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Holography with Timelike Bulk Hypersurfaces

We propose a new version of holographic principle. This proposal extends the holographic principle based on the lightsheet to the one constraining the entropy passing through bulk hypersurface of timelike geodesics by the boundary area divided by 4G. We give a proof of the proposal in the classical regime based on a simple local entropy condition.


Introduction
The holographic principle may be a key paradigm for the understanding of the true nature of gravity and string theories. As suggested first by 't Hooft and Susskind, it states that the degrees of freedom for the gravitational system reside not in the bulk but in its boundaries [1,2]. Furthermore the number of boundary degrees of freedom per Planck area is constrained not to exceed unity.
This principle clearly contradicts with the current framework of quantum field theories because their numbers of degrees scale like the bulk volume not by the boundary area. These ideas are precisely exemplified in the formulation of AdS/CFT correspondence [3,4]. When the gravity and field theories coexist like our Universe, the UV structure of quantum field theories must be modified drastically in order to have fundamental degrees limited by their boundary areas.
Application of the above holographic principle to more general spacetime including cosmology is problematic [5]. For the flat FRW universe as an example, the matter entropy scales like the coordinate volume and the bulk entropy eventually wins over the area bound for large enough region. The authors of [5] suggested that the entropy crossing past lightlike region generated from the particle horizon should be bounded by the area of the particle horizon. Based on the subsequent works [6], Bousso has given a covariant version of the holographic principle that may be applicable for general backgrounds [7]. It states that the entropy on the lightsheet L is bounded by its boundary area divided by 4G. The lightsheet is defined by the bundle of lightlike geodesics that are orthogonally generated from the boundary surface with nonpositive expansion away from the boundary surface. The shape of the boundary surface may be arbitrary once it is consistent with the nonpositive expansion condition.
More stronger bound is proposed by Flanagan, Marolf and Wald (FMW) [8], asserting that the entropy flux passing through the lightsheet orthogonally generated from one boundary and ending on another boundary is bounded by the difference of the boundary areas divided by 4G.
Interestingly this version can be proven in the semiclassical limit provided certain local conditions on the entropy are assumed [9,10]. The proof is utilizing focusing nature of the lightlike geodesic based on the Raychaudhuri equation.
In this note, we observe that the Raychaudhuri equation is also for the timelike geodesics, which plays an important role in the investigation of the global structures of spacetime including horizons and singularities. We extend the above holography to the case where the bulk hypersurface consists of timelike geodesics. The proposal dictates that the entropy passing through the bulk hypersurface consisting of timelike geodesics orthogonally generated from a codimension two spacelike hypersur-face B and terminating on B ′ , should be bounded by the difference of the boundary area divided by 4G. The proof shall be given in the classical regime with simple well motivated local entropy conditions.
There is a strong supportive argument for the above timelike holography. Consider a d + 1 dimensional geometry of the form, M × S 1 , where M is a d dimensional spacetime and S 1 denotes a circle. Lightlike geodesics in the d+1 dimensions correspond to timelike geodesics in the spacetime M if the momentum along the circle is nonvanishing. This way the lightlike holography in M × S 1 may lead to the timelike holography in the lower dimensional spacetime 1 . This paper is organized as follows. In Section 2, we review the covariant holography proposed by Bousso and extended by FMW. In Section 3, we give the proof (in the classical regime) based on the simple local conditions on the entropy density. Section 4 deals with our proposal on the holography with timelike bulk hypersurface. We discuss a proof of the statement starting from appropriate local conditions on the entropy flux. Last section is devoted for the discussions.

Review of the lightlike holographic principle
For later use, we will review the basic construct of the holographic principle introduced by Bousso and later extended by FMW.
The Bousso's holographic principle compares the entropy on the lightsheet L with its boundary area divided by 4G. The lightsheet is defined by the bundle of lightlike geodesics that are orthogonally generated from the boundary surface with nonpositive expansion away from a spacelikesurface.
The principle then dictates that where S L is the entropy flux passing through the lightsheet and A the boundary area.
The lightsheet consists of collections of lightlike geodesics k a = ∂ ∂λ a emanated orthogonally from a given codimension-two spacelike surface B where λ is the affine parameter. The expansion, describes the changing rate of the logarithm of the cross sectional area of the light rays with respect to λ. The definition of lightsheet requires the expansion away from the boundary surface to be nonpositive. The lightsheet should be terminated if the expansion turns into a positive value or the geodesics hit singularities.
Though the local concept of entropy is not well defined, we introduce the entropy current density s a , which may have a well defined physical meaning in the thermodynamic limit. The entropy flux density s passing through the lightsheet is given by with the vector n a = −k a (+k a ) normal to the future (past) directed lightsheet. The total entropy passing through the lightsheet is the volume integral, where y is the coordinate for the boundary and h(y) is the determinant of the induced metric on the boundary. The area factor A represents the growth of the unit area along the affine parameter and the cross sectional area at λ is then given by More stronger bound, has been proposed by FMW, where S L(B−B ′ ) corresponds to the entropy flux passing through the lightlike surface orthogonally generated from B and ending on B ′ before reaching caustic or singularities. One compares this entropy to the difference of the area of B and B ′ . This version of the entropy bound is the one we are interested in this note.

Conditions for the holography
The latter version of the holography can be shown to hold in the classical regime with certain conditions on the entropy flux density are satisfied.
Here we review the conditions and the proof given recently in Refs. [9,10]. The entropy conditions used are: The second condition is on the initial condition for the entropy and the first corresponds to the local version of the Bekenstein bound as discussed in Ref. [9,13,14].
Based on these two conditions, the proof of the entropy bound is very simple in the semiclassical limit where one may use the Einstein equation, The expansion of the bundle of the lightlike geodesics obeys the Raychaudhuri equation where σ ab and ω ab are the shear and the twist. Since the lightsheets are hypersurface orthogonal, the twist ω ab vanishes.
The FMW bound has a differential form. We note that the local inequality, implies the integral version (7). In particular considering B ′ infinitesimally away from B in terms of the affine parameter, one gets the condition (ii) from the integral version. It is clear that the initial condition on the entropy should hold without any disposal because the part of statement of the holography is the initial condition itself. Since s(0) is not to do with how the boundary surface is curved, the initial condition is not satisfied for an arbitrary boundary surface. Rather at a given initial spacetime point, one needs more curved boundary in order to have the negative of expansion larger than the initial entropy flux density. Therefore the initial condition puts restriction not on the initial entropy density but on the boundary surface. In Ref. [9], the authors insisted θ(0) ≤ 0, which is clearly different from ours.
From the two conditions, the proof of the inequality (10) is quite simple. Using (9), the first condition and the Einstein Equation, we have Hence together with the initial condition, one has related to the Bekenstein bound as discussed recently in Ref. [9,13,14]. We shall not repeat the arguments here. One might try using the following weaker condition; But the expansion is not such a physical quantity that is fixed solely by the spacetime point. Rather it is determined by the point as well as by the shape of the initial surface. On the other hand, s ′ and T ab k a k b are the physical quantities (of course depending on the geodesic k a ) that are nothing to do with the shape of the initial surface. In this respect we view that the relation (13) is not desirable.
The saturation of the above holographic principle requires θ = 0 as well as s ′ = 2πT ab k a k b . The

Holography with timelike sheets
In the discussion of the holography with lightsheets, the Raychaudhuri equation plays an important role. The Raychaudhuri equation is, however, not just for the bundle of lightlike geodesics but also for timelike geodesics. In this respect here we discuss whether one may extend the formulation of holography to the case of timelike bulk hypersurface consisting of timelike geodesics.
Let us consider bundle of timelike geodesics orthogonally generated from a codimension two spacelike hypersurface B and terminating on B ′ before reaching caustics or singularities. We shall denote the timelike geodesic by ξ a = (∂/∂τ ) a (with ξ a ξ a = −1) where τ is the proper time along the geodesic. We introduce the entropy flux density s by where n a is the normal vector to the timelike bulk hypersurface. The entropy on the timelike bulk hypersurface is then where y is the coordinate for the boundary and B represents the growth of the unit area along τ given by with the area expansion θ A . Note that there are two possible choices for the normal vector field n a . We shall always take the one that makes S T (B−B ′ ) nonnegative. Infinitesimally away from the timelike hypersurface to the normal direction, we extend the geodesics such that geodesics are still hypersurface orthogonal. The hypersurface orthogonality condition is rather strict and fix the extension in the infinitesimal region. Note that, in the timelike case, θ = ∇ a k a describes not the area expansion but the volume expansion rate. We introduce the coordinate vector m a = (∂/∂s) a whose direction agrees with n a at the point on the timelike surface where we evaluate the expansion rate. By straightforward computation [15], one may show that where θ m denotes the expansion of the normal vector, d dτ ln |m|. The timelike bulk hypersurface is further restricted by the initial condition and the condition on the expansion of the vector m a θ m (τ ) ≥ 0 .
We shall call the resulting hypersurface as "timelike sheet". At first sight, the latter condition appears too restrictive. However, we note that the effect of tidal force in general makes size of body grow when it is attracted toward a distribution of mass. This implies that θ m is nonnegative rather generically for the geodesics of massive objects attracted toward the mass distribution. We shall give detailed examples of this elsewhere [15]. Now the proposal is simple. The entropy passing through the timelike sheet should be constrained by the difference of the boundary area divided by 4G; where T (B − B ′ ) denotes the timelike sheet.
For the proof (valid in the classical regime), we shall use the local entropy condition where dots represent derivatives with respect to τ and T = T a a . As in the lightlike case, the initial condition (18) is not for the initial entropy flux but for the choice of the initial boundary surface.
The holography here is much stronger version than the previous ones, and implies the lightlike holography we discussed before. Namely if the holography is used for the timelike sheet that consists of almost lightlike geodesics, the lightlike holography follows as the limiting case. Therefore the timelike holography needs a stronger requirement on the local entropy flux. Considering the almost lightlike geodesics, the expansion of the normal vector, θ m , approaches zero and the condition (iii) implies the condition (i).
The proof of the above in the classical regime is straightforward. The Raychaudhuri equation where we used the Einstein equation. Since geodesics are hypersurface orthogonal, the twist ω ab is zero automatically.
Again we note that the differential form of the holography, implies the integral version, where the second inequality follows from the condition (19).
Using (21) and the condition (iii), we have Together with the initial condition, one has Like the case of the lightlike holography, so called spacelike projection theorem holds if certain conditions are met [7]. If the future directed timelike bulk hypersurface is complete i.e. B is the only boundary, the entropy on the spacelike bulk enclosed by the boundary would be limited by the boundary area divided by 4G. This is a consequence of the generalized second law. Therefore if any future directed timelike sheet is complete, the original 't Hooft version of the holographic principle follows.
Main difference between the lightlike and the timelike holography lies in the form of the relevant energy conditions. For the lightlike case, the null energy condition, is relevant. Let us consider a diagonalizable energy-momentum tensor that takes the form where (t a , x On the other hand, the energy condition related to the the timelike holography is the strong energy condition T ab ξ a ξ b + 1 D − 2 T ≥ 0 for all timelike ξ a . For the above form of the energy momentum tensor, the strong energy condition implies (D − 3)ρ + i p i ≥ 0 and ρ + p i ≥ 0 for i = 1, · · · , D − 1 .
It is generically believed that, for any classically reasonable matter, the energy density T ab ξ a ξ b ≥ 0 for any timelike vector ξ a . This energy condition is referred as weak energy condition. In term of the energy density and momentum, the weak energy condition implies ρ ≥ 0 and ρ + p i ≥ 0 for i = 1, · · · , D − 1 . Finally let us illustrate one specific example in which the timelike holography leads to a stronger statement than the lightlike holography. We take the case where the future-directed timelike sheet is complete before reaching singularity with the initial conditions on the expansion respected. (See Fig. 2.) Using the lightlike and the timelike holography, one has S L ≤ 1 4G A and S T ≤ 1 4G A with A being the boundary area. In addition, by the generalized second law, S L ≤ S T . Thus, one gets This is clearly a stronger statement than the one from the lightlike holography i.e. S L ≤ 1 4G A.

Discussions
In this note, we propose a new version of holographic principle which compares the entropy passing through a bulk hypersurface consisting of timelike geodesics generated from a boundary surface to the boundary area divided by 4G. We discuss the corresponding entropy condition closely related to the Bekenstein bound as a generalization of the lightlike case. Based on the condition, we prove our proposal in the classical regime. It may be quite interesting to derive our timelike holography from the lightlike holography of one higher dimensions via the dimensional reduction with a nonvanishing Kaluza-Klein momentum.
Our proposal is stronger than the lightlike case because the lightlike case can be obtained by a limiting procedure in which the velocity field along the timelike geodesics approaches the light velocity arbitrarily closely.
One may wonder whether further extension of the holography to the one based on spacelike geodesics is possible or not. However the spacelike version, if possible, will lead to the 't Hooft version of holography for general backgrounds, for which we already gave the flat FRW universe as a counterexample.
There are number of points to be studied further. Here we do not discuss about the possible quantum modification of the timelike holographic principle. In the lightlike case, an attempt is made in Ref. [10]. The Raychaudhuri equation also plays an essential role for the proof of the singularity theorem. As we discussed, the presence of singularity limits lightsheets and timelike sheets. Singularities seem to have some connections with the entropy of the system. In presence of the cosmological singularities like the big bang, the meaning of the generalized second law is not quite clear. Namely the entropy of certain region may not have any operational meaning because the part of the region may lie outside of the observer's particle horizon. Moreover, the notion of time loses its meaning at the big bang or other singularities. Can we talk about then the fate of degrees of freedom or entropy there? The holographic principle may shed some light on how to tackle the singularities.