==== Front Entropy (Basel) Entropy (Basel) entropy Entropy 1099-4300 MDPI 33287065 10.3390/e22111297 entropy-22-01297 Article Null Wave Front and Ryu–Takayanagi Surface Tsujimura Jun * https://orcid.org/0000-0003-2596-4650Nambu Yasusada Department of Physics, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan; nambu@gravity.phys.nagoya-u.ac.jp * Correspondence: tsujimura.jun@a.mbox.nagoya-u.ac.jp 14 11 2020 11 2020 22 11 129725 9 2020 12 11 2020 © 2020 by the authors.2020Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).The Ryu–Takayanagi formula provides the entanglement entropy of quantum field theory as an area of the minimal surface (Ryu–Takayanagi surface) in a corresponding gravity theory. There are some attempts to understand the formula as a flow rather than as a surface. In this paper, we consider null rays emitted from the AdS boundary and construct a flow representing the causal holographic information. We present a sufficient and necessary condition that the causal information surface coincides with Ryu–Takayanagi surface. In particular, we show that, in spherical symmetric static spacetimes with a negative cosmological constant, wave fronts of null geodesics from a point on the AdS boundary become extremal surfaces and therefore they can be regarded as the Ryu–Takayanagi surfaces. In addition, from the viewpoint of flow, we propose a wave optical formula to calculate the causal holographic information. Ryu–Takayanagi surfacenull geodesicswave frontextremal surfacewave optics ==== Body 1. Introduction It is well known that the entanglement entropy (EE) of conformal field theory (CFT) can calculate in a corresponding gravity theory by the Ryu–Takayanagi (RT) formula [1,2] in AdS/CFT correspondence [3,4]. In general, although the EE of quantum field theory is not easy to calculate, the RT formula tells that the EE SA of a region A in CFT can calculated as the area of the minimal bulk surface MA homologous to A (RT surface): (1) SA=AreaMA4GN, where GN is the Newton constant of gravitation. This relation promotes informational theoretical analysis of AdS/CFT correspondence. By regarding the geometry of a bulk as a tensor network, it implements quantum error correcting code of boundary CFT [5] or MERA [6], subregion correspondence, which is proposed for a reduced density matrix [7]. From this point of view, it is better to regard the RT formula as a flow proposed by the paper [8]. The authors introduced “bit threads” which are an equivalent concept to the RT surface geometrically. The bit threads are defined as a bounded divergenceless vector field vμ (2) ∇μvμ=0,v≤C, and it maximizes its flux on a boundary area A. The property that the maximal total flux of vμ through the area A is equal to the area of the RT surface is proved by the max-flow min-cut theorem [8]. The bit threads give an intuitive picture that a vector field carrying information of the boundary propagates in the bulk, and the bulk region stores information of the boundary. Considering the bulk reconstruction, the bit threads have better properties than the RT surface although it is geometrically equivalent concept to the RT surface. Recall that the RT surface cannot reach the neighborhood of the black hole horizon. On the other hand, the bit threads can probe such a bulk regions. See other benefits of considering the bit threads for example [9]. As a similar concept of holographic EE, a causal holographic information χA [10] was defined by (3) χA=AreaΞA4GN, where ΞA is the causal information surface of A defined as follows. Let us consider the bulk domain of influence of the boundary domain of dependence of a region A. Then, the causal information surface ΞA is the intersection of the future and past directed null surfaces characterizing such a bulk domain of influence. In a static system, we can regard the causal information surface ΞA as a null wave front anchored on ∂A. A causal holographic information χA is known as an upper bound of EE of A. That is, SA≤χA as discussed in [10]. When does the equality SA=χA hold? In this paper, we consider a sufficient and necessary condition to hold it from a viewpoint of flow. By its definition, the causal holographic information χA has a natural flow as a null geodesic congruence. Since RT surface is extremal surface [11,12], we are to clarify the condition for a null wavefront to be extremized. We will see that a null wavefront becomes an extremal surface if and only if the shear of the corresponding null congruence vanishes. For example, in spherical symmetric static spacetimes, owing to its axisymmetry of the configuration, wave fronts of null rays emitted from a point on the AdS boundary are extremal surfaces as long as they propagate in the vacuum region. If SA=χA, we can regard the wave front as the RT surface and the corresponding null rays as bit threads. In this picture, we can calculate the EE of CFT by counting the number of such null rays. This method is also valid for wave optical calculation using the flux of a massless scalar field. The flux based calculation method suggests a picture that information prepared on the boundary side spreads to the bulk as null rays. The structure of this paper is as follows. In Section 2, we demonstrate the correspondence between the RT surface and the wave front of the null rays in the BTZ spacetime. In Section 3, we state the detail of our proposal and show it in spherically symmetric static spacetimes with a negative cosmological constant. In Section 4, we introduce the flux formula to calculate the causal holographic information by counting the number of null rays. Finally, Section 5 is devoted to the conclusions. 2. Null Wave Front and RT Surface In this section, before going to discuss the general situation, we demonstrate that wave fronts of null rays are the RT surface in the BTZ spacetime. 2.1. Ryu–Takayanagi Surface We derive the equation of the RT surface in the BTZ spacetime [13] (4) ds2=−r2ℓAdS2−Mdt2+r2ℓAdS2−M−1dr2+r2dθ2,−π≤θ≤π, where M is the mass of the black hole and ℓAdS is the AdS radius. We prepare a region (arc) −θℓ≤θ≤θℓ on the AdS boundary and consider a line anchored to the boundary of this region. The RT surface extremizes the following line area (length) on a constant time slice: (5) Arear,drdθ=∫−θℓθℓdθr2ℓAdS2−M−1drdθ2+r2. The equation of the RT surface r=rRT(θ) is the solution of the Euler–Lagrange equation obtained by variation of Area[r,dr/dθ] with respect to r, and it is (6) rRT(θ)=MrminsechMθM−rmin2/ℓAdS2tanh2Mθ, where rmin:=rRT(θ=0) denotes the minimum of r (see Figure 1). Note that θℓ=θ(r=∞)=(1/M)arctanh(MℓAdS/rmin). The entanglement entropy of CFT on the AdS boundary for an arc |θ|≤θℓ is obtained by substituting (6) into (5): (7) AreaMℓ=2ℓAdSlog2ℓAdSϵMsinhMθℓ+O(ϵ), where the cutoff is introduced by ϵ:=ℓAdS2/r(r→∞). Now let us consider CFT with inverse temperature β on S1. The circumference of the circle is assumed to be C and we prepare an arc |θ|≤θℓ with the arc length ℓ=Cθℓ/π on it. Then, it is possible to write down (7) using only CFT quantities. By dividing Equation (7) with 4GN, using the Brown–Henneaux formula c=3ℓAdS/(2GN) [14] and AdS/CFT dictionary β/C=1/M, we obtain the correct EE formula of thermal state of CFT on S1 [15,16] after rescaling the cutoff ϵ: (8) SA=c3logβπϵsinhπℓβ. 2.2. Null Rays and Wave Front We consider null rays emitted from a point on the AdS boundary and their wave fronts. Our purpose is to find out the relation between wave fronts of null rays and the RT surface. We consider null rays in the spherically symmetric static spacetime (9) ds2=−f(r)dt2+dr2f(r)+r2dΩd−12, where d denotes spatial dimension and dΩd−12 is the line element of the unit sphere Sd−1. We introduce coordinates on Sd−1 as (10) x1x2x3⋮xd−2xd−1=cosψ1sinψ1cosψ2sinψ1sinψ2⋮sinψ1sinψ2⋯sinψd−2cosψd−1sinψ1sinψ2⋯sinψd−2sinψd−1 with 0≤ψ1,⋯ψd−2≤π,0≤ψd−1≤2π. The line element on Sd−1 is (11) dΩd−12=dψ12+sin2ψ1(dψ22+sin2ψ2(dψ32+⋯⋯)). As is well known, in static spherically symmetric spacetimes, trajectories of null geodesics stay on a spatial two-dimensional plane. Thus, we can fix coordinate values of ψ2,⋯ψd−1 and assume the following (2+1)-dimensional metric to investigate wave fronts of null rays emitted from a point: (12) ds2=−f(r)dt2+dr2f(r)+r2dθ2,θ:=ψ1. In a static spacetime, a wave front of null rays emitted from a point source is defined as a t= constant section of null congruences as depicted in Figure 2, which forms a (d−1)-dimensional surface. Due to the axial symmetry of the configuration, a wave front of null rays is represented as a curve in (r,θ) space in the present situation. The tangent vector of a null ray is (13) kμ=(kt,kr,kθ)=dtdλ,drdλ,dθdλ,−fdtdλ2+1fdrdλ2+r2dθdλ2=0, where λ is the affine parameter. This spacetime has two Killing vectors related to translation of t and θ directions and there exist two conserved charges ω:=f(r)dtdλ,pθ:=r2dθdλ. Combining with Equation (13), we obtain a trajectory of a null ray as (14) θ(r)=θ0±∫r0rbr′21−f(r′)b2r′2−12dr′, (15) t(r)=t0±∫r0r1f(r′)1−f(r′)b2r′2−12dr′, (16) λ(r)=λ0±1ω∫r0r1−f(r′)b2r′2−12dr′, where (t0,θ0,λ0)=(t(r0),θ(r0),λ(r0)) and the impact parameter b:=pθ/ω is introduced. The sign ± in front of the integral corresponds to the sign of dr/dλ. For the (2+1)-dimensional BTZ spacetime (4), we can demonstrate explicitly that wave fronts of null rays are the RT surfaces. We obtain equations of null geodesic from (14) and (15) with (t0,r0,θ0)=(0,∞,0): (17) θ(r)=1Mlogr2−b2r2/ℓAdS2−M+bMr1−b2/ℓAdS2, (18) t(θ)=ℓAdSMarctanhℓAdSbtanhMθ. It is easy to derive a trajectory of a null ray r=rNG(θ,b) with an impact parameter b from (17). On the other hand, the equation of a wave front r=rWF(θ,t) at a fixed t is derived by eliminating the parameter b from (17) and (18). After all, (19) rNG(θ,b)=Mb1−b2/ℓAdS2cschMθ, (20) rWF(θ,t)=MℓAdScothMt/ℓAdSsechMθ1−coth2Mt/ℓAdStanh2Mθ. For the special case M=−1, the spacetime reduces to the pure AdS. Figure 3 and Figure 4 show null rays and their wave fronts in the pure AdS spacetime and the BTZ black hole spacetime, respectively. Note that Equation (20) of the wave front is the same as Equation (6) of the RT surface by identifying rmin=MℓAdScothMt/ℓAdS, which represents the elapsed time of a null ray traveling from r=∞ to r=rmin. Indeed, this quantity is obtained by taking b=0 in the equation of the null ray (15): (21) t=∫∞rmindrr2/ℓAdS2−M=ℓAdSMarctanhMℓAdSrmin. Therefore, we have confirmed that wave fronts of null rays emitted from the AdS boundary coincide with the RT surfaces in the BTZ spacetime. Note that, for sufficient elapse of time after emission of null rays, a self-intersection of the wave front occurs. Then, one might consider that the identification of the wave front as the RT surface becomes ambiguous. However, we do not have to consider such a situation because the “subregion” passed by null rays become bigger than whole boundary region, and then RT formula makes no sense. 3. Null Wave Front and Extremal Surface In this section, based on the observation in the previous section for the BTZ spacetime, we show the following proposition for spherically symmetric static spacetimes with a cosmological constant (no matter fields). Proposition 1. Wave fronts of shear free null congruence are extremal surfaces in static spacetime when the affine parameter of null rays goes to infinity. Corollary 1. In static spacetimes with a negative cosmological constant, wave fronts of null rays emitted from the AdS boundary are extremal surfaces if and only if the shear of the null congruence vanishes. We adopt the metric (12) with coordinates xμ=(t,r,θ,⋯). Let ξμ=(∂t)μ be the time-like Killing vector, kμ=dxμ/dλ be the tangent vector of null geodesics. We introduce the projection tensor Pμν=gμν−ξ−2ξμξν=diag(0,f,1/r2,⋯) onto a constant time slice. We denote the tangent vector of null geodesics projected onto the hypersurface as k˜i=Pijkj=(kr,kθ,0,⋯). The conserved quantity associated with the Killing vector is ω=−ξμkμ=fkt and the norm of the spatial vector k˜i is k˜ik˜i=ω2/f. We prove the proposition by using the fact that the extremal surface is a surface with zero mean curvature. The mean curvature H of a wave front of null rays on a constant time slice is defined by (22) H:=Din˜i,n˜i=k˜ik˜=f1/2ω(kr,kθ,0,⋯), where n˜i is the unit normal vector of the wave front and Di=Pij∇j=(∇r,∇θ,⋯) is the covariant derivative on a constant time slice. Then, (23) H=Dif1/2ωki=1f−1/2h∂if1/2ωf−1/2hki=f1/2ωh∂r(hkr)+∂θ(hkθ), where h=rd−1 comes from determinant of the metric on Sd−1. On the other hand, the expansion of a null congruence is (24) Θ=∇μkμ=1h∂μ(hkμ)=1h∂r(hkr)+∂θ(hkθ). Therefore, H=(f1/2/ω)Θ and the mean curvature H of a wave front is proportional to the expansion of the null geodesic congruence. The expansion Θ along a null geodesic obeys the Raychaudhuri equation (25) dΘdλ=−Θ2d−1−Rμνkμkν. In the present case, as the congruence of null geodesics has axial symmetry, the shear and the rotation of the congruence do not appear in this equation. For vacuum spacetimes with a cosmological constant, the term with the Ricci curvature disappears. Then, the solution of Equation (25) is Θ(λ)=(d−1)/(λ−λ0), where λ0 is the affine parameter at the source. Thus, the expansion goes to zero as the affine parameter goes to infinity, and the mean curvature of the wave front is zero and is the extremal surface. Therefore, the proposition is proved. As an example of this proposition, let us consider a wave front in the Minkowski spacetime. A spherical wave front emitted from a point source placed at the spatial infinity becomes plane wave, which is zero mean curvature surface in the Minkowski spacetime. However, in this case, the coordinate time (15) becomes infinite when a wave front of null rays arrives at an observer. Asymptotically AdS spacetimes are peculiar because they have the timelike boundary. We consider the pure AdS spacetime of which metric function is given by f(r)=1+r2/ℓAdS2. As f≈r2/ℓAdS2 in the vicinity of the AdS boundary, the affine parameter of null rays (16) from the AdS boundary r0=ℓAdS2/ϵ,ϵ→0 diverges as (26) λ(r)≈1ω∫rℓAdS2/ϵ1−b2ℓAdS2−1/2dr=ℓAdS2/ϵ−rω1−b2/ℓAdS2→∞. On the other hand, the coordinate time (15) converges as (27) t(r)≈∫rℓAdS2/ϵdrℓAdS2r21−b2ℓAdS2−1/2=ℓAdS21−b2/ℓAdS21r. This property also holds for general asymptotically AdS spacetimes because they have the same metric in the vicinity of the AdS boundary as the pure AdS spacetime. After all, we conclude that, for static spherical symmetric asymptotically AdS spacetimes, wave fronts of a null geodesic congruence emitted from a point source on the AdS boundary are extremal surfaces. We can see that the condition for SA=χA is that the shear of null congruence vanishes. To satisfy this, we need the strong symmetry for the spacetime and the wavefront. 4. Flux Formula Based on the idea presented in the previous section, we can understand null rays as a natural flow characterizing the EE of the dual CFT if SA=χA. Hence, a congruence of null rays is one of the bit threads described in Section 1. This makes us conceive a picture that null rays propagate in the bulk with information of the AdS boundary. This picture suggests that the EE can be calculable by counting the number of null rays. In this section, we reformulate the RT formula in terms of the wave optics. Concepts of wave fronts and the flux of null rays are naturally derived as the eikonal limit of wave optics. As an application of wave optics to black hole spacetimes, Refs. [17,18,19] investigate image formation of the photon sphere of black holes. In this paper, we focus on the structure of wave fronts of a massless scalar field. For the monochromatic massless scalar field with time dependence ∝e−iωt, we present wave patterns in Figure 5 and Figure 6 (see details in the Appendix A. We also show a wave pattern for SA≠χA case in Appendix A). They show wave fronts from a point wave source on the AdS boundary (see Figure 3 and Figure 4 for corresponding wave fronts in the geometrical optics). For the massless scalar field ϕ(xμ) obeying the Klein–Gordon equation ϕ=(−g)−1∂μ(−ggμν∂νϕ)=0, the WKB form of the wave function is (28) ϕ(xμ)=a(xμ)expiS(xμ), where a and S are real functions. In the eikonal limit, they obey (29) gμν∇μS∇νS=0, (30) ∇μ(a2∇μS)=0. The Equation (29) is the Hamilton–Jacobi (HJ) equation and Equation (30) represents conservation of the Klein–Gordon current Jμ=(1/2i)(ϕ∗∂μϕ−ϕ∂μϕ∗). In terms of the wave vector kμ=∂μS, which defines the tangent of null rays, (31) gμνkμkν=0,∇μ(a2kμ)=0. For the stationary case, the phase function S can be written as S=−ωt+W(r,θ), (32) k˜i=(kr,kθ,0,⋯)=fWr,1r2Wθ,0,⋯,k˜ik˜i=ω2f, (33) f−1/2Dia2f1/2k˜i=0. Here, k˜i represents the tangent vector of null rays projected on a constant time slice. We can write the solution of (30) as (34) a(λ,χ)=a(λ0,χ)exp−12∫λ0λdλΘ(λ),Θ=∇μkμ, where the integral is along a null ray (with respect to the affine parameter λ) and χ denotes a coordinate distinguishing different geodesics. As the expansion of null congruence from the AdS boundary is zero, the amplitude a(λ,χ) is conserved along a null ray and independent of λ. Furthermore, for a point source isotropically emitting null rays, a is independent of χ and can assume to be constant. Thus, (33) implies (35) Din˜i=0,n˜i=f1/2ωk˜i,n˜in˜i=1, and n˜i is divergenceless normalized vector field. A vector field vμ=(0,n˜i) is one realization of the bit threads satisfying Equation (2). Notice that this construction highly depends on the stationarity of the spacetime. The wave front is the surface with the unit normal n˜i, and is the extremal surface. The number of null rays passing through the wave front EA, which is the extremal surface homologous to the region A on the AdS boundary, is (36) Area(EA)=∫EAn˜idΣi,dΣi=n˜ihdd−1σ, where h denotes determinant of the induced metric on EA. Now let us consider the setup shown as Figure 7. We prepare a screen A(ϵ) which is r= constant surface in the bulk. For the regularization, the screen is placed at r=ℓAdS2/ϵ near the AdS boundary. Because n˜i is a divergence free vector field, Equation (36) equals (37) Area(EA)=∫A(ϵ)n˜idΣi=1ω∫A(ϵ)JidΣi. This is a formula for area of the RT surface in terms of flux integration of null rays on the screen A(ϵ). As the Klein–Gordon current Ji/ω=f1/2k˜i/ω represents the number density of null rays, we can regard the Klein–Gordon current as a representation of the amount of information propagating in the bulk spacetime from the AdS boundary. As a demonstration, we evaluate the right-hand side of this relation for the BTZ spacetime. By fixing the radial coordinate as r=ℓAdS2/ϵ in Equation (19), the impact parameter b on the screen is (38) b=ℓAdS2sinhMθϵ2M+ℓAdS2sinh2Mθ. From Equation (13), the radial component of the tangent vector of the null ray is (39) k˜rω=1−fb2r21/2=1−r2ℓAdS2−Mb2r2, and, on the screen, (40) k˜rωA(ϵ)=McoshMθM+(ℓAdS2/ϵ2)sinh2Mθ. The area element on the screen is (41) dΣrA(ϵ)=rnrdθA(ϵ)=rf−1/2dθ, where nr is the radial component of the unit normal to the screen. Thus, (42) f1/2k˜rωdΣrA(ϵ)=r1−fb2r21/2dθ=MℓAdScosh(Mθ)dθsinh2(Mθ)+Mϵ2/ℓAdS2. Therefore, (37) becomes (43) ∫−θℓθℓf1/2k˜idΣiω=∫−θℓθℓdθMℓAdScoshMθMϵ2/ℓAdS2+sinh2Mθ=ℓAdSlogsinh2(Mθℓ)+Mϵ2/ℓAdS2+sinh(Mθℓ)sinh2(Mθℓ)+Mϵ2/ℓAdS2−sinh(Mθℓ)=2ℓAdSlogℓAdS(ϵ/2)MsinhMθℓ+O(ϵ), and reproduces the “area” of the RT surface (7). Dividing by 4GN, this result correctly reproduces the EE of CFT (8). Therefore, we can regard a null geodesic congruence as one realization of the bit threads. 5. Conclusions In this paper, we show that wave fronts of null rays emitted from a point on the AdS boundary are extremal surfaces in static spherical symmetric spacetimes and clarify the condition that the causal holographic information coincides with the holographic entanglement entropy. If they coincide, the RT surface can be understood as a wave front, and null rays naturally define a flow characterizing the amount of the EE of CFT. Hence, such a flow can be regarded as the bit threads. As we assumed a point source on the AdS boundary, the shape of a region on the AdS boundary (entangling surface) becomes spherical because the boundary of the region is a wave front on the AdS boundary. However, by superposing point sources, it is possible to construct an extremal surface homologous to a region with arbitrary shapes on the AdS boundary by considering the envelope of wave fronts from each point sources. Thus, the method presented in this paper may be applicable to the plateaux problem [20,21] with non-trivial shapes of an entangling surface and to further understanding of the property of the holographic EE. Acknowledgments Y. N. was supported in part by JSPS KAKENHI Grant No. 19K03866. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Author Contributions J.T. and Y.N. contributed equally to this work. All authors have read and agreed to the published version of the manuscript. Funding This research received no external funding. Conflicts of Interest The authors declare no conflict of interest. Appendix A. Massless Scalar Field in AdS Spacetimes We consider the solution of the Klein–Gordon equation ϕ=0 in the AdS spacetime with the metric (9). Assuming the axially symmetric and stationary configuration of the scalar field ϕ=e−iωtϕ˜(r,θ), ϕ˜ obeys the following Helmholtz type equation: (A1) ∂2ϕ˜∂r2+d−1r+f′f∂ϕ˜∂r+ω2f2ϕ˜+1fr2sind−2θ∂∂θsind−2θ∂ϕ˜∂θ=0. Assuming ϕ˜=R(r)Φ(θ), (A2) R″+d−1r+f′fR′+ω2f2−m(m+d−2)fr2R=0, (A3) Φθθ+(d−2)cotθΦθ+m(m+d−2)Φ=0, and Φ=Cmd/2−1(cosθ) (Gegenbauer polymonial). For d=2, Φ=eimθ,m∈Z and for d=3, Φ=Pm(cosθ),m∈Z+. We consider d=2 case. For the normalized radial function limr→∞Rm(r)=1, the solution of the Klein–Gordon equation with a point source at the AdS boundary is represented as (A4) ϕ(r,θ)=∑m=−∞m=∞eimθRm(r), and this wave function gives limr→∞ϕ∝δ(θ) and satisfies the boundary condition with a point wave source at the AdS boundary. For the BTZ spacetime, the solution satisfying the ingoing boundary condition at the black hole horizon is given by (A5) Rm=Γ(a)Γ(b)Γ(c)ξ−iℓAdSω2MFi2M(ℓAdSω−m),i2M(ℓAdSω+m),1+iℓAdSωM,ξ,a=1−i2M(ℓAdSω−m),b=1+i2M(ℓAdSω−m),c=1+iℓAdSωM, where F is Gauss’s hypergeometric function and ξ=1−MℓAdS2/r2. The Figure 5 and Figure 6 are obtained by taking sum in (A4) up to mmax∼200. In the following, we will present an example of SA≠χA. We consider the 4+1 dimensional AdS spacetime with the Poincaré patch (A6) ds2=ℓAdS2r2dr2+r2ℓAdS2(−dt2+dw2+dx2+dy2), and consider a subregion A on the AdS boundary {−ℓ≤w≤ℓ,(x,y)∈R2}, which is a band shape region. The corresponding solution of the Klein–Gordon equation with a wave source in placed on the line w=0 is represented as (A7) ϕ(r,w,x,y)=∫−∞∞dpwz2N2(ω2−pw2z)ei(pww−ωt) where z=ℓAdS/r2 and N2 is the Bessel function of the second kind. Taking the eikonal limit, the wavefronts are obtained as w=±ℓ2−z2,(x,y)∈R2. Notice that this wavefront does not coincide with the RT surface since the wavefronts are not axisymmetric and then the corresponding null congruence has the sheer. Actually, the RT surfaces of A are (A8) w=±xd+11−(x/ℓ)2d(d+1)ℓ2d−x2d2F112,d+12d,123+1d,x2dℓ2d. The RT surface coincides with the causal information surface if and only if d=2. Figure A1 depicts the wavefronts projected on t=const. slice (the causal information surfaces) and the RT surfaces corresponding to the subregion A. Figure A1 Wave pattern of the monochromatic massless scalar field in the AdS5 spacetime with the Poincaré patch (real part of ϕ,ω=4/3). RT surface anchoring ∂A(ℓ=10) is shown as a solid red line. The corresponding wave front (causal information surface) is shown as a solid yellow line. Figure 1 The RT surface in the BTZ spacetime (M=ℓAdS=1) with coordinates (x,y)=(ρcosθ,ρsinθ), ρ:=ℓAdSarctanr/ℓAdS. Each blue line is parametrized by rmin=1.01,1.05,1.2,2.0,5.0. For large interval 2θℓ on the AdS boundary, dotted lines become minimal surfaces. Figure 2 Left panel: Axisymmetric configurations of congruence of null rays emitted from a point source on the AdS boundary. The right panel: the wavefronts projected onto a t=const. time slice. The coordinate x→ denotes d−1 dimensional space as Equation (10). The blue, red, and green lines represent the wavefronts rWF(t), the null rays and subregion A(t), respectively. Figure 3 Null rays (dotted red lines) and wave fronts (blue lines) in the pure AdS2+1 spacetime (M=−1,ℓAdS=1). Figure 4 Null rays (dotted red lines) emitted from (r,θ)=(∞,0) and their wave fronts (blue lines) in the BTZ spacetime (left panel: M=ℓAdS=1, right panel: M=0.1,ℓAdS=1). Figure 5 Wave pattern of the monochromatic massless scalar field with ω=20 in the AdS spacetime. The real part of ϕ/|ϕ| is shown. Figure 6 Wave pattern of the massless scalar field with ω=20 (Re ϕ/|ϕ|) in the BTZ spacetime with M=1 (left panel) and M=0.1 (right panel). Figure 7 Null rays (red dotted lines) emitted from a point on the AdS boundary pass through the screen A(ϵ) placed at r=ℓAdS2/ϵ (dotted line). As the null rays are orthogonal to the wave front (blue line), the number of null rays is proportional to the area of the RT surface. ==== Refs References 1. Ryu S. Takayanagi T. Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence Phys. Rev. Lett. 2006 96 181602 10.1103/PhysRevLett.96.181602 16712357 2. Ryu S. Takayanagi T. Aspects of Holographic Entanglement Entropy JHEP 2006 8 045 10.1088/1126-6708/2006/08/045 3. Gubser S. Klebanov I. Polyakov A. Gauge theory correlators from non-critical string theory Phys. Lett. B 1998 428 105 114 10.1016/S0370-2693(98)00377-3 4. Witten E. Anti de Sitter space and holography Adv. Theor. Math. Phys. 1998 2 253 291 10.4310/ATMP.1998.v2.n2.a2 5. Pastawski F. Yoshida B. Harlow D. Preskill J. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence JHEP 2015 6 149 10.1007/JHEP06(2015)149 6. Swingle B. Entanglement renormalization and holography Phys. Rev. D 2012 86 065007 10.1103/PhysRevD.86.065007 7. Czech B. Karczmarek J.L. Nogueira F. Raamsdonk M.V. The gravity dual of a density matrix Class. Quantum Gravity 2012 29 155009 10.1088/0264-9381/29/15/155009 8. Freedman M. Headrick M. Bit threads and holographic entanglement Commun. Math. Phys. 2017 352 407 438 10.1007/s00220-016-2796-3 9. Agón C.A. Cáceres E. Pedraza J.F. Bit threads, Einstein’s equations and bulk locality arXiv 2020 2007.07907 10. Hubeny V.E. Rangamani M. Causal Holographic Information JHEP 2012 6 114 10.1007/JHEP06(2012)114 11. Fursaev D.V. Proof of the holographic formula for entanglement entropy JHEP 2006 9 018 10.1088/1126-6708/2006/09/018 12. Lewkowycz A. Maldacena J. Generalized gravitational entropy JHEP 2013 8 090 10.1007/JHEP08(2013)090 13. Bañados M. Teitelboim C. Zanelli J. Black hole in three-dimensional spacetime Phys. Rev. Lett. 1992 69 1849 1851 10.1103/PhysRevLett.69.1849 10046331 14. Brown J.D. Henneaux M. Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity Commun. Math. Phys. 1986 104 207 226 10.1007/BF01211590 15. Calabrese P. Cardy J.L. Entanglement entropy and quantum field theory J. Stat. Mech. 2004 0406 P06002 10.1088/1742-5468/2004/06/P06002 16. Calabrese P. Cardy J. Entanglement entropy and conformal field theory J. Phys. 2009 A42 504005 10.1088/1751-8113/42/50/504005 17. Kanai K. Nambu Y. Viewing black holes by waves Class. Quantum Gravity 2013 30 175002 10.1088/0264-9381/30/17/175002 18. Nambu Y. Noda S. Wave optics in black hole spacetimes: The Schwarzschild case Class. Quantum Gravity 2016 33 075001 10.1088/0264-9381/33/7/075011 19. Hashimoto K. Kinoshita S. Murata K. Imaging black holes through the AdS/CFT correspondence Phys. Rev. D 2020 101 66018 10.1103/PhysRevD.101.066018 20. Habeny V. Maxfield H. Rangamani M. Tonni E. Holographic entanglement plateaux JHEP 2013 8 092 10.1007/JHEP08(2013)092 21. Freibogel B. Jefferson R. Mosk L.K.B. Yang I.-S. Casting shadows on holographic reconstruction Phys. Rev. D 2015 91 086013 10.1103/PhysRevD.91.086013