The holographic principle
Abstract
There is strong evidence that the area of any surface limits the information content of adjacent spacetime regions, at 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 lightsheets, 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 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. We survey some successes and challenges in implementing the holographic principle.
Contents
 I Introduction
 II Entropy bounds from black holes

III Towards a holographic principle
 III.1 Degrees of freedom
 III.2 Fundamental system
 III.3 Complexity according to local field theory
 III.4 Complexity according to the spherical entropy bound
 III.5 Why local field theory gives the wrong answer
 III.6 Unitarity and a holographic interpretation
 III.7 Unitarity and black hole complementarity
 III.8 Discussion
 IV A spacelike entropy bound?
 V The covariant entropy bound
 VI The dynamics of lightsheets
 VII Applications and examples
 VIII The holographic principle
 IX Holographic screens and holographic theories
 A General relativity
I Introduction
i.1 A principle for quantum gravity
Progress in fundamental physics has often been driven by the recognition of a new principle, a key insight to guide the search for a successful theory. Examples include the principles of relativity, the equivalence principle, and the gauge principle. Such principles lay down general properties that must be incorporated into the laws of physics.
A principle can be sparked by contradictions between existing theories. By judiciously declaring which theory contains the elements of a unified framework, a principle may force other theories to be adapted or superceded. The special theory of relativity, for example, reconciles electrodynamics with Galilean kinematics at the expense of the latter.
A principle can also arise from some newly recognized pattern, an apparent law of physics that stands by itself, both uncontradicted and unexplained by existing theories. A principle may declare this pattern to be at the core of a new theory altogether.
In Newtonian gravity, for example, the proportionality of gravitational and inertial mass in all bodies seems a curious coincidence that is far from inevitable. The equivalence principle demands that this pattern must be made manifest in a new theory. This led Einstein to the general theory of relativity, in which the equality of gravitational and inertial mass is built in from the start. Because all bodies follow geodesics in a curved spacetime, things simply couldn’t be otherwise.
The holographic principle belongs in the latter class. The unexplained “pattern”, in this case, is the existence of a precise, general, and surprisingly strong limit on the information content of spacetime regions. This pattern has come to be recognized in stages; its present, most general form is called the covariant entropy bound. The holographic principle asserts that this bound is not a coincidence, but that its origin must be found in a new theory.
The covariant entropy bound relates aspects of spacetime geometry to the number of quantum states of matter. This suggests that any theory that incorporates the holographic principle must unify matter, gravity, and quantum mechanics. It will be a quantum theory of gravity, a framework that transcends general relativity and quantum field theory.
This expectation is supported by the close ties between the covariant entropy bound and the semiclassical properties of black holes. It has been confirmed—albeit in a limited context—by recent results in string theory.
The holographic principle conflicts with received wisdom; in this sense, it also belongs in the former class. Conventional theories are local; quantum field theory, for example, contains degrees of freedom at every point in space. Even with a short distance cutoff, the information content of a spatial region would appear to grow with the volume. The holographic principle, on the other hand, implies that the number of fundamental degrees of freedom is related to the area of surfaces in spacetime. Typically, this number is drastically smaller than the field theory estimate.
Thus, the holographic principle calls into question not only the fundamental status of field theory but the very notion of locality. It gives preference, as we shall see, to the preservation of quantummechanical unitarity.
In physics, information can be encoded in a variety of ways: by the quantum states, say, of a conformal field theory, or by a lattice of spins. Unfortunately, for all its precise predictions about the number of fundamental degrees of freedom in spacetime, the holographic principle betrays little about their character. The amount of information is strictly determined, but not its form. It is interesting to contemplate the notion that pure, abstract information may underlie all of physics. But for now, this austerity frustrates the design of concrete models incorporating the holographic principle.
Indeed, a broader caveat is called for. The covariant entropy bound is a compelling pattern, but it may still prove incorrect or merely accidental, signifying no deeper origin. If the bound does stem from a fundamental theory, that relation could be indirect or peripheral, in which case the holographic principle would be unlikely to guide us to the core ideas of the theory. All that aside, the holographic principle is likely only one of several independent conceptual advances needed for progress in quantum gravity.
At present, however, quantum gravity poses an immense problem tackled with little guidance. Quantum gravity has imprinted few traces on physics below the Planck energy. Among them, the information content of spacetime may well be the most profound.
The direction offered by the holographic principle is impacting existing frameworks and provoking new approaches. In particular, it may prove beneficial to the further development of string theory, widely (and, in our view, justly) considered the most compelling of present approaches.
This article will outline the case for the holographic principle whilst providing a starting point for further study of the literature. The material is not, for the most part, technical. The main mathematical aspect, the construction of lightsheets, is rather straightforward. In order to achieve a selfcontained presentation, some basic material on general relativity has been included in an appendix.
In demonstrating the scope and power of the holographic correspondence between areas and information, our ultimate task is to convey its character as a law of physics that captures one of the most intriguing aspects of quantum gravity. If the reader is led to contemplate the origin of this particular pattern nature has laid out, our review will have succeeded.
i.2 Notation and conventions
Throughout this paper, Planck units will be used:
(1) 
where is Newton’s constant, is Planck’s constant, is the speed of light, and is Boltzmann’s constant. In particular, all areas are measured in multiples of the square of the Planck length,
(2) 
The Planck units of energy density, mass, temperature, and other quantities are converted to cgs units, e.g., in Wald (1984), whose conventions we follow in general. For a small number of key formulas, we will provide an alternate expression in which all constants are given explicitly.
We consider spacetimes of arbitrary dimension , unless noted otherwise. In explicit examples we often take for definiteness. The appendix fixes the metric signature and defines “surface”, “hypersurface”, “null”, and many other terms from general relativity. The term “lightsheet” is defined in Sec. V.
i.3 Outline
In Sec. II, we review Bekenstein’s (1972) notion of black hole entropy and the related discovery of upper bounds on the entropy of matter systems. Assuming weak gravity, spherical symmetry, and other conditions, one finds that the entropy in a region of space is limited by the area of its boundary.^{1}^{1}1The metaphorical name of the principle (’t Hooft, 1993) originates here. In many situations, the covariant entropy bound dictates that all physics in a region of space is described by data that fit on its boundary surface, at one bit per Planck area (Sec. VI.3.1). This is reminiscent of a hologram. Holography is an optical technology by which a threedimensional image is stored on a twodimensional surface via a diffraction pattern. (To avoid any confusion: this linguistic remark will remain our only usage of the term in its original sense.) From the present point of view, the analogy has proven particularly apt. In both kinds of “holography”, light rays play a key role for the imaging (Sec. V). Moreover, the holographic code is not a straightforward projection, as in ordinary photography; its relation to the threedimensional image is rather complicated. (Most of our intuition in this regard has come from the AdS/CFT correspondence, Sec. IX.2.) Susskind’s (1995) quip that the world is a “hologram” is justified by the existence of preferred surfaces in spacetime, on which all of the information in the universe can be stored at no more than one bit per Planck area (Sec. IX.3). Based on this “spherical entropy bound”, ’t Hooft (1993) and Susskind (1995b) formulated a holographic principle. We discuss motivations for this radical step.
The spherical entropy bound depends on assumptions that are clearly violated by realistic physical systems. A priori there is no reason to expect that the bound has universal validity, nor that it admits a reformulation that does. Yet, if the number of degrees of freedom in nature is as small as ’t Hooft and Susskind asserted, one would expect wider implications for the maximal entropy of matter.
In Sec. IV, however, we demonstrate that a naive generalization of the spherical entropy bound is unsuccessful. The “spacelike entropy bound” states that the entropy in a given spatial volume, irrespective of shape and location, is always less than the surface area of its boundary. We consider four examples of realistic, commonplace physical systems, and find that the spacelike entropy bound is violated in each one of them.
In light of these difficulties, some authors, forgoing complete generality, searched instead for reliable conditions under which the spacelike entropy bound holds. We review the difficulties faced in making such conditions precise even in simple cosmological models.
Thus, the idea that the area of surfaces generally bounds the entropy in enclosed spatial volumes has proven wrong; it can be neither the basis nor the consequence of a fundamental principle. This review would be incomplete if it failed to stress this point. Moreover, the ease with which the spacelike entropy bound (and several of its modifications) can be excluded underscores that a general entropy bound, if found, is no triviality. The counterexamples to the spacelike bound later provide a useful testing ground for the covariant bound.
Inadequacies of the spacelike entropy bound led Fischler and Susskind (1998) to a bound involving light cones. The covariant entropy bound (Bousso, 1999a), presented in Sec. V, refines and generalizes this approach. Given any surface , the bound states that the entropy on any lightsheet of will not exceed the area of . Lightsheets are particular hypersurfaces generated by light rays orthogonal to . The light rays may only be followed as long as they are not expanding. We explain this construction in detail.
After discussing how to define the entropy on a lightsheet, we spell out known limitations of the covariant entropy bound. The bound is presently formulated only for approximately classical geometries, and one must exclude unphysical matter content, such as large negative energy. We conclude that the covariant entropy bound is welldefined and testable in a vast class of solutions. This includes all thermodynamic systems and cosmologies presently known or considered realistic.
In Sec. VI we review the geometric properties of lightsheets, which are central to the operation of the covariant entropy bound. Raychaudhuri’s equation is used to analyse the effects of entropy on lightsheet evolution. By construction, a lightsheet is generated by light rays that are initially either parallel or contracting. Entropic matter systems carry mass, which causes the bending of light.
This means that the light rays generating a lightsheet will be focussed towards each other when they encounter entropy. Eventually they selfintersect in a caustic, where they must be terminated because they would begin to expand. This mechanism would provide an “explanation” of the covariant entropy bound if one could show that the mass associated with entropy is necessarily so large that lightsheets focus and terminate before they encounter more entropy than their initial area.
Unfortunately, present theories do not impose an independent, fundamental lower bound on the energetic price of entropy. However, Flanagan, Marolf, and Wald (2000) were able to identify conditions on entropy density which are widely satisfied in nature and which are sufficient to guarantee the validity of the covariant entropy bound. We review these conditions.
The covariant bound can also be used to obtain sufficient criteria under which the spacelike entropy bound holds. Roughly, these criteria can be summarized by demanding that gravity be weak. However, the precise condition requires the construction of lightsheets; it cannot be formulated in terms of intrinsic properties of spatial volumes.
The event horizon of a black hole is a lightsheet of its final surface area. Thus, the covariant entropy bound includes to the generalized second law of thermodynamics in black hole formation as a special case. More broadly, the generalized second law, as well as the Bekenstein entropy bound, follow from a strengthened version of the covariant entropy bound.
In Sec. VII, the covariant entropy bound is applied to a variety of thermodynamic systems and cosmological spacetimes. This includes all of the examples in which the spacelike entropy bound is violated. We find that the covariant bound is satisfied in each case.
In particular, the bound is found to hold in strongly gravitating regions, such as cosmological spacetimes and collapsing objects. Aside from providing evidence for the general validity of the bound, this demonstrates that the bound (unlike the spherical entropy bound) holds in a regime where it cannot be derived from black hole thermodynamics.
In Sec. VIII, we arrive at the holographic principle. We note that the covariant entropy bound holds with remarkable generality but is not logically implied by known laws of physics. We conclude that the bound has a fundamental origin. As a universal limitation on the information content of Lorentzian geometry, the bound should be manifest in a quantum theory of gravity. We formulate the holographic principle and list some of its implications. The principle poses a challenge for local theories. It suggests a preferred role for null hypersurfaces in the classical limit of quantum gravity.
In Sec. IX we analyze an example of a holographic theory. Quantum gravity in certain asymptotically Antide Sitter spacetimes is fully defined by a conformal field theory. The latter theory contains the correct number of degrees of freedom demanded by the holographic principle. It can be thought of as living on a kind of holographic screen at the boundary of spacetime and containing one bit of information per Planck area.
Holographic screens with this information density can be constructed for arbitrary spacetimes—in this sense, the world is a hologram. In most other respects, however, global holographic screens do not generally support the notion that a holographic theory is a conventional field theory living at the boundary of a spacetime.
At present, there is much interest in finding more general holographic theories. We discuss the extent to which string theory, and a number of other approaches, have realized this goal. A particular area of focus is de Sitter space, which exhibits an absolute entropy bound. We review the implications of the holographic principle in such spacetimes.
i.4 Related subjects and further reading
The holographic principle has developed from a large set of ideas and results, not all of which seemed mutually related at first. This is not a historical review; we have aimed mainly at achieving a coherent, modern perspective on the holographic principle. We do not give equal emphasis to all developments, and we respect the historical order only where it serves the clarity of exposition. Along with length constraints, however, this approach has led to some omissions and shortcomings, for which we apologize.
We have chosen to focus on the covariant entropy bound because it can be tested using only quantum field theory and general relativity. Its universality motivates the holographic principle independently of any particular ansatz for quantum gravity (say, string theory) and without additional assumptions (such as unitarity). It yields a precise and general formulation.
Historically, the idea of the holographic principle was tied, in part, to the debate about information loss in black holes^{2}^{2}2See, for example, Hawking (1976b, 1982), Page (1980, 1993), Banks, Susskind, and Peskin (1984), ’t Hooft (1985, 1988, 1990), Polchinski and Strominger (1994), Strominger (1994). and to the notion of black hole complementarity.^{3}^{3}3See, e.g., ’t Hooft (1991), Susskind, Thorlacius, and Uglum (1993), Susskind (1993b), Stephens, ’t Hooft, and Whiting (1994), Susskind and Thorlacius (1994). For recent criticism, see Jacobson (1999). Although we identify some of the connections, our treatment of these issues is far from comprehensive. Reviews include Thorlacius (1995), Verlinde (1995), Susskind and Uglum (1996), Bigatti and Susskind (2000), and Wald (2001).
Some aspects of what we now recognize as the holographic principle were encountered, at an early stage, as features of string theory. (This is as it should be, since string theory is a quantum theory of gravity.) In the infinite momentum frame, the theory admits a lowerdimensional description from which the gravitational dynamics of the full spacetime arises nontrivially (Giles and Thorn, 1977; Giles, McLerran, and Thorn, 1978; Thorn, 1979, 1991, 1995, 1996; Klebanov and Susskind, 1988). Susskind (1995b) placed this property of string theory in the context of the holographic principle and related it to black hole thermodynamics and entropy limitations.
Some authors have traced the emergence of the holographic principle also to other approaches to quantum gravity; see Smolin (2001) for a discussion and further references.
By tracing over a region of space one obtains a density matrix. Bombelli et al. (1986) showed that the resulting entropy is proportional to the boundary area of the region. A more general argument was given by Srednicki (1993). Gravity does not enter in this consideration. Moreover, the entanglement entropy is generally unrelated to the size of the Hilbert space describing either side of the boundary. Thus, it is not clear to what extent this suggestive result is related to the holographic principle.
This is not a review of the AdS/CFT correspondence (Maldacena, 1998; see also Gubser, Klebanov, and Polyakov, 1998; Witten, 1998). This rich and beautiful duality can be regarded (among its many interesting aspects) as an implementation of the holographic principle in a concrete model. Unfortunately, it applies only to a narrow class of spacetimes of limited physical relevance. By contrast, the holographic principle claims a far greater level of generality—a level at which it continues to lack a concrete implementation.
We will broadly discuss the relation between the AdS/CFT correspondence and the holographic principle, but we will not dwell on aspects that seem particular to AdS/CFT. (In particular, this means that the reader should not expect a discussion of every paper containing the word “holographic” in the title!) A detailed treatment of AdS/CFT would go beyond the purpose of the present text. An extensive review has been given by Aharony et al. (2000).
The AdS/CFT correspondence is closely related to some recent models of our 3+1 dimensional world as a defect, or brane, in a 4+1 dimensional AdS space. In the models of Randall and Sundrum (1999a,b), the gravitational degrees of freedom of the extra dimension appear on the brane as a dual field theory under the AdS/CFT correspondence. While the holographic principle can be considered a prerequisite for the existence of such models, their detailed discussion would not significantly strengthen our discourse. Earlier seminal papers in this area include Hořava and Witten (1996a,b).
A number of authors (e.g., Brustein and Veneziano, 2000; Verlinde, 2000; Brustein, Foffa, and Veneziano, 2001; see Cai, Myung, and Ohta, 2001, for additional references) have discussed interesting bounds which are not directly based on the area of surfaces. Not all of these bounds appear to be universal. Because their relation to the holographic principle is not entirely clear, we will not attempt to discuss them here. Applications of entropy bounds to string cosmology (e.g., Veneziano, 1999a; Bak and Rey, 2000b; Brustein, Foffa, and Sturani, 2000) are reviewed by Veneziano (2000).
The holographic principle has sometimes been said to exclude certain physically acceptable solutions of Einstein’s equations because they appeared to conflict with an entropy bound. The covariant bound has exposed these tensions as artifacts of the limitations of earlier entropy bounds. Indeed, this review bases the case for a holographic principle to a large part on the very generality of the covariant bound. However, the holographic principle does limit the applicability of quantum field theory on cosmologically large scales. It calls into question the conventional analysis of the cosmological constant problem (Cohen, Kaplan, and Nelson, 1999; Hořava, 1999; Banks, 2000a; Hořava and Minic, 2000; Thomas, 2000). It has also been applied to the calculation of anisotropies in the cosmic microwave background (Hogan, 2002a,b). The study of cosmological signatures of the holographic principle may be of great value, since it is not clear whether more conventional imprints of shortdistance physics on the early universe are observable even in principle (see, e.g., Kaloper et al., 2002, and references therein).
Most attempts at implementing the holographic principle in a unified theory are still in their infancy. It would be premature to attempt a detailed review; some references are given in Sec. IX.4.
Other recent reviews overlapping with some or all of the topics covered here are Bigatti and Susskind (2000), Bousso (2000a), ’t Hooft (2000b), Bekenstein (2001) and Wald (2001). Relevant textbooks include Hawking and Ellis (1973); Misner, Thorne, and Wheeler (1973); Wald (1984, 1994); Green, Schwarz, and Witten (1987); and Polchinski (1998).
Ii Entropy bounds from black holes
This section reviews black hole entropy, some of the entropy bounds that have been inferred from it, and their relation to ’t Hooft’s (1993) and Susskind’s (1995b) proposal of a holographic principle.
The entropy bounds discussed in this section are “universal” (Bekenstein, 1981) in the sense that they are independent of the specific characteristics and composition of matter systems. Their validity is not truly universal, however, because they apply only when gravity is weak.
We consider only Einstein gravity. For black hole thermodynamics in higherderivative gravity, see, e.g., Myers and Simon (1988), Jacobson and Myers (1993), Wald (1993), Iyer and Wald (1994, 1995), Jacobson, Kang, and Myers (1994), and the review by Myers (1998).^{4}^{4}4Abdalla and CorreaBorbonet (2001) have commented on entropy bounds in this context.
ii.1 Black hole thermodynamics
The notion of black hole entropy is motivated by two results in general relativity.
ii.1.1 Area theorem
The area theorem (Hawking, 1971) states that the area of a black hole event horizon never decreases with time:
(3) 
Moreover, if two black holes merge, the area of the new black hole will exceed the total area of the original black holes.
For example, an object falling into a Schwarzschild black hole will increase the mass of the black hole, .^{5}^{5}5This assumes that the object has positive mass. Indeed, the assumptions in the proof of the theorem include the null energy condition. This condition is given in the Appendix, where the Schwarzschild metric is also found. Hence the horizon area, in , increases. On the other hand, one would not expect the area to decrease in any classical process, because the black hole cannot emit particles.
The theorem suggests an analogy between black hole area and thermodynamic entropy.
ii.1.2 Nohair theorem
Work of Israel (1967, 1968), Carter (1970), Hawking (1971, 1972), and others, implies the curiously named nohair theorem: A stationary black hole is characterized by only three quantities: mass, angular momentum, and charge.^{6}^{6}6Proofs and further details can be found, e.g., in Hawking and Ellis (1973), or Wald (1984). This form of the theorem holds only in . Gibbons, Ida, and Shiromizu (2002) have recently given a uniqueness proof for static black holes in . Remarkably, Emparan and Reall (2001) have found a counterexample to the stationary case in . This does not affect the present argument, in which the no hair theorem plays a heuristic role.
Consider a complex matter system, such as a star, that collapses to form a black hole. The black hole will eventually settle down into a final, stationary state. The nohair theorem implies that this state is unique.
From an outside observer’s point of view, the formation of a black hole appears to violate the second law of thermodynamics. The phase space appears to be drastically reduced. The collapsing system may have arbitrarily large entropy, but the final state has none at all. Different initial conditions will lead to indistinguishable results.
A similar problem arises when a matter system is dropped into an existing black hole. Geroch has proposed a further method for violating the second law, which exploits a classical black hole to transform heat into work; see Bekenstein (1972) for details.
ii.1.3 Bekenstein entropy and the generalized second law
Thus, the nohair theorem poses a paradox, to which the area theorem suggests a resolution. When a thermodynamic system disappears behind a black hole’s event horizon, its entropy is lost to an outside observer. The area of the event horizon will typically grow when the black hole swallows the system. Perhaps one could regard this area increase as a kind of compensation for the loss of matter entropy?
Based on this reasoning, Bekenstein (1972, 1973, 1974) suggested that a black hole actually carries an entropy equal to its horizon area, , where is a number of order unity. In Sec. II.1.4 it will be seen that (Hawking, 1974):
(4) 
[In full, .] The entropy of a black hole is given by a quarter of the area of its horizon in Planck units. In ordinary units, it is the horizon area divided by about m.
Moreover, Bekenstein (1972, 1973, 1974) proposed that the second law of thermodynamics holds only for the sum of black hole entropy and matter entropy:
(5) 
For ordinary matter systems alone, the second law need not hold. But if the entropy of black holes, Eq. (4), is included in the balance, the total entropy will never decrease. This is referred to as the generalized second law or GSL.
The content of this statement may be illustrated as follows. Consider a thermodynamic system , consisting of wellseparated, noninteracting components. Some components, labeled , may be thermodynamic systems made from ordinary matter, with entropy . The other components, , are black holes, with horizon areas . The total entropy of is given by
(6) 
Here, is the total entropy of all ordinary matter. is the total entropy of all black holes present in .
Now suppose the components of are allowed to interact until a new equilibrium is established. For example, some of the matter components may fall into some of the black holes. Other matter components might collapse to form new black holes. Two or more black holes may merge. In the end, the system will consist of a new set of components and , for which one can again compute a total entropy, . The GSL states that
(7) 
What is the microscopic, statistical origin of black hole entropy? We have learned that a black hole, viewed from the outside, is unique classically. The BekensteinHawking formula, however, suggests that it is compatible with independent quantum states. The nature of these quantum states remains largely mysterious. This problem has sparked sustained activity through various different approaches, too vast in scope to sketch in this review.
However, one result stands out because of its quantitative accuracy. Recent developments in string theory have led to models of limited classes of black holes in which the microstates can be identified and counted (Strominger and Vafa, 1996; for a review, see, e.g., Peet, 2000). The formula was precisely confirmed by this calculation.
ii.1.4 Hawking radiation
Black holes clearly have a mass, . If Bekenstein entropy, , is to be taken seriously, then the first law of thermodynamics dictates that black holes must have a temperature, :
(8) 
Indeed, Einstein’s equations imply an analogous “first law of black hole mechanics” (Bardeen, Carter, and Hawking, 1973). The entropy is the horizon area, and the surface gravity of the black hole, , plays the role of the temperature:
(9) 
For a definition of , see Wald (1984); e.g., a Schwarzschild black hole in has .
It may seem that this has taken the thermodynamic analogy a step too far. After all, a blackbody with nonzero temperature must radiate. But for a black hole this would seem impossible. Classically, no matter can escape from it, so its temperature must be exactly zero.
This paradox was resolved by the discovery that black holes do in fact radiate via a quantum process. Hawking (1974, 1975) showed by a semiclassical calculation that a distant observer will detect a thermal spectrum of particles coming from the black hole, at a temperature
(10) 
For a Schwarzschild black hole in , this temperature is , or about Kelvin divided by the mass of the black hole in grams. Note that such black holes have negative specific heat.
The discovery of Hawking radiation clarified the interpretation of the thermodynamic description of black holes. What might otherwise have been viewed as a mere analogy (Bardeen, Carter, and Hawking, 1973) was understood to be a true physical property. The entropy and temperature of a black hole are no less real than its mass.
In particular, Hawking’s result affirmed that the entropy of black holes should be considered a genuine contribution to the total entropy content of the universe, as Bekenstein (1972, 1973, 1974) had anticipated. Via the first law of thermodynamics, Eq. (8), Hawking’s calculation fixes the coefficient in the Bekenstein entropy formula, Eq. (4), to be .
A radiating black hole loses mass, shrinks, and eventually disappears unless it is stabilized by charge or a steady influx of energy. Over a long time of order , this process converts the black hole into a cloud of radiation. (See Sec. III.7 for the question of unitarity in this process.)
It is natural to study the operation of the GSL in the two types of processes discussed in Sec. II.1.2. We will first discuss the case in which a matter system is dropped into an existing black hole. Then we will turn to the process in which a black hole is formed by the collapse of ordinary matter. In both cases, ordinary entropy is converted into horizon entropy.
A third process, which we will not discuss in detail, is the Hawking evaporation of a black hole. In this case, the horizon entropy is converted back into radiation entropy. This type of process was not anticipated when Bekenstein (1972) proposed black hole entropy and the GSL. It is all the more impressive that the GSL holds also in this case (Bekenstein, 1975; Hawking, 1976a). Page (1976) has estimated that the entropy of Hawking radiation exceeds that of the evaporated black hole by 62%.
ii.2 Bekenstein bound
When a matter system is dropped into a black hole, its entropy is lost to an outside observer. That is, the entropy starts at some finite value and ends up at zero. But the entropy of the black hole increases, because the black hole gains mass, and so its area will grow. Thus it is at least conceivable that the total entropy, , does not decrease in the process, and that therefore the generalized second law of thermodynamics, Eq. (5), is obeyed.
Yet it is by no means obvious that the generalized second law will hold. The growth of the horizon area depends essentially on the mass that is added to the black hole; it does not seem to care about the entropy of the matter system. If it were possible to have matter systems with arbitrarily large entropy at a given mass and size, the generalized second law could still be violated.
The thermodynamic properties of black holes developed in the previous subsection, including the assignment of entropy to the horizon, are sufficiently compelling to be considered laws of nature. Then one may turn the above considerations around and demand that the generalized second law hold in all processes. One would expect that this would lead to a universal bound on the entropy of matter systems in terms of their extensive parameters.
For any weakly gravitating matter system in asymptotically flat space, Bekenstein (1981) has argued that the GSL implies the following bound:
(11) 
[In full, ; note that Newton’s constant does not enter.] Here, is the total massenergy of the matter system. The circumferential radius is the radius of the smallest sphere that fits around the matter system (assuming that gravity is sufficiently weak to allow for a choice of time slicing such that the matter system is at rest and space is almost Euclidean).
We will begin with an argument for this bound in arbitrary spacetime dimension that involves a strictly classical analysis of the Geroch process, by which a system is dropped into a black hole from the vicinity of the horizon. We will then show, however, that a purely classical treatment is not tenable. The extent to which quantum effects modify, or perhaps invalidate, the derivation of the Bekenstein bound from the GSL is controversial. The gist of some of the pertinent arguments will be given here, but the reader is referred to the literature for the subtleties.
ii.2.1 Geroch process
Consider a weakly gravitating stable thermodynamic system of total energy . Let be the radius of the smallest sphere circumscribing the system. To obtain an entropy bound, one may move the system from infinity into a Schwarzschild black hole whose radius, , is much larger than but otherwise arbitrary. One would like to add as little energy as possible to the black hole, so as to minimize the increase of the black hole’s horizon area and thus to optimize the tightness of the entropy bound. Therefore, the strategy is to extract work from the system by lowering it slowly until it is just outside the black hole horizon, before one finally drops it in.
The mass added to the black hole is given by the energy of the system, redshifted according to the position of the center of mass at the dropoff point, at which the circumscribing sphere almost touches the horizon. Within its circumscribing sphere, one may orient the system so that its center of mass is “down”, i.e., on the side of the black hole. Thus the center of mass can be brought to within a proper distance from the horizon, while all parts of the system remain outside the horizon. Hence, one must calculate the redshift factor at radial proper distance from the horizon.
The Schwarzschild metric is given by
(12) 
where
(13) 
defines the redshift factor, (Myers and Perry, 1986). The black hole radius is related to the mass at infinity, , by
(14) 
where sphere. The black hole has horizon area is the area of a unit
(15) 
Let be the radial coordinate distance from the horizon:
(16) 
Near the horizon, the redshift factor is given by
(17) 
to leading order in . The proper distance is related to the coordinate distance as follows:
(18) 
Hence,
(19) 
ii.2.2 Unruh radiation
The above derivation of the Bekenstein bound, by a purely classical treatment of the Geroch process, suffers from the problem that it can be strengthened to a point where it yields an obviously false conclusion. Consider a system in a rectangular box whose height, , is much smaller than its other dimensions. Orient the system so that the small dimension is aligned with the radial direction, and the long dimensions are parallel to the horizon. The minimal distance between the center of mass and the black hole horizon is then set by the height of the box, and will be much smaller than the circumferential radius. In this way, one can “derive” a bound of the form
(23) 
The right hand side goes to zero in the limit of vanishing height, at fixed energy of the box. But the entropy of the box does not go to zero unless all of its dimensions vanish. If only the height goes to zero, the vertical modes become heavy and have to be excluded. But entropy will still be carried by light modes living in the other spatial directions.
Unruh and Wald (1982, 1983) have pointed out that a system held at fixed radius just outside a black hole horizon undergoes acceleration, and hence experiences Unruh radiation (Unruh, 1976). They argued that this quantum effect will change both the energetics (because the system will be buoyed by the radiation) and the entropy balance in the Geroch process (because the volume occupied by the system will be replaced by entropic quantum radiation after the system is dropped into the black hole). Unruh and Wald concluded that the Bekenstein bound is neither necessary nor sufficient for the operation of the GSL. Instead, they suggested that the GSL is automatically protected by Unruh radiation as long as the entropy of the matter system does not exceed the entropy of unconstrained thermal radiation of the same energy and volume. This is plausible if the system is indeed weakly gravitating and if its dimensions are not extremely unequal.
Bekenstein (1983, 1994a), on the other hand, has argued that Unruh radiation merely affects the lowest layer of the system and is typically negligible. Only for very flat systems, Bekenstein (1994a) claims that the UnruhWald effect may be important. This would invalidate the derivation of Eq. (23) in the limit where this bound is clearly incorrect. At the same time, it would leave the classical argument for the Bekenstein bound, Eq. (22), essentially intact. As there would be an intermediate regime where Eq. (23) applies, however, one would not expect the Bekenstein bound to be optimally tight for nonspherical systems.
The question of whether the GSL implies the Bekenstein bound remains controversial (see, e.g., Bekenstein, 1999, 2001; Pelath and Wald, 1999; Wald, 2001; Marolf and Sorkin, 2002).
The arguments described here can also be applied to other kinds of horizons. Davies (1984) and Schiffer (1992) considered a Geroch process in de Sitter space, respectively extending the UnruhWald and the Bekenstein analysis to the cosmological horizon. Bousso (2001) has shown that the GSL implies a Bekensteintype bound for dilute systems in asymptotically de Sitter space, with the assumption of spherical symmetry but not necessarily of weak gravity. In this case one would not expect quantum buoyancy to play a crucial role.
ii.2.3 Empirical status
Independently of its logical relation to the GSL, one can ask whether the Bekenstein bound actually holds in nature. Bekenstein (1981, 1984) and Schiffer and Bekenstein (1989) have made a strong case that all physically reasonable, weakly gravitating matter systems satisfy Eq. (11); some come within an order of magnitude of saturation. This empirical argument has been called into question by claims that certain systems violate the Bekenstein bound; see, e.g., Page (2000) and references therein. Many of these counterexamples, however, fail to include the whole gravitating mass of the system in . Others involve questionable matter content, such as a very large number of species (Sec. II.3.4). Bekenstein (2000c) gives a summary of alleged counterexamples and their refutations, along with a list of references to more detailed discussions. If the Bekenstein bound is taken to apply only to complete, weakly gravitating systems that can actually be constructed in nature, it has not been ruled out (Flanagan, Marolf, and Wald, 2000; Wald, 2001).
The application of the bound to strongly gravitating systems is complicated by the difficulty of defining the radius of the system in a highly curved geometry. At least for spherically symmetric systems, however, this is not a problem, as one may define in terms of the surface area. A Schwarzschild black hole in four dimensions has . Hence, its Bekenstein entropy, , exactly saturates the Bekenstein bound (Bekenstein, 1981). In , black holes come to within a factor of saturating the bound (Bousso, 2001).
ii.3 Spherical entropy bound
Instead of dropping a thermodynamic system into an existing black hole via the Geroch process, one may also consider the Susskind process, in which the system is converted to a black hole. Susskind (1995b) has argued that the GSL, applied to this transformation, yields the spherical entropy bound
(24) 
where is a suitably defined area enclosing the matter system.
The description of the Susskind process below is influenced by the analysis of Wald (2001).
ii.3.1 Susskind process
Let us consider an isolated matter system of mass and entropy residing in a spacetime . We require that the asymptotic structure of permits the formation of black holes. For definiteness, let us assume that is asymptotically flat. We define to be the area of the circumscribing sphere, i.e., the smallest sphere that fits around the system. Note that is welldefined only if the metric near the system is at least approximately spherically symmetric. This will be the case for all spherically symmetric systems, and for all weakly gravitating systems, but not for strongly gravitating systems lacking spherical symmetry. Let us further assume that the matter system is stable on a timescale much greater than . That is, it persists and does not expand or collapse rapidly, so that the timedependence of will be negligible.
The system’s mass must be less than the mass of a black hole of the same surface area. Otherwise, the system could not be gravitationally stable, and from the outside point of view it would already be a black hole. One would expect that the system can be converted into a black hole of area by collapsing a shell of mass onto the system.^{7}^{7}7This assumes that the shell can actually be brought to within without radiating or ejecting shell mass or system mass. For two large classes of systems, Bekenstein (2000a,b) obtains Eq. (24) under weaker assumptions.
Let the shell be wellseparated from the system initially. Its entropy, , is nonnegative. The total initial entropy in this thermodynamic process is given by
(25) 
The final state is a black hole, with entropy
(26) 
By the generalized second law of thermodynamics, Eq. (5), the initial entropy must not exceed the final entropy. Since is obviously nonnegative, Eq. (24) follows.
ii.3.2 Relation to the Bekenstein bound
Thus, the spherical entropy bound is obtained directly from the GSL via the Susskind process. Alternatively, and with similar limitations, one can obtain the same result from the Bekenstein bound, if the latter is assumed to hold for strongly gravitating systems. The requirement that the system be gravitationally stable implies in four dimensions. From Eq. (11), one thus obtains:
(27) 
This shows that the spherical entropy bound is weaker than the Bekenstein bound, in situations where both can be applied.
The spherical entropy bound, however, is more closely related to the holographic principle. It can be cast in a covariant and general form (Sec. V). An interesting open question is whether one can reverse the logical direction and derive the Bekenstein bound from the covariant entropy bound under suitable assumptions (Sec. VI.3.2).
In , gravitational stability and the Bekenstein bound imply only (Bousso, 2001). The discrepancy may stem from the extrapolation to strong gravity and/or the lack of a reliable calibration of the prefactor in the Bekenstein bound.
ii.3.3 Examples
The spherical entropy bound is best understood by studying a number of examples in four spacetime dimensions. We follow ’t Hooft (1993) and Wald (2001).
It is easy to see that the bound holds for black holes. By definition, the entropy of a single Schwarzschild black hole, , precisely saturates the bound. In this sense, a black hole is the most entropic object one can put inside a given spherical surface (’t Hooft, 1993).
Consider a system of several black holes of masses , in . Their total entropy will be given by
(28) 
From the point of view of a distant observer, the system must not already be a larger black hole of mass . Hence, it must be circumscribed by a spherical area
(29) 
Hence, the spherical entropy bound is satisfied with room to spare.
Using ordinary matter instead of black holes, it turns out to be difficult even to approach saturation of the bound. In order to obtain a stable, highly entropic system, a good strategy is to make it from massless particles. Rest mass only enhances gravitational instability without contributing to the entropy. Consider, therefore, a gas of radiation at temperature , with energy , confined in a spherical box of radius . We must demand that the system is not a black hole: . For an orderofmagnitude estimate of the entropy, we may neglect the effects of selfgravity and treat the system as if it lived on a flat background.
The energy of the ball is related to its temperature as
(30) 
where is the number of different species of particles in the gas. The entropy of the system is given by
(31) 
Hence, the entropy is related to the size and energy as
(32) 
Gravitational stability then implies that
(33) 
In order to compare this result to the spherical entropy bound, , recall that we are using Planck units. For any geometric description to be valid, the system must be much larger than the Planck scale:
(34) 
A generous estimate for the number of species in nature is . Hence, is much smaller than for all but the smallest, nearly Planck size systems, in which the present approximations cannot be trusted in any case. For a gas ball of size , the spherical entropy bound will be satisfied with a large factor, , to spare.
ii.3.4 The species problem
An interesting objection to entropy bounds is that one can write down perfectly welldefined field theory Lagrangians with an arbitrarily large number of particle species (Sorkin, Wald, and Zhang, 1981; Unruh and Wald, 1982). In the example of Eq. (33), a violation of the spherical entropy bound for systems up to size would require
(35) 
For example, to construct a counterexample of the size of a proton, one would require . It is trivial to write down a Lagrangian with this number of fields. But this does not mean that the entropy bound is wrong.
In nature, the effective number of matter fields is whatever it is; it cannot be tailored to the specifications of one’s favorite counterexample. The spherical bound is a statement about nature. If it requires that the number of species is not exponentially large, then this implication is certainly in good agreement with observation. At any rate it is more plausible than the assumption of an exponentially large number of light fields.
Indeed, an important lesson learned from black holes and the holographic principle is that nature, at a fundamental level, will not be described by a local field theory living on some background geometry (Susskind, Thorlacius, and Uglum, 1993).
The spherical entropy bound was derived from the generalized second law of thermodynamics (under a set of assumptions). Could one not, therefore, use the GSL to rule out large ? Consider a radiation ball with massless species, so that . The system is transformed to a black hole of area by a Susskind process. However, Wald (2001) has shown that the apparent entropy decrease is irrelevant, because the black hole is catastrophically unstable. In Sec. II.1.4, the time for the Hawking evaporation of a black hole was estimated to be in . This implicitly assumed a small number of radiated species. But for large , one must take into account that the radiation rate is actually proportional to . Hence, the evaporation time is given by
(36) 
With , one has . The time needed to form a black hole of area is at least of order , so the black hole in question evaporates faster than it forms.
Wald’s analysis eliminates the possibility of using the GSL to exclude large for the process at hand. But it produces a different, additional argument against proliferating the number of species. Exponentially large would render black holes much bigger than the Planck scale completely unstable. Let us demand, therefore, that superPlanckian black holes be at least metastable. Then cannot be made large enough to construct a counterexample from Eq. (33). From a physical point of view, the metastability of large black holes seems a far more natural assumption than the existence of an extremely large number of particle species.
Further arguments on the species problem (of which the possible renormalization of Newton’s constant with has received particular attention) are found in Bombelli et al. (1986), Bekenstein (1994b, 1999, 2000c), Jacobson (1994), Susskind and Uglum (1994, 1996), Frolov (1995), Brustein, Eichler, and Foffa (2000), Veneziano (2001), Wald (2001), and Marolf and Sorkin (2002).
Iii Towards a holographic principle
iii.1 Degrees of freedom
How many degrees of freedom are there in nature, at the most fundamental level? The holographic principle answers this question in terms of the area of surfaces in spacetime. Before reaching this rather surprising answer, we will discuss a more traditional way one might have approached the question. Parts of this analysis follow ’t Hooft (1993) and Susskind (1995b).
For the question to have meaning, let us restrict to a finite region of volume and boundary area . Assume, for now, that gravity is not strong enough to blur the definition of these quantities, and that spacetime is asymptotically flat. Application of the spherical entropy bound, Eq. (24), will force us to consider the circumscribing sphere of the region. This surface will coincide with the boundary of the region only if the boundary is a sphere, which we shall assume.
In order to satisfy the assumptions of the spherical entropy bound we also demand that the metric of the enclosed region is not strongly timedependent, in the sense described at the beginning of Sec. II.3.1. In particular, this means that will not be a trapped surface in the interior of a black hole.
Let us define the number of degrees of freedom of a quantummechanical system, , to be the logarithm of the dimension of its Hilbert space :
(37) 
Note that a harmonic oscillator has with this definition. The number of degrees of freedom is equal (up to a factor of ) to the number of bits of information needed to characterize a state. For example, a system with 100 spins has states, degrees of freedom, and can store 100 bits of information.
iii.2 Fundamental system
Consider a spherical region of space with no particular restrictions on matter content. One can regard this region as a quantummechanical system and ask how many different states it can be in. In other words, what is the dimension of the quantum Hilbert space describing all possible physics confined to the specified region, down to the deepest level?
Thus, our question is not about the Hilbert space of a specific system, such as a hydrogen atom or an elephant. Ultimately, all these systems should reduce to the constituents of a fundamental theory. The question refers directly to these constituents, given only the size^{8}^{8}8The precise nature of the geometric boundary conditions is discussed further in Sec. V.3. of a region. Let us call this system the fundamental system.
How much complexity, in other words, lies at the deepest level of nature? How much information is required to specify any physical configuration completely, as long as it is contained in a prescribed region?
iii.3 Complexity according to local field theory
In the absence of a unified theory of gravity and quantum fields, it is natural to seek an answer from an approximate framework. Suppose that the “fundamental system” is local quantum field theory on a classical background spacetime satisfying Einstein’s equations (Birrell and Davies, 1982; Wald, 1994). A quantum field theory consists of one or more oscillators at every point in space. Even a single harmonic oscillator has an infinitedimensional Hilbert space. Moreover, there are infinitely many points in any volume of space, no matter how small. Thus, the answer to our question appears to be . However, so far we have disregarded the effects of gravity altogether.
A finite estimate is obtained by including gravity at least in a crude, minimal way. One might expect that distances smaller than the Planck length, cm, cannot be resolved in quantum gravity. So let us discretize space into a Planck grid and assume that there is one oscillator per Planck volume. Moreover, the oscillator spectrum is discrete and bounded from below by finite volume effects. It is bounded from above because it must be cut off at the Planck energy, GeV. This is the largest amount of energy that can be localized to a Planck cube without producing a black hole. Thus, the total number of oscillators is (in Planck units), and each has a finite number of states, . (A minimal model one might think of is a Planckian lattice of spins, with .) Hence, the total number of independent quantum states in the specified region is
(38) 
The number of degrees of freedom is given by
(39) 
This result successfully captures our prejudice that the degrees of freedom in the world are local in space, and that, therefore, complexity grows with volume. It turns out, however, that this view conflicts with the laws of gravity.
iii.4 Complexity according to the spherical entropy bound
Thermodynamic entropy has a statistical interpretation. Let be the thermodynamic entropy of an isolated system at some specified value of macroscopic parameters such as energy and volume. Then is the number of independent quantum states compatible with these macroscopic parameters. Thus, entropy is a measure of our ignorance about the detailed microscopic state of a system. One could relax the macroscopic parameters, for example by requiring only that the energy lie in some finite interval. Then more states will be allowed, and the entropy will be larger.
The question at the beginning of this section was “How many independent states are required to describe all the physics in a region bounded by an area ?” Recall that all thermodynamic systems should ultimately be described by the same underlying theory, and that we are interested in the properties of this “fundamental system”. We are now able to rephrase the question as follows: “What is the entropy, , of the ‘fundamental system’, given that only the boundary area is specified?” Once this question is answered, the number of states will simply be , by the argument given in the previous paragraph.
In Sec. II.3 we obtained the spherical entropy bound, Eq. (24), from which the entropy can be determined without any knowledge of the nature of the “fundamental system”. The bound,
(40) 
makes reference only to the boundary area; it does not care about the microscopic properties of the thermodynamic system. Hence, it applies to the “fundamental system” in particular. A black hole that just fits inside the area has entropy
(41) 
so the bound can clearly be saturated with the given boundary conditions. Therefore, the number of degrees of freedom in a region bounded by a sphere of area is given by
(42) 
the number of states is
(43) 
We assume that all physical systems are larger than the Planck scale. Hence, their volume will exceed their surface area, in Planck units. (For a proton, the volume is larger than the area by a factor of ; for the earth, by .) The result obtained from the spherical entropy bound is thus at odds with the much larger number of degrees of freedom estimated from local field theory. Which of the two conclusions should we believe?
iii.5 Why local field theory gives the wrong answer
We will now argue that the field theory analysis overcounted available degrees of freedom, because it failed to include properly the effects of gravitation. We assume and neglect factors of order unity. (In the gist of the discussion is unchanged though some of the powers are modified.)
The restriction to a finite spatial region provides an infrared cutoff, precluding the generation of entropy by long wavelength modes. Hence, most of the entropy in the field theory estimate comes from states of very high energy. But a spherical surface cannot contain more mass than a black hole of the same area. According to the Schwarzschild solution, Eq. (12), the mass of a black hole is given by its radius. Hence, the mass contained within a sphere of radius obeys
(44) 
The ultraviolet cutoff imposed in Sec. III.3 reflected this, but only on the smallest scale (). It demanded only that each Planck volume must not contain more than one Planck mass. For larger regions this cutoff would permit , in violation of Eq. (44). Hence our cutoff was too lenient to prevent black hole formation on larger scales.
For example, consider a sphere of radius cm, or in Planck units. Suppose that the field energy in the enclosed region saturated the naive cutoff in each of the Planck cells. Then the mass within the sphere would be . But the most massive object that can be localized to the sphere is a black hole, of radius and mass .
Thus, most of the states included by the field theory estimate are too massive to be gravitationally stable. Long before the quantum fields can be excited to such a level, a black hole would form.^{9}^{9}9Thus, black holes provide a natural covariant cutoff which becomes stronger at larger distances. It differs greatly from the fixed distance or fixed energy cutoffs usually considered in quantum field theory. If this black hole is still to be contained within a specified sphere of area , its entropy may saturate but not exceed the spherical entropy bound.
Because of gravity, not all degrees of freedom that field theory apparently supplies can be used for generating entropy, or storing information. This invalidates the field theory estimate, Eq. (39), and thus resolves the apparent contradiction with the holographic result, Eq. (42).
Note that the present argument does not provide independent quantitative confirmation that the maximal entropy is given by the area. This would require a detailed understanding of the relation between entropy, energy, and gravitational backreaction in a given system.
iii.6 Unitarity and a holographic interpretation
Using the spherical entropy bound, we have concluded that degrees of freedom are sufficient to fully describe any stable region in asymptotically flat space enclosed by a sphere of area . In a field theory description, there are far more degrees of freedom. However, we have argued that any attempt to excite more than of these degrees of freedom is thwarted by gravitational collapse. From the outside point of view, the most entropic object that fits in the specified region is a black hole of area , with degrees of freedom.
A conservative interpretation of this result is that the demand for gravitational stability merely imposes a practical limitation for the information content of a spatial region. If we are willing to pay the price of gravitational collapse, we can excite more than degrees of freedom—though we will have to jump into a black hole to verify that we have succeeded. With this interpretation, all the degrees of freedom of field theory should be retained. The region will be described by a quantum Hilbert space of dimension .
The following two considerations motivate a rejection of this interpretation. Both arise from the point of view that physics in asymptotically flat space can be consistently described by a scattering matrix. The Smatrix provides amplitudes between initial and final asymptotic states defined at infinity. Intermediate black holes may form and evaporate, but as long as one is not interested in the description of an observer falling into the black hole, an Smatrix description should be satisfactory from the point of view of an observer at infinity.
One consideration concerns economy. A fundamental theory should not contain more than the necessary ingredients. If is the amount of data needed to describe a region completely, that should be the amount of data used. This argument is suggestive; however, it could be rejected as merely aesthetical and gratuitously radical.
A more compelling consideration is based on unitarity. Quantummechanical evolution preserves information; it takes a pure state to a pure state. But suppose a region was described by a Hilbert space of dimension , and suppose that region was converted to a black hole. According to the Bekenstein entropy of a black hole, the region is now described by a Hilbert space of dimension . The number of states would have decreased, and it would be impossible to recover the initial state from the final state. Thus, unitarity would be violated. Hence, the Hilbert space must have had dimension to start with.
The insistence on unitarity in the presence of black holes led ’t Hooft (1993) and Susskind (1995b) to embrace a more radical, “holographic” interpretation of Eq. (42).
Holographic principle (preliminary formulation). A region with boundary of area is fully described by no more than degrees of freedom, or about 1 bit of information per Planck area. A fundamental theory, unlike local field theory, should incorporate this counterintuitive result.
iii.7 Unitarity and black hole complementarity
The unitarity argument would be invalidated if it turned out that unitarity is not preserved in the presence of black holes. Indeed, Hawking (1976b) has claimed that the evaporation of a black hole—its slow conversion into a cloud of radiation—is not a unitary process. In semiclassical calculations, Hawking radiation is found to be exactly thermal, and all information about the ingoing state appears lost. Others (see Secs. I.4, IX.1) argued, however, that unitarity must be restored in a complete quantum gravity theory.
The question of unitarity of the Smatrix arises not only when a black hole forms, but again, and essentially independently, when the black hole evaporates. The holographic principle is necessary for unitarity at the first stage. But if unitarity were later violated during evaporation, it would have to be abandoned, and the holographic principle would lose its basis.
It is not understood in detail how Hawking radiation carries away information. Indeed, the assumption that it does seems to lead to a paradox, which was pointed out and resolved by Susskind, Thorlacius, and Uglum (1993). When a black hole evaporates unitarily, the same quantum information would seem to be present both inside the black hole (as the original matter system that collapsed) and outside, in the form of Hawking radiation. The simultaneous presence of two copies appears to violate the linearity of quantum mechanics, which forbids the “xeroxing” of information.
One can demonstrate, however, that no single observer can see both copies of the information. Obviously an infalling observer cannot escape the black hole to record the outgoing radiation. But what prevents an outside observer from first obtaining, say, one bit of information from the Hawking radiation, only to jump into the black hole to collect a second copy?
Page (1993) has shown that more than half of a system has to be observed to extract one bit of information. This means that an outside observer has to linger for a time compared to the evaporation time scale of the black hole ( in ) in order to gather a piece of the “outside data”, before jumping into the black hole to verify the presence of the same data inside.
However, the second copy can only be observed if it has not already hit the singularity inside the black hole by the time the observer crosses the horizon. One can show that the energy required for a single photon to evade the singularity for so long is exponential in the square of the black hole mass. In other words, there is far too little energy in the black hole to communicate even one bit of information to an infalling observer in possession of outside data.
The apparent paradox is thus exposed as the artifact of an operationally meaningless, global point of view. There are two complementary descriptions of black hole formation, corresponding to an infalling and and an outside observer. Each point of view is selfconsistent, but a simultaneous description of both is neither logically consistent nor practically testable. Black hole complementarity thus assigns a new role to the observer in quantum gravity, abandoning a global description of spacetimes with horizons.
Further work on black hole complementarity includes ’t Hooft (1991), Susskind (1993a,b, 1994), Stephens, ’t Hooft, and Whiting (1994), Susskind and Thorlacius (1994), Susskind and Uglum, (1994). Aspects realized in string theory are also discussed by Lowe, Susskind, and Uglum (1994), Lowe et al. (1995); see Sec. IX.1. For a review, see, e.g., Thorlacius (1995), Verlinde (1995), Susskind and Uglum (1996), and Bigatti and Susskind (2000).
Together, the holographic principle and black hole complementarity form the conceptual core of a new framework for black hole formation and evaporation, in which the unitarity of the Smatrix is retained at the expense of locality.^{10}^{10}10In this sense, the holographic principle, as it was originally proposed, belongs in the first class discussed in Sec. I.1. However, one cannot obtain its modern form (Sec. VIII) from unitarity. Hence we resort to the covariant entropy bound in this review. Because the bound can be tested using conventional theories, this also obviates the need to assume particular properties of quantum gravity in order to induce the holographic principle.
In the intervening years, much positive evidence for unitarity has accumulated. String theory has provided a microscopic, unitary quantum description of some black holes (Strominger and Vafa, 1996; see also Callan and Maldacena, 1996; Sec. IX.1). Moreover, there is overwhelming evidence that certain asymptotically Antide Sitter spacetimes, in which black holes can form and evaporate, are fully described by a unitary conformal field theory (Sec. IX.2).
Thus, a strong case has been made that the formation and evaporation of a black hole is a unitary process, at least in asymptotically flat or AdS spacetimes.
iii.8 Discussion
In the absence of a generally valid entropy bound, the arguments for a holographic principle were incomplete, and its meaning remained somewhat unclear. Neither the spherical entropy bound, nor the unitarity argument which motivates its elevation to a holographic principle, are applicable in general spacetimes.
An Smatrix description is justified in a particle accelerator, but not in gravitational physics. In particular, realistic universes do not permit an Smatrix description. (For recent discussions see, e.g., Banks, 2000a; Fischler, 2000a,b; Bousso, 2001a; Fischler et al., 2001; Hellerman, Kaloper, and Susskind, 2001.) Even in spacetimes that do, observers don’t all live at infinity. Then the question is not so much whether unitarity holds, but how it can be defined.
As black hole complementarity itself insists, the laws of physics must also describe the experience of an observer who falls into a black hole. The spherical entropy bound, however, need not apply inside black holes. Moreover, it need not hold in many other important cases, in view of the assumptions involved in its derivation. For example, it does not apply in cosmology, and it cannot be used when spherical symmetry is lacking. In fact, it will be seen in Sec. IV that the entropy in spatial volumes can exceed the boundary area in all of these cases.
Thus, the holographic principle could not, at first, establish a general correspondence between areas and the number of fundamental degrees of freedom. But how can it point the way to quantum gravity, if it apparently does not apply to many important solutions of the classical theory?
The AdS/CFT correspondence (Sec. IX.2), holography’s most explicit manifestation to date, was a thing of the future when the holographic principle was first proposed. So was the covariant entropy bound (Secs. V–VII), which exposes the apparent limitations noted above as artifacts of the original, geometrically crude formulation. The surprising universality of the covariant bound significantly strengthens the case for a holographic principle (Sec. VIII).
As ’t Hooft and Susskind anticipated, the conceptual revisions required by the unitarity of the Smatrix have proven too profound to be confined to the narrow context in which they were first recognized. We now understand that areas should generally be associated with degrees of freedom in adjacent spacetime regions. Geometric constructs that precisely define this relation—lightsheets—have been identified (Fischler and Susskind, 1998; Bousso, 1999a). The holographic principle may have been an audacious concept to propose. In light of the intervening developments, it has become a difficult one to reject.
Iv A spacelike entropy bound?
The heuristic derivation of the spherical entropy bound rests on a large number of fairly strong assumptions. Aside from suitable asymptotic conditions, the surface has to be spherical, and the enclosed region must be gravitationally stable so that it can be converted to a black hole.
Let us explore whether the spherical entropy bound, despite these apparent limitations, is a special case of a more general entropy bound. We will present two conjectures for such a bound. In this section, we will discuss the spacelike entropy bound, perhaps the most straightforward and intuitive generalization of Eq. (24). We will present several counterexamples to this bound and conclude that it does not have general validity. Turning to a case of special interest, we will find that it is difficult to precisely define the range of validity of the spacelike entropy bound even in simple cosmological spacetimes.
iv.1 Formulation
One may attempt to extend the scope of Eq. (24) simply by dropping the assumptions under which it was derived (asymptotic structure, gravitational stability, and spherical symmetry). Let us call the resulting conjecture the spacelike entropy bound: The entropy contained in any spatial region will not exceed the area of the region’s boundary. More precisely, the spacelike entropy bound is the following statement (Fig. 1):
Let be a compact portion of a hypersurface of equal time in the spacetime .^{11}^{11}11Here is used both to denote a spatial region, and its volume. Note that we use more careful notation to distinguish a surface () from its area (). Let be the entropy of all matter systems in . Let be the boundary of and let be the area of the boundary of . Then
(45) 
iv.2 Inadequacies
The spacelike entropy bound is not a successful conjecture. Eq. (45) is contradicted by a large variety of counterexamples. We will begin by discussing two examples from cosmology. Then we will turn to the case of a collapsing star. Finally, we will expose violations of Eq. (45) even for all isolated, spherical, weakly gravitating matter systems.
iv.2.1 Closed spaces
It is hardly necessary to describe a closed universe in detail to see that it will lead to a violation of the spacelike holographic principle. It suffices to assume that the spacetime contains a closed spacelike hypersurface, . (For example, there are realistic cosmological solutions in which has the topology of a threesphere.) We further assume that contains a matter system that does not occupy all of , and that this system has nonzero entropy .
Let us define the volume to be the whole hypersurface, except for a small compact region outside the matter system. Thus, . The boundary of coincides with the boundary of . Its area can be made arbitrarily small by contracting to a point. Thus one obtains , and the spacelike entropy bound, Eq. (45), is violated.
iv.2.2 The Universe
On large scales, the universe we inhabit is well approximated as a threedimensional, flat, homogeneous and isotropic space, expanding in time. Let us pick one homogeneous hypersurface of equal time, . Its entropy content can be characterized by an average “entropy density”, , which is a positive constant on . Flatness implies that the geometry of is Euclidean . Hence, the volume and area of a twosphere grow in the usual way with the radius:
(46) 
The entropy in the volume is given by
(47) 
Recall that we are working in Planck units. By taking the radius of the sphere to be large enough,
(48) 
one finds a volume for which the spacelike entropy bound, Eq. (45), is violated (Fischler and Susskind, 1998).
iv.2.3 Collapsing star
Next, consider a spherical star with nonzero entropy . Suppose the star burns out and undergoes catastrophic gravitational collapse. From an outside observer’s point of view, the star will form a black hole whose surface area will be at least , in accordance with the generalized second law of thermodynamics.
However, we can follow the star as it falls through its own horizon. From collapse solutions (see, e.g., Misner, Thorne, and Wheeler, 1973), it is known that the star will shrink to zero radius and end in a singularity. In particular, its surface area becomes arbitrarily small: . By the second law of thermodynamics, the entropy in the enclosed volume, i.e., the entropy of the star, must still be at least . Once more, the spacelike entropy bound fails (Easther and Lowe, 1999).
As in the previous two examples, this failure does not concern the spherical entropy bound, even though spherical symmetry may hold. We are considering a regime of dominant gravity, in violation of the assumptions of the spherical bound. In the interior of a black hole, both the curvature and the timedependence of the metric are large.
iv.2.4 Weakly gravitating system
The final example is the most subtle. It shows that the spacelike entropy bound can be violated by the very systems for which the spherical entropy bound is believed to hold: spherical, weakly gravitating systems. This is achieved merely by a nonstandard coordinate choice that breaks spherical symmetry and measures a smaller surface area.
Consider a weakly gravitating spherical thermodynamic system in asymptotically flat space. Note that this class includes most thermodynamic systems studied experimentally; if they are not spherical, one redefines their boundary to be the circumscribing sphere.
A coordinateindependent property of the system is its world volume, . For a stable system with the spatial topology of a threedimensional ball (), the topology of is given by (Fig. 2).
The volume of the ball of gas, at an instant of time, is geometrically the intersection of the world volume with an equal time hypersurface :
(49) 
The boundary of the volume is a surface given by
(50) 
The time coordinate , however, is not uniquely defined. One possible choice for is the proper time in the rest frame of the weakly gravitating system (Fig. 2a). With this choice, and are metrically a ball and a sphere, respectively. The area and the entropy were calculated in Sec. II.3.3 for the example of a ball of gas. They were found to satisfy the spacelike entropy bound, Eq. (45).
From the point of view of general relativity, there is nothing special about this choice of time coordinate. The laws of physics must be covariant, i.e., invariant under general coordinate transformations. Thus Eq. (45) must hold also for a volume associated with a different choice of time coordinate, . In particular, one may choose the const hypersurface to be rippled like a fan. Then its intersection with , , will be almost null almost everywhere, like the zigzag line circling the worldvolume in Fig. 2b. The boundary area so defined can be made arbitrarily small (Jacobson, 1999; Flanagan, Marolf, and Wald, 2000; Smolin, 2001).^{12}^{12}12The following construction exemplifies this for a spherical system. Consider the spatial sphere defined by and parametrized by standard spherical coordinates . Divide into segments of longitude defined by
How is this possible? After all, the spherical entropy bound should hold for this system, because it can be converted into a spherical black hole of the same area. However, this argument implicitly assumed that the boundary of a spherically symmetric system is a sphere (and therefore agrees with the horizon area of the black hole after the conversion). With the nonstandard time coordinate , however, the boundary is not spherically symmetric, and its area is much smaller than the final black hole area. (The latter is unaffected by slicing ambiguities because a black hole horizon is a null hypersurface.)
iv.3 Range of validity
In view of these problems, it is clear that the spacelike entropy bound cannot be maintained as a fully general conjecture holding for all volumes and areas in all spacetimes. Still, the spherical entropy bound, Eq. (24), clearly holds for many systems that do not satisfy its assumptions, suggesting that those assumptions may be unnecessarily restrictive.
For example, the earth is part of a cosmological spacetime that is not, as far as we know, asymptotically flat. However, the earth does not curve space significantly. It is well separated from other matter systems. On time and distance scales comparable to the earth’s diameter, the universe is effectively static and flat. In short, it is clear that the earth will obey the spacelike entropy bound.^{13}^{13}13Pathological slicings such as the one in Sec. IV.2.4 must still be avoided. Here we define the earth’s surface area by the natural slicing in its approximate Lorentz frame.
The same argument can be made for the solar system, and even for the milky way. As we consider larger regions, however, the effects of cosmological expansion become more noticable, and the flat space approximation is less adequate. An important question is whether a definite line can be drawn. In cosmology, is there a largest region to which the spacelike entropy bound can be reliably applied? If so, how is this region defined? Or does the bound gradually become less accurate at larger and larger scales?^{14}^{14}14The same questions can be asked of the Bekenstein bound, Eq. (11). Indeed, Bekenstein (1989), who proposed its application to the past light cone of an observer, was the first to raise the issue of the validity of entropy bounds in cosmology.
Let us consider homogeneous, isotropic universes, known as FriedmannRobertsonWalker (FRW) universes (Sec. VII.1). Fischler and Susskind (1998) abandoned the spacelike formulation altogether (Sec. V.1). For adiabatic FRW universes, however, their proposal implied that the spacelike entropy bound should hold for spherical regions smaller than the particle horizon (the future light cone of a point at the big bang).
Restriction to the particle horizon turns out to be sufficient for the validity of the spacelike entropy bound in simple flat and open models; thus, the problem in Sec. IV.2.2 is resolved. However, it does not prevent violations in closed or collapsing universes. The particle horizon area vanishes when the light cone reaches the far end of a closed universe—this is a special case of the problem discussed in Sec. IV.2.1. An analogue of the problem of Sec. IV.2.3 can arise also. Generally, closed universes and collapsing regions exhibit the greatest difficulties for the formulation of entropy bounds, and many authors have given them special attention.
Davies (1988) and Brustein (2000) proposed a generalized second law for cosmological spacetimes. They suggested that contradictions in collapsing universes may be resolved by augmenting the area law with additional terms. Easther and Lowe (1999) argued that the second law of thermodynamics implies a holographic entropy bound, at least for flat and open universes, in regions not exceeding the Hubble horizon.^{15}^{15}15The Hubble radius is defined to be , where is the scale factor of the universe; see Eq. (71) below. Similar conclusions were reached by Veneziano (1999b), Kaloper and Linde (1999), and Brustein (2000).
Bak and Rey (2000a) argued that the relevant surface is the apparent horizon, defined in Sec. VII.1.2. This is a minor distinction for typical flat and open universes, but it avoids some of the difficulties with closed universes.^{16}^{16}16Related discussions also appear in Dawid (1999) and Kalyana Rama (1999). The continued debate of the difficulties of the FischlerSusskind proposal in closed universes (Wang and Abdalla, 1999, 2000; Cruz and Lepe, 2001) is, in our view, rendered nugatory by the covariant entropy bound.
The arguments for bounds of this type return to the Susskind process, the gedankenexperiment by which the spherical entropy bound was derived (Sec. II.3.1). A portion of the universe is converted to a black hole; the second law of thermodynamics is applied. One then tries to understand what might prevent this gedankenexperiment from being carried out.
For example, regions larger than the horizon are expanding too rapidly to be converted to a black hole—they cannot be “held together” (Veneziano, 1999b). Also, if a system is already inside a black hole, it can no longer be converted to one. Hence, one would not expect the bound to hold in collapsing regions, such as the interior of black holes or a collapsing universe (Easther and Lowe, 1999; Kaloper and Linde, 1999).
This reasoning does expose some of the limitations of the spacelike entropy bound (namely those that are illustrated by the explicit counterexamples given in Sec. IV.2.1 and IV.2.3). However, it fails to identify sufficient conditions under which the bound is actually reliable. Kaloper and Linde (1999) give counterexamples to any statement of the type “The area of the particle (apparent, Hubble) horizon always exceeds the entropy enclosed in it” (Sec. VII.1.6).
In the following section we will introduce the covariant entropy bound, which is formulated in terms of lightsheets. In Sec. VII we will present evidence that this bound has universal validity. Starting from this general bound, one can find sufficient conditions under which a spacelike formulation is valid (Sec. VI.3.1, VII.1.7). However, the conditions themselves will involve the lightsheet concept in an essential way. Not only is the spacelike formulation less general than the lightsheet formulation; the range of validity of the former cannot be reliably identified without the latter.
We conclude that the spacelike entropy bound is violated by realistic matter systems. In cosmology, its range of validity cannot be intrinsically defined.
V The covariant entropy bound
In this section we present a more successful generalization of Eq. (24): the covariant entropy bound.
There are two significant formal differences between the covariant bound and the spacelike bound, Eq. (45). The spacelike formulation starts with a choice of spatial volume . The volume, in turn, defines a boundary , whose area is then claimed to be an upper bound on , the entropy in . The covariant bound proceeds in the opposite direction. A codimension 2 surface serves as the starting point for the construction of a codimension 1 region . This is the first formal difference. The second is that is a null hypersurface, unlike which is spacelike.
More precisely, is a lightsheet. It is constructed by following light rays that emanate from the surface , as long as they are not expanding. There are always at least two suitable directions away from (Fig. 3). When light rays selfintersect, they start to expand. Hence, lightsheets terminate at focal points.
The covariant entropy bound states that the entropy on any lightsheet of a surface will not exceed the area of :
(51) 
We will give a more formal definition at the end of this section.
We begin with some remarks on the conjectural nature of the bound, and we mention related earlier proposals. We will explain the geometric construction of lightsheets in detail, giving special attention to the considerations that motivate the condition of nonexpansion (). We give a definition of entropy on lightsheets, and we discuss the extent to which the limitations of classical general relativity are inherited by the covariant entropy bound. We then summarize how the bound is formulated, applied, and tested. Parts of this section follow Bousso (1999a).
v.1 Motivation and background
There is no fundamental derivation of the covariant entropy bound. We present the bound because there is strong evidence that it holds universally in nature. The geometric construction is welldefined and covariant. The resulting entropy bound can be saturated, but no example is known where it is exceeded.
In Sec. VI.2 plausible relations between entropy and energy are shown to be sufficient for the validity of the bound. But these relations do not at present appear to be universal or fundamental. In special situations, the covariant entropy bound reduces to the spherical entropy bound, which is arguably a consequence of black hole thermodynamics. But in general, the covariant entropy bound cannot be inferred from black hole physics; quite conversely, the generalized second law of thermodynamics may be more appropriately regarded as a consequence of the covariant bound (Sec. VI.3.2).
The origin of the bound remains mysterious. As discussed in the introduction, this puzzle forms the basis of the holographic principle, which asserts that the covariant entropy bound betrays the number of degrees of freedom of quantum gravity (Sec. VIII).
Aside from its success, little motivation for a lightlike formulation can be offered. Under the presupposition that some general entropy bound waits to be discovered, one is guided to light rays by circumstantial evidence. This includes the failure of the spacelike entropy bound (Sec. IV), the properties of the Raychaudhuri equation (Sec. VI.1), and the loss of a dynamical dimension in the light cone formulation of string theory (Sec. I.4).
Whatever the reasons, the idea that light rays might be involved in relating a region to its surface area—or, rather, relating a surface area to a lightlike “region”—arose in discussions of the holographic principle from the beginning.
Susskind (1995b) suggested that the horizon of a black hole can be mapped, via light rays, to a distant, flat holographic screen, citing the focussing theorem (Sec. VI.1) to argue that the information thus projected would satisfy the holographic bound. Corley and Jacobson (1996) pointed out that the occurrence of focal points, or caustics, could invalidate this argument, but showed that one causticfree family of light rays existed in Susskind’s example. They further noted that both past and future directed families of light rays can be considered.
Fischler and Susskind (1998) recognized that a lightlike formulation is crucial in cosmological spacetimes, because the spacelike entropy bound fails. They proposed that any spherical surface in FRW cosmologies (see Sec. VII.1) be related to (a portion of) a light cone that comes from the past and ends on . This solved the problem discussed in Sec. IV.2.2 for flat and open universes but not the problem of small areas in closed or recollapsing universes (see Secs. IV.2.1, IV.2.3).
The covariant entropy bound (Bousso, 1999a) can be regarded as a refinement and generalization of the FischlerSusskind proposal. It can be applied in arbitrary spacetimes, to any surface regardless of shape, topology, and location. It considers all four null directions orthogonal to without prejudice. It introduces a new criterion, the contraction of light rays, both to select among the possible lightlike directions and to determine how far the light rays may be followed. For any , there will be at least two “allowed” directions and hence two lightsheets, to each of which the bound applies individually.
v.2 Lightsheet kinematics
Compared to the previously discussed bounds, Eqs. (24) and (45), the nontrivial ingredient of the covariant entropy bound lies in the concept of lightsheets. Given a surface, a lightsheet defines an adjacent spacetime region whose entropy should be considered. What has changed is not the formula, , but the prescription that determines where to look for the entropy that enters that formula. Let us discuss in detail how lightsheets are constructed.
v.2.1 Orthogonal null hypersurfaces
A given surface possesses precisely four orthogonal null directions (Fig. 3). They are sometimes referred to as future directed ingoing, future directed outgoing, past directed ingoing, and past directed outgoing, though “in” and “out” are not always useful labels. Locally, these directions can be represented by null hypersurfaces that border on . The are generated by the past and the future directed light rays orthogonal to , on either side of .
For example, suppose that is the wall of a (spherical) room in approximately flat space, as shown in Fig. 3, at . (We must keep in mind that denotes a surface at some instant of time.) Then the future directed light rays towards the center of the room generate a null hypersurface , which looks like a light cone. A physical way of describing is to imagine that the wall is lined with light bulbs that all flash up at . As the light rays travel towards the center of the room they generate .
Similarly, one can line the outside of the wall with light bulbs. Future directed light rays going to the outside will generate a second null hypersurface . Finally, one can also send light rays towards the past. (We might prefer to think of these as arriving from the past, i.e., a light bulb in the center of the room flashed at an appropriate time for its rays to reach the wall at .) In any case, the past directed light rays orthogonal to will generate two more null hypersurfaces and .
In Fig. 3, the two ingoing cones and , and the two outgoing “skirts”, and , are easily seen to be null and orthogonal to . However, the existence of four null hypersurfaces bordering on is guaranteed in Lorentzian geometry independently of the shape and location of . They are always uniquely generated by the four sets of surfaceorthogonal light rays.
At least two of the four null hypersurfaces will be selected as lightsheets, according to the condition of nonpositive expansion discussed next.
v.2.2 Lightsheet selection
Let us return to the example where is the wall of a spherical room. If gravity is weak, one would expect that the area of will be a bound on the entropy in the room (Sec. II.3.1). Clearly, cannot be related in any way to the entropy in the infinite region outside the room; that entropy could be arbitrarily large. It appears that we should select or as lightsheets in this example, because they correspond to our intuitive notion of “inside”.
The question is how to generalize this notion. It is obvious that one should compare an area only to entropy that is in some sense “inside” the area. However, consider a closed universe, in which space is a threesphere. As Sec. IV.2.1 has illustrated, we need a criterion that prevents us from considering the large part of the threesphere to be “inside” a small twosphere .
What we seek is a local condition, which will select whether some direction away from is an inside direction. This condition should reduce to the intuitive, global notion—inside is where infinity is not—where applicable. An analogy in Euclidean space leads to a useful criterion, the contraction condition.
Consider a convex closed surface of codimension one and area in flat Euclidean space, as shown in Fig. 4a. Now construct all the geodesics intersecting orthogonally. Follow each geodesic an infinitesimal proper distance to one of the two sides of . The set of points thus obtained will span a similarly shaped surface of area . If , let us call the chosen direction the “inside”. If , we have gone “outside”.
Unlike the standard notion of “inside”, the contraction criterion does not depend on any knowledge of the global properties of and of the space it is embedded in. It can be applied independently to arbitrarily small pieces of the surface. One can always construct orthogonal geodesics and ask in which direction they contract. It is local also in the orthogonal direction; the procedure can be repeated after each infinitesimal step.
Let us return to Lorentzian signature, and consider a codimension 2 spatial surface . The contraction criterion cannot be used to find a spatial region “inside” . There are infinitely many different spacelike hypersurfaces containing . Which side has contracting area could be influenced by the arbitrary choice of .
However, the four null directions away from are uniquely defined. It is straightforward to adapt the contraction criterion to this case. Displacement by an infinitesimal spatial distance is meaningless for light rays, because two points on the same light ray always have distance zero. Rather, an appropriate analogue to length is the affine parameter along the light ray (see the Appendix). Pick a particular direction . Follow the orthogonal null geodesics away from for an infinitesimal affine distance . The points thus constructed span a new surface of area . If , then the direction will be considered an “inside” direction, or lightsheet direction.
By repeating this procedure for , one finds all null directions that point to the “inside” of in this technical sense. Because the light rays generating opposite pairs of null directions (e.g., and ) are continuations of each other, it is clear that at least one member of each pair will be considered inside. If the light rays are locally neither expanding nor contracting, both members of a pair will be called “inside”. Hence, there will always be at least two lightsheet directions. In degenerate cases, there may be three or even four.
Mathematically, the contraction condition can be formulated thus:
(52) 
where is an affine parameter for the light rays generating and we assume that increases in the direction away from . is the value of on . The expansion, , of a family of light rays is discussed in detail in Sec. VI.1. It can be understood as follows. Consider a bunch of infinitesimally neighboring light rays spanning a surface area . Then
(53) 
As in the Euclidean analogy, this condition can be applied to each infinitesimal surface element separately and so is local. Crucially, it applies to open surfaces as well as to closed ones. This represents a significant advance in the generality of the formulation.
For oddly shaped surfaces or very dynamical spacetimes, it is possible for the expansion to change sign along some . For example, this will happen for smooth concave surfaces in flat space. Because of the locality of the contraction criterion, one may split such surfaces into pieces with constant sign, and continue the analysis for each piece separately. This permits us to assume henceforth without loss of generality that the surfaces we consider have continuous lightsheet directions.
For the simple case of the spherical surface in Minkowski space, the condition (53) reproduces the intuitive answer. The area is decreasing in the and directions—the past and future directed light rays going to the center of the sphere. We will call any such surface, with two lightsheet directions on the same spatial side, normal.
In highly dynamical geometries, the expansion or contraction of space can be the more important effect on the expansion of light rays. Then it will not matter which spatial side they are directed at. For example, in an expanding universe, areas get small towards the past, because the big bang is approached. A sufficiently large sphere will have two past directed lightsheets, but no future directed ones. A surface of this type is called antitrapped. Similarly, in a collapsing universe or inside a black hole, space can shrink so rapidly that both lightsheets are future directed. Surfaces with this property are trapped.
In a Penrose diagram (Appendix), a sphere is represented by a point. The four orthogonal null directions correspond to the four legs of an “X” centered on this point. Lightsheet directions can be indicated by drawing only the corresponding legs (Bousso 1999a). Normal, trapped, and antitrapped surfaces are thus denoted by wedges of different orientation (see Figs. 5, 7a, and 8).
v.2.3 Lightsheet termination
From now on we will consider only inside directions, , where runs over two or more elements of . For each , a lightsheet is generated by the corresponding family of light rays. In the example of the spherical surface in flat space, the lightsheets are cones bounded by , as shown in Fig. 3.
Strictly speaking, however, there was no particular reason to stop at the tip of the cone, where all light rays intersect. On the other hand, it would clearly be desastrous to follow the light rays arbitrarily far. They would generate another cone which would grow indefinitely, containing unbounded entropy. One must enforce, by some condition, that the lightsheet is terminated before this happens. In all but the most special cases, the light rays generating a lightsheet will not intersect in a single point, so the condition must be more general.
A suitable condition is to demand that the expansion be nonpositive everywhere on the lightsheet, and not only near :
(54) 
for all values of the affine parameter on the lightsheet.
By construction (Sec. V.2.2) the expansion is initially negative or zero on any lightsheet. Raychaudhuri’s equation guarantees that the expansion can only decrease.. (This will be shown explicitly in Sec. VI.1.) The only way can become positive is if light rays intersect, for example at the tip of the light cone. However, it is not necessary for all light rays to intersect in the same point. By Eq. (53), the expansion becomes positive at any caustic, that is, any place where a light ray crosses an infinitesimally neighboring light ray in the lightsheet (Fig. 4b).
Thus, Eq. (54) operates independently of any symmetries in the setup. It implies that lightsheets end at caustics.^{17}^{17}17If the null energy condition (Appendix) is violated, the condition (54) can also terminate lightshees at noncaustic points. In general, each light ray in a lightsheet will have a different caustic point, and the resulting caustic surfaces can be very complicated. The case of a light cone is special in that all light rays share the same caustic point at the tip. An ellipsoid in flat space will have a selfintersecting lightsheet that may contain the same object more than once (at two different times). Gravitational backreaction of matter will make the caustic surfaces even more involved.
Nonlocal selfintersections of light rays do not lead to violations of the contraction condition, Eq. (54). That is, the lightsheet must be terminated only where a light ray intersects its neighbor, but not necessarily when it intersects another light ray coming from a different portion of the surface . One can consider modifications of the lightsheet definition where any selfintersection terminates the lightsheet (Tavakol and Ellis, 1999; Flanagan, Marolf, and Wald, 2000). Since this modification can only make lightsheets shorter, it can weaken the resulting bound. However, in most applications, the resulting lightsheets are easier to calculate (as Tavakol and Ellis, in particular, have stressed) and still give useful bounds.^{18}^{18}18Low (2002) has argued that the future directed lightsheets in cosmological spacetimes can be made arbitrarily extensive by choosing a closed surface containing sufficiently flat pieces. Low concludes that the covariant entropy bound is violated in standard cosmological solutions, unless it is modified to terminate lightsheets also at nonlocal selfintersections.—This reasoning overlooks that any surface element with local curvature radius larger than the apparent horizon possesses only past directed lightsheets (Bousso, 1999a; see Sec. VII.1.2). Independently of the particular flaw in Low’s argument, the conclusion is also directly invalidated by the proof of Flanagan, Marolf, and Wald (2000). (This is just as well, as the modification advocated by Low would not have solved the problem; nonlocal intersections can be suppressed by considering open surfaces.)
v.3 Defining entropy
v.3.1 Entropy on a fixed lightsheet
The geometric construction of lightsheets is welldefined. But how is “the entropy on a lightsheet”, , determined? Let us begin with an example where the definition of is obvious. Suppose that is a sphere around an isolated, weakly gravitating thermodynamic system. Given certain macroscopic constraints, for example an energy or energy range, pressure, volume, etc., the entropy of the system can be computed either thermodynamically, or statistically as the logarithm of the number of accessible quantum states.
To good approximation, the two lightsheets of are a past and a future light cone. Let us consider the future directed lightsheet. The cone contains the matter system completely (Fig. 2c), in the same sense in which a surface contains the system completely (Fig. 2a). A lightsheet is just a different way of taking a snapshot of a matter system—in light cone time. (In fact, this comes much closer to how the system is actually observed in practice.) Hence, the entropy on the lightsheet is simply given by the entropy of the matter system.
A more problematic case arises when the lightsheet intersects only a portion of an isolated matter system, or if there simply are no isolated systems in the spacetime. A reasonable (statistical) working definition was given by Flanagan, Marolf, and Wald (2000), who demanded that long wavelength modes which are not fully contained on the lightsheet should not be included in the entropy.
In cosmological spacetimes, entropy is well approximated as a continuous fluid. In this case, is the integral of the entropy density over the lightsheet (Secs. VI.2, VII.1).
One would expect that the gravitational field itself can encode information perturbatively, in the form of gravitational waves. Because it is difficult to separate such structure from a ‘‘background metric’’, we will not discuss this case here.^{19}^{19}19Flanagan, Marolf, and Wald (2000) pointed out that perturbative gravitational entropy affects the lightsheet by producing shear, which in turn accelerates the focussing of light rays (Sec. VI). This suggests that the inclusion of such entropy will not lead to violations of the bound. Related research is currently pursued by Bhattacharya, Chamblin, and Erlich (2002).
We have formulated the covariant entropy bound for matter systems in classical geometry and have not made provisions for the inclusion of the semiclassical Bekenstein entropy of black holes. There is evidence, however, that the area of event horizons can be included in . However, in this case the lightsheet must not be continued to the interior of the black hole. The Bekenstein entropy of the black hole already contains the information about objects that fell inside; it must not be counted twice (Sec. III.7).
v.3.2 Entropy on an arbitrary lightsheet
So far we have treated the lightsheet of as a fixed null hypersurface, e.g., in the example of an isolated thermodynamic system. Different microstates of the system, however, correspond to different distributions of energy. This is a small effect on average, but it does imply that the geometry of lightsheets will vary with the state of the system in principle.
In many examples, such as cosmological spacetimes, one can calculate lightsheets in a largescale, averaged geometry. In this approximation, one can estimate while holding the lightsheet geometry fixed.
In general, however, one can at best hold the surface fixed,^{20}^{20}20We shall take this to mean that the internal metric of the surface is held fixed. It may be possible to relax this further, for example by specifying only the area along with suitable additional restrictions. but not the lightsheet of . We must consider to be the entropy on any lightsheet of . Sec. VII.2.3, for example, discusses the collapse of a shell onto an apparent black hole horizon. In this example, a part of the spacetime metric is known, including and the initial expansions of its orthogonal light rays. However, the geometry to the future of is not presumed, and different configurations contributing to the entropy lead to macroscopically different future lightsheets.
In a static, asymptotically flat space the specification of a spherical surface reduces to the specification of an energy range. The enclosed energy must lie between zero and the mass of a black hole that fills in the sphere. Unlike most other thermodynamic quantities such as energy, however, the area of surfaces is welldefined in arbitrary geometries.
In the most general case, one may specify only a surface but no information about the embedding of in any spacetime. One is interested in the entropy of the “fundamental system” (Sec. III.2), i.e., the number of quantum states associated with the lightsheets of in any geometry containing . This leaves too much freedom for Eq. (51) to be checked explicitly. The covariant entropy bound essentially becomes the full statement of the holographic principle (Sec. VIII) in this limit.
v.4 Limitations
Here we discuss how the covariant entropy bound is tied to a regime of approximately classical spacetimes with reasonable matter content. The discussion of the “species problem” (Sec. II.3.4) carries over without significant changes and will not be repeated.
v.4.1 Energy conditions
In Sec. II.3.3 we showed that the entropy of a ball of radiation is bounded by , and hence is less than its surface area. For larger values of the entropy, the mass of the ball would exceed its radius, so it would collapse to form a black hole. But what if matter of negative energy was added to the system? This would offset the gravitational backreaction of the gas without decreasing its entropy. The entropy in any region could be increased at will while keeping the geometry flat.
This does not automatically mean that the holographic principle (and indeed, the generalized second law of thermodynamics) is wrong. A way around the problem might be to show that instabilities develop that will invalidate the setup we have just suggested. But more to the point, the holographic principle is expected to be a property of the real world. And to a good approximation, matter with negative mass does not exist in the real world.^{21}^{21}21We discuss quantum effects and a negative cosmological constant below.
Einstein’s general relativity does not restrict matter content, but tells us only how matter affects the shape of spacetime. Yet, of all the types of matter that could be added to a Lagrangian, few actually occur in nature. Many would have pathological properties or catastrophic implications, such as the instability of flat space.
In a unified theory underlying gravity and all other forces, one would expect that the matter content is dictated by the theory. String theory, for example, comes packaged with a particular field content in its perturbative limits. However, there are many physically interesting spacetimes that have yet to be described in string theory (Sec. IX.1), so it would be premature to consider only fields arising in this framework.
One would like to test the covariant entropy bound in a broad class of systems, but we are not interested whether the bound holds for matter that is entirely unphysical. It is reasonable to exclude matter whose energy density appears negative to a light ray, or which permits the superluminal transport of energy.^{22}^{22}22This demand applies to every matter component separately (Bousso, 1999a). This differs from the role of energy conditions in the singularity theorems (Hawking and Ellis, 1973), whose proofs are sensitive only to the total stress tensor. The above example shows that the total stress tensor can be innocuous when components of negative and positive mass are superimposed. An interesting question is whether instabilities lead to a separation of components, and thus to an eventual violation of energy conditions on the total stress tensor. We would like to thank J. Bekenstein and A. Mayo for raising this question. In other words, let us demand the null energy condition as well as the causal energy condition. Both conditions are spelled out in the Appendix, Eqs. (132) and (133). They are believed to be satisfied classically by all physically reasonable forms of matter.^{23}^{23}23The dominant energy condition has sometimes been demanded instead of Eq. (132) and Eq. (133). It is a stronger condition that has the disadvantage of excluding a negative cosmological constant (Bousso, 1999a).—One can also ask whether, in a reversal of the logical direction, entropy bounds can be used to infer energy conditions that characterize physically acceptable matter (Brustein, Foffa, and Mayo, 2002).
Negative energy density is generally disallowed by these conditions, with the exception of a negative cosmological constant. This is desirable, because a negative cosmological constant does not lead to instabilities or other pathologies. It may well occur in the universe, though it is not currently favored by observation. Unlike other forms of negative energy, a negative cosmological constant cannot be used to cancel the gravitational field of ordinary thermodynamic systems, so it should not lead to difficulties with the holographic principle.
Quantum effects can violate the above energy conditions. Casimir energy, for example, can be negative. However, the relation between the magnitude, size, and duration of such violations is severely constrained (see, e.g., Ford and Roman, 1995, 1997, 1999; Flanagan, 1997; Fewster and Eveson, 1998; Fewster, 1999; further references are found in Borde, Ford, and Roman, 2001). Even where they occur, their gravitational effects may be overcompensated by those of ordinary matter. It has not been possible so far to construct a counterexample to the covariant entropy bound using quantum effects in ordinary matter systems.
v.4.2 Quantum fluctuations
What about quantum effects in the geometry itself? The holographic principle refers to geometric concepts such as area, and orthogonal light rays. As such, it can be applied only where spacetime is approximately classical. This contradicts in no way its deep relation to quantum gravity, as inferred from the quantum aspects of black holes (Sec. II) and demonstrated by the AdS/CFT correspondence (Sec. IX.2).
In the real world, is fixed, so the regime of classical geometry is generically found in the limit of low curvature and large distances compared to the Planck scale, Eq. (2). Setting to would not only be unphysical; as Lowe (1999) points out, it would render the holographic bound, , trivial.
Lowe (1999) has argued that a naive application of the bound encounters difficulties when effects of quantum gravity become important. With sufficient fine tuning, one can arrange for an evaporating black hole to remain in equilibrium with ingoing radiation for an arbitarily long time. Consider the future directed outgoing lightsheet of an area on the black hole horizon. Lowe claims that this lightsheet will have exactly vanishing expansion and will continue to generate the horizon in the future, as it would in a classical spacetime. This would allow an arbitrarily large amount of ingoing radiation entropy to pass through the lightsheet, in violation of the covariant entropy bound.
If a lightsheet lingers in a region that cannot be described by classical general relativity without violating energy conditions for portions of the matter, then it is outside the scope of the present formulation of the covariant entropy bound. The study of lightsheets of this type may guide the exploration of semiclassical generalizations of the covariant entropy bound. For example, it may be appropriate to associate the outgoing Hawking radiation with a negative entropy flux on this lightsheet (Flanagan, Marolf, and Wald, 2000).^{24}^{24}24More radical extensions have been proposed by Markopoulou and Smolin (1999) and by Smolin (2001).
However, Bousso (2000a) argued that a violation of the covariant entropy bound has not been demonstrated in Lowe’s example. In any realistic situation small fluctuations in the energy density of radiation will occur. They are indeed inevitable if information is to be transported through the lightsheet. Thus the expansion along the lightsheet will fluctuate. If it becomes positive, the lightsheet must be terminated. If it fluctuates but never becomes positive, then it will be negative on average. In that case an averaged version of the focussing theorem implies that the light rays will focus within a finite affine parameter.
The focussing is enhanced by the term in Raychaudhuri’s equation, (64), which contributes to focussing whenever fluctuates about zero. Because of these effects, the lightsheets considered by Lowe (1999) will not remain on the horizon, but will collapse into the black hole. New families of light rays continually move inside to generate the event horizon. It is possible to transport unlimited entropy through the black hole horizon in this case, but not through any particular lightsheet.
v.5 Summary
In any dimensional Lorentzian spacetime , the covariant entropy bound can be stated as follows.
Let be the area of an arbitrary dimensional spatial surface (which need not be closed). A dimensional hypersurface is called a lightsheet of if is generated by light rays which begin at , extend orthogonally away from , and have nonpositive expansion,
(55) 
everywhere on . Let be the entropy on any lightsheet of . Then
(56) 
Let us restate the covariant entropy bound one more time, in a constructive form most suitable for applying and testing the bound, as we will in Sec. VII.

Pick any dimensional spatial surface , and determine its area . There will be four families of light rays projecting orthogonally away from : .

Usually additional information is available, such as the macroscopic spacetime metric everywhere or in a neighborhood of .^{25}^{25}25The case where no such information is presumed seems too general to be practally testable; see the end of Sec. V.3.2. Then the expansion of the orthogonal light rays can be calculated for each family. Of the four families, at least two will not expand (). Determine which.

Pick one of the nonexpanding families, . Follow each light ray no further than to a caustic, a place where it intersects with neighboring light rays. The light rays form a dimensional null hypersurface, a lightsheet .

Determine the entropy of matter on the lightsheet , as described in Sec. V.3.1.^{26}^{26}26In particular, one may wish to include in quantum states which do not all give rise to the same macroscopic spacetime geometry, keeping fixed only the intrinsic geometry of . In this case, step 3 has to be repeated for each state or class of states with different geometry. Then denotes the collection of all the different lightsheets emanating in the th direction.

The quantities and can then be compared. The covariant entropy bound states that the entropy on the lightsheet will not exceed a quarter of the area: . This must hold for any surface , and it applies to each nonexpanding null direction, , separately.
The first three steps can be carried out most systematically by using geometric tools which will be introduced at the beginning of Sec. VI.1. In simple geometries, however, they often require little more than inspection of the metric.
The lightsheet construction is welldefined in the limit where geometry can be described classically. It is conjectured to be valid for all physically realistic matter systems. In the absence of a fundamental theory with definite matter content, the energy conditions given in Sec. V.4.1 approximately delineate the boundaries of an enormous arena of spacetimes and matter systems, in which the covariant entropy bound implies falsifiable, highly nontrivial limitations on information content.
In particular, the bound is predictive and can be tested by observation, in the sense that the entropy and geometry of real matter systems can be determined (or, as in the case of large cosmological regions, at least estimated) from experimental measurements.
Vi The dynamics of lightsheets
Entropy requires energy. In Sec. III.5, this notion gave us some insight into a mechanism underlying the spherical entropy bound. Let us briefly repeat the idea. When one tries to excite too many degrees of freedom in a spherical region of fixed boundary area , the region becomes very massive and eventually forms a black hole of area no larger than . Because of the second law of thermodynamics, this collapse must set in before the entropy exceeds . Of course, it can be difficult to verify this quantitatively for a specific system; one would have to know its detailed properties and gravitational backreaction.
In this section, we identify a related mechanism underlying the covariant entropy bound. Entropy costs energy, energy focusses light, focussing leads to the formation of caustics, and caustics prevent lightsheets from going on forever. As before, the critical link in this argument is the relation between entropy and energy. Quantitatively, it depends on the details of specific matter systems and cannot be calculated in general. Indeed, this is one of the puzzles that make the generality of the covariant entropy bound so striking.
In many situations, however, entropy can be approximated by a local flow of entropy density. With plausible assumptions on the relation between the entropy and energy density, which we review, Flanagan, Marolf, and Wald (2000) proved the covariant entropy bound.
We also present the spacelike projection theorem, which identifies conditions under which the covariant bound implies a spacelike bound (Bousso, 1999a).
vi.1 Raychaudhuri’s equation and the focussing theorem
A family of light rays, such as the ones generating a lightsheet, is locally characterized by its expansion, shear, and twist, which are defined as follows.
Let be a surface of spatial dimensions, parametrized by coordinates , . Pick one of the four families of light rays that emanate from into the past and future directions to either side of (Fig. 3). Each light ray satisfies the equation for geodesics (Appendix):
(57) 
where is an affine parameter. The tangent vector is defined by
(58) 
and satisfies the null condition . The light rays generate a null hypersurface parametrized by coordinates . This can be rephrased as follows. In a neighborhood of , each point on is unambiguously defined by the light ray on which it lies () and the affine distance from ().
Let be the null vector field on that is orthogonal to and satisfies . (This means that has the same time direction as and is tangent to the orthogonal light rays constructed on the other side of .) The induced dimensional metric on the surface is given by
(59) 
In a similar manner, an induced metric can be found for all other spatial crosssections of .
The null extrinsic curvature,
(60) 
contains information about the expansion, , shear, , and twist, , of the family of light rays, :
(61)  
(62)  
(63) 
Note that all of these quantities are functions of .
At this point, one can inspect the initial values of on . Where they are positive, one must discard and choose a different null direction for the construction of a lightsheet.
The Raychaudhuri equation describes the change of the expansion along the light rays:
(64) 
For a surfaceorthogonal family of light rays, such as , the twist vanishes (Wald, 1984). The final term, , will be nonpositive if the null energy condition is satisfied by matter, which we assume (Sec. V.4.1). Then the right hand side of the Raychaudhuri equation is manifestly nonpositive. It follows that the expansion never increases.
By solving the differential inequality
(65) 
one arrives at the focussing theorem:^{27}^{27}27In the context of the AdS/CFT correspondence (Sec. IX.2), the role of focussing theorem in the construction of lightsheets has been related to the ctheorem (Balasubramanian, Gimon, and Minic, 2000; Sahakian, 2000a,b). If the expansion of a family of light rays takes the negative value at any point , then will diverge to at some affine parameter .
The divergence of indicates that the crosssectional area is locally vanishing, as can be seen from Eq. (53). As discussed in Sec. V.2.3, this is a caustic point, at which infinitesimally neighboring light rays intersect.
By construction, the expansion on lightsheets is zero or negative. If it is zero, the focussing theorem does not apply. For example, suppose that is a portion of the plane in Minkowski space: . Then each lightsheet is infinitely large, with everywhere vanishing expansion: . However, this is correct only if the spacetime is exactly Minkowski, with no matter or gravitational waves. In this case the lightsheets contain no entropy in any case, so their infinite size leads to no difficulties with the covariant entropy bound.
If a lightsheet encounters any matter (or more precisely, if anywhere on the lightsheet), then the light rays will be focussed according to Eq. (64). Then the focussing theorem applies, and it follows that the light rays will eventually form caustics, forcing the lightsheet to end. This will happen even if no further energy is encountered by the light rays, though it will occur sooner if there is additional matter.
If we accept that entropy requires energy, we thus see at a qualitative level that entropy causes light rays to focus. Thus, the presence of entropy hastens the termination of lightsheets. Quantitatively, it appears to do so at a sufficient rate to protect the covariant entropy bound, but slowly enough to allow saturation of the bound. This is seen in many examples, including those studied in Sec. VII. The reason for this quantitative behavior is not yet fundamentally understood. (This just reformulates, in terms of lightsheet dynamics, the central puzzle laid out in the introduction and reiterated in Sec. VIII.)
vi.2 Sufficient conditions for the covariant entropy bound
Flanagan, Marolf, and Wald (2000; henceforth in this section, FMW) showed that the covariant entropy bound is always satisfied if certain assumptions about the relation between entropy density and energy density are made. In fact, they proved the bound under either one of two sets of assumptions. We will state these assumptions and discuss their plausibility and physical significance. We will not reproduce the two proofs here.
The first set of conditions are no easier to verify, in any given spacetime, than the covariant entropy bound itself. Lightsheets have to be constructed, their endpoints found, and entropy can be defined only by an analysis of modes. The first set of conditions should therefore be regarded as an interesting reformulation of the covariant entropy bound, which may shed some light on its relation to the Bekenstein bound, Eq. (11).
The second set of conditions involves relations between locally defined energy and entropy densities only. As long as the entropy content of a spacetime admits a fluid approximation, one can easily check whether these conditions hold. In such spacetimes, the second FMW theorem obviates the need to construct all lightsheets and verify the bound for each one.
Neither set of conditions is implied by any fundamental law of physics. The conditions do not apply to some physically realistic systems (which nevertheless obey the covariant entropy bound). Furthermore, they do not permit macroscopic variations of spacetime, precluding a verification of the bound in its strongest sense (Sec. V.3.2).
Thus, as FMW point out, the two theorems do not constitute a fundamental explanation of the covariant entropy bound. By eliminating a large class of potential counterexamples, they do provide important evidence for the validity of the covariant entropy bound. The second set can significantly shortcut the verification of the bound in cosmological spacetimes. Moreover, the broad validity of the FMW hypotheses may itself betray an aspect of an underlying theory.
vi.2.1 The first FMW theorem
The first set of assumptions is

Associated with each lightsheet in spacetime there is an entropy flux 4vector whose integral over is the entropy flux through .

The inequality
(66) holds everywhere on . Here is the value of the affine parameter at the endpoint of the lightsheet.
The entropy flux vector is defined nonlocally by demanding that only modes that are fully captured on contribute to the entropy on . Modes that are partially contained on do not contribute. This convention recognizes that entropy is a nonlocal phenomenon. It is particularly useful when lightsheets penetrate a thermodynamic system only partially, as discussed in Sec. V.3.1.
This set of assumptions can be viewed as a kind of “light ray equivalent” of Bekenstein’s bound, Eq. (11), with the affine parameter playing the role of the circumferential radius. However, it is not clear whether one should expect this condition to be satisfied in regions of dominant gravity. Indeed, it does not apply to some weakly gravitating systems (Sec. VI.3.2).
FMW were actually able to prove a stronger form of the covariant entropy bound from the above hypotheses. Namely, suppose that the lightsheet of a surface of area is constructed, but the light rays are not followed all the way to the caustics. The resulting lightsheet is, in a sense, shorter than necessary, and one would expect that the entropy on it, , will not saturate the bound. The final area spanned by the light rays, , will be less than but nonzero (Fig. 4b).
FMW showed, with the above assumptions, that a tightened bound results in this case:
(67) 
Note that this expression behaves correctly in the limit where the lightsheet is maximized [; one recovers Eq. (56)] and minimized (; there is no lightsheet and hence no entropy).
vi.2.2 The second FMW theorem
Through a rather nontrivial proof, FMW showed that the covariant entropy bound can also be derived from a second set of assumptions, namely:

The entropy content of spacetime is well approximated by an absolute entropy flux vector field .

For any null vector , the inequalities
(68) (69) hold at everywhere in the spacetime.
These assumptions are satisfied by a wide range of matter systems, including Bose and Fermi gases below the Planck temperature. It is straightforward to check that all of the adiabatically evolving cosmologies investigated in Sec. VII.1 conform to the above conditions. Thus, the second FMW theorem rules out an enormous class of potential counterexamples, obviating the hard work of calculating lightsheets. (We will find lightsheets in simple cosmologies anyway, both in order to gain intuition about how the lightsheet formulation works in cosmology, and also because this analysis is needed for the discussion of holographic screens in Sec. IX.3.)
Generally speaking, the notion of an entropy flux assumes that entropy can be treated as a kind of local fluid. This is often a good approximation, but it ignores the nonlocal character of entropy and does not hold at a fundamental level.
vi.3 Relation to other bounds and to the GSL
vi.3.1 Spacelike projection theorem
We have seen in Sec. IV.2 that the spacelike entropy bound does not hold in general. Taking the covariant entropy bound as a general starting point, one may derive other, more limited formulations, whose regimes of validity are defined by the assumptions entering the derivation. Here we use the lightsheet formulation to recover the spacelike entropy bound, Eq. (45), along with precise conditions under which it holds. By imposing further conditions, even more specialized bounds can be obtained; an example valid for certain regions in cosmological spacetimes is discussed in Sec. VII.1.7 below.
Spacelike projection theorem (Bousso, 1999a). Let be a closed surface. Assume that permits at least one future directed lightsheet . Moreover, assume that is complete, i.e., is its only boundary (Fig. 6). Let be the entropy in a spatial region enclosed by on the same side as . Then
(70) 
Proof. Independently of the choice of (i.e., the choice of a time coordinate), all matter present on will pass through . The second law of thermodynamics implies the first inequality, the covariant entropy bound implies the second.
What is the physical significance of the assumptions made in the theorem? Suppose that the region enclosed by is weakly gravitating. Then we may expect that all assumptions of the theorem are satisfied. Namely, if did not have a future directed lightsheet, it would be antitrapped—a sign of strong gravity. If had other boundaries, this would indicate the presence of a future singularity less than one lightcrossing time from —again, a sign of strong gravity.
Thus, for a closed, weakly gravitating, smooth surface we may expect the spacelike entropy bound to be valid. In particular, the spherical entropy bound, deemed necessary for the validity of the GSL in the Susskind process, follows from the covariant bound. This can be see by inspecting the assumptions in (Sec. II.3), which guarantee that the conditions of the spacelike projection theorem are satisfied.
vi.3.2 Generalized second law and Bekenstein bound
In fact, FMW showed that the covariant bound implies the GSL directly for any process of black hole formation, such as the Susskind process (Sec. II.3.1).
Consider a surface of area on the event horizon of a black hole. The past directed ingoing light rays will have nonpositive expansion; they generate a lightsheet. The lightsheet contains all the matter that formed the black hole. The covariant bound implies that . Hence, the generalized second law is satisfied for the process in which a black hole is newly formed from matter.
Next, let us consider a more general process, the absorption of a matter system by an existing black hole. This includes the Geroch process (Sec. II.2.1). Does the covariant bound also imply the GSL in this case?
Consider a surface on the event horizon after the matter system, of entropy , has fallen in, and follow the pastingoing lightrays again. The lightrays are focussed by the energy momentum of the matter. “After” proceeding through the matter system, let us terminate the lightsheet. Thus the lightsheet contains precisely the entropy . The rays will span a final area (which is really the initial area of the event horizon before the matter fell in).
According to an outside observer, the Bekenstein entropy of the black hole has increased by , while the matter entropy has been lost. According to the “strengthened form” of the covariant entropy bound considered by FMW, Eq. (67), the total entropy has not decreased. The original covariant bound, Eq. (56), does not by itself imply the generalized second law of thermodynamics, Eq. (5), in this process.
Eq. (67) can also be used to derive a version of Bekenstein’s bound, Eq. (11)—though, unfortunately, a version that is too strong. Consider the lightsheet of an approximately flat surface of area , bounding one side of a rectangular thermodynamic system. With suitable timeslicing, the surface can be chosen to have vanishing null expansion, .
With assumptions on the average energy density and the equation of state, Raychaudhuri’s equation can be used to estimate the final area of the lightsheet where it exits the opposite side of the matter system. The strengthened form of the covariant entropy bound, Eq. (67) then implies the bound given in Eq. (23). However, for very flat systems this bound can be violated (Sec. II.2.2)!
Hence, (67) cannot hold in the same generality that is claimed for the original covariant entropy bound, Eq. (56).^{28}^{28}28It follows that the first FMW hypotheses do not hold in general. An earlier counterexample to Eq. (67), and hence to these hypotheses, was given by Guedens (2000). However, the range of validity of Eq. (67) does appear to be extremely broad. In view of the significance of its implications, it will be important to better understand its scope.
We conclude that the covariant entropy bound implies the spherical bound in its regime of validity, defines a range of validity for the spacelike bound, and implies the GSL for black hole formation processes. The strengthened form of the covariant bound given by FMW, Eq. (67), implies the GSL for absorption processes and, under suitable assumptions, yields Bekenstein’s bound [though in a form that demonstates that Eq. (67) cannot be universally valid].
The result of this section suggest that the holographic principle (Sec. VIII) will take a primary role in the complex of ideas we have surveyed. It may come to be viewed as the logical origin not only of the covariant entropy bound, but also of more particular laws that hold under suitable conditions, such as the spherical entropy bound, Bekenstein’s bound, and the generalized second law of thermodynamics.
Vii Applications and examples
In this section, the covariant entropy bound is applied to a variety of matter systems and spacetimes. We demonstrate how the lightsheet formulation evades the various difficulties encountered by the spacelike entropy bound (Sec. IV.2).
We apply the bound to cosmology and verify explicitly that it is satisfied in a wide class of universes. No violations are found during the gravitational collapse of a star, a shell, or the whole universe, though the bound can be saturated.
vii.1 Cosmology
vii.1.1 FRW metric and entropy density
FriedmannRobertsonWalker (FRW) metrics describe homogeneous, isotropic universes, including, to a good degree of approximation, the portion we have seen of our own universe. Often the metric is expressed in the form
(71) 
We will find it more useful to use the conformal time and the comoving coordinate :
(72) 
In these coordinates the FRW metric takes the form
(73) 
Here , , and , , correspond to open, flat, and closed universes respectively. Relevant Penrose diagrams are shown in Figs. 5 and 7a.
In cosmology, the entropy is usually described by an entropy density , the entropy per physical volume:
(74) 
For FRW universes, depends only on time. We will assume, for now, that the universe evolves adiabatically. Thus, the physical entropy density is diluted by cosmological expansion:
(75) 
The comoving entropy density is constant in space and time.
vii.1.2 Expansion and apparent horizons
Let us verify that the covariant entropy bound is satisfied for each lightsheet of any spherical surface . The first step is to identify the lightsheet directions. We must classify each sphere as trapped, normal, or antitrapped (Sec. V.2.2). Let us therefore compute the initial expansion of the four families of light rays orthogonal to an arbitrary sphere characterized by some value of .
We take the affine parameter to agree locally with and use Eq. (53). Differentiation with respect to () is denoted by a dot (prime). Instead of labelling the families , it will be more convenient to use the notation , where the first sign refers to the time () direction of the light rays and the second sign denotes whether they are directed at larger or smaller values of .
For the future directed families one finds
(76) 
The expansion of the past directed families is given by
(77) 
Note that the first term in Eq. (76) is positive when the universe expands and negative if it contracts. The term diverges when , i.e., near singularities. The second term is given by (; ) for a closed (flat; open) universe. It diverges at the origin (), and for a closed universe it also diverges at the opposite pole ().
The signs of the four quantities depend on the relative strength of the two terms. The quickest way to classify surfaces is to identify marginal spheres, where the two terms are of equal magnitude.
The apparent horizon is defined geometrically as a sphere at which at least one pair of orthogonal null congruences have zero expansion. It satisfies the condition
(78) 
which can be used to identify its location as a function of time. There is one solution for open and flat universes. For a closed universe, there are generally two solutions, which are symmetric about the equator [].
The proper area of the apparent horizon is given by
(79) 
Using Friedmann’s equation,
(80) 
one finds
(81) 
where is the energy density of matter.
At any time , the spheres that are smaller than the apparent horizon,
(82) 
are normal. (See the end of Sec. V.2.2 for the definitions of normal, trapped, and antitrapped surfaces.) Because the second term dominates in the expressions for the expansion, the cosmological evolution has no effect on the lightsheet directions. The two lightsheets will be a past and a future directed family going to the same spatial side. In a flat or open universe, they will be directed towards (Fig. 5). In a closed universe, the lightsheets of a normal sphere will be directed towards the nearest pole, or (Fig. 7a).
For spheres greater than the apparent horizon
(83) 
the cosmological term dominates in the expressions for the expansion. Then there are two cases. Suppose that , i.e., the universe is expanding. Then the spheres are antitrapped. Both lightsheets are past directed, as indicated by a wedge opening to the bottom in the Penrose diagram. If and , then both future directed families will have negative expansion. This case describes trapped spheres in a collapsing universe. They are denoted by a wedge opening to the top (Fig. 7a).
vii.1.3 Lightsheets vs. spatial volumes
We have now classified all spherical surfaces in all FRW universes according to their lightsheet directions. Before proceeding to a detailed calculation of the entropy contained on the lightsheets, we note that the violations of the spacelike entropy bound identified in Secs. IV.2.1 and IV.2.2 do not apply to the covariant bound.
The area of a sphere at is given by
(84) 
To remind ourselves that the spacelike entropy bound fails in cosmology, let us begin by comparing this area to the entropy enclosed in the spatial volume defined by at equal time . With our assumption of adiabaticity, this depends only on :
(85) 
For a flat universe [], the area grows like but the entropy grows like