- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

The holographic principle

Raphael Bousso*

Raphael Bousso*

  • Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

  • *Electronic address: bousso@itp.ucsb.edu
PDF

Rev. Mod. Phys. 74, 825 – Published 5 August, 2002

DOI: https://doi.org/10.1103/RevModPhys.74.825

Abstract

There is strong evidence that the area of any surface limits the information content of adjacent spacetime regions, at 1.4×1069 bits per square meter. This article reviews the developments that have led to the recognition of this entropy bound, placing special emphasis on the quantum properties of black holes. The construction of light sheets, which associate relevant spacetime regions to any given surface, is discussed in detail. This article explains how the bound is tested, and its validity is demonstrated in a wide range of examples. A universal relation between geometry and information is thus uncovered. It has yet to be explained. The holographic principle asserts that its origin must lie in the number of fundamental degrees of freedom involved in a unified description of spacetime and matter. It must be manifest in an underlying quantum theory of gravity. This article surveys some successes and challenges in implementing the holographic principle.

View figure in article

References (214)

  1. Abdalla, E., and L. A. Correa-Borbonet, 2001, “Aspects of higher order gravity and holography,” e-print hep-th/0109129.
  2. Achucarro, A., and P. K. Townsend, 1986, “A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories,” Phys. Lett. B 180, 89.
  3. Aharony, O., S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, 2000, “Large N field theories, string theory and gravity,” Phys. Rep. 323, 183.
  4. Bak, D., and S.-J. Rey, 2000a, “Cosmic holography,” Class. Quantum Grav. 17, L83.
  5. Bak, D., and S.-J. Rey, 2000b, “Holographic principle and string cosmology,” Class. Quantum Grav. 17, L1.
  6. Balasubramanian, V., J. de Boer, and D. Minic, 2001, “Mass, entropy and holography in asymptotically de Sitter spaces,” e-print hep-th/0110108.
  7. Balasubramanian, V., E. G. Gimon, and D. Minic, 2000, “Consistency conditions for holographic duality,” J. High Energy Phys. 05, 014.
  8. Balasubramanian, V., P. Hořava, and D. Minic, 2001, “Deconstructing de Sitter,” J. High Energy Phys. 05, 043.
  9. Balasubramanian, V., and P. Kraus, 1999, “Spacetime and the holographic renormalization group,” Phys. Rev. Lett. 83, 3605.
  10. Banks, T., 1998, “Matrix theory,” Nucl. Phys. B, Proc. Suppl. 67, 180.
  11. Banks, T., 1999, “TASI lectures on Matrix theory,” e-print hep-th/9911068.
  12. Banks, T., 2000a, “Cosmological breaking of supersymmetry or little Lambda goes back to the future II,” e-print hep-th/0007146.
  13. Banks, T., 2000b, “On isolated vacua and background independence,” e-print hep-th/0011255.
  14. Banks, T., 2000c, “Supersymmetry and spacetime,” talk given at String Theory at the Millenium, Caltech, January 2000, http://quark.theory.caltech.edu/people/rahmfeld/Banks/fsl.html
  15. Banks, T., and M. Dine, 2001, “Dark energy in perturbative string cosmology,” J. High Energy Phys. 10, 012.
  16. Banks, T., and W. Fischler, 2001a, “An holographic cosmology,” e-print hep-th/0111142.
  17. Banks, T., and W. Fischler, 2001b, “M-theory observables for cosmological space-times,” e-print hep-th/0102077.
  18. Banks, T., W. Fischler, S. H. Shenker, and L. Susskind, 1997, “M theory as a matrix model: A conjecture,” Phys. Rev. D 55, 5112.
  19. Banks, T., and L. Susskind, 1996, “The number of states of two dimensional critical string theory,” Phys. Rev. D 54, 1677.
  20. Banks, T., L. Susskind, and M. E. Peskin, 1984, “Difficulties for the evolution of pure states into mixed states,” Nucl. Phys. B 244, 125.
  21. Bardeen, J. M., B. Carter, and S. W. Hawking, 1973, “The four laws of black hole mechanics,” Commun. Math. Phys. 31, 161.
  22. Bekenstein, J. D., 1972, “Black holes and the second law,” Lett. Nuovo Cimento Soc. Ital. Fis. 4, 737.
  23. Bekenstein, J. D., 1973, “Black holes and entropy,” Phys. Rev. D 7, 2333.
  24. Bekenstein, J. D., 1974, “Generalized second law of thermodynamics in black hole physics,” Phys. Rev. D 9, 3292.
  25. Bekenstein, J. D., 1975, “Statistical black-hole thermodynamics,” Phys. Rev. D 12, 3077.
  26. Bekenstein, J. D., 1981, “A universal upper bound on the entropy to energy ratio for bounded systems,” Phys. Rev. D 23, 287.
  27. Bekenstein, J. D., 1983, “Entropy bounds and the second law for black holes,” Phys. Rev. D 27, 2262.
  28. Bekenstein, J. D., 1984, “Entropy content and information flow in systems with limited energy,” Phys. Rev. D 30, 1669.
  29. Bekenstein, J. D., 1989, “Is the cosmological singularity thermodynamically possible?,” Int. J. Theor. Phys. 28, 967.
  30. Bekenstein, J. D., 1994a, “Do we understand black hole entropy?,” e-print gr-qc/9409015.
  31. Bekenstein, J. D., 1994b, “Entropy bounds and black hole remnants,” Phys. Rev. D 49, 1912.
  32. Bekenstein, J. D., 1999, “Non-Archimedean character of quantum buoyancy and the generalized second law of thermodynamics,” e-print gr-qc/9906058.
  33. Bekenstein, J. D., 2000a, “Holographic bound from second law,” e-print gr-qc/0007062.
  34. Bekenstein, J. D., 2000b, “Holographic bound from second law of thermodynamics,” Phys. Lett. B 481, 339.
  35. Bekenstein, J. D., 2000c, “On Page’s examples challenging the entropy bound,” e-print gr-qc/0006003.
  36. Bekenstein, J. D., 2001, “Quantum information and quantum black holes,” e-print gr-qc/0107049.
  37. Bhattacharya, T., A. Chamblin, and J. Erlich, 2002, unpublished.
  38. Bigatti, D., and L. Susskind, 1997, “Review of matrix theory,” e-print hep-th/9712072.
  39. Bigatti, D., and L. Susskind, 2000, “TASI lectures on the holographic principle,” e-print hep-th/0002044.
  40. Birrell, N. D., and P. C. W. Davies, 1982, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England).
  41. Bombelli, L., R. K. Koul, J. Lee, and R. D. Sorkin, 1986, “A quantum source of entropy for black holes,” Phys. Rev. D 34, 373.
  42. Borde, A., L. H. Ford, and T. A. Roman, 2001, “Constraints on spatial distributions of negative energy,” e-print gr-qc/0109061.
  43. Bousso, R., 1999a, “A covariant entropy conjecture,” J. High Energy Phys. 07, 004.
  44. Bousso, R., 1999b, “Holography in general space-times,” J. High Energy Phys. 06, 028.
  45. Bousso, R., 2000a, “The holographic principle for general backgrounds,” Class. Quantum Grav. 17, 997.
  46. Bousso, R., 2000b, “Positive vacuum energy and the N-bound,” J. High Energy Phys. 11, 038.
  47. Bousso, R., 2001, “Bekenstein bounds in de Sitter and flat space,” J. High Energy Phys. 04, 035.
  48. Bousso, R., and L. Randall, 2001, “Holographic domains of Anti-de Sitter space,” e-print hep-th/0112080.
  49. Bousso, R., O. DeWolfe, and R. C. Myers, 2002, e-print hep-th/0205080.
  50. Brown, J. D., and M. Henneaux, 1986, “Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity,” Commun. Math. Phys. 104, 207.
  51. Brustein, R., 2000, “The generalized second law of thermodynamics in cosmology,” Phys. Rev. Lett. 84, 2072.
  52. Brustein, R., D. Eichler, and S. Foffa, 2000, “The shortest scale of quantum field theory,” e-print hep-th/0009063.
  53. Brustein, R., S. Foffa, and A. E. Mayo, 2002, “Causal entropy bound for non-singular cosmologies,” Phys. Rev. D 65, 024004.
  54. Brustein, R., S. Foffa, and R. Sturani, 2000, “Generalized second law in string cosmology,” Phys. Lett. B 471, 352.
  55. Brustein, R., S. Foffa, and G. Veneziano, 2001, “CFT, holography, and causal entropy bound,” Phys. Lett. B 507, 270.
  56. Brustein, R., and G. Veneziano, 2000, “A causal entropy bound,” Phys. Rev. Lett. 84, 5695.
  57. Cai, R.-G., Y. S. Myung, and N. Ohta, 2001, “Bekenstein bound, holography and brane cosmology in charged black hole background,” Class. Quantum Grav. 18, 5429.
  58. Callan, C. G., and J. Maldacena, 1996, “D-brane approach to black hole quantum mechanics,” Nucl. Phys. B 472, 591.
  59. Carlip, S., 1995, “Lectures on (2+1) dimensional gravity,” e-print gr-qc/9503024.
  60. Carneiro da Cunha, B. G., 2002, “Inflation and holography in string theory,” Phys. Rev. D 65, 026001.
  61. Carter, B., 1970, “An axisymmetric black hole has only two degrees of freedom,” Phys. Rev. Lett. 26, 331.
  62. Cataldo, M., N. Cruz, S. del Campo, and S. Lepe, 2001, “Holographic principle and the dominant energy condition for Kasner type metrics,” Phys. Lett. B 509, 138.
  63. Cohen, A. G., D. B. Kaplan, and A. E. Nelson, 1999, “Effective field theory, black holes, and the cosmological constant,” Phys. Rev. Lett. 82, 4971.
  64. Corley, S., and T. Jacobson, 1996, “Focusing and the holographic hypothesis,” Phys. Rev. D 53, 6720.
  65. Cruz, N., and S. Lepe, 2001, “Closed universes can satisfy the holographic principle in three dimensions,” Phys. Lett. B 521, 343.
  66. Davies, P. C. W., 1984, “Mining the universe,” Phys. Rev. D 30, 737.
  67. Davies, P. C. W., 1987, “Cosmological horizons and the generalized second law of thermodynamics,” Class. Quantum Grav. 4, L225.
  68. Davies, P. C. W., 1988, “Cosmological horizons and entropy,” Class. Quantum Grav. 5, 1349.
  69. Dawid, R., 1999, “A speculative remark on holography,” Phys. Lett. B 451, 19.
  70. Dyson, L., J. Lindesay, and L. Susskind, 2002, “Is there really a de Sitter/CFT duality,” e-print hep-th/0202163.
  71. Easther, R., and D. A. Lowe, 1999, “Holography, cosmology and the second law of thermodynamics,” e-print hep-th/9902088.
  72. Emparan, R., and H. S. Reall, 2001, “A rotating black ring in five dimensions,” e-print hep-th/0110260.
  73. Fabinger, M., 2001, “Virtual reality in the world of holograms,” e-print hep-th/0104111.
  74. Fewster, C. J., 2000, “A general worldline quantum inequality,” Class. Quantum Grav. 17, 1897.
  75. Fewster, C. J., and S. P. Eveson, 1998, “Bounds on negative energy densities in flat spacetime,” Phys. Rev. D 58, 084010.
  76. Fischler, W., 2000a, unpublished.
  77. Fischler, W., 2000b, “Taking de Sitter seriously,” talk given at Role of Scaling Laws in Physics and Biology (Celebrating the 60th Birthday of Geoffrey West), Santa Fe, December 2000.
  78. Fischler, W., A. Kashani-Poor, R. McNees, and S. Paban, 2001, “The acceleration of the universe, a challenge for string theory,” J. High Energy Phys. 07, 003.
  79. Fischler, W., and L. Susskind, 1998, “Holography and cosmology,” e-print hep-th/9806039.
  80. Flanagan, E. E., 1997, “Quantum inequalities in two dimensional Minkowski spacetime,” Phys. Rev. D 56, 4922.
  81. Flanagan, E. E., D. Marolf, and R. M. Wald, 2000, “Proof of classical versions of the Bousso entropy bound and of the Generalized Second Law,” Phys. Rev. D 62, 084035.
  82. Ford, L. H., and T. A. Roman, 1995, “Averaged energy conditions and quantum inequalities,” Phys. Rev. D 51, 4277.
  83. Ford, L. H., and T. A. Roman, 1997, “Restrictions on negative energy density in flat spacetime,” Phys. Rev. D 55, 2082.
  84. Ford, L. H., and T. A. Roman, 1999, “The quantum interest conjecture,” Phys. Rev. D 60, 104018.
  85. Frolov, V. P., 1995, “Why the entropy of a black hole is A/4,” Phys. Rev. Lett. 74, 3319.
  86. Gao, S., and R. M. Wald, 2000, “Theorems on gravitational time delay and related issues,” Class. Quantum Grav. 17, 4999.
  87. Gibbons, G. W., and S. W. Hawking, 1977, “Cosmological event horizons, thermodynamics, and particle creation,” Phys. Rev. D 15, 2738.
  88. Gibbons, G. W., D. Ida, and T. Shiromizu, 2002, “Uniqueness and non-uniqueness of static vacuum black holes in higher dimensions,” e-print gr-qc/0203004.
  89. Giles, R., and C. B. Thorn, 1977, “A lattice approach to string theory,” Phys. Rev. D 16, 366.
  90. Giles, R., L. D. McLerran, and C. B. Thorn, 1978, “The string representation for a field theory with internal symmetry,” Phys. Rev. D 17, 2058.
  91. Green, M. B., J. H. Schwarz, and E. Witten, 1987, Superstring Theory (Cambridge University Press, Cambridge, UK).
  92. Gubser, S. S., I. R. Klebanov, and A. M. Polyakov, 1998, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428, 105.
  93. Guedens, R., 2000, unpublished.
  94. Hawking, S. W., 1971, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. 26, 1344.
  95. Hawking, S. W., 1972, “Black holes in general relativity,” Commun. Math. Phys. 25, 152.
  96. Hawking, S. W., 1974, “Black hole explosions,” Nature (London) 248, 30.
  97. Hawking, S. W., 1975, “Particle creation by black holes,” Commun. Math. Phys. 43, 199.
  98. Hawking, S. W., 1976a, “Black holes and thermodynamics,” Phys. Rev. D 13, 191.
  99. Hawking, S. W., 1976b, “Breakdown of predictability in gravitational collapse,” Phys. Rev. D 14, 2460.
  100. Hawking, S. W., 1982, “The unpredictability of quantum gravity,” Commun. Math. Phys. 87, 395.
  101. Hawking, S. W., and G. F. R. Ellis, 1973, The Large Scale Structure of Space-time (Cambridge University Press, Cambridge, England).
  102. Hellerman, S., N. Kaloper, and L. Susskind, 2001, “String theory and quintessence,” J. High Energy Phys. 06, 003.
  103. Hogan, C. J., 2002a, “Holographic discreteness of inflationary perturbations,” e-print astro-ph/0201020.
  104. Hogan, C. J., 2002b, “Observing-quanta on a cosmic scale,” e-print astro-ph/0201021.
  105. Hořava, P., 1999, “M-theory as a holographic field theory,” Phys. Rev. D 59, 046004.
  106. Hořava, P., and D. Minic, 2000, “Probable values of the cosmological constant in a holographic theory,” Phys. Rev. Lett. 85, 1610.
  107. Hořava, P., and E. Witten, 1996a, “Eleven-dimensional supergravity on a manifold with boundary,” Nucl. Phys. B 475, 94.
  108. Hořava, P., and E. Witten, 1996b, “Heterotic and type I string dynamics from eleven dimensions,” Nucl. Phys. B 460, 506.
  109. Israel, W., 1967, “Event horizons in static vacuum space-times,” Phys. Rev. 164, 1776.
  110. Israel, W., 1968, “Event horizons in static electrovac space-times,” Commun. Math. Phys. 8, 245.
  111. Iyer, V., and R. M. Wald, 1994, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50, 846.
  112. Iyer, V., and R. M. Wald, 1995, “A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes,” Phys. Rev. D 52, 4430.
  113. Jacobson, T., 1994, “Black hole entropy and induced gravity,” e-print gr-qc/9404039.
  114. Jacobson, T., 1995, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. 75, 1260.
  115. Jacobson, T., 1999, “On the nature of black hole entropy,” e-print gr-qc/9908031.
  116. Jacobson, T., G. Kang, and R. C. Myers, 1994, “On black hole entropy,” Phys. Rev. D 49, 6587.
  117. Jacobson, T., and R. C. Myers, 1993, “Black hole entropy and higher curvature interactions,” Phys. Rev. Lett. 70, 3684.
  118. Kaloper, N., M. Kleban, A. E. Lawrence, and S. Shenker, 2002, “Signatures of short distance physics in the cosmic microwave background,” e-print hep-th/0201158.
  119. Kaloper, N., and A. Linde, 1999, “Cosmology vs. holography,” e-print hep-th/9904120.
  120. Kalyana Rama, S., 1999, “Holographic principle in the closed universe: A resolution with negative pressure matter,” Phys. Lett. B 457, 268.
  121. Kalyana Rama, S., and T. Sarkar, 1999, “Holographic principle during inflation and a lower bound on density fluctuations,” Phys. Lett. B 450, 55.
  122. Karch, A., and L. Randall, 2001, “Locally localized gravity,” J. High Energy Phys. 05, 008.
  123. Klebanov, I. R., and L. Susskind, 1988, “Continuum strings from discrete field theories,” Nucl. Phys. B 309, 175.
  124. Linde, A. D., 1990, Particle Physics and Inflationary Cosmology (Harwood, Chur, Switzerland).
  125. Low, R. J., 2002, “Light sheets and the covariant entropy conjecture,” Class. Quantum Grav. 19, L1.
  126. Lowe, D. A., 1999, “Comments on a covariant entropy conjecture,” J. High Energy Phys. 10, 026.
  127. Lowe, D. A., J. Polchinski, L. Susskind, L. Thorlacius, and J. Uglum, 1995, “Black hole complementarity versus locality,” Phys. Rev. D 52, 6997.
  128. Lowe, D. A., L. Susskind, and J. Uglum, 1994, “Information spreading in interacting string field theory,” Phys. Lett. B 327, 226.
  129. Maldacena, J., 1998, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231.
  130. Markopoulou, F., and L. Smolin, 1998, “Quantum geometry with intrinsic local causality,” Phys. Rev. D 58, 084032.
  131. Markopoulou, F., and L. Smolin, 1999, “Holography in a quantum spacetime,” e-print hep-th/9910146.
  132. Marolf, D., and R. Sorkin, 2002, “Perfect mirrors and the self-accelerating box paradox,” e-print hep-th/0201255.
  133. Mena Marugan, G. A., and S. Carneiro, 2001, “Holography and the large number hypothesis,” e-print gr-qc/0111034.
  134. Misner, C. W., K. S. Thorne, and J. A. Wheeler, 1973, Gravitation (Freeman, New York).
  135. Myers, R. C., 1998, “Black holes in higher curvature gravity,” e-print gr-qc/9811042.
  136. Myers, R. C., and M. J. Perry, 1986, “Black holes in higher dimensional space-times,” Ann. Phys. (Leipzig) 172, 304.
  137. Myers, R. C., and J. Z. Simon, 1988, “Black hole thermodynamics in Lovelock gravity,” Phys. Rev. D 38, 2434.
  138. Page, D. N., 1976, “Particle emission rates from a black hole. II. Massless particles from a rotating hole,” Phys. Rev. D 14, 3260.
  139. Page, D. N., 1980, “Is black hole evaporation predictable?,” Phys. Rev. Lett. 44, 301.
  140. Page, D. N., 1993, “Expected entropy of a subsystem,” Phys. Rev. Lett. 71, 1291.
  141. Page, D. N., 2000, “Subsystem entropy exceeding Bekenstein’s bound,” e-print hep-th/0007237.
  142. Peet, A. W., 2000, “TASI lectures on black holes in string theory,” e-print hep-th/0008241.
  143. Peet, A. W., and J. Polchinski, 1999, “UV/IR relations in AdS dynamics,” Phys. Rev. D 59, 065011.
  144. Pelath, M. A., and R. M. Wald, 1999, “Comment on entropy bounds and the generalized second law,” Phys. Rev. D 60, 104009.
  145. Penrose, R., and M. A. H. MacCallum, 1972, “Twistor theory: An approach to the quantization of fields and space-time,” Phys. Rep. 6, 241.
  146. Perlmutter, S., et al., 1999, “Measurements of Omega and Lambda from 42 high-redshift supernovae,” Astrophys. J. 517, 565.
  147. Polchinski, J., 1995, “Dirichlet-branes and Ramond-Ramond charges,” Phys. Rev. Lett. 75, 4724.
  148. Polchinski, J., 1998, String Theory (Cambridge University Press, Cambridge, UK).
  149. Polchinski, J., and A. Strominger, 1994, “A possible resolution of the black hole information puzzle,” Phys. Rev. D 50, 7403.
  150. Randall, L., and R. Sundrum, 1999a, “An alternative to compactification,” Phys. Rev. Lett. 83, 4690.
  151. Randall, L., and R. Sundrum, 1999b, “A large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. 83, 3370.
  152. Riess, A. G., et al., 1998, “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” Astron. J. 116, 1009.
  153. Sahakian, V., 2000a, “Comments on D-branes and the renormalization group,” J. High Energy Phys. 05, 011.
  154. Sahakian, V., 2000b, “Holography, a covariant c-function and the geometry of the renormalization group,” Phys. Rev. D 62, 126011.
  155. Schiffer, M., 1992, “The possible role of event horizons in quantum gravity,” Gen. Relativ. Gravit. 24, 705.
  156. Schiffer, M., and J. D. Bekenstein, 1989, “Proof of the quantum bound on specific entropy for free fields,” Phys. Rev. D 39, 1109.
  157. Smolin, L., 2001, “The strong and weak holographic principles,” Nucl. Phys. B 601, 209.
  158. Sorkin, R. D., R. M. Wald, and Z. J. Zhang, 1981, “Entropy of self-gravitating radiation,” Gen. Relativ. Gravit. 13, 1127.
  159. Spradlin, M., and A. Volovich, 2001, “Vacuum states and the S-matrix in dS CFT,” e-print hep-th/0112223.
  160. Srednicki, M., 1993, “Entropy and area,” Phys. Rev. Lett. 71, 666.
  161. Stephens, C. R., G. ’t Hooft, and B. F. Whiting, 1994, “Black hole evaporation without information loss,” Class. Quantum Grav. 11, 621.
  162. Strominger, A., 1994, “Unitary rules for black hole evaporation,” e-print hep-th/9410187.
  163. Strominger, A., 2001a, “The dS/CFT correspondence,” J. High Energy Phys. 10, 034.
  164. Strominger, A., 2001b, “Inflation and the dS/CFT correspondence,” J. High Energy Phys. 11, 049.
  165. Strominger, A., and C. Vafa, 1996, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett. B 379, 99.
  166. Susskind, L., 1993a, “Some speculations about black hole entropy in string theory,” e-print hep-th/9309145.
  167. Susskind, L., 1993b, “String theory and the principles of black hole complementarity,” Phys. Rev. Lett. 71, 2367.
  168. Susskind, L., 1994, “Strings, black holes and Lorentz contraction,” Phys. Rev. D 49, 6606.
  169. Susskind, L., 1995a, “Particle growth and BPS saturated states,” e-print hep-th/9511116.
  170. Susskind, L., 1995b, “The world as a hologram,” J. Math. Phys. 36, 6377.
  171. Susskind, L., and L. Thorlacius, 1994, “Gedanken experiments involving black holes,” Phys. Rev. D 49, 966.
  172. Susskind, L., L. Thorlacius, and J. Uglum, 1993, “The stretched horizon and black hole complementarity,” Phys. Rev. D 48, 3743.
  173. Susskind, L., and J. Uglum, 1994, “Black hole entropy in canonical quantum gravity and superstring theory,” Phys. Rev. D 50, 2700.
  174. Susskind, L., and J. Uglum, 1996, “String physics and black holes,” Nucl. Phys. B, Proc. Suppl. 45BC, 115.
  175. Susskind, L., and E. Witten, 1998, “The holographic bound in Anti-de Sitter space,” e-print hep-th/9805114.
  176. ’t Hooft, G., 1985, “On the quantum structure of a black hole,” Nucl. Phys. B 256, 727.
  177. ’t Hooft, G., 1988, in Quantum Gravity, edited by M. A. Markov, V. A. Berezin, and V. P. Frolov (World Scientific Press, Singapore), pp. 551–567.
  178. ’t Hooft, G., 1990, “The black hole interpretation of string theory,” Nucl. Phys. B 335, 138.
  179. ’t Hooft, G., 1991, “The black hole horizon as a quantum surface,” Phys. Scr. T36, 247.
  180. ’t Hooft, G., 1993, “Dimensional reduction in quantum gravity,” e-print gr-qc/9310026.
  181. ’t Hooft, G., 1999, “Quantum gravity as a dissipative deterministic system,” e-print gr-qc/9903084.
  182. ’t Hooft, G., 2000a, “Determinism and dissipation in quantum gravity,” e-print hep-th/0003005.
  183. ’t Hooft, G., 2000b, “The holographic principle,” e-print hep-th/0003004.
  184. ’t Hooft, G., 2001a, “Determinism in free bosons,” e-print hep-th/0104080.
  185. ’t Hooft, G., 2001b, “How does god play dice? (Pre-) determinism at the Planck scale,” e-print hep-th/0104219.
  186. ’t Hooft, G., 2001c, “Quantum mechanics and determinism,” e-print hep-th/0105105.
  187. Tavakol, R. K., and G. Ellis, 1999, “On holography and cosmology,” Phys. Lett. B 469, 37.
  188. Thomas, S., 2000, “Holographic vacuum energy,” e-print hep-th/0010145.
  189. Thorlacius, L., 1995, “Black hole evolution,” Nucl. Phys. B, Proc. Suppl. 41, 245.
  190. Thorn, C. B., 1979, “Quark confinement in the infinite momentum frame,” Phys. Rev. D 19, 639.
  191. Thorn, C. B., 1991, “Reformulating string theory with the 1/N expansion,” e-print hep-th/9405069.
  192. Thorn, C. B., 1995, “Calculating the rest tension for a polymer of string bits,” Phys. Rev. D 51, 647.
  193. Thorn, C. B., 1996, “Substructure of string,” e-print hep-th/9607204.
  194. Unruh, W. G., 1976, “Notes on black hole evaporation,” Phys. Rev. D 14, 870.
  195. Unruh, W. G., and R. M. Wald, 1982, “Acceleration radiation and generalized second law of thermodynamics,” Phys. Rev. D 25, 942.
  196. Unruh, W. G., and R. M. Wald, 1983, “Entropy bounds, acceleration radiation, and the generalized second law,” Phys. Rev. D 27, 2271.
  197. van de Bruck, C., 2000, “On gravity, holography and the quantum,” e-print gr-qc/0001048.
  198. van Nieuwenhuizen, P., 1985, “d=3 conformal supergravity and Chern-Simons terms,” Phys. Rev. D 32, 872.
  199. Veneziano, G., 1999a, “Entropy bounds and string cosmology,” e-print hep-th/9907012.
  200. Veneziano, G., 1999b, “Pre-bangian origin of our entropy and time arrow,” e-print hep-th/9902126.
  201. Veneziano, G., 2000, “String cosmology: The pre-big bang scenario,” e-print hep-th/0002094.
  202. Veneziano, G., 2001, “Large-N bounds on, and compositeness limit of, gauge and gravitational interactions,” e-print hep-th/0110129.
  203. Verlinde, E., 1995, “Black hole evaporation and complementarity,” e-print hep-th/9503120.
  204. Verlinde, E., 2000, “On the holographic principle in a radiation dominated universe,” e-print hep-th/0008140.
  205. Wald, R. M., 1984, General Relativity (The University of Chicago Press, Chicago).
  206. Wald, R. M., 1993, “Black hole entropy in Noether charge,” Phys. Rev. D 48, 3427.
  207. Wald, R. M., 1994, Quantum Field Theory in Curved Space-time and Black Hole Thermodynamics (The University of Chicago Press, Chicago).
  208. Wald, R. M., 2001, “The thermodynamics of black holes,” Living Rev. Relativ. 4, 6.
  209. Wang, B., and E. Abdalla, 1999, “Holography in (2+1)-dimensional cosmological models,” Phys. Lett. B 466, 122.
  210. Wang, B., and E. Abdalla, 2000, “Holography and the generalized second law of thermodynamics in (2+1)-dimensional cosmology,” Phys. Lett. B 471, 346.
  211. Wang, B., E. Abdalla, and T. Osada, 2000, “Entropy and holography constraints for inhomogeneous universes,” Phys. Rev. Lett. 85, 5507.
  212. Witten, E., 1988, “(2+1)-dimensional gravity as an exactly soluble system,” Nucl. Phys. B 311, 46.
  213. Witten, E., 1998, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253.
  214. Witten, E., 2001, “Quantum gravity in de Sitter space,” e-print hep-th/0106109.

Outline

Information

Rev. Mod. Phys. 74, 825– Published 5 August, 2002

    © 2002 American Physical Society

    Sign In to Your Journals Account

    Filter

    Filter

    Article Lookup

    Enter a citation