Null wave front as Ryu-Takayanagi surface

The Ryu-Takayanagi formula provides the entanglement entropy of quantum field theory as an area of the minimal surface (Ryu-Takayangi 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 propose that null rays emitted from the AdS boundary can be regarded as such a flow. 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, based on the viewpoint of flow, we propose a wave optical formula to calculate the holographic entanglement entropy.


I. 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 S A of a region A in CFT can calculated as the area of the minimal bulk surface M A homologous to A (RT surface): where G N 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 subregion correspondence which is proposed for 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 equivalent concept to the RT surface geometrically. The bit threads are defined as a bounded divergenceless vector field v µ ∇ µ v µ = 0, |v| ≤ C, (2) and it maximizes its flux on a boundary area A. The property that the maximal total flux of v µ 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.
Although the concept of bit threads is inspirational, as mentioned by [8], the RT surface has infinitely many equivalent bit threads. Moreover, it is non-trivial task to construct bit threads practically and to calculate the EE of CFT by it due to its dependence of global structure of spacetimes. Therefore, as one of the interpretations of the RT surface respecting the viewpoint of the flow, we propose an interpretation that the RT surfaces are wave fronts of null rays emitted from a point on the AdS boundary. In particular, we prove that in spherical symmetric static spacetimes, owing to its axisymmetry of the configuration, such wave fronts of null rays are extremal surfaces as long as they propagates in the vacuum region.
As the RT surface is the extremal surface [9,10], thus the wave front can be considered as the RT surface. On the other hand, we can naturally understand 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 EE of CFT by counting the number of null rays. Finally, Section 5 is devoted to summary.

II. 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.

A. Ryu-Takayanagi surface
We derive the equation of the RT surface in the BTZ spacetime [11] where M is the mass of the black hole and AdS is the AdS radius. We prepare a region (arc) −θ ≤ θ ≤ θ 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: The equation of the RT surface r = r RT (θ) is the solution of the Euler-Lagrange equation obtained by variation of Area[r, dr/dθ] with respect to r, and it is where r min := r RT (θ = 0) denotes the minimum of r (see Fig. 1). Note that θ = θ(r = The entanglement entropy of CFT on the AdS boundary for an arc |θ| ≤ θ is obtained by substituting (5) into (4) : where the cutoff is introduced by := 2 AdS /r (r → ∞). Now let us consider CFT with inverse temperature β on S 1 . The circumference of the circle is assumed to be C and we prepare an arc |θ| ≤ θ with the arc length = Cθ /π on it. Then it is possible to write down (6) using only CFT quantities. By dividing Eq. (6) with 4G N , using the Brown-Henneaux formula c = 3 AdS /(2G N ) [12] and AdS/CFT dictionary β/C = 1/ √ M , we obtain the correct EE formula of thermal state of CFT on S 1 [13,14] after rescaling the cutoff : B. 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 where d denotes spatial dimension and dΩ 2 d−1 is the line element of the unit sphere S d−1 . We introduce coordinates on S d−1 as 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 ψ 2 , · · · ψ d−1 and assume the following (2+1)-dimensional metric to investigate wave fronts of null rays emitted from a point: x < l a t e x i t s h a 1 _ b a s e 6 4 = " a / D 4 y e I N y 8 K l d 1 u t B E m h P w 0 x q a J q E 5 L Z Q i z + g W 8 W F K w U X 4 m c I 4 g + 4 8 B P E p Y I b F 0 7 S I K g o T s i 9 c 8 / M m b n n j u a Y h i e J H i N K W 3 t H Z 1 e 0 O 9 b T 2 9 c / E B 8 c W v P s m q u L k m 6 b t r u h q Z 4 w D U u U p C F N s e G 4 Q q 1 q p l j X K k t + f L 0 u X M + w r V X Z c M R W V d 2 z j F 1 D V y V D + c Z 2 P E l p C i z x 0 8 m E T h K h 5 e z 4 H T a x A x s 6 a q h C w I J k 3 4 Q K j 7 8 y M i A 4 j G 2 h y Z j L n h H E B Q 4 R Y 2 6 N s w R n q I x W e N 3 j U z l E L T 7 7 N b 2 A r X M X k 3 + X m Q m k 6 I G u 6 Y X u 6 Y a e 6 P 3 X W k 2 u 0 G B c 8 u p 3 n e L 4 f o D 4 9 / C C W 6 R + 5 f r 9 f R 0 N 3 r V W X + F s D x y N F N / + w a v y L r n b J + 9 P x R K 7 m A u U G q z c C R D / D f Q W v 3 5 w 9 l J c K K S a E 3 R J z 6 z + g h 7 p l v V b 9 V f 9 K i 8 K 5 4 j x + D L f h / X T K U 2 n 5 9 O U n 0 l m F 8 x q a J q E 5 L Z Q i z + g W 8 W F K w U X 4 m c I 4 g + 4 8 B P E p Y I b F 0 7 S I K g o T s i 9 c 8 / M m b n n j u a Y h i e J H i N K W 3 t H Z 1 e 0 O 9 b T 2 9 c / E B 8 c W v P s m q u L k m 6 b t r u h q Z 4 w D U u U p C F N s e G 4 Q q 1 q p l j X K k t + f L 0 u X M + w r V X Z c M R W V d 2 z j F 1 D V y V D + c Z 2 P E l p C i z x 0 8 m E T h K h 5 e z 4 H T a x A x s 6 a q h C w I J k 3 4 Q K j 7 8 y M i A 4 j G 2 h y Z j L n h H E B Q 4 R Y 2 6 N s w R n q I x W e N 3 j U z l E L T 7 7 N b 2 A r X M X k 3 + X m Q m k 6 I G u 6 Y X u 6 Y a e 6 P 3 X W k 2 u 0 G B c 8 u p 3 n e L 4 f o D 4 9 / C C W 6 R + 5 f r 9 f R 0 N 3 r V W X + F s D x y N F N / + w a v y L r n b J + 9 P x R K 7 m A u U G q z c C R D / D f Q W v 3 5 w 9 l J c K K S a E 3 R J z 6 z + g h 7 p l v V b 9 V f 9 K i 8 K 5 4 j x + D L f h / X T K U 2 n 5 9 O U n 0 l m F 8 x q a J q E 5 L Z Q i z + g W 8 W F K w U X 4 m c I 4 g + 4 8 B P E p Y I b F 0 7 S I K g o T s i 9 c 8 / M m b n n j u a Y h i e J H i N K W 3 t H Z 1 e 0 O 9 b T 2 9 c / E B 8 c W v P s m q u L k m 6 b t r u h q Z 4 w D U u U p C F N s e G 4 Q q 1 q p l j X K k t + f L 0 u X M + w r V X Z c M R W V d 2 z j F 1 D V y V D + c Z 2 P E l p C i z x 0 8 m E T h K h 5 e z 4 H T a x A x s 6 a q h C w I J k 3 4 Q K j 7 8 y M i A 4 j G 2 h y Z j L n h H E B Q 4 R Y 2 6 N s w R n q I x W e N 3 j U z l E L T 7 7 N b 2 A r X M X k 3 + X m Q m k 6 I G u 6 Y X u 6 Y a e 6 P 3 X W k 2 u 0 G B c 8 u p 3 n e L 4 f o D 4 9 / C C W 6 R + 5 f r 9 f R 0 N 3 r V W X + F s D x y N F N / + w a v y L r n b J + 9 P x R K 7 m A u U G q z c C R D / D f Q W v 3 5 w 9 l J c K K S a E 3 R J z 6 z + g h 7 p l v V b 9 V f 9 K i 8 K 5 4 j x + D L f h / X T K U 2 n 5 9 O U n 0 l m F 8    In this metric with coordinates x µ = (t, r, θ), a wave front of null rays emitted from a point source is defined as a t = constant section of null congruences, 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

v P M + Q w = " > A A A C o 3 i c j V H L S s N Q E D 3 G V 6 2 P V t 0 I b s R S c S F l K o K P V d G N C x e 1 D x W 0 l C R e a z B N Q n J b q M U f E N z a h S s F F + J n C O I P u P A T x G U F N y 6 c p E F Q U Z y Q e + e e m T N z z x 3 N M Q 1 P E j 1 1 K d 0 9 v X 3 9 k Y H o 4 N D w S C w + O r b l 2 T V X F 0 X d N m 1 3 R 1 M 9 Y R q W K E p D m m L H c Y V a 1 U y x r R 2 t + f H t u n A 9 w 7 Y K s u G I U l W t W M a B o a u S o b w s z 5 f j C U p R Y F M / n X T o J B B a 1 o 7 f Y w / 7 s K G j h i o E L E j 2 T a j w + N t F G g S H s R K a j L n s G U F c 4 A R R 5 t Y 4 S 3 C G y u g R r x U + 7 Y a o x W e /
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) dt dλ , p θ := r 2 dθ dλ .
The sign ± in front of the integral corresponds to the sign of dr/dλ.
For the (2 + 1)-dimensional BTZ spacetime (3), we can demonstrate explicitly that wave fronts of null rays are the RT surfaces. We obtain equations of null geodesic from (13) and (14) with (t 0 , r 0 , θ 0 ) = (0, ∞, 0): It is easy to derive a trajectory of a null ray r = r NG (θ, b) with an impact parameter b from (16). On the other hand, the equation of a wave front r = r WF (θ, t) at a fixed t is derived by eliminating the parameter b from (16) and (17). After all, 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. We adopt the metric (11) with coordinates x µ = (t, r, θ, · · · ). Let ξ µ = (∂ t ) µ be the timelike Killing vector, k µ = dx µ /dλ be the tangent vector of null geodesics. We introduce the projection tensor P µν = g µν − ξ −2 ξ µ ξ ν = diag(0, f, 1/r 2 , · · · ) onto a constant time slice. We denote the tangent vector of null geodesics projected onto the hypersurface as . The conserved quantity associated with the Killing vector is ω = −ξ µ k µ = f k t and the norm of the spatial vectork i isk ik i = ω 2 /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 whereñ i is the unit normal vector of the wave front and D i = P i j ∇ j = (∇ r , ∇ θ , · · · ) is the covariant derivative on a constant time slice. Then, where √ h = r d−1 comes from determinant of the metric on S d−1 . On the other hand, the expansion of a null congruence is Therefore H = (f 1/2 /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 In the present case, as the congruence of null geodesics has axial symmetry, the shear and 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 (14) 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 + r 2 / 2 AdS . As f ≈ r 2 / 2 AdS in the vicinity of the AdS boundary, the affine parameter of null rays (15) from the AdS boundary r 0 = 2 AdS / , → 0 diverges as On the other hand, the coordinate time (14) 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.

IV. 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. Hence a congruence of null rays is one of the bit threads described in Section I. 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, papers [15][16][17] 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 Fig. 5 and Fig. 6 (see detail in Appendix). They show wave fronts from a point wave source on the AdS bounary (see Fig. 3 and Fig. 4 for corresponding wave fronts in the geometrical optics). For the massless scalar field φ(x µ ) obeying the Klein-Gordon equation φ = ( √ −g) −1 ∂ µ ( √ −gg µν ∂ ν φ) = 0, the WKB form of the wave function is where a and S are real functions. In the eikonal limit, they obey g µν ∇ µ S∇ ν S = 0, ∇ µ (a 2 ∇ µ S) = 0.
For the stationary case, the phase function S can be written as S = −ωt + W (r, θ), k i = (k r , k θ , 0, · · · ) = f W r , 1 r 2 W θ , 0, · · · ,k ik i = Here,k i represents the tangent vector of null rays projected on a constant time slice. We can write the solution of (29) as a(λ, χ) = a(λ 0 , χ) exp − 1 2 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 (32) implies andñ i is divergenceless normalized vector field. The wave front is the surface with the unit normalñ i , and is the extremal surface. The number of null rays passing through the wave front E A , which is the extremal surface homologous to the region A on the AdS boundary, is where √ h denotes determinant of the induced metric on E A . Now let us consider the setup shown as Fig. 7. We prepare a screen A( ) which is r =constant surface in the bulk. For the regularization, the screen is placed at r = 2 AdS / near the AdS boundary. and on the screen,k r ω 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 [18,19] with non-trivial shapes of an entangling surface and to further understanding of property of the holographic EE.