bring lunches, either something from home or carry-out.  A favorite for carry-out was the “pink bag shop”, which was actually Maison Blanc (“Blanc” after the proprietor, Raymond Blanc).   The meetings would sometimes go long, in which case we would break to join the Institute’s regular tea downstairs.  One of the rules of the meetings concerned the graduate students’ questions.  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.  Well, one Friday, a new-ish graduate student asked, “What’s renormalization?”  At first, the room went quiet.  Then various people took a turn at trying to explain it rigorously.  I don’t remember exactly, but that Friday meeting probably went long.

While in Oxford, I happened to attend the British Mathematics Colloquium with Penrose.  Besides the talks, a small number of publishers and vendors had booths or tables set up.  One vendor had various puzzles, many of them were of the string and rings on a stick variety.  Penrose looked at the table.  After a minute or two, he pointed at one and said, “Well that one’s different – isn’t it?”  I asked how it was different.  Penrose replied, “It’s the only one that can’t be done.”  The vendor confirmed it, and proceeded to have a hearty conversation with Penrose.

If you look up Roger Penrose in MathSciNet, you will see that we have papers by him in various subjects, but mostly in relativity and gravitational theory, followed by quantum theory.  Penrose is famous for many things in both mathematics and physics.  By far his most cited paper, however, is his paper on generalized inverses:

MR0069793
Penrose, R.
A generalized inverse for matrices.
Proc. Cambridge Philos. Soc. 51 (1955), 406–413.
(Reviewer: O. Taussky-Todd)

This is also know as the Moore-Penrose pseudoinverse.  The generalized inverse is defined as the unique solution of a certain set of equations involving the matrix and its conjugate transpose.  There is a function hard-coded into MATLAB, called pinv, that computes it (using SVD).  We have 421 citations for it.  Google Scholar has 4766.  I was unable to find the paper in Web of Science.  It seems to be from before their coverage starts.

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.  The canonical place is his two-volume book with Wolfgang Rindler:

MR0776784
Penrose, Roger(4-OX)Rindler, Wolfgang(1-TX-P)
Spinors and space-time. Vol. 1.
Two-spinor calculus and relativistic fields. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1984. x+458 pp. ISBN: 0-521-24527-3
83-02 (53A50 53B50 53C80 83Cxx)

MR0838301
Penrose, Roger(4-OX)Rindler, Wolfgang(1-TXD)
Spinors and space-time. Vol. 2.
Spinor and twistor methods in space-time geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1986. x+501 pp. ISBN: 0-521-25267-9
83Cxx (32L25 53A50 53C80 81D25 81D27 83-02 83C60)

Penrose has written some other very good books.  His books on consciousness are well-known, and created a stir.  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’s take on some of the hot topics in modern physics.  I would be hard-pressed to name better treatments for the first and second parts of the book.

MR2116746
Penrose, Roger(4-OX)
A complete guide to the laws of the universe. Alfred A. Knopf, Inc., New York, 2005. xxviii+1099 pp. ISBN: 0-679-45443-8
83-02 (00A05 00A79 81-02)
Reviewed by Peter R. Law

The publication of the podcast was timed to coincide with Roger Penrose’s birthday (August 8).  I didn’t get around to listening to it until August 27.  So, belated happy birthday, Roger Penrose!

Below are the texts of some reviews of Penrose’s work from MathSciNet.

## Some reviews of Penrose’s works

MR0069793
Penrose, R.
A generalized inverse for matrices.
Proc. Cambridge Philos. Soc. 51 (1955), 406–413.
09.0X

A generalized inverse (g.i.) is introduced for arbitrary (possibly rectangular) matrices $A$ with complex elements. It is the unique solution $X$ of the four equations $AXA=X$$XAX=X$$(AX)^\ast=AX$$(XA)^\ast=XA. [Abstract rings with inverses satisfying axa=a had been studied by v. Neumann, Proc. Nat. Acad. Sci. U. S. A. 22, 707–713 (1936).] The g.i. is used to formulate a necessary and sufficient condition for the solvability of the matrix equation AXB=C and of the system AX=C$$XB=D$ and 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 $A$ to be expressed in a simple manner in terms of the principal idempotents even for non-normal matrices.

Reviewed by O. Taussky-Todd

MR0172678
Penrose, Roger
Gravitational collapse and space-time singularities.
Phys. Rev. Lett. 14 (1965), 57–59.
83.53

It is shown that even in the absence of spherical symmetry the occurrence and persistence of “trapped surfaces” 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 “singularities”, 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.

Reviewed by P. G. Bergmann

MR0776784
Penrose, Roger(4-OX)Rindler, Wolfgang(1-TX-P)
Spinors and space-time. Vol. 1.
Two-spinor calculus and relativistic fields. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1984. x+458 pp. ISBN: 0-521-24527-3
83-02 (53A50 53B50 53C80 83Cxx)

The space-times considered in general relativity are 4-dimensional manifolds with metrics of signature $+\,-\,-\,-$. On such a manifold there is, at least locally, a 2-complex-dimensional vector bundle ${\scr S}_A$, 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 ${\scr S}_A$ is the most primitive distinction between space-times and general Lorentzian manifolds.

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.

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 ${\scr S}_A$. This is a clear account of some rather sophisticated geometry.

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.

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.

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. 14 (1973), 874–881; MR0323287]. 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.

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 173 (1939), 211–232; MR0001173; H. A. Buchdahl, Nuovo Cimento (10) 25 (1962), 486–496; MR0145855] 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.

Lastly, there is an appendix describing Penrose’s “musical” notation for index manipulation, which is useful for calculations involving lots of index permutation.

MR0838301
Penrose, Roger(4-OX)Rindler, Wolfgang(1-TXD)
Spinors and space-time. Vol. 2.
Spinor and twistor methods in space-time geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1986. x+501 pp. ISBN: 0-521-25267-9
83Cxx (32L25 53A50 53C80 81D25 81D27 83-02 83C60)

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 ${\bf M},\ {\bf R}^4$ with 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 ${\bf M}$ (called the Poincaré group, $P(1,3)$).

The Poincaré group is a subgroup of the conformal group $C(1,3)$ of diffeomorphisms preserving the metric up to scale. (More precisely, $C(1,3)$ acts on the conformal completion $\widetilde{{\bf M}}$ of ${\bf M}$.) 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, $C(1,3)$ is a simple Lie group and so has nicer mathematical properties than $P(1,3)$.

The identity-connected component of $C(1,3)$ is 4-1 covered by $\textrm{SU}(2,2)$. The defining representation of this group is twistor space ${\bf T}\simeq {\bf C}^4$, and all other finite-dimensional representations can be obtained from tensor powers of ${\bf T}$. In particular, the conformal geometry of ${\bf M}$ can be studied by means of twistors, and its metrical geometry by twistors and reducing $\textrm{SU}(2,2)$ to (a covering group of the identity-connected component of) $P(1,3)$. 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.

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 $\bf P$, the Ward construction, which gives a one-to-one correspondence between self-dual Yang-Mills fields and certain bundles over $\bf P$, Penrose’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’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.

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.

In Chapter 6, twistors are introduced as spinor fields satisfying the “twistor equation”. Such fields are “square roots” 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 ${\bf T}$ 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.

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), 75–79; ibid. No. 21 (1986), 49–55; ibid. No. 21 (1986), 56–57; addendum, ibid. No. 22 (1986), 43; ibid. No. 21 (1986), 57–58; ibid. No. 22 (1986), 26–27; ibid. No. 22 (1986), 28–31; ibid. No. 22 (1986), 41–42] and his dissertation [“Twistors in curved space-time”, Ph.D. Thesis, Oxford Univ., Oxford, 1985].

Chapter 8 is an algebraic classification of curvature spinors and a detailed analysis of the associated homogeneous functions on spin space.

Chapter 9 discusses the conformal compactification of space-time and some of its applications. The two spin structures on $\widetilde{{\bf M}}$ are described, and their relation to twistor geometry and the “Grgin phenomenon”. 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’s proposed definitions for the momentum $P_a$ and angular momentum $M_{ab}$ enclosed 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 $M_{ab}$ is real and $P_a$ is 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 $\scr I^+$. 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.

Lastly, there is an appendix on spinors in arbitrary (finite) dimensions.