First Law of Entanglement Entropy in Ads Black Hole Backgrounds

PHYSICAL REVIEW D 104, 026004 (2021)

First law of entanglement entropy in AdS black hole backgrounds

† Akihiro Ishibashi1,* and Kengo Maeda2, 1Department of Physics and Research Institute for Science and Technology, Kindai University, Higashi-Osaka 577-8502, Japan 2Faculty of Engineering, Shibaura Institute of Technology, Saitama 330-8570, Japan

(Received 12 April 2021; accepted 10 June 2021; published 8 July 2021)

The first law for entanglement entropy in CFT in an odd-dimensional asymptotically AdS black hole is studied by using the AdS=CFT duality. The entropy of CFT considered here is due to the entanglement between two subsystems separated by the horizon of the AdS black hole, which itself is realized as the conformal boundary of a black droplet in even-dimensional global AdS bulk spacetime. In (2 þ 1)- dimensional CFT, the first law is shown to be always satisfied by analyzing a class of metric perturbations of the exact solution of a 4-dimensional black droplet. In (4 þ 1)-dimensions, the first law for CFT is shown to hold under the Neumann boundary condition at a certain bulk hypersurface anchored to the conformal boundary of the boundary AdS black hole. From the boundary view point, this Neumann condition yields there being no energy flux across the boundary of the boundary AdS black hole. Furthermore, the asymptotic geometry of a 6-dimensional small AdS black droplet is constructed as the gravity dual of our (4 þ 1)-dimensional CFT, which exhibits a negative energy near the spatial infinity, as expected from vacuum polarization.

DOI: 10.1103/PhysRevD.104.026004

I. INTRODUCTION the boundary theory [5], being consistent with the basic idea [6]. In some sense, the first law of the entanglement The entanglement entropy of a quantum field in black entropy could be a guiding principle for a consistent hole backgrounds has attracted much attention as the key formulation of quantum gravity, just like the first law of concept toward understanding the origin of the Bekenstein- black hole thermodynamics. Motivated by this, we holo- Hawking entropy (for review, see e.g., [1]). Although it is in graphically explore the first law of the entanglement general a Herculean task to calculate an entropy in a curved entropy for two subsystems of CFT separated by a black background, it is tractable to compute the entropy for hole horizon in asymptotically AdS spacetime. strongly coupled conformal fields in the holographic In the holographic proof of the first law [5], the Noether setting, or the anti–de Sitter (AdS) conformal field theory charge formula [7] plays an essential role. This is because (CFT) correspondence, by using the Ryu-Takayanagi for- the entanglement entropy for any ball-shaped spatial region mula [2]. In this formula, the entropy calculation for a in flat AdS boundary corresponds, through a conformal subsystem is reduced to a much simpler task of calculating transformation, to the horizon area of a zero-mass hyper- the area of a minimal or extremal surface anchored to the bolic black hole in the bulk. Therefore it is inferred that the boundary of the asymptotically AdS gravity dual. Noether charge formula can also be applied to the deriva- In the framework of the AdS=CFT duality, the first law tion of the first law of the entanglement entropy of a of the vacuum entanglement entropy was shown for any strongly coupled CFT in black hole backgrounds. In the ball-shaped subregions when the AdS boundary is flat standard Noether charge derivation [7] of the first law of the Minkowski spacetime [3,4]. Conversely, the bulk linearized black hole thermodynamics in D-dimensions, one first Einstein equations around the pure AdS spacetime can be considers a (D − 1)-dimensional spacelike hypersurface derived when the first law is satisfied for the subregions in which has two (D − 2)-dimensional boundaries, one at the black hole horizon (bifurcate surface) and the other at *[email protected] the spatial infinity, and then computes the Noether charges † [email protected] at these two boundaries. For asymptotically flat black holes, one can impose a natural boundary condition for Published by the American Physical Society under the terms of physical fields at the spatial infinity to be consistent with the Creative Commons Attribution 4.0 International license. the asymptotic flatness and black hole no-hair property, but Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, for asymptotically AdS black holes, one needs to be more and DOI. Funded by SCOAP3. careful to consider possible boundary conditions at the AdS

2470-0010=2021=104(2)=026004(14) 026004-1 Published by the American Physical Society AKIHIRO ISHIBASHI and KENGO MAEDA PHYS. REV. D 104, 026004 (2021) infinity. In the holographic setting, further care may be needed since one first extends the spacelike hypersurface into the (D þ 1)-dimensional bulk spacetime to have a D-dimensional spacelike hypersurface and, if the extended hypersurface ends at a bulk boundary (e.g., black hole horizon isolated from the boundary black hole), then one also has to take into account the additional contribution to the Noether charge from such a bulk boundary. In this paper, we examine the first law of the entanglement entropy of odd-dimensional conformal field theories (CFT) in AdS black hole backgrounds by using the Noether charge formula in the holographic setting. Namely, we consider a D (odd)-dimensional AdS black hole which is realized as a part of the conformal boundary of the (D þ 1)-asymptotically AdS bulk spacetime. Then, as FIG. 1. A schematic diagram of a time slice of our black droplet 1 briefly mentioned above, we evaluate Noether charges at in asymptotically AdS bulk spacetime of (D þ )-dimension. The black circle describes a (D − 1)-dimensional sphere as a time two (D − 2)-boundaries (i.e., the boundary black hole slice of the D-dimensional conformal boundary of the Einstein horizon and the boundary AdS infinity) as well as those static cylinder or an asymptotically, globally AdS spacetime in − 1 at (D )-bulk boundaries. (D þ 1)-dimension. The conformal boundary of our black droplet For simplicity, we assume that the bulk geometry itself is a D-dimensional asymptotically AdS black hole (illus- contains neither horizons in the IR isolated from the trated in Fig. 2), whose (D − 1)-dimensional time slice (dashed AdS boundary nor black funnels in the Hartle-Hawking blue half-circle) therefore covers only half of the (D − 1)-dimen- state, which connect two or more boundary black holes sional sphere. Note that the boundary D-dimensional black hole [14]. So, the only possible bulk solution dual to the has two asymptotic regions (corresponding to A and B of Fig. 2), boundary CFT in an AdS black hole background is the but the lower half circle (dashed blue half-circle) shows only one Σ black droplet solution [14], in which the bulk horizon is of them, say the region A of Fig. 2. We take our hypersurface as the lower half of this time slice (shaded region) enclosed by part connected to the single boundary black hole. The black of the AdS boundary (part of the dashed blue curve, which droplet solutions with a boundary black hole in asymp- corresponds to the region A), black droplet horizon (black curve) totically flat spacetime were eagerly constructed with and χ ¼ π=2 (red dotted line). The contribution to the Noether numerics, motivated by the investigation of the Hawking charge arises from part (red dotted line, χ ¼ π=2)of∂Σ, as well radiation in strongly coupled field theories [8–13] (see also as from the bulk horizon and the lower half of the AdS boundary. analytic solutions for lower-dimensional black droplets As we will discuss, we consider a reflection boundary condition [14] or perturbative analytic solutions [15,16]). at χ ¼ π=2 (red dotted line). In Fig. 1, we show an example of the black droplet solution in which the bulk horizon is connected to a the entanglement entropy is satisfied in (2 þ 1)-dimen- boundary black hole in an asymptotically AdS spacetime. sional CFT by perturbing the black droplet solution. In 2 1 In ( þ )-dimensional field theory, we calculate the Sec. IV, we derive the Noether charge on the equatorial perturbation of the black droplet solution with Banados- plane and explore the boundary condition in which the first Teitelboim-Zanelli (BTZ) black hole on the AdS boundary law is satisfied in (4 þ 1)-dimensional CFT. We also show [14]. (For a brane model, the first law of the entanglement that the energy flux across the AdS boundary is zero when entropy was shown [17].) It is shown that the first law of the the first law is satisfied. In Sec. V, a black droplet solution 4 1 entanglement entropy is always satisfied. In ( þ )-dimen- with a boundary AdS black hole is perturbatively con- sional field theory, we show that the first law is satisfied for structed from the global AdS spacetime. Section VI is the Neumann boundary condition at the equatorial plane in devoted to summary and discussions. In the Appendix, we Fig. 1. In the field theory side, the condition corresponds to briefly discuss the Noether charge formula (2.4) for general the no energy flux condition across the timelike null perturbations in (5 þ 1)-dimensional bulk metric (4.2). infinity. Under the boundary condition, we analytically construct the asymptotic geometry of such a black droplet solution from the perturbation of the global AdS spacetime. II. PRELIMINARIES: HOLOGRAPHIC SETTING It is shown that negative energy appears in asymptotic AND NOETHER CHARGE FORMULA region and it decays with r−3 power in the radial coordinate In this section we explain our setup and the strategy r. This could be understood as a vacuum polarization effect. before exploring the first law of the entanglement entropy. The rest of our paper is organized as follows. In Sec. II, Our holographic setup is the following: We consider a we briefly review the Noether charge formula [7] and (D þ 1)-dimensional asymptotically (locally) anti–de Sitter explain our setup. In Sec. III, we show that the first law of (AlAdS) spacetime ðM;gMNÞ which satisfies the vacuum

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spacetime singularity where here and hereafter we set the (D þ 1)-dimensional gravitational constant equal to one and we omit the indices AdS black M1M2 when writing differential forms, as in the lhs δ hole horizon AdS above. For an arbitrary small deviation gMN from the bulk null Infinity geometry ðM;gMNÞ, the symplectic potential D-form Θ is defined as B C S 1 A Θðg; δgÞ¼ ϵ gNJgLKð∇ δg − ∇ δg Þ: D 16π NM1M2MD L JK J LK ð2:2Þ

As shown in Ref. [7], the exterior derivative of the variation of the Noether charge is expressed by the symplectic potential Θ as FIG. 2. The Penrose diagram of an AdS black hole on the boundary theory. The Noether charge generically appears on the dδQ ¼ dðξ · ΘÞ; ð2:3Þ bulk hypersurface anchored to D. where the “centered dot” in the rhs of Eq. (2.3) denotes the Einstein equations and whose conformal boundary contraction of the Killing vector field ξM into the first index ð∂M; g˜μνÞ is locally conformally mapped to the D-dimen- of the form Θ. Integrating this over a D-dimensional sional static Einstein cylinder. On the conformal boundary spacelike hypersurface Σ, we obtain ∂M, we consider a D-dimensional static, asymptotically AdS black hole, whose spatial section is conformally Z embedded in the half of the (D − 1)-sphere—i.e., a spatial ðδQ − ξ · Θðg; δgÞÞ ¼ 0; ð2:4Þ section of the static Einstein cylinder—as shown by the ∂Σ dashed blue curve in Fig. 1. On such a D-dimensional AdS black hole realized in our holographic setting, we consider where ∂Σ is the (D − 1)-dimensional boundary of the the entanglement entropy for the subregion A on a space- hypersurface Σ. Applying this equation to a black hole like hypersurface S separated by the horizon (see Fig. 2). spacetime with a bifurcate Killing horizon with respect to The entropy is obtained by tracing over the degrees of ξM, one can obtain the first law. freedom located in the subregion B. In the Fefferman-Graham expansion, the bulk metric is From the bulk point of view, there is a static black droplet represented by solution in which the horizon is connected to the boundary horizon of the boundary AdS black hole. We assume that 2 2 M N L 2 μ ν ds ¼ g dx dx ¼ ðdz þ g˜μνðz;xÞdx dx Þ; there are no bulk black holes disconnected from the AdS MN z2 boundaries or black funnels connecting two or more ˜ ˜ 2 ˜ D ˜ boundary horizons. According to the Ryu-Takayanagi gμνðz;xÞ¼gð0ÞμνðxÞþz gð2ÞμνðxÞþþz gðDÞμνðxÞþ; formula [2], the entanglement entropy between A and B xM ¼ðz;xμÞðμ ¼ 0;1;…;D− 1Þ; ð2:5Þ is obtained by the area of the bulk minimal surface anchored to the bifurcate surface C between A and B. where L is the AdS curvature length. Since the bulk spacetime is static, there is a bifurcate For D ¼ 3 and D ¼ 5 cases we are interested in, there is Killing horizon in the bulk anchored to the bifurcate surface no conformal anomaly, and hence there is no logarithmic C on the AdS boundary. So, the entanglement entropy term in (2.5) [18]. Since the background bulk solution gMN corresponds to the area of the bifurcate surface of the is static, there is a timelike Killing vector field, ð∂ ÞM on Killing horizon of the black droplet solution. t M, and let us take ξM ¼ð∂ ÞM. We assume that δg is a We examine the first law of entanglement entropy in the t MN solution to the linearized equations of motion satisfying the boundary field theory by applying the Noether charge Dirichlet boundary condition at the conformal boundary of formula to the bulk Einstein gravity in a (D 1)- þ the bulk AdS M, dimensional AlAdS spacetime [7]. Let ξM be a Killing vector field with respect to the bulk metric g . The Noether MN δ˜ 0 charge (D − 1)-form Q associated with ξM is given by gð0Þμν ¼ : ð2:6Þ

1 Since g˜ μνð0

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D−2 2 δgμν ¼ Oðz Þ: ð2:7Þ 2 2 dρ 2 2 2 2 ds4 ¼ −FðρÞdτ þ þ ρ ðdθ þ sin θdΦ Þ; FðρÞ When ∂Σ is the bifurcation surface of a bulk black hole, 2 3 ρ μ ρ0 the second term in Eq. (2.4) disappears on ∂Σ and the first FðρÞ¼ þ 1 − ; μ ¼ þ ρ0; ð3:1Þ L2 ρ L2 term yields the variation of the black hole area (entropy). In the black droplet solutions, the horizon extends to the AdS boundary, and hence the area diverges towards the boun- where ρ0 is the horizon radius. The double Wick rotation dary. Note however that the variation should be finite for the Dirichlet boundary condition (2.6). When ∂Σ is taken as ˜ a part of the boundary of Σ at the AdS conformal boundary, τ ¼ iχ; Φ ¼ it ð3:2Þ the formula (2.4) provides the variation of the holographic energy E [19], defined by and the coordinate transformation r˜ ≔ cos θ yields Z the following warped product type metric including μ ν E ¼ Tμνξ n ; ð2:8Þ 2-dimensional de Sitter spacetime, ∂Σ ρ2 ˜2 where Tμν is the stress-energy tensor of the boundary field 2 2 d 2 2 ˜2 dr μ ds ¼ FðρÞdχ þ þ ρ −ð1 − r˜ Þdt þ : theory, n is the unit normal vector to ∂Σ, and the volume FðρÞ 1 − r˜2 element of ∂Σ with respect to the conformal metric g 0 is ð Þ ð3:3Þ understood in the integral. Applying the holographic stress- energy formula [18], Tμν is given in terms of the coefficient ˜ χ ˜ ˜ gðDÞμν as By the rescaling of the coordinates ð ; r; tÞ,

D−1 DL r0 r0 L Tμν ¼ g˜ μν; ð2:9Þ ˜ ˜ χ φ 16π ðDÞ t ¼ 2 t; r ¼ ; ¼ r0 ; ð3:4Þ l r l in odd dimensional cases. Our setup for holographic derivation of the first law for one obtains the following bubble solution: entanglement entropy in CFT is as follows. As our spatial hypersurface Σ for evaluating the Noether charge formula, 2 2 2 2 2 2 2 2 2 dρ ρ r r − r l dr L r FðρÞ we take the lower half of the time slice enclosed by part of 2 0 − 0 2 φ2 ds ¼ ρ þ 2 2 2 dt þ 2 − 2 þ ρ2 d ; the AdS boundary, black droplet horizon and χ ¼ π=2 Fð Þ l r l r r0 (shaded region in Fig. 1). Our hypersurface Σ has the ð3:5Þ asymptotic boundary S ⊂ ∂Σ, which itself admits its boundary (D − 2)-sphere denoted by D in Fig. 2. Thus, 1 the hypersurface Σ admits the boundaries and corners. We where φ circle has a period 2π. The S circle along ∂φ will impose the reflecting (Neumann) boundary condition shrinks to zero size at the bubble radius, ρ ¼ ρ0,i.e., at χ ¼ π=2 (red dotted line in Fig. 1). We will discuss in Fðρ0Þ¼0. By imposing a regularity condition at the Sec. IV that this condition is necessary for the purpose of bubble radius, r0 is expressed by ρ0 as avoiding any additional contributions to the first law from the upper side half-moon-shaped region. In the next 2ρ0lL section, in more concrete setting, we consider the r0 ¼ 3ρ2 2 : ð3:6Þ Noether charge at the boundaries ∂Σ and also study an 0 þ L additional contribution to the Noether charge from the corner. Note that the boundary metric [i.e., 3-dimensional metric in the square brackets in Eq. (3.5)] is conformal to the static III. THE FIRST LAW OF ENTANGLEMENT BTZ black hole with the horizon radius r ¼ r0 [20]. Note IN THE EXACT DROPLET SOLUTION IN also that in the 4-dimensional bulk, there is a bulk horizon (3 + 1)-DIMENSION which extends from the bubble radius ρ0, deep inside the bulk spacetime to the boundary BTZ black hole horizon. In A. The exact droplet solution in (3 + 1)-dimension this way, this bulk metric describes a black droplet as Let us first quickly review the exact solution of a black illustrated in Fig. 1. droplet [14] constructed by the analytic continuation of the Let us change the coordinates in Eq. (3.5) into the (3 þ 1)-Schwarzschild-AdS solution, Fefferman-Graham coordinate system (2.5) by

026004-4 FIRST LAW OF ENTANGLEMENT ENTROPY IN ADS BLACK … PHYS. REV. D 104, 026004 (2021) 1 1 2 3 Now we assume that our perturbations hMN satisfy the ¼ z þ α2ðηÞz þ α3ðηÞz þ ; ρðη;zÞ ηL symmetry along ∂φ. Under the gauge choice (3.9), the 2 4 variation of the Noether charge (2.1) and the symplectic rðη;zÞ¼r0ðη þ β2ðηÞz þ β4ðηÞz þÞ; two form (2.2) for the Killing vector ξ ¼ ∂t are calculated η2 − 2 μ ∞ α − α − for t ¼ const: and r ¼ surface for the metric (3.5) as 2 ¼ 4 η3 ; 3 ¼ 6 2η4 ; L L Z Z 2 4 1 η − 1 η − 1 a i tt rr β β δQ ¼ ϵρφ ðh − h Þg g ∂ g 2 ¼ ; 4 ¼ 3 ; : ð3:7Þ 32π tr a i r tt 2η 8η r¼∞ Z 1 − ϵ tt rr ∂ − ∂ Then, we obtain the stress-energy tensor (2.9) on the 16π ρφtrg g ð thrt rhttÞ; ð3:10Þ boundary field theory by

2 2 and μLr0ðη − 1Þ T ¼ − ; tt 16πl4η3 Z Z 1 μ rr tt L ξ · Θ ¼ − ϵρφ g g Tηη ¼ ; 16π tr 16πη3ðη2 − 1Þ r¼∞ 1 μ 2 i a Lr0 × hi ∂rgtt − gtt∂rha þ ∂thrt − ∂rhtt ; Tφφ ¼ − : ð3:8Þ 2 8πl2η ð3:11Þ Since the BTZ black hole is in odd-dimensions, the trace μ anomaly vanishes; i.e., Tμ ¼ 0, as seen in Eq. (3.8). So, a ρ φ where we denote the partial traces by ha ¼ hρ þ hφ ; while even-dimensional black droplet solutions have infin- i t r h ¼ h þ h . Note that as usual, the indices of hμν are ite energy due to the anomaly [21], the holographic energy i t r E in Eq. (2.8) converges to a finite value. When one applies raised and lowered by the background metric gμν and its ν ≔ λν the formula (2.4) to the black droplet solution (3.5), one inverse, e.g., hμ hμλg . Then, the combination of should note that the 2-dimensional surface ∂Σ consists of Eq. (3.10) and (3.11) yields 0 three parts: a part at the AdS boundary, z ¼ , a part of the Z Z bulk bifurcate Killing horizon, r ¼ r0, and a part of the 1 1 δQ−ξ Θ ϵ rr tt a∂ − ∂ a timelike surface at r ∞. In the next subsection we ð · Þ¼ ρφtrg g ha rgtt gtt rha : ¼ ∞ 16π 2 evaluate Eq. (2.4) on part of the timelike surface at infinity. r¼ ð3:12Þ B. Perturbation of the (3 + 1)-dimensional black droplet solution With the condition (3.9) and the perturbed vacuum Einstein a equations, we can find that the partial trace ha vanishes in We consider metric perturbations hMN ≔ δgMN of the black droplet solution (3.5). For convenience we 4-dimensional bulk [22], i.e., express our coordinate system as xM ¼ðya;xiÞ with a i a φ ρ 0 y ≔ ðρ; φÞ;x ≔ ðt; rÞ and denote by γ˜ij the 2-dimensional ha ¼ hφ þ hρ ¼ : ð3:13Þ de Sitter metric spanned by xi in (3.5). We can impose the following gauge condition: Thus, the rhs of Eq. (3.12) vanishes on t ¼ const: and r ¼ ∞ surface. It follows from the holographic interpretation 0 ∝ γ˜ hai ¼ ;hij ij: ð3:9Þ [23] that the first law of the entanglement entropy S is satisfied: This condition may immediately be justified by, e.g., applying the scalar-type perturbations of [22] to the δ δ 4-dimensional Schwarzschild-AdS solution (3.1) with the E ¼ T S ð3:14Þ i scalar harmonics Skðx Þ and taking the gauge fa ¼ 0; HT ¼ 0 in the terminology of [22]. Note however that in the black droplet solution, where T denotes the temper- when applying the formalism of [22] to the present case, the ature computed from the surface gravity of the bifurcate i scalar “harmonics” Skðx Þ corresponds to the solutions of Killing horizon. This result is peculiar to the 4-dimensional the Klein-Gordon equation on the Lorentzian sphere (i.e., bulk case, because Eq. (3.13) is satisfied only for D ¼ 3 de Sitter space with the metric γ˜ij), rather than the standard case. In the other dimensions, we need to impose boundary eigenfunction of the laplace operator with respect to the conditions for bulk metric perturbations to derive the first unit two-sphere metric. law, as we will see in the next section.

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IV. THE FIRST LAW IN (4 + 1)-DIMENSIONAL gravity dual, just mentioned above—we derive the first law FIELD THEORY IN ADS BLACK HOLE under the no flux condition. BACKGROUND In this section, we examine the first law of the entangle- A. The energy flux on the perturbed AdS geometry ment entropy for CFT in D ¼ 5-dimensions by using the We consider, as our background, the (5 þ 1)-dimen- holographic principle and applying the Noether charge sional AdS bulk with the metric of the form: formula as in the previous section. For the use of the holographic ideas, it would be most convenient if there is, ρ2 ρ2 2 − ρ 2 d ρ2 Ω2 ρ ≔ 1 ds6 ¼ F0ð Þdt þ þ d 4;F0ð Þ þ 2 ; as the gravity dual, any known exact solution of the F0ðρÞ L (5 þ 1)-dimensional black droplet that possesses the boundary D ¼ 5-dimensional metric ð4:2Þ where dΩ2 denotes the standard metric of the 4-dimen- 2 2 4 2 2 dr 2 2 M r sional unit sphere. We use the perturbation formalism of ds ¼ −fðrÞdt þ þ r dΩ3;fðrÞ ≔ 1 − þ ; 5 fðrÞ r2 L2 [24], which is most convenient for the current purpose and ð4:1Þ is also useful in the next section. Accordingly we use coordinates ya ≔ ðt; ρÞ to express the 2-dimensional AdS metric gab given by the first two terms in (4.2), and angular Ω2 ≔ σ θm θn i m where d 3 mnd d is the metric of 3-dimensional coordinates x ¼ðχ; θ Þ with 0 ≤ χ ≤ π to the metric γij of θm 1 Ω unit sphere with angular coordinates (m ¼ , 2, 3). the 4-dimensional unit sphere ð4Þ: Unfortunately there is no such an analytic solution available in (5 þ 1)-dimension. But, fortunately, with the 2 i j 2 2 2 dΩ4 ¼ γijdx dx ≔ dχ þ sin χdΩ3 Noether charge formula at our hand, all we need for our χ2 2 χσ θm θn purpose is the asymptotic behavior of the gravity dual near ¼ d þ sin mnd d : ð4:3Þ the boundary rather than the entire structure of the χ explicitly written exact solution. In fact, in the D ¼ 3 case Note that the coordinate is later changed to r. examined in III B, we have the exact gravity dual (3.5) On this background, we consider the metric perturba- available, but we need not use the explicit expression of tions, which can in general be classified into three types: the scalar-, the vector-, and the tensor-type according to (3.5) to derive the first law (3.14) for the boundary CFT. Ω Therefore, in this section, we simply assume that the their tensorial behavior on the 4-dimensional sphere ð4Þ as explained in [24]. The tensor-type perturbations are char- desired gravity dual exists. We will attempt to justify this δ assumption by perturbatively constructing a 6-dimensional acterized as the transverse and trace-free part of gij, and in δ 0 black droplet solution in the next section. particular, gtχ ¼ , and therefore there is no contribution Since our 5-dimensional boundary (4.1) itself is an to the energy flux Ttr from the tensor-type perturbations. asymptotically AdS spacetime with a 4-dimensional time- We discuss the vector-type perturbations in the Appendix. like surface as its conformal boundary [that is, r → ∞ in In the following we focus on the scalar-type metric (4.1): see point D in Fig. 2 and also compare ρ ¼ ∞; perturbations. S χ ¼ π=2 in Fig. 1], by analyzing the asymptotic behavior of It is convenient to introduce the harmonic functions ðlÞ Ω the gravity dual, we can holographically study the asymp- on the unit four-sphere ð4Þ, which satisfy totic fall-off behavior of our 5-dimensional CFT at i S 3 S 0 0 1 2 … large distances r → ∞. By doing so we can estimate the D Di ðlÞ þ lðl þ Þ ðlÞ ¼ ;l¼ ; ; ; ; ð4:4Þ energy flux—if it exists—across the 4-dimensional boundary r → ∞ and discuss conditions—if necessary—for the with Di being the covariant derivative with respect to the γ first law to hold. In Subsec. IVA, we holographically study metric ij. Then the scalar-type metric perturbations can be S possible energy flux at the boundary r → ∞ and derive the expressed in terms of ðlÞ as condition for having no energy flux across the boundary. X X δ S δ S For this purpose, it is sufficient to examine the metric gab ¼ HðlÞab ðlÞ; gai ¼ HðlÞaDi ðlÞ; 5 1 perturbations of the ( þ )-dimensional pure AdS space- l l time since the asymptotic behavior of the gravity dual near X 1 δ γ S − γ k S the boundary (ρ → ∞; χ → π=2 of Fig. 1) should be gij ¼ HðlÞL ij ðlÞ þ HðlÞT DiDj 4 ijD Dk ðlÞ; approximately well described by metric perturbations of l the pure AdS background spacetime near the corresponding ð4:5Þ boundary [see also Eq. (4.2) below]. Then, in Subsec. IV B, by considering metric variations of the (5 þ 1)-dimensional where HðlÞab, HðlÞa, HðlÞL, and HðlÞT are functions of the static black droplet—which is assumed to exist as the coordinates ya ¼ðt; ρÞ. Hereafter, for brevity we omit the

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l a1 a2 indices ( ) labeling the mode, unless otherwise stated. We Φ a0 : : ¼ þ ρ þ ρ2 þ ð4 12Þ introduce the following perturbation variables Z and Zab: l l 3 The boundary condition (2.6) imposes a0 ¼ 0 and sets the 4 ð þ Þ 2ρ aρ Z ¼ HL þ 4 HT þ ðD ÞXa ; integral constants in Eqs. (4.10) to zero. Then the stress- energy tensor on the boundary theory (2.9) is proportional 2 3 ρ to the coefficient a1. Zab ¼ ðHab þ DaXb þ DbXaÞþ4 Zgab; The perturbed metric is transformed to the Fefferman- 1 H X ≔−H ρ2D T ; : Graham coordinate (2.5) by a a þ 2 a ρ2 ð4 6Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r2=L2 L2ðL2 − r2Þ where D is the covariant derivative with respect to the ρ ¼ 1 þ z2 a z 4ðL2 þ r2Þ 2-dimensional unperturbed metric gab. Note that the two 4 4 − 4 2 2 4 variables Z and Zab are gauge-invariant. The perturbed L ðL L r þ r Þ 4 þ 2 2 2 z þ ; Einstein equations can be expressed in terms of Z and Zab 16ðL þ r Þ and furthermore reduced to the following equation for a 2 2 4 þ L z single scalar field Φ on the 2-dimensional AdS spacetime L tan χ ¼ r : ð4:13Þ 4 − L2z2 with the metric gab: Then, expanding the metric component δg as a series in z, 2 þ lðl þ 3Þ tr DaD Φ − Φ 0; : 3 a ρ2 ¼ ð4 7Þ Ttr in (2.9) can be read from the coefficient of z as

a1 in terms of which Z and Z are expressed as T ∼ ∂ S: ð4:14Þ ab tr ð1 þ r2=L2Þ3=2 r 1 Z ¼ D D − g ðρ2ΦÞ; Therefore in the boundary theory side, the energy flux ab a b L2 ab across the AdS boundary r ¼ ∞ (see χ → π=2 in Fig. 1) 2 a a − ρ2Φ becomes Z ¼ Za ¼ D Da 2 ð Þ: ð4:8Þ L rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 ∼ μ 1 r 3 Ω ∼ 2∂ S Now, using the gauge degrees of freedom, we can choose J lim Ttμn þ 2r d 3dt a1 lim r r ; r→∞ L r→∞ the following gauge: ð4:15Þ

Hρ ¼ Hρρ ¼ Htρ ¼ 0: ð4:9Þ where nμ is the unit normal to the timelike AdS boundary ∞ In this gauge, Xρ, Xt, and HT are determined by the second hypersurface at r ¼ . From the harmonic equation (4.4), and third lines in Eqs. (4.6) as S behaves near the boundary as Z 1 ρ pffiffiffiffiffiffi 3 B ffiffiffiffiffiffi d − Zffiffiffiffiffiffi S ≃ A : : Xρ ¼ p 2 F0Zρρ p ; þ ð4 16Þ 2 F0 ρ 4 F0 r Z dρ 2 X F0 Z ρ iωρ Xρ ; By Eq. (4.15), the condition for no energy flux is t ¼ ρ2 ð t þ Þ Z F0 ρ 2 d B ¼ 0: ð4:17Þ H 2ρ Xρ; : T ¼ ρ2 ð4 10Þ In terms of the angle coordinate χ, this condition is 2 2 where as previously introduced F0 ¼ 1 þ ρ =L . From the equivalent to imposing third line of Eqs. (4.6), Ht is expressed as lim ∂χS ¼ 0; ð4:18Þ 1 χ→π 2 − − ω = Ht ¼ Xt 2 i HT; ð4:11Þ for the harmonic functions. It is easily checked from where we have assumed that all the metric functions depend Eq. (4.4) that this is satisfied only for even l. In the next on t as e−iωt. subsection, we will check that the first law of the entan- Let us expand the master variable Φ near the AdS glement entropy is satisfied under the above no-energy flux boundary ρ → ∞ as conditions.

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B. The first law of the 6-dimensional hμν ¼ δgμν. For simplicity, we assume that the perturbed black droplet solution metric satisfies SOð4Þ symmetry, in which the only As mentioned before, throughout this section, we assume off-diagonal term appeared is htχ. Since our background that there is a static black droplet solution whose conformal boundary metric is the 5-dimensional static black hole, boundary is the D ¼ 5 asymptotically AdS, static SOð4Þ such a SOð4Þ symmetric perturbation corresponds to symmetric black hole with the metric gμνðz ¼ 0;xÞ given by monopole or mass perturbation for the boundary (4.1). Such a black droplet solution should admit the black hole. We briefly comment on possible contri- Fefferman-Graham expansion (2.5) with D ¼ 5 metric butions to the Noether charge formula from general 4 gμνðz; xÞ conformal to the boundary metric (4.1). In this perturbations without SOð Þ symmetry in the subsection, as the radial coordinate in the D ¼ 5-boundary, Appendix. we use χ instead of r, so that xμ ¼ðt; χ; θmÞ. On our assumptions, the Noether charge (2.1) and Now, to derive the first law applying the Noether the symplectic form (2.2) are evaluated at t ¼ const: and charge formula, let us consider metric perturbations χ ¼ π=2 hypersurface as

Z Z 1 1 tt χχ t χ μ δQ ¼ − ϵz123tχg g ∂thχt − ∂χhtt þð∂χgttÞ ht þ hχ − hμ ; ð4:19Þ χ¼π=2 16π 2 Z Z 1 1 1 ξ Θ − ϵ χχ tt∂ t tt∂ − l mn∂ ½ · z123 ¼ z123tχg g thχt þ ht g χgtt hm g χgnl χ π 2 16π 2 2 ¼ = 1 χ tt∂ mn∂ − tt∂ − mn∂ n ml∂ þ 2 hχ ðg χgtt þ g χgmnÞ g χhtt g χhmn þ hm g χgnl ; ð4:20Þ

m where we emphasize that the indices l, m, n represent the angular coordinates on the unit three-sphere: θ ∈ Ω3. Each term includes the off-diagonal term δgtχ, but this vanishes in the combination of Eqs. (4.19) and (4.20) as follows:

Z Z 1 1 1 1 χχ tt m mn l χ mn mn ðδQ − ξ · ΘÞ¼ ϵz123tχg − g hm ∂χgtt − g hm ∂χgnl − hχ g ∂χgmn þ g ∂χhmn : ð4:21Þ χ¼π=2 16π 2 2 2

From the expression above (4.21), we immediately find that where ∇μ is the covariant derivative with respect to the ˜ if both the background and perturbed metrics satisfy the conformal boundary metric gð0Þμν in Eq. (2.5) under the Neumann boundary condition boundary condition (2.6). The χ-component of (4.23) is written as

lim ∂χgμν ¼ lim ∂χhμν ¼ 0; ð4:22Þ ν α ν ν α χ→π=2 χ→π=2 ∂νδTχ − Γ νχδTα þ Γ ναδTχ ¼ 0: ð4:24Þ

ν t Since the only nonzero off-diagonal term of δTμ is δTχ (or then Eq. (4.21) vanishes. This also shows the first law of the δT χ), and the background metric is static diagonal as entanglement entropy (3.14) in the D ¼ 5 case, as in the t assumed, the last two terms in Eq. (4.24) must vanish on D ¼ 3 case. χ ¼ π=2 by the boundary condition (4.22). Thus, As shown in the previous subsection, the latter condition Eq. (4.24) leads to corresponds to the no-energy flux condition (4.18) or (4.17) for the scalar-type perturbations. As for the vector- and ∇ δ ν ∂ δ t ∂ δ χ 0 limπ ν Tχ ¼ limπð t Tχ þ χ Tχ Þ¼ : ð4:25Þ tensor-type perturbations, it is not obvious from the above χ→2 χ→2 expression alone that one has no contributions to (4.21). Nevertheless, one can show, at least for the present SOð4Þ The boundary condition (4.22) implies limχ→π=2 ∂χg˜μν ¼ ˜ symmetric case, that the energy flux caused by the limχ→π=2 ∂χhμν ¼ 0. The fifth-order of the Fefferman- perturbation hμνðz; xÞ vanishes on χ ¼ π=2 as follows: ˜ δ˜ Graham expansion of the perturbed metric hð5Þμν ¼ gð5Þμν The perturbed boundary stress energy tensor δTμν satisfies ˜ satisfies ∂χh 5 μν ¼ 0 at χ ¼ π=2, and therefore it follows the conservation law [18] ð Þ from the holographic stress-energy tensor formula (2.9) that χ ˜ ˜ limχ→π 2 ∂χδTχ ¼ 0. Then, since ∂ hμν ¼ −iωhμν on the ∇ δ ν 0 = t ν Tμ ¼ ; ð4:23Þ linearized perturbation, Eq. (4.25) yields

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t lim δTχ ¼ 0: ð4:26Þ In the static perturbation with ω ¼ 0, Ztρ ¼ Htρ ¼ Ht ¼ χ→π=2 Xt ¼ 0. Adopting the gauge choice

This is nothing but the no energy flux condition, as observed 0 in Eq. (4.17). Hρ ¼ Hρρ ¼ ; ð5:2Þ

V. THE PERTURBATIVE APPROACH IN Xρ, HT are determined by the gauge-invariant variables (5 + 1)-DIMENSIONAL BLACK DROPLET Zρρ, Z by Eqs. (4.10). One can always set SOLUTION H lim tt ¼ 0 ð5:3Þ In this section, we will construct the asymptotic geom- ρ→∞ ρ2 etry of the 6-dimensional black droplet solution under the boundary condition (4.22) in order to justify our analysis by choosing the integral constant in the first line in based on the holographic method in the last section. We Eqs. (4.10) appropriately. To construct the boundary metric will also examine the energy density of the boundary field (5.1) by perturbing the bulk metric (4.2), we find that the theory. When the boundary black hole mass parameter M in angle coordinate χ in Eq. (4.2) and the radial coordinate r of Eq. (4.1) is small compared with the AdS scale, i.e., the boundary metric (5.1) should be related as M ≪ L2, the asymptotic metric can be constructed from the static perturbation of the pure AdS spacetime (4.2).As dr shown below, a negative energy appears near the asymp- dχ ¼ : ð5:4Þ Lf r totic region, inconsistent with the D ¼ 3 result (3.8).We ð Þ also show that the holographic energy (2.8) is finite. This is because there is no trace anomaly in odd-dimensions, being This is integrated and approximately expressed at large r as different from the even-dimensional case [21]. 2 3 4 We require that the conformal boundary metric of the χ r 1 − ML ML tan ¼ 4 þ 6 þ : ð5:5Þ perturbed bulk metric coincides with the metric (4.1) L 5r 35r asymptotically in the sense that 2 Then, hrr ¼ δgrr at Oðρ Þ can be expanded as 1 2 − 2 1 −6 2 ds5 ¼ fðrÞdt þ ð þ Oðr ÞÞdr 1 X α fðrÞ hrr k lim 2 ¼ 2 2 : ð5:6Þ 2 −6 2 ρ→∞ ρ L f r r2k þ r ð1 þ Oðr ÞÞdΩ3 ; ð5:1Þ ð Þ k¼0 where fðrÞ¼1 − M=r2 þ r2=L2 as previously introduced Therefore the leading order of the bulk perturbed metric by (4.1). with Oðρ2Þ becomes

ρ2 ρ2 X α 2 2 ≃ d 1 k dr − 2 2 2 χ Ω2 ρ0 ds6 þ 2 þ 2k þ fðrÞf dt þ L ðsin þ hLÞd 3g þ Oð Þ; ð5:7Þ F0 ρ L f r f r ð Þ ð Þ k¼0 r ð Þ

where hL is the function of χ, arising from the perturbation. which satisfy the boundary condition (4.22). From now on If both the conditions we show the indices (l) labeling the mode. The analytic solution of Eq. (4.7) that satisfies the regularity at ρ ¼ 0 is given by the hypergeometric function α0 ¼ α1 ¼ α2 ¼ 0; ð5:8Þ 2 2 F with the argument F0ðρÞ¼1 þ ρ =L (see [24]): and ρ − 1 1þl=2 2 2 1 1 Φ F0ð Þ l þ l þ ðlÞ ¼ a0ðlÞ · F ; ; ; F0ðρÞ 2 2 2 F0ðρÞ 2 2 2 −4 fðrÞL ðsin χ þ h Þ¼r þ Oðr Þ; ð5:9Þ 3 2 L Γðlþ Þ 1 l þ 3 l þ 3 3 1 − 2 2 pffiffiffiffiffiffiffiffiffiffiffiffi · F ; ; ; : lþ2 2 2 2 ρ Γð 2 Þ F0ðρÞ F0ð Þ are satisfied, our bulk metric is conformal to the 5- dimensional Schwarzschild-AdS metric (5.1),uptoOðr−4Þ. ð5:10Þ As shown below, both the conditions (5.8) and (5.9) are satisfied for a suitable superposition of the first three even This equation enables us to describe a1ðlÞ by a0ðlÞ as S 2 mode harmonic functions, ðlÞ (l ¼ , 4, 6) in Eq. (4.4), follows:

026004-9 AKIHIRO ISHIBASHI and KENGO MAEDA PHYS. REV. D 104, 026004 (2021) lþ3 2 Γð 2 Þ where Eq. (4.4) is used in the first line. As for the second a1 l ¼ −2L a0 l : ð5:11Þ ð Þ lþ2 ð Þ line, we first write Zab;Zin Eqs. (4.8) and Xρ in Eq. (4.10) Γð 2 Þ using Eq. (5.10) and obtain the expression of HL, HT, S which we then insert in the first line. Note that when The first three even mode harmonic functions ðlÞ are given by evaluating Xρ, one has to integrate the rhs of the first line of Eq. (4.10) and choose the integration constant so that the 3 S χ 5 2 χ − 1 condition (5.3) is satisfied, as mentioned before. Then, one ð2Þð Þ¼4 ð cos Þ; finds in the leading order, 5 S χ 1 − 14 2 χ 21 4 χ ð4Þð Þ¼8 ð cos þ cos Þ; a0ðlÞ 3 7 X ρ ∼− þ Oð1=ρ Þ: ð5:15Þ S χ −5 135 2 χ − 495 4 χ 429 6 χ ðlÞ ρ ð6Þð Þ¼64 ð þ cos cos þ cos Þ: ð5:12Þ Together with By Eqs. (4.10), the leading term of HðlÞT becomes

2 H ¼ c ρ2 þ Oð1Þ; ð5:13Þ ρ ðlÞT ðlÞ Z ∼ 4a0 þ Oð1Þ; ð5:16Þ ðlÞ ðlÞ L2 where cðlÞ is the integral constant for HðlÞT in Eqs. (4.10). Then, from (4.5), we find that δgχχ is asymptotically, one obtains ρ → ∞, described by X lðl þ 3Þ 2 hχχ H H S χ l l 3 Z ρ ¼ ðlÞL þ 4 ðlÞT ðlÞð Þ ð þ Þ − 2ρ aρ ∼ 3 1 2 4 6 HL þ HT ¼ ðD ÞXa a0ðlÞ 2 þ Oð Þ; l¼ ; ; 4 4 L ∂2S χ ð5:17Þ þ HðlÞT χ ðlÞð Þ X 3 a0ðlÞ 2 2 ¼ S þ c ∂χS ρ þ Oð1Þ; ð5:14Þ which provides the expression of (5.14). This hχχ can be L2 ðlÞ ðlÞ ðlÞ l¼2;4;6 expanded as a series in 1=r by using Eq. (5.5). The result is

hχχ 5 3 lim ¼ ð48c 2 − 112c 4 þ 189c 6 Þ − ð48a0 2 − 40a0 4 þ 35a0 6 Þ ρ→∞ ρ2 32 ð Þ ð Þ ð Þ 64L2 ð Þ ð Þ ð Þ 5 144 − 336 567 − 16 2 12 − 154 567 þ 64 2 f a0ð2Þ a0ð4Þ þ a0ð6Þ L ð cð2Þ cð4Þ þ cð6ÞÞg r 5 2 1 þ f−72a0 2 þ 420a0 4 − 1323a0 6 þ L ð96c 2 − 2576c 4 þ 19089c 6 Þg þ O : ð5:18Þ 32r4 ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ r6

X 3 From this expansion, the coefficients αk in Eq. (5.6) can be a0ðlÞ 2 h ¼ sin χS þ c sin χ cos χS χ : determined. Then, under the condition (5.8), each coef- L L2 ðlÞ ðlÞ ðlÞ; l¼2;4;6 ficient cðlÞ can be obtained as ð5:20Þ 864 240 245 a0ð2Þ þ a0ð4Þ þ a0ð6Þ cð2Þ ¼ 2 ; 5280L Substituting Eq. (5.5) into Eq. (5.9), we determine each −144 240 115 2 a0ð2Þ þ a0ð4Þ þ a0ð6Þ coefficient a0ðlÞ (l ¼ , 4, 6) in terms of the small parameter c 4 ¼ ; ð Þ 2080L2 M as 576a0 2 þ 600a0 4 − 6895a0 6 − ð Þ ð Þ ð Þ cð6Þ ¼ 90090 2 : ð5:19Þ 4 16 64 L − M − M − M a0ð2Þ ¼ 297 ;a0ð4Þ ¼ 585 ;a0ð6Þ ¼ 5005 : Similarly, hL in Eq. (5.7) is expanded in terms of the ð5:21Þ harmonic functions SðlÞ as

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Therefore the asymptotic bulk geometry is completely VI. SUMMARY AND DISCUSSIONS determined by the coefficients. In this paper, we have derived the first law of the Due to the gauge choice (5.2), the coordinate trans- entanglement entropy for two subsystems separated by formation an AdS black hole for odd-dimensional CFTs by using the holographic method and applying the Noether charge L 1 ρ ¼ − z ð5:22Þ formula. We have seen that the nonvanishing contribution 2 z to our Noether charge generically arises not only at the bifurcate horizon and the asymptotic infinity but also at a leads to the Fefferman-Graham coordinate (2.5) in the spacelike hypersurface which terminates at the “equatorial μ original conformally static Einstein frame, x ¼ðt; χ; θmÞ, plane" on the boundary static Einstein universe (see the ˜ m ¼ 1, 2, 3. Then, under the boundary condition (5.3), htt dotted red line in Fig. 1). This is because the boundary AdS in Eqs. (4.5) can be expanded as a series in z, black hole covers only one-half of the global AdS boundary. X 4 − 3 3 3 For (2 þ 1)-dimensional CFT, we have, as its gravity ˜ lðl þ Þ 1 − 3 2 htt ¼ 2 8 þ 8 ½ lðl þ Þz dual, the exact black droplet solution [14], and therefore by l¼2;4;6 L perturbing it, we have holographically shown that the first Γ 3þl 2 4lðl þ 3Þ ð 2 Þ 3 4 law is satisfied without imposing any additional conditions. þ z a0 S þ Oðz Þ; ð5:23Þ 5 lþ2 ðlÞ ðlÞ As for (4 1)-dimensional CFT, restricted to the SO 4 Γð 2 Þ þ ð Þ symmetric perturbations, we have shown in Sec. IV that the where we used Eq. (5.11). So, according to the AdS=CFT first law is satisfied when there is no energy flux across the timelike conformal boundary, which corresponds to the dictionary, the energy density Ttt in the static Einstein universe can be read off from the z3 coefficient as Neumann boundary condition on the spacelike bulk hypersurface at χ ¼ π=2. We give a brief discussion on the analysis of generic perturbations in the Appendix. We 3285L2M π 2 ≃− χ − believe that an analysis for general perturbations without Ttt 1024 þ O 2 ð5:24Þ SOð4Þ symmetry would also show the first law, but in the present paper, we have not been able to fully clarify combined with the result (5.21). Note that the this issue. Schwarzschild-AdS metric on the boundary (5.1) is We would like to emphasize again that in our holo- obtained by a conformal transformation from the static graphic setup, in order to compute the Noether charge, we Einstein frame by have taken Σ as the lower half shaded region in Fig. 1 and pffiffiffiffiffiffiffiffiffi imposed, in particular, the reflection (Neumann) boundary 2σ σ χ π 2 gμν → e gμν;e¼ fðrÞ: ð5:25Þ condition at ¼ = . The reflection boundary condition at χ ¼ π=2 is necessary to avoid additional contributions to Then, the stress-energy tensor is obtained from the trans- the first law from the upper side of Fig. 1. One may think of Σ formation what happens if one takes as the entire time slice. For example, if Σ is taken as the union of the single black −3σ droplet in the lower half and the regular (no black droplet) Tμν → e Tμν: ð5:26Þ region in the upper half of Fig. 1, then the reflection condition at χ ¼ π=2 is no longer satisfied, and hence there The asymptotic energy density in the 5-dimensional is no guarantee that the first law can be derived. First of all, Schwarzschild-AdS metric is obtained by if Σ is taken as the entire time slice, then the corresponding boundary CFT would live in a spatially compact universe, → −3=2 ∼ −3 Ttt TttfðrÞ r : ð5:27Þ rather than in an asymptotically AdS boundary spacetime, and the notion of a boundary black hole itself is no longer Thus, the holographic energy (2.8) becomes finite. clearly defined. Since the negative energy density (5.24) in the asymp- In Sec. V, under the Neumann boundary condition for totic region is proportional to the mass parameter M in no-energy flux, we have constructed the asymptotic geom- the Schwarzschild AdS black hole (4.1), this reflects the etry of a small black droplet solution from scalar-type of the curvature of the black hole. So, one may say that the linear metric perturbations of the pure AdS spacetime. For negative energy density is caused by the vacuum polari- this purpose, we have expanded the metric perturbations by zation effect, as explained in [9]. This behavior is quali- scalar harmonics on the sphere and also set the boundary tatively the same as the case of 5-dimensional small black black hole mass M as the small parameter. To satisfy the droplet numerical solutions [21] with an AdS black hole on Neumann boundary condition, only even modes of the the boundary. scalar harmonics are permitted.

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In Ref. [25], quantization of conformally coupled scalar ACKNOWLEDGMENTS fields in anti–de Sitter spacetime was considered in various We would like to thank Takashi Okamura for useful schemes. One is the “transparent” boundary condition in discussions. This work was supported in part by JSPS which each positive frequency classical solution propagates KAKENHI Grants No. 17K05451, No. 20K03975 (K. M.) beyond the timelike boundary at spatial infinity, and hence and No. 15K05092, No. 20K03938 (A. I.). the Klein-Gordon inner product is defined on the whole Einstein static cylinder, by adding another “virtual” AdS APPENDIX: NOETHER CHARGE FORMULA spacetime. In this case, observers living in an asymptotically FOR GENERAL PERTURBATIONS OF 6- AdS spacetime would recognize that some information or DIMENSIONAL ADS energy is lost through the timelike boundary. Another boundary condition is the “reflective” boundary condition In this Appendix we consider the Noether charge in which each positive frequency classical solution is formula (2.4) for general perturbations in (5 þ 1)- reflected at the timelike boundary at infinity, and therefore dimensional pure AdS bulk background metric (4.2). information or energy loss does not occur on the boundary. Note that although the perturbation analysis in Sec. IV B In this paper, by adapting the latter “reflective” boundary is restricted to SOð4Þ symmetric case, the background condition, we have shown the first law of the entanglement geometry treated in Sec. IV B is the black droplet solution, entropy for D ¼ 5 CFT in asymptotically AdS spacetime. which we assume to exist. In this sense the analysis in Conversely, if the first law is satisfied in asymptotically AdS Sec. IV B is more relevant to our purpose for obtaining the spacetime, the “reflective” boundary condition is derived for first law than the analysis below. the holographic stress-energy tensor. On the other hand, in As introduced in sec. IVA, we use the coordinates the D ¼ 3 case, the Noether charge at the infinity is always ya ≔ ðt; ρÞ in the 2-dimensional AdS spacetime and the zero, being irrespective of the boundary conditions in the angular coordinates xi ¼ðχ; θmÞ. We denote our Killing holographic model we consider. It would be interesting to vector field by ξM ¼ðξa; ξiÞ. The integrand of the Noether explore whether the first law of the entanglement entropy is charge formula (2.4) is in the present case a 4-form on the Ω γ satisfied for various quantizations of a free scalar field in 4-sphere ð ð4Þ; ijÞ. For generic metric perturbations BTZ background. δgMN ¼ hMN ¼ðhab;hai;hijÞ,wehave

1 1 Dbρ δQ − ξ · Θðg; δgÞ¼− ϵ ab hgac − hac D ξb − hajD ξb þ haξj 16π i1i4 2 c j ρ j 1 ρ hb ha a b − b a ξc a j − b j ξj þ 2 ðD hc D hcÞ þ 2 D ρ D ρ D ρ Dbρ − ξa D hbc D hbj n c hbc − hk − Dbh : c þ j þ ρ ρ k ðA1Þ

X ð1Þ The tensor-type metric perturbations h are tangential δg ¼ 0; δg ¼ H V ; MN ab ai ðlvÞa ðlvÞi Ω γ a l to ð ð4Þ; ijÞ with no components along y direction, and X v Ω γ ð1Þ behave as a transverse-traceless tensor on ð ð4Þ; ijÞ.Itis δg ¼ H ðD V − D V Þ: ðA3Þ ij TðlvÞ i ðlvÞj j ðlvÞi straightforward to see that the 4-form (A1) vanishes for lv tensor-type perturbations. Now let us consider the vector-type perturbations. We Hereafter we omit the mode indices ðlvÞ for notational introduce the vector harmonics on Ω 4 , which satisfy ð Þ simplicity. Note that since the gauge transformation of the vector-type perturbations is generated by a vector field ð1Þ ð1Þ Xi ¼ X V i with a scalar X on the 2-dimensional AdS i V 3 − 1 V 0 iV 0 D Di ðlvÞj þ½lvðlv þ Þ ðlvÞj ¼ ;DðlvÞi ¼ ; spacetime, one can impose, for instance, the gauge con- 1 dition Hð Þ 0. lv ¼ 1; 2; 3; …: ðA2Þ T ¼ The gauge invariant variables for the vector-type perturbations are given by

The vector-type metric perturbations can be expanded in ð1Þ terms of V as ð1Þ 2 HT ðlvÞi Z H − ρ D : a ¼ a a ρ2 ðA4Þ

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This variable can be expressed by a single master variable So far, we have not yet used the property that ξM is the ΦV on the 2-dimensional AdS spacetime spanned by ðt; ρÞ as Killing vector. In Sec. IVA, we are concerned with the static black hole background and our Killing vector is 1 ξa∂ ∂ ∂ ξi 0 Z ¼ ϵ Dbðρ2Φ Þ; ðA5Þ a ¼ = t, ¼ . Thus, in this case, one can immedi- a ρ2 ab V ately see that the above formula (A9) vanishes. Now suppose that a rotating black droplet solution be ϵ where ab is the metric compatible volume element on the available, and let us consider ξi ≠ 0. In that case, one 2-dimensional AdS spacetime gab. The master variable should be able to choose ξi as a linear combination of the ΦV satisfies Ω rotational symmetry generators of ð4Þ. It is known that for the dipole moment l ¼ 1, V i itself is a Killing vector field 2 2 þ l ðl þ 3Þ v DaD Φ − v v Φ 0: Ω ξi V i a V 2 þ ρ2 V ¼ ðA6Þ on ð4Þ, and therefore one can take one of such so that L i ξ V i ≠ 0. Then, one should, in principle, be able to find a Killing vector field ξM∂ ¼ ∂=∂t þ Ω ∂=∂ψ which The normalizable solution ΦV which also satisfies the M ðAÞ ðAÞ regularity condition in thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bulk is given by Eq. (154) of becomes tangent to the null generators of the Killing horizon of the black droplet solution, where Ω are [24] with ν ¼ 3=2; σ ¼ lvðlv þ 3Þþ9=4. (Note that for ðAÞ the normalizable solution, the frequency ω is quantized to be certain constants corresponding to the angular velocity ψ ω ¼ ∓ð2m þ 1 þ ν þ σÞ, m ¼ 0; 1; 2; ….) Then, from of the black hole, and ðAÞ angular Killing parameters for Eq. (154) of [24], one can see the asymptotic behavior of ξi. For such a Killing vector field ξM, the second-term of rhs ΦV as of the above formula (A9) should provide the term proportional to the variation of the angular momenta. In fact, b1 b2 Φ : A7 substituting (A8) into (A9) gives a nonvanishing term V ¼ ρ2 þ ρ4 þ ð Þ Ω ω2 2 6 2 proportional to ðAÞð L b1 þ b2=L Þ. For the stationary case ω ¼ 0, this can be viewed as adding an angular ρ It then follows that at large , momentum to the background black hole by the dipole perturbation. 2 2 ∼− b2=L 1 ρ5 ∼− ω 2 b1 b2 The analysis of generic perturbations of the scalar-type Zt 3 þ Oð = Þ;Zρ i L 4 þ 6 þ : ρ ρ ρ appears to be much more involved with more terms in the ðA8Þ Noether charge formula (A1). If we can evaluate the integration of the above formula (A1) for both the vector- For the vector-type perturbations, the Noether charge type and scalar-type perturbations, not only at ρ → ∞; formula (A1) is expressed in terms of the gauge-invariant χ → π=2 but also at the Killing horizon of the (hypothetical) variable (A5) as rotating black droplet as well as other relevant boundaries, then we expect that the first law with work term should be δQ − ξ · Θðg; δgÞ obtained. To fully verify this argument is, however, beyond the scope of the present paper where we consider only linear 1 Z ¼ − ϵ ab − a V iD ξ þ D Z ξiV : ðA9Þ perturbations and their asymptotic behavior. 16π i1i4 ρ2 i b ½a b i

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