Penrose and the 2020 Physics Nobel Prize

JoséNatário (Instituto Superior Técnico, Lisbon)

Lisboa, December 2020 Roger Penrose

• Born in Colchester, United Kingdom, 1931.

• PhD in Mathematics (“Tensor methods in algebraic geome- try”), Cambridge, 1958.

• Known for Penrose-Moore inverse, Penrose triangle, Pen- rose diagrams, Penrose singularity theorem, cosmic censorship conjecture, Penrose inequality, Penrose process, Penrose tilings, twistor theory.

• Physics Nobel Prize, 2020, “for the discovery that black hole formation is a robust prediction of the general theory of rel- ativity”. Singularities in 1965

• Oppenheimer-Snyder collapse is spherically symmetric, so a singularity at the center is expected.

• Known black hole solutions have singularities, but are highly symmetric.

• Lifshitz and Khalatnikov (1963): singularities are not generic. The theorem

If:

• (M,g) globally hyperbolic (connected) spacetime.

• Null energy condition is satisﬁed.

• M contains a compact trapped surface Σ.

Then:

• Either any Cauchy hypersurface S is compact or (M,g) is future null geodesically incomplete. The proof

• Σ compact trapped, the Raychaudhuri equation and the null energy condition imply that ∂J+(Σ) is ruled by null geodesics which are either incomplete or develop a conjugate point, and so have ﬁnite aﬃne length.

• In the latter case, ∂J+(Σ) is compact. Since there is a natural projection π : ∂J+(Σ) → S which is continuous and open, its image is both open and closed, hence coincides with S. Therefore S is also compact. Example: the Schwarzschild solution

Example: the de Sitter universe Example: the Schwarzschild-de Sitter solution Cosmic censorship

• Theorem proves geodesic incompleteness.

• Conjecture: geodesic incompleteness arises from a curvature singularity.

• Conjecture (weak cosmic censorship): curvature singularities must be hidden by an event horizon.

• Both these conjectures are widely regarded as being true, but have not yet been proved. VOLUME 14, NUMBER $ PHYSICAL REVIEW LETTERS 18 JANUARY 1965

~ I. S. Shklovsky, Cosmic Radio %aves {Harvard Holland Publishing Company, Amsterdam, 1955}. University Press, Cambridge, Massachusetts, 1960), 30E. N. Parker, Space Sci. Rev. 1, 62 (1962}. pp. 271-316. M. Neugebauer and C. %. Snyder, Science 138, C. Hazard, M. B. Mackey, and A. J. Simmins, 1095 (1962}. Nature 197, 1037 (1963). A. Bonetti, H. S. Bridge, A. J. Lazarus, B. Ros- ~8M. Schmidt, Nature 197, 1040 {1963). si, and F. Scherb, J. Geophys. Res. 69, 4017 (1963}. 2 A. T. Moffet, Science 146, 764 (1964). E. J. Smith, L. Davis, P. J. Coleman, and C. P. 2~M. Ryle and A. Sandage, Astrophys. J. 139, 419 Sonett, Science 139, 1099 {1962). (1964}. 34N. F. Ness, C. S. Scearce, and J. B. Seek, J. Geo- I. S. Shk]ovsky, reference 17, pp. 369, 372. phys. Res. 69, 3531 (1964). 3E. Fermi, Phys. Rev. 75, 1169 {1949). 35P. J. Coleman, L. Davis, and C. P. Sonett, Phys. 24E. N. Parker, Phys. Rev. 109, 1328 {1958). Rev. Letters 5, 43 (1960). T. H. Stix, Plasma Physics Laboratory Report 36D. A. Bryant, T. L. Cline, U. D. Desai, and F. B. No. MATT-239, Princeton University, 1964 (unpub- McDonald, J. Geophys. Res. 67, 4983 (1962). lished). A similar increase in the low-energy proton in- 26R. W. Fredricks, F. L. Scarf, and W. Bernstein, tensity, in coincidence with the passage of a blast to be published. wave, was observed on 12 November 1960 t J. F. V. Sarabhai, J. Geophys. Res. 68, 1555 (1963). Steljes, H. Carmichael, and K. G. McCracken, J. E. N. Parker, Astrophys. J. 133, 1014 (1961); Geophys. Res. 66, 1363 (1961}]. The increases pro- Interplanetarjj Dynamical Processes (John %iley ton intensity may have been partly or wholly of in- @ Sons, Inc. , New York, 1963}. terplanetary origin, but the increase was of such T. Gold, Gas Dynamics of Cosmic Clouds, edited a form that it is not possible to rule out a solar ori- by H. C. van de Hulst and J. M. Burgers (North- gin.

GRAVITATIONAL COLLAPSE AND SPACE- TIME SINGULARITIES Roger Penrose Department of Mathematics, Birkbeck College, London, England {Received 18 December 1964)

The discovery of the quasistellar radio sources measured by local comoving observers, the has stimulated renewed interest in the question body passes within its Schwarzschild radius of gravitational collapse. It has been suggested r = 2m. (The densities at which this happens by some authors' that the enormous amounts need not be enormously high if the total mass of energy that these objects apparently emit is large enough. ) To an outside observer the may result from the collapse of a mass of the contraction to ~ = 2m appears to take an infinite order of (10'-10')MC, to the neighborhood of time. Nevertheless, the existence of a singu- its Schwarzschild radius, accompanied by a larity presents a serious problem for any com- violent release of energy, possibly in the form plete discussion of the physics of the interior of gravitational radiation. The detailed math- region. ematical discussion of such situations is dif- The question has been raised as to whether ficult since the full complexity of general rela- this singularity is, in fact, simply a proper- tivity is required. Consequently, most exact ty of the high symmetry assumed. The mat- calculations concerned with the implications ter collapses radially inwards to the single of gravitational collapse have employed the point at the center, so that a resulting space- simplifying assumption of spherical symme- time catastrophe there is perhaps not surpris- try. Unfortunately, this precludes any detailed ing. Could not the presence of perturbations discussion of gravitational radiation —which which destroy the spherical symmetry alter requires at least a quadripole structure. the situation drastically? The recent rotating The general situation with regard to a spher- solution of Kerr' also possesses a physical ically symmetrical body is well known. ' For singularity, but since a high degree of sym- a sufficiently great mass, there is no final metry is still present (and the solution is al- equilibrium state. %'hen sufficient thermal gebraically special), it might again be argued energy has been radiated away, the body con- that this is not representative of the general tracts and continues to contract until a physi- situation. 4 Collapse without assumptions of cal singularity is encountered at r = 0. As symmetry will be discussed here. VOLUME 14, NUMBER $ PHYSICAL REVIEW LETTERS 18 JPNUPRY 1965

Consi'd er thee time development of a Cauchy Indee,d th e Kerr solutions with m &a angular hypersurface C' rerepresenting an initial ma tter momentum ma ) all possess trapped surfac distribu'b t1on. %e may assume Einstein s fie whereas th osse for which m ~a do not..' The equations an d suitable equations of state gov- argument w111 bee to show that the existencee oof erning the ma tt er. In fact the only assump- a trapped sur ace 1m„lies —irrespective of sym- made here about these equations of sta e metry —thata singularities necessarilyil develop. will be the non-negative definiteness o The existence of a singular1ty cann never be energy express1on (withw' or without cosmologi- inferre,d however,ow without an assump tionio such cal terms.c. Suupposeose this matter distribution as completeness for the manifold underr ccon- undergoes gravitational collapse in a way w ic, si eration. It will be necessary, here, to sup- at first, qualitatively resembles thee spherical- th t the manifold M+,4, which'ch is the future ' t al case. It will be shown that, time developmentdeve opm of an initial Cauchy yp after a certain critical condition has been u- sur face C~ (past boundary of the M+4 rreg'ion), filled, deviations from sphericalal symmetry is in fact null complete into the future.re. The cannot prevent space-tim e sin gu larities from var lou s assumptions are, more precise y, as ar18 1ng. If, ass seems justifiable, actual phys- follows: (i) M+~ is a nonsingular (+---+---) Rie- 0 ~ ~ 1ca1 singularities 1n space-tim- ime are not to be ifold for which the null half-cones permitted to occur, the conclus1on wouould ap- form two separate systems ("past" an u r ' pear inescapa bl e th at inside such a collapsing (ii) Every nu ge e oob'ect]ec aat least one of the following holds: into thee futureu to arbitrarily large affine param- (a) Negative local energy occurs. ster values (null completeness . ux stein's equations are v1olated. ,c, The s ace- timelike or null geodesic in M+4 can bee extended time manifold is incomplete. ' (d} The concept f e-time loses its meaning at very high — curvatures possibly because of quantum phe-e- nomena. ' In fact (a}, (b), (c), d are some- what interrelated, the distinction being partly one of attitude of mind. Before examining the asymmetrical case, consider a spherically symmetrical matter distribution of finite radius in C' which col- 1 ses symmetrically. The empty region surrounding the matter will, in th1s case, Schwarzsc h 1ld field, and we can conveniently use the metric ds' = 2drdv+dU'-(I-2m/r r'(d8'+ sin-'8dcp') with an advanced time pa- rameter v to describe it. The situation is ' depictedd 1n F'1g.. 1. Note that an exterior observer will always see matter outside r =2m, the collapse through r =2m to the singularity at r =0 being invisible to him. Aft er the matter has contracted within r =2m, S' = a space lik e spheree (f =const,=, 2m &r= cson)t can be found in the empty region surrounding the matter. This sphere is an example of what will be called here a trapped surface —defined

T' with the property that the two systems of null eodesics which meet T' orthogonally con- verge locally in future directions at T'. ly trapped surfaces will still exist if the matter FIG. 1. Spherically symmetrical collapse (one region has no sharp boundary or if spherica s ace dimension surpressed). The diagram essen- tially also serves for the discussion of thee as ym met- tions from the above situation are not too great. rical case.

58 VOtUMZ 14, NUMSZR ) PHYSICAL REVIEW LETTERS 18 JANUARY 1965

into the past until it meets C, (Cauchy hyper- -C' induced by p, when s =0 must also be uni- surface condition). (iv) At every point of M+~, ty. The impossibility of this follows from the all timelike vectors t& satisfy (—R „+~g noncompactness of C'. -gg )t&t~ ~ 0 (non-negativeness of local ener- Full details of this and other related results jLL V gy). (v) There exists a trapped surface F' in will be given elsewhere. M+4. It will be shown here, in outline, that (i), ~ ~ ~, (v) are together inconsistent. Let E be the set of points in M+4 which can ~F. Hoyle and W. A. Fowler, Monthly Notices Roy. be connected to T' by a smooth timelike curve Astron. Soc. 125, 169 (1963); F. Hoyle, W. A. Fow- leading into the future from T'. Let 8' be the ler, G. R. Burbidge, and E. M. Burbidge, Astrophys. boundary of F . Local considerations show J. 139, 909 (1964); W. A. Fowler, Rev. Mod. Phys. 36, 545 (1964); Ya. B. Zel'dovich and I. D. Novikov, that B' is null where it is nonsingular, being Dokl. Akad. Nauk SSSR 155, 1033 (1964) [translation: generated by the null geodesic segments which — T' Soviet Phys. Doklady 9, 246 (1964)]; I. S. Shklov- meet orthogonally at a past endpoint and skii and N. S. Kardashev, Dokl. Akad. Nauk SSSR have a future endpoint if this is a singularity 155, 1039 (1964) [translation: Soviet Phys. -Dok- (on a caustic or crossing region) of B'. I.et lady 9, 252 (1964)]; Ya. B. Zel'dovich and M. A. lP' (subject to I& ~l =0), p (=-qlP' &), and l&l Podurets, Dokl. Akad. Nauk SSSR 156, 57 (1964) [translation: Soviet Phys. —Doklady 373 Also (=[zl(&.~)l »~- &(I&.&)']'"j be, respectively, 9, (1964)]. a future-pointing tangent vector, the conver- various articles in the Proceedings of the 1963 Dallas Conference on Gravitational Collapse (University gence, and the shear for these null geodesics, " of Chicago Press, Chicago, Illinois, 1964). and let A be a corresponding infinitesimal area B'. 2J. R. Oppenheimer and H. Snyder, Phys. Rev. of cross section of Then [(A'").&lP).~l 56, 455 (1939). See also J. A. Wheeler, in Relati- =-(A p). lP=-A ([cr~ +4) ~0 where 4' vity, Groups and Topology, edited by C. deWitt and = -&R&~lPI~ [~0 by (iv)]. Since T' is trapped, B. deWitt (Gordon and Breach Publishers, Inc. , p& 0 at T', whence A becomes zero at a finite New York, 1964); and reference 1. 3R. affine distance to the future of T' on each null P. Kerr, Phys. Rev. Letters 11, 237 (1963). See also E. M. Lifshitz and I. M. Khalatnikov, geodesic. Each geodesic thus encounters a B' Advan. Phys. 12, 185 (1963). caustic. Hence is compact (closed), being See also P. G. Bergmann, Phys. Rev. Letters generated by a compact system of finite seg- 12, 139 (1964). ments. We may approximate Bs arbitrarily The negative energy of a "Q field" may be invoked closely by a smooth, closed, spacelike hyper- to avoid singularities: F. Hoyle and J. V. Narlikar, surface B'*. Let K' denote the set of pairs Proc. Roy. Soc. (London) A278, 465 (1964). How- it is difficult to see how even the PCB'* and Define con- ever, presence (P, s) with 0 s -1. a "bounce" K4-M+4 of negative energy could lead to an effective tinuous map p.. where, for fixed P, if local causality is to be maintained. p((P, s)j is the past geodesic segment normal YThe "I'm all right, Jack" philosophy with regard to B'* at P = pOP, I)] and meeting C' [as it to the singularities would be included under this must, by (iii)] in the point p,((P, 0)). At each heading j point Q of p(K'}, we can define the degree d(Q) D. Finkelstein, Phys. Rev. 110, 965 (1959). The case m & a is interesting in that here a singu- of p. to be the number of points of K4 which larity is visible to an outside observer. Whether map to Q (correctly counted). Over any region or not "visible" singularities inevitably arise under not containing the image of a boundary point circumstances is an question B~*, appropriate intriguing of E~, d(Q) will be constant. Near p is not covered by the present discussion. 1-1, so d(Q) =1. It follows that d(Q) =1 near For the notation, etc. , see E. Newman and R. Pen- Cs also, whence the degree of the map Bs rose, J. Math. Phys. 3, 566 (1962).