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The Penrose Inequality

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Book cover The Einstein Equations and the Large Scale Behavior of Gravitational Fields

Abstract

In 1973, R. Penrose presented an argument that the total mass of a space-time which contains black holes with event horizons of total area A should be at least \( \sqrt {A/16\pi} \). An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. This inequality was first established by G. Huisken and T. Ilmanen in 1997 for a single black hole and then by one of the authors (HB) in 1999 for any number of black holes. The two approaches use two different geometric flow techniques and are described here. We further present some background material concerning the problem at hand, discuss some applications of Penrose-type inequalities, as well as the open questions remaining.

Research supported in part by NSF grants #DMS-0206483 and #DMS-9971960 and by the Erwin Schrödinger Institute, Vienna.

Partially supported by a Polish Research Committee grant 2 PO3B 073 24, by the Erwin SchrOdinger Institute, Vienna, and by a travel grant from the Vienna City Council.

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References

  1. L.F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B195 (1982), 76–96.

    Article  ADS  Google Scholar 

  2. L. Andersson and P.T. Chrukiel, On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”, Dissert. Math. 355 (1996), 1–100.

    Google Scholar 

  3. L. Andersson, P.T. ChruAciel, and H. Friedrich, On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einsteins field equations, Commun. Math. Phys. 149 (1992), 587–612.

    Article  ADS  MATH  Google Scholar 

  4. L. Andersson and M. Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Annals of Global Anal. and Geom. 16 (1998), 1–27, dg-ga/9707017.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Arnowitt, S. Deser, and C.W. Misner, The dynamics of general relativity, Gravitation (L. Witten, ed.), Wiley, N.Y., 1962, pp. 227–265, gr-qc/0405109.

    Google Scholar 

  6. R. Bartnik, The mass of an asymptotically flat manifold, Comm Pure Appl. Math. 39 (1986), 661–693.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989), 2346–2348.

    Article  MathSciNet  ADS  Google Scholar 

  8. R. Bartnik, Energy in general relativity, Tsing Hua Lectures on Geometry and Analysis (S.T. Yau, ed.), International Press, 1997, http://www.ise.canberra.edu.au/mathstat/StaffPages/Robert2.htm.

    Google Scholar 

  9. R. Bartnik and P.T. Chruściel, Boundary value problems for Dirac-type equations, (2003), math. DG/0307278.

    Google Scholar 

  10. H. Baum, Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom. 7 (1989), 205–226.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Bergqvist, On the Penrose inequality and the role of auxiliary spinor fields, Class. Quantum Gray. 14 (1997), 2577–2583.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. H. Bondi, M.G.J. van der Burg, and A.W.K. Metzner, Gravitational waves in general relativity VII: Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London A 269 (1962), 21–52.

    Article  ADS  MATH  Google Scholar 

  13. H.L. Bray, Global inequalities, lectures given at the Cargese Summer School on 50 years of the Cauchy problem in general relativity, August 2002, online at fanfreluche.math.univ-tours.fr.

    Google Scholar 

  14. H.L. Bray, The Penrose inequality in general relativity and volume comparison theorem involving scalar curvature, Ph.D. thesis, Stanford University, 1997.

    Google Scholar 

  15. H.L. Bray, Proof of the Riemannian Penrose conjecture using the positive mass theorem, Jour. Diff. Geom. 59 (2001), 177–267, math.DG/9911173.

    MathSciNet  MATH  Google Scholar 

  16. H.L. Bray, Black holes and the Penrose inequality in general relativity, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 257–271.

    Google Scholar 

  17. H.L. Bray, Black Holes,Geometric Flows, and the Penrose Inequality in General Rel- ativity,Notices of the AMS 49 (2002), 1372–1381.

    MathSciNet  MATH  Google Scholar 

  18. H.L. Bray and K. Iga, Superharmonic functions in R n and the Penrose inequality in general relativity, Comm. Anal. Geom. 10 (2002), 999–1016.

    MathSciNet  MATH  Google Scholar 

  19. H.L. Bray and A. Neves, Classification of prime 3-manifolds with Yamabe invariant greater than RP 3, Annals of Math. (2003), in press.

    Google Scholar 

  20. H.L. Bray and R.M. Schoen, Recent proofs of the Riemannian Penrose conjecture,Current developments in mathematics, 1999 (Cambridge, MA), Int. Press, Somerville, MA, 1999, pp. 1–36.

    Google Scholar 

  21. Y. Choquet-Bruhat, Positive-energy theorems, Relativity, groups and topology, II (Les Houches, 1983) (B.S. deWitt and R. Stora, eds.), North-Holland, Amsterdam, 1984, pp. 739–785.

    Google Scholar 

  22. U. Christ and J. Lohkamp, in preparation (2003).

    Google Scholar 

  23. D. Christodoulou and S.-T. Yau, Some remarks on the quasi-local mass, Cont. Math. 71 (1988), 9–14.

    Article  MathSciNet  Google Scholar 

  24. P.T. Chruściel, Black holes, Proceedings of the Tubingen Workshop on the Conformal Structure of Space-times, H. Friedrich and J. Frauendiener, Eds., Springer Lecture Notes in Physics 604, 61–102 (2002), gr-qc/0201053.

    Chapter  Google Scholar 

  25. D. Christodoulou, A remark on the positive energy theorem, Class. Quantum Gray. 33 (1986), L115–L121.

    Google Scholar 

  26. D. Christodoulou, Boundary conditions at spatial infinity from a Hamiltonian point of view, Topological Properties and Global Structure of Space-Time (P. Bergmann and V. de Sabbata, eds.), Plenum Press, New York, 1986, pp. 49–59, URL http://www.phys.univ-tours.fr/-piotr/scans/-piotr/scans.

  27. D. Christodoulou, On the invariant mass conjecture in general relativity, Commun Math. Phys. 120 (1988), 233–248.

    Article  ADS  Google Scholar 

  28. P.T. Chruśiel and E. Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. de France. 94 (2003), 1–103, gr-qc/0301073.

    Google Scholar 

  29. P.T. Chruściel, E. Delay, G. Galloway, and R. Howard, Regularity of horizons and the area theorem, Annales Henri Poincaré 2 (2001), 109–178, gr-qc/0001003.

    Article  ADS  MATH  Google Scholar 

  30. P.T. Chruściel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific Jour. Math. 212 (2003), 231–264, dg-ga/0110035.

    Article  Google Scholar 

  31. P.T. Chruściel, J. Jezierski, and S. Lgski, The Trautman-Bondi mass of hyperboloidal initial data sets, (2003), gr-qc/0307109.

    Google Scholar 

  32. P.T. Chruściel and G. Nagy, The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times,Adv. Theor. Math. Phys. 5 (2002), 697–754, grqc/0110014.

    Google Scholar 

  33. P.T. Chruściel and W. Simon, Towards the classification of static vacuum spacetimes with negative cosmological constant, Jour. Math. Phys. 42 (2001), 1779–1817, grqc/0004032.

    Article  ADS  MATH  Google Scholar 

  34. J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214 (2000), 137–189.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. J. Corvino and R. Schoen, On the asymptotics for the vacuum Einstein constraint equations,gr-qc/0301071, 2003.

    Google Scholar 

  36. S. Dain, Trapped surfaces as boundaries for the constraint equations, (2003), grqc/0308009.

    Google Scholar 

  37. V. I. Denisov and V. O. Solov’ev, The energy determined in general relativity on the basis of the traditional Hamiltonian approach does not have physical meaning, Theor. and Math. Phys. 56 (1983), 832–838, English translation, original pagination 301–314.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. P.A.M. Dirac, The theory of gravitation in Hamiltonian form, Proc. Roy. Soc. London A246 (1958), 333–343.

    MathSciNet  ADS  Google Scholar 

  39. J. Frauendiener, On the Penrose inequality, Phys. Rev. Lett. 87 (2001), 101101, gr-qc/0105093.

    Article  MathSciNet  ADS  Google Scholar 

  40. H. Friedrich, Cauchy problem for the conformal vacuum field equations in general relativity, Commun Math. Phys. 91 (1983), 445–472.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. H. Friedrich, Einstein equations and conformal structure: Existence of anti-de-Sitter-type space-times, Jour. Geom. and Phys. 17 (1995), 125–184.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. G.J. Galloway, K. Schleich, D.M. Witt, and E. Woolgar, Topological censorship and higher genus black holes, Phys. Rev. D60 (1999), 104039, gr-qc/9902061.

    MathSciNet  ADS  Google Scholar 

  43. R. Geroch, Energy extraction, Ann. New York Acad. Sci. 224 (1973), 108–117.

    Article  ADS  Google Scholar 

  44. G.W. Gibbons, Collapsing shells and the isoperimetric inequality for black holes, Class. Quantum Gray. 14 (1997), 2905–2915, hep-th/9701049.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. G.W. Gibbons, Gravitational entropy and the inverse mean curvature flow, Class. Quantum Gray. 16 (1999), 1677–1687.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. G.W. Gibbons, S.W. Hawking, G.T. Horowitz, and M.J. Perry, Positive mass theorem for black holes, Commun. Math. Phys. 88 (1983), 295–308.

    Article  MathSciNet  ADS  Google Scholar 

  47. S.W. Hawking, Gravitational radiation in an expanding universe, Jour. Math. Phys. 9 (1968), 598–604.

    Article  ADS  Google Scholar 

  48. S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973.

    Book  MATH  Google Scholar 

  49. S.A. Hayward, Quasilocalization of Bondi-Sachs energy loss, Class. Quantum Gray. 11 (1994), 3037–3048, gr-qc/9405071.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. M. Herzlich, A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds, Commun. Math. Phys. 188 (1997), 121–133.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. G.W. Gibbons, The positive mass theorem for black holes revisited, Jour. Geom. Phys. 26 (1998), 97–111.

    Article  MathSciNet  Google Scholar 

  52. G.W. Gibbons, Minimal spheres, the Dirac operator and the Penrose inequality, Séminaire Théorie Spectrale et Géométrie (Institut Fourier, Grenoble) 20 (2002), 9–16.

    Google Scholar 

  53. G.T. Horowitz, The positive energy theorem and its extensions, Asymptotic behavior of mass and spacetime geometry (F. Flaherty, ed.), Springer Lecture Notes in Physics, vol. 202, Springer Verlag, New York, 1984.

    Chapter  Google Scholar 

  54. G.T. Horowitz and R.C. Myers, The AdS/CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D59 (1999), 026005 (12 pp.).

    MathSciNet  Google Scholar 

  55. G. Huisken and T. Ilmanen, A note on inverse mean curvature flow,(1997), Proceedings of the Workshop on Nonlinear Partial Differential Equations (Saitama University, Sept. 1997), available from Saitama University.

    Google Scholar 

  56. G. Huisken, The Riemannian Penrose inequality, Int. Math. Res. Not. 20 (1997), 1045–1058.

    Article  MathSciNet  Google Scholar 

  57. G. Huisken, The inverse mean curvature flow and the Riemannian Penrose inequality, Jour. Diff. Geom. 59 (2001), 353–437, URL http://www.math.miu.edur~ilmanen~ilmanen.

    MathSciNet  MATH  Google Scholar 

  58. P.S. Jang and R.M. Wald, The positive energy conjecture and the cosmic censor hypothesis, J. Math. Phys. 18 (1977), 41–44.

    Article  MathSciNet  ADS  Google Scholar 

  59. J. Jezierski, Positivity of mass for certain space-times with horizons, Class. Quantum Gray. 6 (1989), 1535–1539.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. J. Jezierski, Perturbation of initial data for spherically symmetric charged black hole and Penrose conjecture, Acta Phys. Pol. B 25 (1994), 1413–1417.

    MathSciNet  MATH  Google Scholar 

  61. J. Kánnár, Hyperboloidal initial data for the vacuum Einstein equations with cosmological constant, Class. Quantum Gray. 13 (1996), 3075–3084.

    Article  MATH  Google Scholar 

  62. F. Kottler, Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie, Annalen der Physik 56 (1918), 401–462.

    Article  ADS  MATH  Google Scholar 

  63. M. Ludvigsen and J.A.G. Vickers, An inequality relating the total mass and the area of a trapped surface in general relativity, Jour. Phys. A: Math. Gen. 16 (1983), 3349–3353.

    Article  MathSciNet  ADS  Google Scholar 

  64. E. Malec, M. Mars, and W. Simon, On the Penrose inequality for general horizons, Phys. Rev. Lett. 88 (2002), 121102, gr-qc/0201024.

    Article  MathSciNet  ADS  Google Scholar 

  65. E. Malec and N. O Murchadha, Trapped surfaces and the Penrose inequality in spherically symmetric geometries, Phys. Rev. D49 (1994), 6931–6934.

    ADS  Google Scholar 

  66. E. Malec and K. Roszkowski, Comment on the Herzlich’s proof of the Penrose inequality, Acta Phys. Pol. B29 (1998), 1975–1978, gr-qc/9806035.

    MathSciNet  ADS  Google Scholar 

  67. D. Maxwell, Solutions of the Einstein constraint equations with apparent horizon boundary,(2003), gr-qc/0307117.

    Google Scholar 

  68. W.H. Meeks, III and S.T. Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), 441–484.

    Article  MathSciNet  MATH  Google Scholar 

  69. T. Parker and C. Taubes, On Witten’s proof of the positive energy theorem, Commun. Math. Phys. 84 (1982), 223–238.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  70. R. Penrose, Gravitational collapse - the role of general relativity, Riv. del Nuovo Cim. (numero speziale) 1 (1969), 252–276.

    ADS  Google Scholar 

  71. R. Schoen and S.-T. Yau, Positivity of the total mass of a general space-time, Phys. Rev. Lett. 43 (1979), 1457–1459.

    Article  MathSciNet  ADS  Google Scholar 

  72. R. Schoen, Proof of the positive mass theorem, Comm. Math. Phys. 65 (1979), 45–76.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  73. R. Schoen, Proof of the positive mass theorem II, Comm. Math. Phys. 79 (1981), 231–260.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  74. W. De Sitter, On the curvature of space, Proc. Kon. Ned. Akad. Wet. 20 (1917), 229–243.

    Google Scholar 

  75. K.P. Tod, The hoop conjecture and the Gibbons-Penrose construction of trapped surfaces, Class. Quantum Gray. 9 (1992), 1581–1591.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. A. Trautman, King’s College lecture notes on general relativity, May-June 1958, mimeographed notes; reprinted in Gen. Rel. Gray. 34 721–762 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  77. A. Trautman, Radiation and boundary conditions in the theory of gravitation, Bull. Acad. Pol. Sci., Série sci. math., astr. et phys. VI (1958), 407–412.

    MathSciNet  Google Scholar 

  78. R.M. Wald, General relativity, University of Chicago Press, Chicago, 1984.

    MATH  Google Scholar 

  79. X. Wang, Mass for asymptotically hyperbolic manifolds, Jour. Diff. Geom. 57 (2001), 273–299.

    MATH  Google Scholar 

  80. E. Witten, A simple proof of the positive energy theorem, Commun. Math. Phys. 80 (1981), 381–402.

    Article  MathSciNet  ADS  Google Scholar 

  81. X. Zhang, A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3 manifolds I,(2001), preprint.

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, 2-179, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Hubert L. Bray

  2. Département de Mathématiques, Faculté des Sciences, Parc de Grandmont, F-37200, Tours, France

    Piotr T. Chruściel

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  1. Hubert L. Bray
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  2. Piotr T. Chruściel
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Editors and Affiliations

  1. Laboratoire de Mathématiques et de Physique Théorique, CNRS UMR 6083, Avenue Grammont, 37200, Tours, France

    Piotr T. Chruściel

  2. Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14467, Golm, Germany

    Helmut Friedrich

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Bray, H.L., Chruściel, P.T. (2004). The Penrose Inequality. In: Chruściel, P.T., Friedrich, H. (eds) The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7953-8_2

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  • DOI: https://doi.org/10.1007/978-3-0348-7953-8_2

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