The holographic principle Raphael Bousso Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, U.S.A.∗ 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. We review 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. We explain how the bound is tested and demonstrate its validity in a wide range of examples. A universal relation between geometry and information is thus uncovered. It has yet to be ex- plained. 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 man- ifest in an underlying quantum theory of gravity. We survey some successes and challenges in implementing the holographic principle. Contents A. Motivation and background 19 B. Light-sheet kinematics 20 I. Introduction 2 1. Orthogonal null hypersurfaces 20 A. A principle for quantum gravity 2 2. Light-sheet selection 20 B. Notation and conventions 3 3. Light-sheet termination 21 C. Outline 3 C. Defining entropy 22 D. Related subjects and further reading 4 1. Entropy on a fixed light-sheet 22 2. Entropy on an arbitrary light-sheet 23 II. Entropy bounds from black holes 6 D. Limitations 23 A. Black hole thermodynamics 6 1. Energy conditions 23 1. Area theorem 6 2. Quantum fluctuations 24 2. No-hair theorem 6 E. Summary 25 3. Bekenstein entropy and the generalized second law 6 VI. The dynamics of light-sheets 25 4. Hawking radiation 7 A. Raychaudhuri’s equation and the focussing theorem 26 B. Bekenstein bound 8 B. Sufficient conditions for the covariant entropy bound 27 1. Geroch process 8 1. The first FMW theorem 27 2. Unruh radiation 9 2. The second FMW theorem 28 3. Empirical status 9 C. Relation to other bounds and to the GSL 28 C. Spherical entropy bound 10 1. Spacelike projection theorem 28 1. Susskind process 10 2. Generalized second law and Bekenstein bound 28 2. Relation to the Bekenstein bound 10 3. Examples 10 VII. Applications and examples 29 4. The species problem 11 A. Cosmology 29 1. FRW metric and entropy density 29 III. Towards a holographic principle 12 2. Expansion and apparent horizons 29 A. Degrees of freedom 12 3. Light-sheets vs. spatial volumes 31 4. Solutions with fixed equation of state 31 arXiv:hep-th/0203101v2 29 Jun 2002 B. Fundamental system 12 C. Complexity according to local field theory 12 5. Flat universe 31 D. Complexity according to the spherical entropy bound12 6. Non-adiabatic evolution and mixed equations of E. Why local field theory gives the wrong answer 13 state 33 F. Unitarity and a holographic interpretation 13 7. A cosmological corollary 33 G. Unitarity and black hole complementarity 14 B. Gravitational collapse 33 H. Discussion 15 1. Collapsing universe 34 2. Collapsing star 34 IV. A spacelike entropy bound? 15 3. Collapsing shell 34 A. Formulation 16 C. Nearly null boundaries 35 B. Inadequacies 16 1. Closed spaces 16 VIII. The holographic principle 36 2. The Universe 16 A. Assessment 36 3. Collapsing star 16 B. Formulation 37 4. Weakly gravitating system 17 C. Implications 37 C. Range of validity 18 IX. Holographic screens and holographic theories 38 V. The covariant entropy bound 19 A. String theory and the holographic principle 38 1. A work in progress 38 2. Is string theory holographic? 38 B. AdS/CFT correspondence 39 C. Holographic screens for general spacetimes 41 ∗Electronic address: [email protected] 1. Construction 41 2 2. Properties and implications 41 quantum mechanics. It will be a quantum theory of grav- D. Towards a holographic theory 42 ity, a framework that transcends general relativity and E. Holography in de Sitter space 43 quantum field theory. 1. de Sitter space 43 ± This expectation is supported by the close ties between 2. dS spacetimes 44 3. dS+ spacetimes 45 the covariant entropy bound and the semi-classical prop- 4. Other universes with positive Λ 45 erties of black holes. It has been confirmed—albeit in a limited context—by recent results in string theory. Acknowledgments 45 The holographic principle conflicts with received wis- dom; in this sense, it also belongs in the former class. A. General relativity 45 Conventional theories are local; quantum field theory, References 48 for example, contains degrees of freedom at every point in space. Even with a short distance cutoff, the infor- mation content of a spatial region would appear to grow I. INTRODUCTION with the volume. The holographic principle, on the other hand, implies that the number of fundamental degrees of A. A principle for quantum gravity freedom is related to the area of surfaces in spacetime. Typically, this number is drastically smaller than the field Progress in fundamental physics has often been driven theory estimate. by the recognition of a new principle, a key insight to Thus, the holographic principle calls into question not guide the search for a successful theory. Examples in- only the fundamental status of field theory but the very clude the principles of relativity, the equivalence princi- notion of locality. It gives preference, as we shall see, to ple, and the gauge principle. Such principles lay down the preservation of quantum-mechanical unitarity. general properties that must be incorporated into the In physics, information can be encoded in a variety of laws of physics. ways: by the quantum states, say, of a conformal field A principle can be sparked by contradictions between theory, or by a lattice of spins. Unfortunately, for all its existing theories. By judiciously declaring which theory precise predictions about the number of fundamental de- contains the elements of a unified framework, a principle grees of freedom in spacetime, the holographic principle may force other theories to be adapted or superceded. betrays little about their character. The amount of in- The special theory of relativity, for example, reconciles formation is strictly determined, but not its form. It is electrodynamics with Galilean kinematics at the expense interesting to contemplate the notion that pure, abstract of the latter. information may underlie all of physics. But for now, this A principle can also arise from some newly recognized austerity frustrates the design of concrete models incor- pattern, an apparent law of physics that stands by itself, porating the holographic principle. both uncontradicted and unexplained by existing theo- Indeed, a broader caveat is called for. The covariant ries. A principle may declare this pattern to be at the entropy bound is a compelling pattern, but it may still core of a new theory altogether. prove incorrect or merely accidental, signifying no deeper In Newtonian gravity, for example, the proportional- origin. If the bound does stem from a fundamental the- ity of gravitational and inertial mass in all bodies seems ory, that relation could be indirect or peripheral, in which a curious coincidence that is far from inevitable. The case the holographic principle would be unlikely to guide equivalence principle demands that this pattern must be us to the core ideas of the theory. All that aside, the made manifest in a new theory. This led Einstein to holographic principle is likely only one of several indepen- the general theory of relativity, in which the equality of dent conceptual advances needed for progress in quantum gravitational and inertial mass is built in from the start. gravity. Because all bodies follow geodesics in a curved spacetime, At present, however, quantum gravity poses an im- things simply couldn’t be otherwise. mense problem tackled with little guidance. Quantum The holographic principle belongs in the latter class. gravity has imprinted few traces on physics below the The unexplained “pattern”, in this case, is the existence Planck energy. Among them, the information content of of a precise, general, and surprisingly strong limit on the spacetime may well be the most profound. information content of spacetime regions. This pattern The direction offered by the holographic principle is has come to be recognized in stages; its present, most impacting existing frameworks and provoking new ap- general form is called the covariant entropy bound. The proaches. In particular, it may prove beneficial to the holographic principle asserts that this bound is not a co- further development of string theory, widely (and, in our incidence, but that its origin must be found in a new view, justly) considered the most compelling of present theory. approaches. The covariant entropy bound relates aspects of space- This article will outline the case for the holographic time geometry to the number of quantum states of mat- principle whilst providing a starting point for further ter. This suggests that any theory that incorporates study of the literature. The material is not, for the most the holographic principle must unify matter, gravity, and part, technical. The main mathematical aspect, the con- 3 struction of light-sheets, is rather straightforward. In formulated a holographic principle. We discuss motiva- order to achieve a self-contained presentation, some ba- tions for this radical step. sic material on general relativity has been included in an The spherical entropy bound depends on assumptions appendix. that are clearly violated by realistic physical systems. In demonstrating the scope and power of the holo- A priori there is no reason to expect that the bound graphic correspondence between areas and information, has universal validity, nor that it admits a reformulation our ultimate task is to convey its character as a law of that does.