Boundary of the future of a surface

Author(s): Akers, C; Bousso, R; Halpern, IF; Remmen, GN | Abstract: © 2018 authors. We prove that the boundary of the future of a surface K consists precisely of the points p that lie on a null geodesic orthogonal to K such that between K and p there are no points conjugate to K nor intersections with another such geodesic. Our theorem has applications to holographic screens and their associated light sheets and in particular enters the proof that holographic screens satisfy an area law.


I. THEOREM
In this paper, we prove the following theorem establishing necessary and sufficient conditions for a point to be on the boundary of the future of a surface in spacetime. (An analogous theorem holds for the past of K.) Theorem 1: Let ðM; gÞ be a smooth, 1 globally hyperbolic spacetime and let K be a smooth codimension-two submanifold of M that is compact and acausal. Then a point b ∈ M is on the boundary of the future of K if and only if all of the following statements hold: (i) b lies on a future-directed null geodesic γ that intersects K orthogonally. (ii) γ has no points conjugate to K strictly before b. (iii) γ does not intersect any other null geodesic orthogonal to K strictly between K and b. Theorem 1 enumerates the conditions under which a light ray, launched normally from a surface, can exit the boundary of the future of that surface and enter its chronological future. In essence, this happens only when the light ray either hits another null geodesic launched orthogonally from the surface or when the light ray encounters a caustic, in a sense that will be made precise in terms of special conditions on the deviation vectors for a family of infinitesimally separated geodesics. These two possibilities for the fate of the light ray are illustrated in Fig. 1.
The theorem is useful for characterizing the causal structure induced by spatial surfaces. In particular, if K splits a Cauchy surface into two parts, then Theorem 1 implies that the four orthogonal null congruences fully characterize the associated split of the spacetime into four portions: the future and past of K and the domains of dependence of each of the two spatial sides (see Fig. 2). This is of particular interest when K is a holographic screen [1]. Then some of the orthogonal congruences form light sheets [2] such that the entropy of matter on a light sheet is bounded by the area of K. This relation makes precise the FIG. 1. Possibilities for how a null geodesic orthogonal to a surface can exit the boundary of its future. In this example, a parabolic surface K (blue line) lies in a particular spatial slice. A future-directed null geodesic (red line) is launched orthogonally from p. At q, it encounters a caustic, entering the interior of the future of K (red dashed line). The point q is conjugate to K. Other null geodesics orthogonal to K (black lines) encounter nonlocal intersections with other such geodesics along the green line, where they exit the boundary of the future of K.
notion that the Universe is like a "hologram" [3][4][5] and should be described as such in a quantum gravity theory. Such holographic theories have indeed been identified for a special class of spacetimes [6].
Specifically, Theorem 1 plays a role in the recent proof of a novel area theorem for holographic screens [7,8], where it was assumed without proof. It also enters the analogous derivation of a related generalized second law in cosmology [9] from the quantum focusing conjecture [10].
Although our motivation lies in applications to general relativity and quantum gravity, we stress that the theorem itself is purely a statement about Lorentzian geometry. It does not assume Einstein's equations and so in particular does not assume any conditions on the stress tensor of matter.
Related work. Parts of the "only if" direction of the theorem are a standard textbook result [11], except for (iii), which we easily establish. The "if" direction is nontrivial and takes up the bulk of our proof.
Reference [12] considers the cut locus, i.e., the set of all cut points associated with geodesics starting at some point p ∈ M. Given a geodesic γ originating at p, a future null cut point, in particular, can be defined in terms of the Lorentzian distance function or equivalently as the final point on γ that is in the boundary of the future of p. As shown in Theorem 5.3 of Ref. [12], if q is the future null cut point on γ of p, then either q corresponds to the first future conjugate point of p along γ, or another null geodesic from p intersects γ at q, or both. Our theorem can be viewed as an analogous result for geodesics orthogonal to codimension-two surfaces and a generalization of our theorem implies the result of Ref. [12] as a special case. The codimension-two surfaces treated by our theorem are of significant physical interest due to the important role of holographic screens in the study of quantum gravity (see, e.g., Ref. [13] for very recent results on the coarse-grained black hole entropy). We encountered nontrivial differences in proving the theorem for surfaces. Moreover our condition (ii) places stronger constraints on the associated deviation vector, as we discuss in Sec. II B. 2 The previously known parts of the "only if" direction of Theorem 1 were originally established in the context of proving singularity theorems [16,17]. It would be interesting to see whether Theorem 1 can be used to derive new or stronger results on the formation or the cosmic censorship of spacetime singularities.
Generalizations. As we are only concerned with the causal structure, the metric can be freely conformally rescaled. Thus, a version of Theorem 1 still holds for noncompact K, as long as it is compact in the conformal completion of the spacetime, i.e., in a Penrose diagram. A situation in which this may be of interest is for surfaces anchored to the boundary of anti-de Sitter space.
Furthermore, the theorem can be generalized to surfaces of codimension other than two, but in that case we can say less about the type of conjugate point that orthogonal null geodesics may encounter. We will discuss this further in Sec. III.
Notation. Throughout, we use standard notation for causal structure. A causal curve is one for which the tangent vector is always timelike or null. The causal (respectively, chronological) future of a set S in our spacetime M, denoted by J þ ðSÞ [respectively, I þ ðSÞ] is the set of all q ∈ M such that there exists p ∈ S for which there is a future-directed causal (respectively, timelike) curve in M from p to q. For the past [I − ðSÞ, J − ðSÞ, etc.], similar definitions apply. We will denote the boundary of a set S by _ S. Standard results [11] include that I AE ðSÞ is open and that _ J AE ðSÞ ¼ _ I AE ðSÞ. We will call a set S acausal if there do not exist distinct p; q ∈ S for which there is a causal path in M from p to q. A spacetime is said to be globally hyperbolic if it contains no closed causal curves and if J þ ðpÞ ∩ J − ðqÞ is compact for all p; q ∈ M. Equivalently [18], M has the topology of Σ × R for some Cauchy surface Σ; that is, Σ is a surface for which, for all p ∈ M, every inextendable timelike curve through p intersects Σ exactly once.
Outline. In Sec. II, we review the notion of a conjugate point and establish some useful lemmas. In Sec. III, we prove Theorem 1.
FIG. 2. In this generic Penrose diagram, the codimension-two surface K (black dot) splits a Cauchy surface Σ (dashed line) into two parts Σ in , Σ out . This induces a splitting of the spacetime M into four parts: the past and future of K (red, yellow) and the domains of dependence of Σ in and Σ out (green, blue) [8]. Theorem 1 guarantees that this splitting is fully characterized by the four orthogonal null congruences originating on K (black diagonal lines). 2 Recently, we were made aware of Refs. [14,15], which also generalize the results of Ref. [12] to codimension-two surfaces. Our work goes further in that we more strongly constrain the type of conjugacy to be that of Def. 17. This is crucial for making contact with the notion of points "conjugate to a surface" used in the physics literature, e.g., in Ref. [11].

A. Exponential map
Let ðM; gÞ be a smooth, globally hyperbolic spacetime of dimension n > 2. Thus, M is a manifold with metric g of signature ð−; þ; Á Á Á ; þÞ. (As already noted, we will be concerned only with the causal structure of M, so g need only be known up to conformal transformations.) For p ∈ M, let T p M be the tangent vector space at p and let TM ≡ ⋃ p∈M fpg × T p M be the tangent bundle of M. TM has a natural topology that makes it a manifold of dimension 2n. In the open subsets associated with charts of M, TM is diffeomorphic to open subsets of R 2n , corresponding to n coordinates for the location of p ∈ M and n components of a tangent vector v ∈ T p M. The tangent space of TM at ðp; vÞ is For every ðp; vÞ ∈ TM, there is a unique inextendable geodesic, where a; b ∈ R ∪ f−∞; ∞g, with affine parameter s and tangent vector v ∈ T p M given by the pushforward of d=ds by Restrictions of exp to submanifolds of TM are frequently of interest. To study the congruence of geodesics emanating from a given point, one may restrict to exp p ∶ T p M → M; v ↦ c p;v ð1Þ. Moreover, one can define the differential of exp p , exp pÃ ∶ T v T p M → T c p;v ð1Þ M, which describes how exp p v varies due to small changes in v. See Fig. 3 for an illustration of the exponential map and its differential. In this paper, we will consider a different restriction suited to the study of the geodesics orthogonal to a given spatial surface; we will define the differential in more detail for this restriction below.
Let K ⊂ M be a smooth submanifold. We consider the normal bundle where T p K ⊥ is the two-dimensional tangent vector space perpendicular to K at p. The normal bundle has the structure of an n-dimensional manifold. Its tangent space at ðp; vÞ ∈ NK is Here, T p K is the tangent space of p in the manifold K; that is, T p K is the subspace of T p M normal to T p K ⊥ . Note that T p K is of the same dimension as K. Definition 3.-The surface-orthogonal exponential map is the restriction of exp to NK. Definition 4.-The Jacobian or differential of the exponential map is given by It is a linear map between vectors that captures the response of exp K to small variations in its argument. It is defined by requiring that ðexp KÃ wÞðfÞ ¼ wðf∘exp K Þ for any function f∶ M → R. Note exp KÃ w is the pushforward of w by exp K . If x α are coordinates in an open neighborhood of ðp; vÞ ∈ NK and y β are coordinates in an open neighborhood of exp K ðp; vÞ ∈ M and we write the vectors in coordinate form, w ¼ P w α ð∂=∂x α Þ and exp KÃ w ¼ Pŵ β ð∂=∂y β Þ, then the components are related by the Jacobian matrix, See Fig. 4 for an illustration of exp K , exp KÃ , and the various tangent spaces used in this paper.
Definition 5.-A Jacobian is an isomorphism if it is invertible, i.e., if it has no eigenvectors with eigenvalue zero.
Since ðM; gÞ and K are smooth, exp K is smooth. The inverse function theorem [19] thus implies the following. Definition 9.-Let ∂ s denote the partial derivative with respect to s. It follows from the above definition that the pushforward S ≡ f Ã ð∂ s Þ ∈ TM is tangent to any geodesic γ r . Similarly, R ≡ f Ã ð∂ r Þ ∈ TM is tangent to any curve μ s ∶ Q → M; r ↦ fðr; sÞ at fixed s. For general families of curves, R represents the deviation vector field of the congruence. In the special case of a geodesic congruence, R restricted to any γ r is called a Jacobi field on γ r .
Remark 10.-The Jacobi field R satisfies the geodesic deviation equation on Q, where RðA; BÞ ≡ ½D A ; D B − D ½A;B is the curvature tensor [11,20] and D V ¼ V μ ∇ μ is the covariant derivative, defined with respect to the Levi-Cività connection, along a vector V.
The exponential map can be used to generate a oneparameter family of geodesics and its derivative exp Ã generates the associated Jacobi fields. We first recall the more familiar case of geodesics through a point p, generated by exp p , as follows.
Remark 11.-LetR;Ŝ ∈ T p M and letR andS be the naturally associated constant vector fields in TT p M. 4 Then fðr; sÞ ¼ exp p ½sðŜ þ rRÞ is smooth and defines a oneparameter family of geodesics. Its tangent vector field is S ¼ exp pÃ j sðŜþrRÞ ðS þ rRÞ and its deviation or Jacobi field is R ¼ exp pÃ j sðŜþrRÞ sR. 5 It is clear from this construction that exp p is singular (i.e., exp pÃ fails to be an isomorphism) at sðŜ þ rRÞ if and only if there exists a nontrivial Jacobi field of the geodesic γ r that vanishes at fðr; sÞ and fðr; 0Þ. This establishes the equivalence of two common definitions of conjugacy to a point p.
Remark 12.-A conjugate point in a geodesic congruence with tangent vector k μ corresponds to a caustic, which is a point at which the expansion θ ¼ ∇ μ k μ goes to −∞.
We turn to the case relevant to this paper: a oneparameter family of geodesics orthogonal to a smooth, compact, acausal, codimension-two submanifold K. (For example, K could be a topological sphere at an instant of time.) Subject to this restriction, the map f and vector fields R and S are defined as before, with T p M replaced by T p K ⊥ . One can choose the parameters ðr; sÞ such that fðr; 0Þ ∈ K and fð0; 0Þ ¼ p. The map ν∶ r ↦ ðfðr; 0Þ; Sj ðr;0Þ Þ is a smooth curve in NK with tangent vectorR ∈ TNK. From this curve, the one-parameter family can be recovered as fðr; sÞ ¼ exp fðr;0Þ sSj ðr;0Þ ¼ exp K ðfðr; 0Þ; sSj ðr;0Þ Þ: ð9Þ Remark 13.-We will be interested in the Jacobi field R ≡ f Ã ∂ r only along one geodesic, say γ at r ¼ 0. By Eq. (8) this depends only on the initial data S andR at p. Thus Rj ð0;sÞ will be the same for any curve ν with tangent vectorR at ðp; Sj ð0;0Þ Þ ∈ NK. Conversely, one can extend any givenR at ðp; Sj ð0;0Þ Þ ∈ NK to a (nonunique) one-parameter family of geodesics by picking such a curve ν. We now take advantage of this freedom in order to find an explicit expression for the Jacobi field in terms of exp KÃ . FIG. 4. An illustration of the surface-orthogonal exponential map exp K evaluated at p ∈ K, which takes a vector in T p K ⊥ to a point c p;v ð1Þ in M. Here, as in text, the tangent space at p, T p M, is broken up as a product T p K ⊥ × T p K. Also shown is the Jacobian exp KÃ at v ∈ T p K ⊥ , which takes a vector w ¼ ðw 1 ; w 2 Þ ∈ T p;v NK ¼ T p K × T v T p K ⊥ to a vector in T c p;v ð1Þ M. 4 Concretely, one can first choose a neighborhood U of p diffeomorphic to R n , which exists since M is a manifold, and then choose a map ϕ∶ U → T p M such that the pushforward ϕ Ã is the identity map from T p M to T v T p M for some v; thenR andS can be defined asR ¼ ϕ ÃR andS ¼ ϕ ÃŜ for v ¼R orŜ, respectively. 5 The subscript is the point where the Jacobian map is evaluated. The vector the Jacobian acts on appears to its right. By Eq. (4), one can uniquely decomposeR ¼ ðŘ;RÞ, withŘ ∈ T p K andR ∈ T S T p K ⊥ . Let π be the defining projection of the fiber bundle, π∶ NK → K. Then μ ≡ πðνÞ is a curve on K with tangent vectorŘ at p. Let fðr; 0Þ ¼ μðrÞ.
Further, let Sj ðr;0Þ ∈ T fðr;0Þ K ⊥ be defined by K-normal parallel transport 6 of the vector Sj ð0;0Þ þ rR ∈ T p K ⊥ along μ from p to μðrÞ. HereR ∈ T p K ⊥ is the vector naturally associated withR ∈ T S T p K ⊥ . Similarly, we defineS ∈ T S T p K ⊥ to be the vector naturally associated with Sj ð0;0Þ . Lemma 14: With the above choices and definitions, Eq. (9) yields a suitable one-parameter family of geodesics. The corresponding Jacobi field and tangent vector along γ can be written as We note thatŘ andR encode the initial value and derivative, respectively, of R, in accordance with the initial value problem set up in Remark 13. From Eq. (10), we obtain a criterion for conjugacy equivalent to that of Def. 7.
Remark 15.-In the above notation, the map exp K is singular at ðp; sSj ð0;0Þ Þ ∈ NK if and only if the geodesic γ possesses a nontrivial Jacobi field that vanishes at exp K ðp; sSj ð0;0Þ Þ and is tangent to K at p.
Specifically, our interest lies in null geodesics orthogonal to K. We now show that their conjugate points satisfy an additional criterion on the associated eigenvector of exp KÃ .
Prior to the conjugate point, the map exp KÃ is a linear isomorphism; hence it maps the (1 þ 1)-dimensional subspace T S T p K ⊥ ∋R of T p;S NK into a (1 þ 1)dimensional subspace exp KÃ T S T p K ⊥ of T fð0;1Þ M. This subspace contains both the null tangent vector Sj ð0;sÞ and the component exp KÃ j ðp;sSj ð0;0Þ Þ ð0; sRÞ of the Jacobi field R, which is itself a Jacobi field since our choice of initial datā R was arbitrary. In a (1 þ 1)-dimensional space, the only vectors orthogonal to a null vector S are proportional to S. The general solution to Eq. (8) for a Jacobi field proportional to the tangent vector S is ðα þ βsÞSj ð0;sÞ . Therefore exp KÃ j ðp;sSj ð0;0Þ Þ ð0; sRÞ must have this form for some real constants α, β. At s ¼ 0, exp KÃ j ðp;sSj ð0;0Þ Þ ð0; sRÞ vanishes trivially, so α ¼ 0.
▪ We now define a refinement of the notion of a conjugate point.
Definition 17.-Let γðsÞ be a geodesic orthogonal to K at p, with γð0Þ ¼ p and with conjugate point ðp; vÞ. Then there exists a nontrivial Jacobi field RðsÞ ∈ TM that vanishes at q ¼ exp K ðp; vÞ and is tangent to K at p. We say that q is conjugate to (the surface) K if R is nonvanishing at p.
Remark.-By Lemma 16,Ř ≠ 0, so the Jacobi field associated withR as defined in Eq. (10) does not vanish at p and hence, if ðp; sSj ð0;0Þ Þ ∈ NK is a conjugate point, then the point exp K ðp; sSj ð0;0Þ Þ is conjugate to K for γ null. . The point q 1 is conjugate to the point p 1 , with the Jacobi field illustrated by the red arrows. The point q 2 is conjugate to the surface K (blue line), at the point p 2 , with the Jacobi field illustrated by the green arrows. Geodesics orthogonal to K are shown in black. If a general conjugate point lies along an orthogonal null geodesic, then by Lemma 16 there exists a Jacobi field such that the conjugate point is of the surface type. Hence, this type of conjugacy appears in Theorem 1. 6 Given a vector v ∈ T p K ⊥ , normal parallel transport defines a vector field vðrÞ along μ normal to K such that the normal component of its covariant derivative along μ vanishes, D ⊥ r vðrÞ ¼ 0. Given μðrÞ and the initial vector in T p K ⊥ , vðrÞ is unique by Lemma 4.40 of Ref. [21]. Moreover, we can similarly define the notion of a point conjugate to another point.
Definition 19.-Given a nontrivial Jacobi field R for a segment γ of a geodesic such that R vanishes at p and q, we say that q is conjugate to (the point) p.
See Fig. 5 for an illustration of the two types of conjugate points defined in Defs. 17 and 19.

III. PROOF OF THE THEOREM
We now prove Theorem 1.
Conclusion (iii) follows by contradiction: let γ 0 be a distinct null geodesic orthogonal to K that intersects γ at some point q strictly between b and K. By acausality of K, γ 0 ∩ K is a single point, p 0 , which is distinct from q. Hence, K can be connected to b by a causal curve that is not an unbroken null geodesic, namely, by following γ 0 from p 0 to q and γ from q to b. By Proposition 4.5.10 in Ref. [23], this implies that some r ∈ K can be joined to b by a timelike curve, in contradiction to b ∈İ þ ðKÞ. Hence, no such γ 0 can exist.
The "if" direction of the theorem states that if (i), (ii), (iii) hold, then b ∈İ þ ðKÞ. We will prove the following equivalent statement: If b ∉İ þ ðKÞ satisfies (i), then b will fail to satisfy (ii) or (iii).
Let the geodesic γðsÞ guaranteed by (i) be parametrized so that γð0Þ ¼ p ≡ γ ∩ K and γð1Þ ¼ b. By (i), b ∈ J þ ðKÞ, the causal future of K. By assumption, b ∉İ þ ðKÞ ¼ _ J þ ðKÞ, so it follows that b ∈ I þ ðKÞ, the chronological future of K. Since p ∈İ þ ðKÞ, there exists an s Ã between 0 and 1 where γ leaves the boundary of the future: The point where γ leavesİ þ ðKÞ, q ≡ γðs Ã Þ, lies inİ þ ðKÞ. 7 Thus s Ã < 1. Moreover, s Ã > 0 by the obvious generalization of Proposition 4.5.1 in Ref. [23] and achronality of K. We conclude that Recall that q ¼ γðs Ã Þ is the future-most point on γ that is not in I þ ðKÞ. Let s n be a strictly decreasing sequence of real numbers that converges to s Ã . That is, s n > s Ã and, for n sufficiently large, the points q n ≡ γðs n Þ exist and lie in I þ ðKÞ. Now, since K is acausal and M is globally hyperbolic, there exists a Cauchy surface Σ ⊃ K. Given p 1 , p 2 ∈ M, define Cðp 1 ; p 2 Þ to be the set of all causal curves from p 1 to p 2 . Since by Corollary 6.6 of Ref. [22] CðΣ; q n Þ is compact, it is closed and bounded. Thus, CðK; q n Þ ⊂ CðΣ; q n Þ is bounded. Consider a sequence of curves μ m from K to q n . By Lemma 6.2.1 of Ref. [23], the limit curve μ of fμ m g is causal; since K is compact and thus contains its limit points, μ runs from K to q n , so μ ∈ CðK; q n Þ. Hence, CðK; q n Þ is closed and therefore compact. Since the proper time is an upper semicontinuous function on CðΣ; q n Þ, it attains its maximum over a compact domain, so we conclude in analogy with Theorem 9.4.5 of Ref. [11] that there exists a timelike geodesic γ n that maximizes the proper time from K to q n . By Theorem 9.4.3 of Ref. [11], γ n is orthogonal to K.
By construction, the point q is a convergence point (and hence a limit point) of the sequence fγ n g. By the timereverse of Lemma 6.2.1 of Ref. [23], there exists, through q, a causal limit curve γ 0 of the sequence fγ n g. This curve must intersect K because all γ n intersect K and K is compact. Since γ 0 passes through q ∈İ þ ðKÞ, it must not be smoothly deformable to a timelike curve since I þ ðKÞ is open. Thus, by Theorem 9.3.10 of Ref. [11], γ 0 must be a null geodesic orthogonal to K, so if γ 0 ≠ γ, condition (iii) fails to hold. See Fig. 6 for an illustration.
The only alternative is that γ is the only limit curve of the sequence fγ n g. In this case, fγ n g contains a subsequence whose convergence curve is γ. From now on, let fγ n g denote this subsequence. Orthogonality to K of the γ n implies that we can write q n ¼ exp K ðp n ; v n Þ; where p n ¼ γ n ∩ K, for some vector v n ∈ T p n K ⊥ tangent to γ n . But since q n ∈ γ, we can also write q n ¼ exp K ðp; k n Þ; ð15Þ FIG. 6. Possibilities in the proof. The sequence of timelike geodesics γ n (black) connects K with a sequence of points q n ∈ I þ ðKÞ on the orthogonal null geodesic γ (red) that joins p ∈ K with q, after which γ leaves _ I þ ðKÞ (red dashed). In the case on the left, γ 0 (green) is distinct from γ, so condition (iii) fails. In the case on the right, γ 0 ¼ γ, which we prove corresponds to a failure of condition (ii).
where k n is tangent to γ. Thus, every q n has a nonunique preimage.
By the above construction, the sequences fðp; k n Þg and fðp n ; v n Þg in NK each have ðp; vÞ as their limit point, where q ¼ exp K ðp; vÞ. Hence there exists no open neighborhood O of ðp; vÞ such that exp K is a diffeomorphism of O onto an open neighborhood of q. By Lemma 6, it follows that exp K is singular at ðp; vÞ; i.e., ðp; vÞ is a conjugate point. By Lemma 16 and Remark 18, q is conjugate to K. Thus, condition (ii) fails to hold; again, see Fig. 6.
▪ Remark 20.-The fact that K had codimension two was only important in the last step in the proof of Theorem 1, i.e., going from knowing that ðp; vÞ is a conjugate point to showing that q is conjugate to the surface K. For K a compact, acausal submanifold that is not of codimension two, the steps in the proof of Theorem 1 still establish that ðp; vÞ is a conjugate point in the sense of Def. 7. Moreover, that the corresponding Jacobi field is orthogonal to S remains true without the codimension-two assumption (see the proof of Lemma 16) and the one-parameter family of geodesics is orthogonal to K (because it was defined via normal parallel transport). As a result, the Jacobi field defines a deviation of γ in terms of only orthogonal null geodesics (as proven in, e.g., Corollary 10.40 of Ref. [21]), but in general that will not mean that q is conjugate to the surface K in the sense of Def. 17. Specifically, the Jacobi field is not necessarily nonvanishing at K if K has codimension greater than two.