Proof of Classical Versions of the Bousso Entropy Bound and of The

Home , Raphael Bousso

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[3] GSL is in bound anal- given needed entropy an is type that such (1.1) believe the we form However, of the GSL. ysis of the bound of validity ef- a the size that for box argue finite again to to due fects modified be must formulas buoyancy httepeiemaigo “ of meaning precise the what law second necessary generalized not the is of (1.1) validity bound the the that for [2–4] Subsequently shown horizons. was hole black it all of area surface total iin h on al hntenme fseisof species of number the large when sufficiently fails is particles bound the dition, b / † ∗ ¯h ntecnnclesml,i ses oso httebound the that show to easy is it ensemble, canonical the In eyrcnl,Bknti 5 a sdtefc htthe that fact the used has [5] Bekenstein recently, Very where , ce aeie2sraeo area of surface 2 pacelike to h ocle oorpi principle. holographic so-called the of nt is( ties L hr xssa boueetoyflux entropy absolute an exists there mle h eeaie eodlwof law second generalized the implies a at abr,C 93106 CA Barbara, Santa ia, nsaeieteei netoyflux entropy an is there spacetime in L ti’ qainadstsyn the satisfying and equation stein’s T L n i)aogec ulgeodesic null each along (ii) and , ab s ftost fhptee.The hypotheses. of sets two of r a steitga of integral the is onso eesenadalso and Bekenstein of bounds y stesrs-nrytensor, stress-energy the is k taa Y14853-5001. NY Ithaca, a ) 2 ≤ yo Chicago of ty T ab 44. k a k † ial,i sfrfo clear from far is it Finally, . b h Generalized the f L R / s A (16 ntecnetr ssup- is conjecture the in ” a eeae by generated utsatisfy must over π ¯h 2 G L and ) and , λ is ∗ nad- In . ry posed to be, particularly in curved spacetime; in curved As it stands, the bound (1.3) is ambiguous, since the spacetime, it is also far from clear what “E” means. Nev- precise meaning of the “bounding area”, A, has not been ertheless, a case can be made that the bound (1.1) may spelled out. In particular, note that any world tube can hold for all physically realistic systems found in nature; always be “enclosed” by a two-surface of arbitrarily small see Ref. [6] for further discussion. area, since given any two-surface in spacetime, there ex- More recently, an alternative entropy bound has been ists a two-surface of arbitrarily small area arbitrarily considered: the entropy S inside any region whose bound- close to the original two-surface (obtained by “wiggling” ary has area A must satisfy [7] the original two-surface suitably in spacetime). However, very recently, a specific conjecture of the form (1.3) was S A/4. (1.3) ≤ suggested by Bousso [12,13], who improved an earlier sug- An argument given in Ref. [8] suggests that the bound gestion of Fischler and Susskind [14,15]. Bousso showed (1.3) should follow from the GSL together with the as- that several example spacetimes, including cosmological sumption that the entropy of a black hole counts the models and gravitational collapse spacetimes, are consis- number of possible internal states of the black hole‡. tent with his conjecture. In addition, when E < R, this bound would follow Bousso’s conjecture is as follows. Let (M,gab) be from the original Bekenstein∼ bound (1.1). The inequality a spacetime satisfying Einstein’s equation and also the (1.3), like the bound (1.1), can be violated if the number dominant energy condition [16]. Let B be a connected of massless particle species is allowed to be arbitrarily 2 dimensional spacelike surface in M. Suppose that ka large§. The inequality (1.3) is related to the hypothe- is a smooth null vector field on B which is everywhere sis known as the holographic principle, which states that orthogonal to B. Then the expansion the physics in any spatial region can be fully described θ = ka (1.4) in terms of degrees of freedom living on the boundary of ∇a that region, with one degree of freedom per Planck area of ka is well defined and is independent of how ka is [11,8]. If the holographic principle is correct, then since extended off B. Suppose that θ 0 everywhere on B. the entropy in any region should be bounded above by Let L denote the null hypersurface≤ generated by the null the number of fundamental degrees of freedom in that geodesics starting at B with initial tangent ka, where region, a bound of the form (1.3) should be valid for all each null geodesic is terminated if and only if a caustic systems, including those with strong self-gravity. is reached (where θ ), and otherwise is extended → −∞ as far as possible. Then the entropy flux, SL, through L satisfies vides strong evidence that this failure does not occur in the SL AB/4, (1.5) microcanonical ensemble. ≤ ‡The argument is attributed to Bekenstein in Ref. [8] and where AB is the area of B. goes as follows. If black hole entropy counts the number of There is a close relationship between Bousso’s conjec- internal states of a black hole, then any system having S ≥ ture and the generalized second law. Consider a foliation A/4 is not a black hole. Then, one would expect to be able to of the horizon of a black hole by spacelike two-surfaces make that system into a black hole with area A by collapsing a B(α), where α is a continuous label that increases in sufficiently massive spherical shell of matter around it. In this the future direction (with respect to the time orienta- process, it appears that no entropy escapes, but this means tion used to define the black hole). Let A(α) be the area that we convert an S ≥ A/4 system into a black hole of area of the two surface B(α), and let S(α) be the total en- A, violating the generalized second law. An antecedent to this tropy that crosses the horizon before the 2-surface B(α). argument can be found in Ref. [10]. For a counterargument, Then if one assumes the ordinary second law, the GSL see Ref. [9] is equivalent to the statement that for any α < α we § 1 2 For example, consider N free massless scalar fields in flat have spacetime, in a cube of edge length L with Dirichlet boundary conditions. In the canonical ensemble, the thermal state 1 S(α2) S(α1) [A(α2) A(α1)] . (1.6) with temperature T with T ≪ 1/L has energy E which scales − ≤ 4 − as E ∼ N exp[−π/(LT )]/L and entropy S which scales like On the other hand, Bousso’s entropy bound applied to S ∼ N exp[−π/(LT )]/(LT ). For the system to be weakly the 2-surface B(α)—with ka taken to be the past di- self-gravitating (a necessary condition for the flat-spacetime rected normal to the horizon, so that we have θ 0 analysis to be a good approximation) we must have E = εL ≤ for some ε ≪ 1. Using this restriction to solve for the maxi- on B(α) when the null energy condition is satisfied— mum allowed value of N yields S/L2 ∼ ε/(LT ), which can be demands merely that arbitrarily large. In the microcanonical ensemble, the viola- 1 tion of Eq. (1.3) for sufficiently large N follows immediately S(α) A(α) (1.7) ≤ 4 from the fact that the density of states at a fixed total energy E grows unboundedly with N at fixed L. (Note, however, for all α. Thus, Bousso’s bound implies that the GSL that Casimir energy has been ignored here.) holds for the case when the initial time, α1, is taken to

2 be the time when the black hole is first formed [so that T kakb 0 for all null vectors ka [see Eqs. (1.9) and ab ≥ S(α1)= A(α1) = 0]. In general, however, it is clear that (1.10)–(1.11) below], which is weaker than the dominant the statement (1.7) is weaker than the statement (1.6). energy condition. Thus, our proof of Bousso’s conjecture, This observation motivates a generalization of Bousso’s like the conjecture itself, is limited to “classical regimes” conjecture. Namely, if one allows the geodesics generat- in which local energy conditions are satisfied. ing the hypersurface L to terminate at some spacelike Clearly, in order to derive the bounds (1.5) and (1.8), 2-surface B′ before coming to a caustic or singularity, we must make some assumptions about entropy. The one can replace the conjectured inequality (1.5) by the entropy that the conjecture refers to presumably should condition include gravitational contributions. It seems plausible 1 that any gravitational entropy flux through the null hy- S [A A ′ ] . (1.8) persurface L will be associated with a shearing of that L 4 B B ≤ − hypersurface, which has the same qualitative effect in the It is clear from the above discussion that this more Raychaudhuri equation [see Eq. (2.13) below] as a matter general bound implies both the original Bousso entropy stress-energy flux. Thus, it may be possible to treat grav- bound and the GSL (assuming of course the validity of itational contributions to entropy in a manner similar to the ordinary second law). the matter contributions. However, our present under- In this paper we shall prove Bousso’s entropy bound standing of quantum gravity is not sufficient to attempt (1.5) under two independent sets of hypotheses concern- to meaningfully quantify the gravitational contributions ing the local entropy content of matter. Furthermore, to entropy. Consequently, in our analysis below, we shall under the first set of hypotheses, we will prove the more consider only the matter contributions to entropy. general entropy bound (1.8). We note that proofs of the With regard to the matter contribution to entropy, for GSL that are more general than the proof of this paper both the GSL and the Bousso bound, there is an appar- have previously been given [17]; however, the previous ent tension between the fact that these statements are proofs used specific properties of black-hole spacetimes, supposed to have the status of fundamental laws and the unlike our analysis. fact that entropy is a quantity whose definition is coarse- Finally, we note that, as discussed further at the end of graining dependent. However, this tension is resolved by Sec. III, our results can be generalized straightforwardly noting that the number of degrees of freedom should be to arbitrary spacetime dimensions greater than 2. an upper bound for the entropy S, irrespective of choice of coarse-graining [13]. Equivalently, we may restrict attention to the case where the matter is locally in thermal B. Derivations of entropy bound and of generalized equilibrium (i.e., maximum entropy density for its given second law: framework, viewpoint and assumptions energy density); if the bound holds in this case, it must hold in all cases. The starting point for our derivation of the entropy We shall proceed by assuming that a phenomenologi- bounds (1.5) and (1.8) is a postulated phenomenological cal description of matter entropy can be given in terms description of entropy, which differs from assumptions of an entropy flux 4-vector sa. We shall then postulate that have been used in the past to derive the GSL [17]. In some properties of sa. In fact, we shall postulate two this section we describe our phenomenological description independent sets of hypotheses on sa, each of which will of entropy and its motivation. be sufficient to prove the bound (1.5); the first set of hy- First, note that one of the hypotheses of Bousso’s con- potheses also will suffice to prove the bound (1.8). Note jecture is the dominant energy condition, which is often that it is not a central goal of this paper to justify our violated by the expected stress energy tensor of matter in hypotheses, although we do discuss some motivations be- semiclassical gravity. Hence the conjecture cannot have low. Instead we shall merely observe that they appear to the status of a fundamental law as it is currently stated, hold in certain regimes. Note also that, at a fundamen- but rather can only be relevant in “classical regimes” tal level, entropy is a non-local quantity and so can be where the dominant energy condition is satisfied [18]. It well described by a entropy flux 4-vector only in certain may be possible to replace the dominant energy condition regimes and over certain scales. This fact is reflected in by a quantum inequality of the type invented by Ford and our hypotheses below. Roman [19–23] to overcome this difficulty∗∗. In this paper we will assume the null convergence condition, that

its area, and in addition it is hard to see how a modified Bousso conjecture incorporating a quantum inequality rather ∗∗Lowe [18] argues that the Bousso conjecture must fail for than a local energy condition could be satisfied. However, this a system consisting of an evaporating black hole accreting at counterexample might be resolved by the fact that it may be just the right rate to balance the Hawking radiation mass appropriate to assign a negative entropy flux at the horizon to loss. For such a system, it would seem that the black hole states with an outgoing Hawking flux, or it might be resolved can accrete an arbitrary amount of entropy without changing by making adjustments to the Bousso conjecture.

3 The first of our two sets of hypotheses is very much in this equation scale the same way under a change of affine the spirit of the original Bekenstein bound (1.1). Sup- parameter. pose that one has a null hypersurface, L, the generators The above set of hypotheses has the property that a of which terminate at a finite value λ∞ of affine parame- the entropy flux, sLka, depends upon L in the sense ter λ. Suppose that one puts matter in a box and drops (described above)− that modes that only partially pass it through L in such a way that the back end of the box through L prior to λ∞ do not contribute to the entropy crosses L at λ∞. Then, if a bound of the nature of Eq. flux. In our second set of hypotheses, we assume the ex- (1.1) holds, the amount of entropy crossing L should be istence of an absolute entropy flux 4-vector sa, which is limited by the energy within the box and the box “size”. independent of the choice of L. We assume that this sa The box size, in turn, would be related to the affine pa- obeys the following purely local, pointwise inequalities rameter at which the front end of the box crossed L. On for any null vector ka: the other hand, suppose that matter flowing through L a 2 a b near λ were not confined by a box. Then there would (sak ) α1 Tabk k (1.10) ∞ ≤ be no “box size restriction” on the entropy flux near λ∞. However, in order to have a larger entropy flux than one and could achieve when using a box, it clearly would be nec- kakb s α T kakb, (1.11) essary to put the matter in a state where the “modes” | ∇a b|≤ 2 ab carrying the entropy “spill over” beyond λ∞. In that ‡‡ where Tab is the stress-energy tensor . Here α1 and α2 case, it is far from clear that the entropy carried by these can be any positive constants that satisfy modes should be credited as arriving prior to λ∞, so that 1/4 1/2 they would count in the entropy flux through L. In other (πα1) + (α2/π) =1. (1.12) words, it seems reasonable to postulate that the entropy flux through L cannot be higher than the case where the [Recall that we are using Planck units with G = c =¯h = matter is placed in a box whose back end crosses L at k = 1.] A specific simple choice of α1 and α2 that satisfy λ∞, and to consider a bound on this entropy flux of the the condition (1.12) is α1 =1/(16π) and α2 = π/4, which general form of Eq. (1.1). are the values quoted in the abstract above. Note that, The above considerations motivate the following hypothesis concerning the entropy flux. We assume that associated with every null surface L there is an entropy a ‡‡ flux 4-vector sL from which one can compute the entropy The stress energy tensor appearing in these inequalities flux through L. Let γ be a null geodesic generator of L, should be interpreted as a macroscopic or averaged stress a a with affine parameter λ and tangent k = (d/dλ) . If γ energy tensor T¯ab, rather than a microscopic stress energy a b is of infinite affine parameter length, then Tabk k = 0 tensor Tab. For example, for an atomic gas, the fundamen- along γ by the focusing theorem [16], and we assume that tal microscopic stress-energy tensor Tab will vary rapidly over a sL = 0 along γ. On the other hand, if γ ends at a finite atomic and nuclear scales, while a suitable averaged macro- †† value, λ∞, of affine parameter, then we assume that scopic stress tensor T¯ab can be taken to vary only over macroscopic scales (like the conventional entropy current sa). Thus a a b s k π(λ∞ λ)T k k . (1.9) our results apply to null surfaces L of an averaged, macro- | L a|≤ − ab scopic metricg ¯ab rather than the physical metric gab [24]. The inequality (1.9) is a direct analog of the original Note that null surfaces ofg ¯ab can differ significantly from the a Bekenstein bound (1.1), with sLka playing the role of null surfaces of gab, since with suitable microscopic sources a b | | S, Tabk k playing the role of E, and λ∞ λ playing (for example cosmic strings) a null surface of gab can be made the role of R. As discussed above, the motivation− for to intersect itself very frequently without the occurrence of the bound (1.9) is essentially the same as that for the caustics. However, the boundary of the future (or past) of bound (1.1). Note that Eq. (1.9) is independent of the the 2-surface B with respect tog ¯ab should be close to the choice of affine parameterization of γ; i.e., both sides of boundary of the future (respectively, past) of B with respect to gab. Thus, if one wishes to work with the exact metric gab, one should presumably replace the null hypersurface, L, in the Bousso conjecture and our generalization (1.8) with a suitable portion of the boundary of the future (or past) of B. ††From Eq. (2.13), our proof also works if we replace the This new formulation of the conjectures should hold whenever hypothesis (1.9) with the weaker hypotheses that Eq. (1.9) or Eqs. (1.10) and (1.11) hold for the macroscopi- a a b ab cally averaged entropy current and macroscopically averaged |sLka|≤ (λ∞ − λ) πTabk k +σ âbσˆ /8 , stress energy tensor. An alternative interpretative framework whereσ âb is the shear tensor [Eq. (9.2.28) of Ref. [16]] as- would be to assume the existence of an “entropy current” sociated with the generators of L. In this context, we can which varies rapidly on the smallest scales that are compati- a interpret sL to be the combined matter and gravitational en- ble with our gradient assumption (1.11) (atomic and nuclear tropy flux, rather than just the matter entropy flux. scales in our example), in which case our result would apply directly to the microscopic metric.

4 like Eq. (1.9), Eqs. (1.10) and (1.11) are independent of pied by N particles. Such a system has a well defined a the choice of scaling of k . Also note that both of our expected stress energy tensor Tâb , whose correspond- sets of hypotheses (1.9) and (1.10)–(1.11) imply the null ing energy density will be of orderh i convergence condition T kakb 0, as mentioned above. ab ≥ N We now turn to a discussion of the physical regimes ρ 4 . (1.13) in which we expect the pointwise assumptions (1.10) and ∼ f λ (1.11) of our second set of hypotheses to be valid. The We now imagine that we are to somehow model such a first assumption (1.10) of our second set of hypothe- system with a smooth entropy flux vector sa. We ex- ses says, roughly speaking, that the entropy density is pect that the total entropy carried by the system should bounded above by the square root of the energy density. be of order N, so that the entropy density s should be One can check that the condition is satisfied for ther- approximately mal equilibrium states of Bose and Fermi gases except N at temperatures above a critical temperature of order s . (1.14) f λ3 the Planck temperature §§. One can also check that for ∼ quantum fields in a box at low temperatures (the exam- Clearly the concept of local entropy flux here cannot ple discussed in Sec. I A above), the condition (1.10) is make sense on scales short compared to the wavelength violated only if the box is Planck size or smaller, or if the λ; only in an averaged sense, on scales comparable to λ number of species is allowed to be very large. Thus, it or larger, does the concept of entropy flux make sense. seems plausible that the bound (1.10) will be universally Thus, the lengthscale = s/ s over which the entropy L |∇ | valid if one assumes a Planck scale cutoff for physics and density varies should be greater than or of the order of if one also assumes a limit to the number of species. Also λ. From the estimates (1.13) and (1.14), the condition one can argue as follows that a bound of the form of Eq. > λ is equivalent to s < ρ, which is essentially our L |∇ | (1.10) should follow from the Bekenstein bound (1.1). assumption∼ (1.11). Hence,∼ our second condition (1.11) Consider a region of space of that is sufficiently small rules out the class of entropy currents sa which vary sig- that (i) the entropy density and energy density are ap- nificantly over scales shorter than λ, allowing only the proximately uniform over the region, and (ii) the region more appropriate sa that vary over scales of a wavelength is weakly self-gravitating so that its total energy E satis- or longer. fies E < R, where R is the size of the region. Then, if S is In summary, we expect our second set of hypotheses the total∼ entropy in the region, the ratio of entropy den- to be valid in regimes where the following conditions are sity squared to energy density is S2/ER3 4π2E/R satisfied: (i) Spacetime structure can be accurately de- ∼ ≤ by Eq. (1.1), which is < 1 as E < R. scribed by a classical metric, gab, and the gravitational The second assumption∼ (1.11)∼ states roughly that the contributions to entropy, other than that from black gradient of the entropy density is bounded above by the holes, are negligible. (ii) The matter entropy can be ac- energy density. For a free, massless boson or fermion gas curately be described by an entropy current sa. In par- in local thermal equilibrium, this condition reduces to the ticular, this condition will be valid in familiar hydrody- condition that the temperature gradient, T , be small namic regimes. (iii) The vacuum energy contributions to compared with T 2, i.e., that the fractional|∇ change| in T the stress-energy tensor are negligible, so that the stress- over a distance 1/T be smaller than unity. This condition energy tensor satisfies classical energy conditions. must be satisfied in order for the notion of local thermal We shall refer to regimes satisfying the above three equilibrium to make sense. conditions as “classical”, even though, in such regimes, In addition, it would appear that condition (1.11) is quantum physics may play an essential role in account- necessary for our entire phenomenological description of ing for the entropy of matter. In classical regimes, our entropy as represented by an 4-current sa to be valid. To hypotheses (1.10) and (1.11) should be valid. We have ar- see this, consider the following illustrative example. Con- gued above that hypothesis (1.9) also should hold. Hence sider a wavepacket mode of a quantum field, where the our arguments show that the Bousso bound (1.5) and wavelength is λ and where the volume occupied by the its generalization (1.8) should hold in classical regimes. wavepacket is fλ3 for some dimensionless factor f > 1. While our arguments do not show that the entropy Consider a state where this wavepacket mode is occu-∼ bounds (1.5) and (1.8) hold at any fundamental level, they do show that any counterexample either must in- volve quantum phenomena in an essential way (in the sense of failure to be in a classical regime), or must vio- §§ Specifically, for a free massless boson gas at temperature T late Eq. (1.9) and/or Eq. (1.10) or Eq. (1.11). the stress energy tensor has the form Tab =(ρ+p)uaub +pgab and the entropy flux vector is sa = σua, where p = ρ/3 and σ = 4ρ/(3T ). It follows that for any null vector ka we have II. DERIVATION OF ENTROPY BOUNDS a 2 a b 2 2 2 (sak ) /Tabk k = 4ρ/(3T ) = 2π gNsT /45, where g is the number of polarization components and Ns is the number of In this section we derive the generalized entropy bound species. (1.8) from the assumption (1.9) and the Bousso bound

5 (1.5) from the assumptions (1.10)–(1.11). is an area-decrease factor associated with the given gen- We start with some definitions and constructions. erator. Similarly, a sufficient condition for the Bousso First, we can without loss of generality take the vector bound (1.5) is that field ka on the 2-surface B to be future directed, since a 1 the conjecture is time reversal invariant. Let l be the a 1 dλ ( sak ) (λ) (2.4) unique vector field on B which is null, future directed, 0 − A ≤ 4 a Z orthogonal to B and which satisfies l ka = 1. We ex- tend both ka and la to L by parallel transport− along the along each generator. null geodesic generators of L. Thus, ka is tangent to each a To prove the lemma, first note that by assumption the geodesic. Then the expansion θ = ak is well defined entropy flux through L is given by on L and independent of how ka is∇ extended off of L, since a SL = s ǫabcd, (2.5) θ = (gab + kalb + kbla) k . (2.1) ZL ∇a b where the orientation on the hypersurface L is that de- By the hypotheses of Bousso’s conjecture and of its gen- termined by the 3-form eralization (1.8), θ is nonpositive everywhere on L. Let Γ 1 2 a x = (x , x ) be any coordinate system on B. Then ǫˆbcd l ǫabcd. (2.6) one{ } obtains a natural coordinate system (λ, x1, x2) on L ≡ in the obvious way, where ka = (d/dλ)a and we take Here we are using the notation of Appendix B of Ref. λ =0 on B. [16] for integrals of differential forms, and we also use For the generator γ which starts at the point xΓ on the abstract index notation of Ref. [16] for all Roman Γ B, let λ∞(x ) be the value of affine parameter (possi- indices throughout the paper. The formula (2.5) applies bly λ∞ = ) at the endpoint of the generator. This for either version of our phenomenological description of ∞ endpoint can either be a caustic (θ = ) or have a entropy; i.e., we can use either sa or sa in the integrand. −∞ L finite expansion θ. We can without loss of generality ex- In Appendix A we derive the following formula for the Γ clude the case λ∞ = , since otherwise we must have integral (2.5) in the coordinate system (λ, x ): a b ∞ Tabk k = 0 and θ = 0 along γ by the focusing theo- λ∞(x) rem, and then either version of our hypotheses implies 2 that sa = 0 along γ, so that there is no contribution to SL = d x det hΓΛ(x) dλ s(λ) (λ). (2.7) B 0 A the entropy flux. Thus, generators of infinite affine pa- Z p Z rameter length make no contribution to the LHS of the Here x (x1, x2) = xΓ, h (x) is the induced 2-metric ≡ ΓΛ inequalities (1.5) and (1.8) while making a non-negative on the 2-surface B, λ∞(x) is the value of affine parameter contribution to the RHS, and so can be ignored. For the at the endpoint of the generator which starts at x, and generators of finite affine parameter length, we can with- a out loss of generality rescale the affine parameter along s sak . (2.8) each generator in order to make the endpoint occur at ≡− λ∞ = 1. Note that s is non-negative for future directed, timelike or null sa, which we expect to be the case. (However, our proof does not require the assumptions that sa be A. Reducing the conjecture to each null geodesic timelike or null and future directed.) Now as discussed generator above, s(λ) = 0 for those generators of infinite affine parameter length, so it follows that

Next, we show that it is sufficient to focus attention ′ 1 on each individual generator of L, one at a time. More 2 SL = d x det hΓΛ(x) dλ s(λ) (λ), (2.9) specifically we have the following lemma. B 0 A Z p Z Lemma: A sufficient condition for the generalized en- ′ 2 where B d x denotes an integral only over those genera- tropy bound (1.8) is that for each null geodesic generator tors of finite affine parameter length. Now we see that if γ of L of finite affine parameter length, we have the conditionR (2.2) is satisfied, then we obtain from Eq. 1 (2.9) that a 1 dλ ( sak ) (λ) [1 (1)] , (2.2) ′ 0 − A ≤ 4 − A 1 Z S d2x det h (x) [1 (1, x)] . (2.10) L ≤ 4 ΓΛ − A where B Z p λ The generalized entropy bound (1.8) now follows from (λ) exp dλθ¯ (λ¯) (2.3) Eq. (2.10), using the fact that the area AB of the 2- A ≡ "Z0 # surface B is given by

6 2 (λ) 0. Hence we have (λ) 0 everywhere on AB = d x det hΓΛ(x), (2.11) A → A ≥ B L. It is convenient to define G(λ)= (λ), from which Z p A ′ it follows from the definition (2.3) of the area-decrease while the area A ′ of the 2-surface B composed of the B factor and from the Raychaudhuri equationp (2.13) that endpoints of the generators is ′′ ′ G (λ) 2 f(λ)= 2 . (2.19) AB′ = d x det hΓΛ(x) (1, x). (2.12) − G(λ) A ZB p It follows that G′′ is negative. Also the expansion θ Similar arguments show that the Bousso bound (1.5) fol- ′ lows from the assumption (2.4). is always negative, and hence G is always negative, so that G is monotonically decreasing, starting at the value G(0) = 1, and ending at some value G(1) with B. Preliminaries 0 G(1) 1. In particular, we have 0 G(λ) 1 for≤ all λ. For≤ those generators which terminate≤ at caus-≤ tics we have G(1) = (1) = 0, but not all generators From the lemma, its sufficient now to prove the con- will terminate at caustics;A some might terminate at the dition (2.2) or, respectively, the condition (2.4) for each auxiliary spacelike 2-surface B′. finite-affine-parameter-length null generator γ of L. For ease of notation we henceforth drop the dependence on Σ the x coordinates in all quantities. Now the twist along C. Proof of the generalized Bousso bound under the each of the null generators will vanish, since it is van- first set of hypotheses ishing initially on the two surface B, and the evolution equation for the twist [16] then implies that it always van- Using the formula (2.14) and the definition G = √ ishes. The Raychaudhuri equation in the relevant case of A vanishing twist can thus be written as we find that the integral (2.18) satisfies 1 dθ 1 2 1 2 = θ + f(λ), (2.13) Iγ dλ (1 λ)f(λ)G(λ) . (2.20) − dλ 2 ≤ 8 − Z0 a b ab where f =8πTabk k +ˆσabσˆ andσ âb is the shear tensor. From the formula (2.19) for f(λ), this can be written as The function f is non-negative by the null convergence 1 condition [which follows from either Eq. (1.9) or Eqs. ′′ (1.10)–(1.11)] sinceσ ˆ σâb 0 always. The assumption Iγ dλ (1 λ)G (λ)G(λ)/4. (2.21) ab ≤− 0 − (1.9) of our first set of hypotheses≥ now implies that Z Now we have 0 G(λ) 1, so we can drop the factor of s(λ) (1 λ)f(λ)/8. (2.14) ≤ ≤ | |≤ − G(λ) in the integrand of Eq. (2.21). Integrating by parts and using the fundamental theorem of calculus now gives Similarly, our second set of hypotheses (1.10) and (1.11) implies that 1 I [G(0) G(1) + G′(0)] . (2.22) γ ≤ 4 − s(λ)2 α¯ f(λ) (2.15) ≤ 1 Now and G(0) G(1) = 1 G(1) 1 (1), (2.23) s′(λ) α¯ f(λ), (2.16) − − ≤ − A | |≤ 2 since G(0) = 1 and G(1) = (1) (1). Also the where third term in Eq. (2.22) is negative.A It≥ follows A that p α¯ =8πα , α¯ =8πα . (2.17) 1 1 2 2 1 Iγ [1 (1)] , (2.24) We define the quantity ≤ 4 − A

1 as required. I dλ s(λ) (λ). (2.18) γ ≡ A Z0 D. Proof of the original Bousso bound under the Our tasks now are to show that Iγ [1 (1)]/4 when ≤ − A second set of hypotheses Eq. (2.14) holds, and that Iγ 1/4 when Eqs. (2.15) and (2.16) hold, using only the deﬁnition≤ (2.3) of the area- decrease factor and the Raychaudhuri equation (2.13). First, note that without loss of generality we can as- Now by assumption any geodesic generator must ter- sume that the function s(λ) is nonnegative. This is be- minate no later than the point (if it exists) at which cause we can replace s(λ) by s(λ) in the integral (2.18) | |

7 without decreasing the value of the integral, and the as- Next, we split the integral (2.18) into a contribution I1 sumptions (2.15) and (2.16) are satisfied by the function from the interval [0, λ0] and a contribution I2 from the s if they are satisfied by s, since s ′ s′ . interval [λ , 1]: | | | | | | ≤ | | 0 We start by fixing a λ1 in (0, 1), the value of which λ0 1 we will pick later. We then choose a λ0 in [0, λ1] which Iγ = s + s = I1 + I2. (2.31) minimizes f in the interval [0, λ1]; i.e., we choose a λ0 0 A λ0 A which satisfies Z Z In the formula for I , we drop the factor of which is 1 A f(λ0) = min f(λ). (2.25) 1, insert a factor of 1 = dλ/dλ, and integrate by parts 0≤λ≤λ1 to≤ obtain [We assume that the function f(λ) is continuous so that ′ I1 I1b + I1. (2.32) this minimum is attained].∗∗∗ We now show that ≤ Here I1b is the boundary term that is generated, given π2 by f(λ0) 2 [1 (1)] . (2.26) ≤ 2λ1 − A I1b = s(λ0)λ0, (2.33) To see this, let θ0(λ) and 0(λ) be the expansion and area-decrease factor that wouldA be obtained by solving and the Raychaudhuri equation (2.13) with f(λ) replaced by λ0 ′ ′ f(λ ) and using the same initial condition θ (0) = θ(0). I = dλ s (λ)λ. (2.34) 0 0 1 − Since f(λ) f(λ ) for 0 λ λ , it is clear that we Z0 ≥ 0 ≤ ≤ 1 must have Similarly we insert a factor of 1 = d(λ 1)/dλ into the − formula for I2 and integrate by parts, which yields (λ) 0(λ) (2.27) A ≤ A ′ I2 = I2b + I2, (2.35) for 0 λ λ1. But the explicit solution of the Ray- chaudhuri≤ equation≤ for θ (λ) and (λ) is where the boundary term is 0 A0 I2b = s(λ0) (λ0)(1 λ0) (2.36) f(λ0) A − θ0(λ)= 2f(λ0) tan (λ + λˆ) (2.28) − "r 2 # and where p 1 and I′ = dλ [s′ (1 λ)+ s ′(1 λ)] . (2.37) 2 A − A − Zλ0 2 f(λ0) ˆ cos 2 (λ + λ) An upper bound on the total integral is now given by the 0(λ)= q , (2.29) relation A 2 f(λ0) ˆ cos 2 λ I I + I + I′ + I′ . (2.38) q γ ≤ 1b 2b 1 2 ˆ ′ ′ where λ is a constant in [0, 1]. Applying the inequality We now proceed to derive bounds on the integrals I1, I2 (2.27) at λ = λ1 now yields and on the total boundary term I1b + I2b. Consider first the total boundary term, which from Eqs. (2.33) and (2.36) is given by 2 f(λ0) (λ1) 0(λ1) cos λ1 . (2.30) A ≤ A ≤ "r 2 # I + I = s(λ )[λ + (1 λ ) (λ )] . (2.39) 1b 2b 0 0 − 0 A 0 Using the inequality sin χ 2χ/π which is valid for 0 Since (λ ) and λ both lie in [0, 1], this is bounded ≥ ≤ A 0 0 χ π/2, and the inequality (1) (λ1), one can above by s(λ0). If we now use our assumption (2.15), we obtain≤ the upper bound (2.26) fromA Eq.≤ A (2.30). find

I + I α¯ f(λ ), (2.40) 1a 1b ≤ 1 0 and using the bound (2.26)p on f(λ0) ﬁnally yields ∗∗∗The proof extends easily to the non-continuous case. If we choose ǫ> 0 and choose λ0 so that √α¯ π I + I 1 [1 (1)]1/2 . (2.41) 1a 1b ≤ √ − A f(λ0)=(1+ ǫ) g.l.b. {f(λ)| 0 ≤ λ ≤ λ1}, 2λ1 ′ Turn now to the integral I1. Inserting the assumption then we can come as close as we please to satisfying the in- ′ equality (2.55). Hence the inequality (2.55) is satisﬁed. (2.16) into the formula (2.34) for I1 and using the formula (2.19) for f(λ) yields

8 λ0 G′′ Also, the last term in Eq. (2.50) is negative. Hence we I′ 2¯α λ dλ. (2.42) 1 ≤− 2 G obtain the upper bound Z0 Since G is a decreasing function, we can replace the I′ 2¯α [1 (1)] . (2.52) 2 ≤ 2 − A 1/G(λ) in the integrand by 1/G(λ0). If we then integrate by parts and use the fundamental theorem of calculus, we Finally we combine Eq. (2.38) with the upper bounds obtain (2.41), (2.48), and (2.52) for the boundary term I1b + I2b ′ ′ and for the integrals I1 and I2 to yield ′ 2¯α2 ′ I1 [G(λ0) 1 G (λ0)λ0] . (2.43) ≤ G(λ0) − − π 1/2 1 Iγ √α¯1 [1 (1)] +2¯α2 [1 (1)] ≤ √2λ1 − A 1 λ1 − A Since G(λ0) 1 this yields − ≤ π 1 1/2 √α¯1 +2¯α2 [1 (1)] . (2.53) ′ ≤ √2λ 1 λ1 − A ′ 2¯α2G (λ0)λ0 1 I1 . (2.44) − ≤− G(λ0) Choosing the value of λ1 that minimizes this upper bound We now show that for all λ in [0, 1], yields 2 ′ G (λ) 1 α¯ [1 (1)] . (2.45) 1 √ 1/2 − G(λ) ≤ 1 λ − A Iγ π + 2¯α2 [1 (1)] . (2.54) − ≤ s r 2  − A To see this, apply the mean value theorem to the function   G over the interval [λ, 1], which yields Using the definition (2.17) of the the parametersα ¯1 and α¯2 together with the assumption (1.12) yields ′ G(1) G(λ)=(1 λ)G (λ∗), (2.46) − − 1 1/2 ′′ Iγ [1 (1)] for some λ∗ in [λ, 1]. But since G is negative by Eq. ≤ 4 − A ′ ′ (2.19), we have G (λ∗) G (λ), and it follows that 1 ≤ , (2.55) ≤ 4 G′(λ) 1 G(1) 1 as required. Note that our proof actually implies the − G(λ) ≤ 1 λ − G(λ) − inequality 1 1 [1 G(1)] [1 (1)] . (2.47) ≤ 1 λ − ≤ 1 λ − A 1 1/2 1/2 − − SL A [AB AB′ ] . (2.56) ≤ 4 B − Here the last inequality follows from G = √ and 0 G 1. Using the relation (2.45), our upper boundA (2.44)≤ This inequality is stronger than the Bousso bound (1.5) ≤ ′ for I1 now yields but weaker than the generalized Bousso bound (1.8). I′ λ λ 1 2¯α 0 2¯α 1 . (2.48) 1 (1) ≤ 2 1 λ ≤ 2 1 λ III. CONCLUSION − A − 0 − 1 ′ Finally, we turn to the integral I2. The second term We have shown that the generalization (1.8) of ′ in the formula (2.37) for I1 is negative and so can be Bousso’s entropy bound is satisfied under the hypothesis dropped. In the first term, we use the formula G = √ , (1.9), and that the original Bousso bound (1.5) holds un- A the formula (2.19) for f(λ) and our gradient assumption der the hypotheses (1.10) – (1.11). While these hypothe- in the form (2.16) to obtain ses are unlikely to represent relations in any fundamental theory, they appear to be satisfied for matter in a certain 1 ′ ′′ semi-classical regime below the Planck scale. As such, I2 2¯α2 dλ G G(1 λ). (2.49) ≤− λ0 − our results rule out a large class of possible counterex- Z amples to Bousso’s conjecture, including cases involving Now since 0 G(λ) 1 for all λ, we can drop the factor gravitational collapse or other strong gravitational inter- ≤ ≤ of G(λ) in the integrand. If we then integrate by parts actions. As with Bousso’s bound, if the holographic prin- and use the fundamental theorem of calculus we obtain ciple is indeed part of a fundamental theory, it may be that the hypotheses discussed here will provide clues to I′ 2¯α [G(λ ) G(1) + (1 λ )G′(λ )] . (2.50) 2 ≤ 2 0 − − 0 0 its formulation. Note that we do not show in this paper that the en- Now since G(λ ) 1 and G = √ we have 0 ≤ A tropy bounds (1.5) and (1.8) can be saturated by entropy 4-currents sa satisfying our assumptions. However, con- G(λ ) G(1) 1 G(1) 1 (1). (2.51) 0 − ≤ − ≤ − A sideration of simple examples shows that the bound (1.5)

9 comes within a factor of order unity of being saturated We start by discussing the relation between tensors by currents satisfying our second set of hypotheses (1.10) on the spacetime M and tensors on the null surface L. and (1.11). Also, a simple scaling argument shows that We introduce the notation that capital roman indices the least upper bound on the ratio SL/AB for currents A, B, C, . . . denote tensors on L, in the sense of the ab- −2 satisfying our assumptions is of the form α2F (α1α2 ) for stract index convention of Ref. [16]. For any 1-form wa −2 some function F . As a result, at fixed α1α2 the least defined on L, we will denote the pullback of wa to L as upper bound depends continuously on α2. This guaran- a tees that there exist some values of α1 and α2, of order wA = PAwa; (A2) unity, such the entropy bound (1.5) is both satisfied and a can be saturated by entropy currents satisfying the in- this defines the operator PA. Since ka is normal to L, a equalities (1.10) and (1.11). A similar statement is true the pullback of ka vanishes, so PAka = 0. Using the null for the bound (1.8) and the hypothesis (1.9). tetrad introduced at the beginning of Sec. II, we can de- In our analysis above, we have taken the dimension of fine a similar mapping between vectors on M and vectors on L. At any point in L, the projection operation spacetime to be 4. However, in an n-dimensional space- P time with n> 2, the Raychaudhuri equation continues to a a a b 2 v (δ + l k )v (A3) take the form (2.13), except that the coefficient of θ on → b b the right side is now 1/(n 2). Consequently, if we define maps the 4-dimensional tangent space T (M) into the G = 1/(n−2) in the n-dimensional− case, an equation of P 3-dimensional tangent space T (L). Thus one can write the formA (2.19) will continue to hold with the factor of P the mapping (A3) as va vA = QAva, which defines 2 on the right side replaced by (n 2). The remainder a the operator QA. Note that→ the vector la is annihilated of our analysis can then be carried− out in direct parallel a by the projection operation (A3), while ka and vectors with the 4-dimensional case. Thus, with suitable adjust- perpendicular to ka and la are unchanged. We define ments to the numerical factors appearing in Eqs. (1.9), kA = QAka = (d/dλ)A, which is the tangent vector in L (1.10), and (1.11), all of our results continue to hold for a the generators of L. all spacetime dimensions greater than 2. Consider now the integrand in the integral (A1). It is proportional to

ACKNOWLEDGMENTS a s k[albecfd], (A4)

We thank Warren Anderson and Raphael Bousso for where ea and f a are spacelike vector fields such that helpful conversations, and the Institute for Theoretical ka,la,ea,f a is an orthonormal basis. When we pull- Physics (ITP) in Santa Barbara, where this work was back{ the 3-form} (A4) to L, all the terms where the index a initiated, for its hospitality. We also thank Jacob Beken- on ka is free and not contracted with s will be annihi- stein for discussions on the validity of the entropy bound lated. Hence without loss of generality we can replace a b a (1.1) at low temperatures. We particularly wish to thank s with (sbk )l . Thus we obtain from Eqs. (A1) and Raf Guedens for bringing to our attention an error in an (2.6) that− earlier version of this manuscript. EF acknowledges the support of the Alfred P. Sloan Foundation. This work SL = s ǫâbc, (A5) has been supported by NSF grants PHY 9407194 to the L ITP, PHY 95-14726 to the University of Chicago, PHY Z where s = s ka and where the 3-form ˆǫ is defined in 9722189 to Cornell University, PHY 97-22362 to to Syra- − a abc cuse University, and by funds provided by Syracuse Uni- Eq. (2.6) above. versity. As a tool for evaluating the integral (A5), we define an induced connection DA on L by

D vB = P c QB vd (A6) APPENDIX A: FORMULA FOR INTEGRAL A A d ∇c OVER NULL HYPERSURFACE d A A b where v is any vector field on M with v = Qb v . One can check that this formula defines a derivative operator In this appendix we derive the formula (2.7) for the on L. Next we note that the pullbackǫ ÂBC of the 3-form integral (2.6) is parallel transported along each null generator of L with respect to the connection (A6): a SL = s ǫabcd (A1) A ZL k DA ǫˆBCD =0. (A7) of the entropy current over the null hypersurface L [cf. a a Γ This follows from the fact that k , l and ǫabcd are par- Eq. (2.5) above], using the coordinate system (λ, x ) = allel transported along each generator with respect to (λ, x1, x2) defined in Sec. II.

10 the 4-dimensional connection†††. Next, consider the Lie [1] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973); 23, 287 A 49 derivative ~kǫˆ ofǫ ÂBC with respect to k . Since the (1981); , 1912 (1994). result is a 3-formL we must have [2] W.G. Unruh and R.M. Wald, Phys. Rev. D 25, 942 (1982). ~kˆǫ ABC = η ǫÂBC, (A8) [3] W.G. Unruh and R.M. Wald, Phys. Rev. D 27, 2271 L (1983). for some scalar field η. We can define a upper index [4] M.A. Pelath and R. M. Wald, Phys. Rev. D 60, 104009 ABC volume formǫ ˆ by the requirement that (1999) (gr-qc/9901032).

ABC [5] J. D. Bekenstein, gr-qc/9906058. ǫˆ ǫÂBC = 3!. (A9) [6] J. D. Bekenstein and M. Schiffer, Int. J. Mod. Phys. C 1, 355 (1990); J. D. Bekenstein, Phys. Rev. D 30 1669 [A definition is terms of raising indices is inapplicable (1984). here since there is no natural non-degenerate metric on ABC [7] G. ’t Hooft, in Quantum Gravity: proceedings, edited by L.] Now contracting both sides of Eq. (A8) withǫ ˆ M.A. Markov, V.A. Berezin, and V.P Frolov (World Sci- and using Eq. (A7) yields entific, Singapore, 1988); J. D. Bekenstein, in Proceed- ings of the VII Marcel Grossman meeting, edited by xxx, η = D kA = ka, (A10) A ∇a (World Scientific, Singapore, 1996), p. 39-58 (also gr- which is just the usual expansion θ. qc/9409015); L. Smolin, gr-qc/9508064; S. Corley and 53 Next, we define a 3-formǫ ˜ on L by demanding that T. Jacobson, Phys. Rev. D , 6720 (1996). ABC [8] L. Susskind, J. Math. Phys. 36, 6377 (1995) (hep- it coincide with ˆǫ on the 2-surface B, and that it be ABC th/9409089). Lie transported along the generators of L. If we write [9] R. M. Wald, “The Thermodynamics of Black Holes”, gr- ǫÂBC = ζǫÃBC , where ζ is a scalar field on L, it follows qc/9912119. from Eq. (A8) with η = θ that [10] J. D. Bekenstein Phys. Rev. D 9, 3292 (1974). [11] G. ’t Hooft, “Dimensional reduction in quantum grav- ζ = θζ. (A11) L~k ity”, gr-qc/9310026. Solving this equation using the definition (2.3) of the [12] R. Bousso, J. High Energy Phys. 07, 004 (1999) (hep- area-decrease factor yields ζ = . Thus we see that th/9905177). 06 the geometrical meaning of the factorA is that it is the [13] R. Bousso, J. High Energy Phys. , 028 (1999) (hep- th/9906022). ratio between the Lie-transported 3-volumeA formǫ ˜ ABC [14] W. Fischler and L. Susskind, “Holography and cosmol- and the parallel transported 3-volume form ˆǫ , where ABC ogy,” hep-th/9806039. in both cases one starts from the 2-surface B. [15] For other comments on extensions of the original Consider now the specific coordinate system (λ, xΓ)= 1 2 Fischler-Susskind bound [14], see R. Dawid, “Holo- (λ, x , x ). In this coordinate system the fact thatǫ ÃBC graphic Cosmology and its Relative Degrees of Freedom,” is Lie transported along the generators translates into hep-th/9907115. ∂ [16] R. M. Wald, General Relativity (The University of ˜ǫλx1x2 (λ, x)=0, (A12) Chicago Press, Chicago, 1984). ∂λ [17] W. Zurek, Phys. Rev. Lett. 49 (1982) 1683; W.H. Zurek so that and K.S. Thorne, Phys. Rev. Lett. 54 (1985) 2171; K.S. Thorne, W.H. Zurek and R.H. Price, in Black ǫ˜λx1x2 (λ, x)=˜ǫλx1x2 (0, x)= det hΓΛ(x), (A13) Holes: The Membrane Paradigm, edited by K.S. Thorne, R.H. Price and D.A. MacDonald (Yale University Press, where ˜ǫλx1x2 (λ, x) denotes one ofp the coordinate compo- New Haven, CT, 1986) p. 280; W.G. Unruh and R.M. Γ nents of the tensorǫ ÃBC in the coordinate system (λ, x ), Wald, Phys. Rev. D 25, 942 (1982); Phys. Rev. D 27, 1 2 x (x , x ) as before, and hΓΛ is the induced 2-metric 2271 (1983); correction: M.J. Radzikowski and W.G. Un- on≡B. It follows that ruh, Phys. Rev. D 37, 3059(E) (1988); R.M. Wald, Quan- tum Field Theory in Curved Spacetime and Black Hole ǫˆ 1 2 (λ, x)= (λ, x) det hΓΛ(x). (A14) Thermodynamics, (University of Chicago Press, Chicago, λx x A 1994); R. Sorkin, Phys. Rev. Lett. 56, 1885 (1986); p Combining this with the formula (A5) for the entropy V.P. Frolov and D.N. Page, Phys. Rev. Lett. 71 (1993) flux finally yields the formula (2.7). 3902. [18] David A. Lowe, J. High Energy Phys. 010, 026 (1999) (hep-th/9907062). [19] L.H. Ford and T.A. Roman, Phys. Rev. D 51, 4277 (1995). [20] L.H. Ford and T.A. Roman, Phys. Rev. D 55, 2082 (1997) (gr-qc/9607003). †††Note however that unlike the situation in the four dimen- [21] E.´ E.´ Flanagan, Phys. Rev. D 56, 4922 (1997). sional setting, in general DA ˆǫBCD 6= 0; i.e., ˆǫABC is not [22] C.J. Fewster and S.P. Eveston, Phys. Rev. D 58, 084010 covariantly constant with respect to DA.

11 (1998) (gr-qc/9805024). [23] L. H. Ford and T. A. Roman, Phys. Rev. D 60, 104018 (1999) (gr-qc/9901074). [24] R. M. Zalaletdinov, Bull. Astr. Soc. India 25, 401 (1997) (gr-qc/9703016).