Bousso's holographic bound

The Bousso bound captures a fundamental relation between quantum information and the geometry of space and time. It appears to be an imprint of a unified theory that combines quantum mechanics with Einstein's general relativity.[1] The study of black hole thermodynamics and the information paradox led to the idea of the holographic principle: the entropy of matter and radiation in a spatial region cannot exceed the Bekenstein-Hawking entropy of the boundary of the region, which is proportional to the boundary area. However, this "spacelike" entropy bound fails in cosmology; for example, it does not hold true in our universe.[2]

Raphael Bousso showed that the spacelike entropy bound is violated more broadly in many dynamical settings. For example, the entropy of a collapsing star, once inside a black hole, will eventually exceed its surface area.[3] Due to relativistic length contraction, even ordinary thermodynamic systems can be enclosed in an arbitrarily small area.[1]

To preserve the holographic principle, Bousso proposed a different law, which does not follow from black hole physics: the covariant entropy bound[3] or Bousso bound.[4][5] Its central geometric object is a lightsheet, defined as a region traced out by non-expanding light-rays emitted orthogonally from an arbitrary surface B. For example, if B is a sphere at a moment of time in Minkowski space, then there are two lightsheets, generated by the past or future directed light-rays emitted towards the interior of the sphere at that time. If B is a sphere surrounding a large region in an expanding universe (an anti-trapped sphere), then there are again two light-sheets that can be considered. Both are directed towards the past, to the interior or the exterior. If B is a trapped surface, such as the surface of a star in its final stages of gravitational collapse, then the lightsheets are directed to the future.

The Bousso bound evades all known counterexamples to the spacelike bound.[1][3] It was proven to hold when the entropy is approximately a local current, under weak assumptions.[4][5][6] In weakly gravitating settings, the Bousso bound implies the Bekenstein bound[7] and admits a formulation that can be proven to hold in any relativistic quantum field theory.[8] The lightsheet construction can be inverted to construct holographic screens for arbitrary spacetimes.[9]

A more recent proposal, the quantum focusing conjecture,[10] implies the original Bousso bound and so can be viewed as a stronger version of it. In the limit where gravity is negligible, the quantum focusing conjecture predicts the quantum null energy condition,[11] which relates the local energy density to a derivative of the entropy. This relation was later proven to hold in any relativistic quantum field theory, such as the Standard Model.[11][12][13][14]


  1. ^ a b c Bousso, Raphael (5 August 2002). "The holographic principle". Reviews of Modern Physics. 74 (3): 825–874. arXiv:hep-th/0203101. Bibcode:2002RvMP...74..825B. doi:10.1103/RevModPhys.74.825. S2CID 55096624.
  2. ^ Fischler, W.; Susskind, L. (1998-06-11). "Holography and Cosmology". arXiv:hep-th/9806039.
  3. ^ a b c Bousso, Raphael (13 August 1999). "A Covariant Entropy Conjecture". Journal of High Energy Physics. 1999 (7): 004. arXiv:hep-th/9905177. Bibcode:1999JHEP...07..004B. doi:10.1088/1126-6708/1999/07/004. S2CID 9545752.
  4. ^ a b Flanagan, Eanna E.; Marolf, Donald; Wald, Robert M. (2000-09-27). "Proof of Classical Versions of the Bousso Entropy Bound and of the Generalized Second Law". Physical Review D. 62 (8): 084035. arXiv:hep-th/9908070. Bibcode:2000PhRvD..62h4035F. doi:10.1103/PhysRevD.62.084035. ISSN 0556-2821. S2CID 7648994.
  5. ^ a b Strominger, Andrew; Thompson, David (2004-08-09). "A Quantum Bousso Bound". Physical Review D. 70 (4): 044007. arXiv:hep-th/0303067. Bibcode:2004PhRvD..70d4007S. doi:10.1103/PhysRevD.70.044007. ISSN 1550-7998. S2CID 18666260.
  6. ^ Bousso, Raphael; Flanagan, Eanna E.; Marolf, Donald (2003-09-03). "Simple sufficient conditions for the generalized covariant entropy bound". Physical Review D. 68 (6): 064001. arXiv:hep-th/0305149. Bibcode:2003PhRvD..68f4001B. doi:10.1103/PhysRevD.68.064001. ISSN 0556-2821. S2CID 119049155.
  7. ^ Bousso, Raphael (2003-03-27). "Light-sheets and Bekenstein's bound". Physical Review Letters. 90 (12): 121302. arXiv:hep-th/0210295. doi:10.1103/PhysRevLett.90.121302. ISSN 0031-9007. PMID 12688865. S2CID 41570896.
  8. ^ Bousso, Raphael; Casini, Horacio; Fisher, Zachary; Maldacena, Juan (2014-08-01). "Proof of a Quantum Bousso Bound". Physical Review D. 90 (4): 044002. arXiv:1404.5635. Bibcode:2014PhRvD..90d4002B. doi:10.1103/PhysRevD.90.044002. ISSN 1550-7998. S2CID 119218211.
  9. ^ Bousso, Raphael (1999-06-28). "Holography in General Space-times". Journal of High Energy Physics. 1999 (6): 028. arXiv:hep-th/9906022. Bibcode:1999JHEP...06..028B. doi:10.1088/1126-6708/1999/06/028. ISSN 1029-8479. S2CID 119518763.
  10. ^ Bousso, Raphael; Fisher, Zachary; Leichenauer, Stefan; Wall, and Aron C. (2016-03-16). "A Quantum Focussing Conjecture". Physical Review D. 93 (6): 064044. arXiv:1506.02669. Bibcode:2016PhRvD..93f4044B. doi:10.1103/PhysRevD.93.064044. ISSN 2470-0010. S2CID 116979904.
  11. ^ a b Bousso, Raphael; Fisher, Zachary; Koeller, Jason; Leichenauer, Stefan; Wall, Aron C. (2016-01-12). "Proof of the Quantum Null Energy Condition". Physical Review D. 93 (2): 024017. arXiv:1509.02542. Bibcode:2016PhRvD..93b4017B. doi:10.1103/PhysRevD.93.024017. ISSN 2470-0010. S2CID 59469752.
  12. ^ Balakrishnan, Srivatsan; Faulkner, Thomas; Khandker, Zuhair U.; Wang, Huajia (September 2019). "A General Proof of the Quantum Null Energy Condition". Journal of High Energy Physics. 2019 (9): 20. arXiv:1706.09432. Bibcode:2019JHEP...09..020B. doi:10.1007/JHEP09(2019)020. ISSN 1029-8479. S2CID 85530291.
  13. ^ Wall, Aron C. (2017-04-10). "A Lower Bound on the Energy Density in Classical and Quantum Field Theories". Physical Review Letters. 118 (15): 151601. arXiv:1701.03196. Bibcode:2017PhRvL.118o1601W. doi:10.1103/PhysRevLett.118.151601. ISSN 0031-9007. PMID 28452547. S2CID 28785629.
  14. ^ Ceyhan, Fikret; Faulkner, Thomas (2019-03-20). "Recovering the QNEC from the ANEC". arXiv:1812.04683 [hep-th].