Holographic chaos, pole-skipping, and regularity
aa r X i v : . [ h e p - t h ] F e b Prog. Theor. Exp. Phys. , 00000 (19 pages)DOI: 10.1093 / ptep/0000000000 Holographic chaos, pole-skipping, andregularity
Makoto Natsuume and Takashi Okamura KEK Theory Center, Institute of Particle and Nuclear Studies, High EnergyAccelerator Research Organization, Tsukuba, Ibaraki, 305-0801, Japan †∗ E-mail: [email protected] Department of Physics, Kwansei Gakuin University, Sanda, Hyogo, 669-1337, Japan ∗ E-mail: [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We investigate the “pole-skipping” phenomenon in holographic chaos. According to thepole-skipping, the energy-density Green’s function is not unique at a special point incomplex momentum plane. This arises because the bulk field equation has two regularnear-horizon solutions at the special point. We study the regularity of two solutions morecarefully using curvature invariants. In the upper-half ω -plane, one solution, which isnormally interpreted as the outgoing mode, is in general singular at the future horizonand produces a curvature singularity. However, at the special point, both solutions areindeed regular. Moreover, the incoming mode cannot be uniquely defined at the specialpoint due to these solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index AdS/CFT correspondence, Black holes in string theory, Relativity
1. Introduction and Summary
In recent years, the quantum many-body chaos attracts much attention. A useful probe ofchaos is out-of-time-ordered correlation function (OTOC). An OTOC shows the early-timeexponential growth for a chaotic system: C ( t, ~x ) = h V ( t, ~x ) W (0) V ( t, ~x ) W (0) i β ≃ − e λ ( t − x/v B ) + · · · , (1)where V and W are generic operators, β is the inverse temperature. λ is the (quantum)Lyapunov exponent and v B is the butterfly velocity. Generically, λ satisfies the bound [13] λ ≤ πT . (2)The AdS/CFT duality or holography [1–4] is a useful tool to study quantum many-bodysystems (see, e.g. , Refs. [5–9]). It is conjectured that a holographic system saturates thebound, or black hole is maximally chaotic [10–13]. † Also at Department of Particle and Nuclear Physics, SOKENDAI (The Graduate University for Advanced Studies),1-1 Oho, Tsukuba, Ibaraki, 305-0801, Japan; Department of Physics Engineering, Mie University, Tsu, 514-8507, Japan. c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. he quantum chaos has been studied using OTOC, but recently it is claimed that thechaotic behavior can be seen even at the level of retarded Green’s functions. This phe-nomenon is known as “pole-skipping” [14, 15] . To motivate the pole-skipping, write C ( t, ~x )in a plane-wave form C ( t, ~x ) ≃ − e − iωt + iqx (3)with purely imaginary values of ( ω, q ): ω = iλ =: ω ⋆ , q = i λv B =: q ⋆ . (4)For Schwarzschild-AdS p +2 black holes [10], λ = 2 πT , v B = p + 12 p . (5)The pole-skipping claims that retarded Green’s function shows a characteristic behavior atthe “special point” in momentum space ( ω ⋆ , q ⋆ ). More explicitly, consider the energy-density2-point function. Any perturbation carries energy, so one would expect to see the chaoticbehavior from the energy-density 2-point function. Generically, one would write the functionas G RT T ( ω, q ) = b ( ω, q ) a ( ω, q ) . (6)The pole-skipping claims that a ( ω ⋆ , q ⋆ ) = b ( ω ⋆ , q ⋆ ) = 0 , (7)and one can locate λ and v B in this way. Then, naively G R = 0 /
0, but more precisely, G R is not uniquely determined at the special point. Near the special point, G R = δω ( ∂ ω b ) ⋆ + δq ( ∂ q b ) ⋆ + · · · δω ( ∂ ω a ) ⋆ + δq ( ∂ q a ) ⋆ + · · · = ( ∂ ω b ) ⋆ + δqδω ( ∂ q b ) ⋆ + · · · ( ∂ ω a ) ⋆ + δqδω ( ∂ q a ) ⋆ + · · · . (8)Then, the Green’s function at the special point is not unique because it depends on the slope δq/δω .The pole-skipping is interesting and is useful. Perviously, one needs to evaluate an OTOCor a 4-point function in order to see a chaotic behavior. But a real-time finite-temperature4-point function is not easy to evaluate even in the AdS/CFT duality. In principle, onecan compute them [18], but such a computation is extremely rare. As a result, OTOChas not been extensively studied in a variety of bulk systems. One approach is to use theWKB approximation which assumes large scaling dimensions for V and W [10]. As anotherapproach, one can compute OTOC in the AdS black hole [19] since one can map the blackhole to the pure AdS via a conformal transformation. But the chaos bound (2) has not beenexplicitly verified in a variety of bulk systems. This situation contrasts with the viscositybound η/s = 1 / (4 π ) [20], which has been extensively verified in a variety of bulk systems.However, the pole-skipping claims that the chaotic behavior can be seen at the level of2-point functions, and these are objects we often compute in the AdS/CFT duality. So,the approach allows us to study the holographic chaos more in details in a variety of bulksystems. However, the pole-skipping has many unanswered questions. Most importantly, itis not clear why the 2-point function has anything to do with the 4-point function. See, e.g. , Refs. [16, 17] for discussion of the pole-skipping from field theory point of view.2/19 .2. Universality and near-horizon physics
It is conjectured that holographic systems saturate the chaos bound. From the bulk point ofview, such a universal behavior is often related to the universal nature of the near-horizonphysics. One well-known example is η/s . So, one would expect that the near-horizon physicsplays an important role in the holographic chaos as well. Ref. [15] studies this issue in thecontext of the pole-skipping.The energy-density 2-point function corresponds to solving the scalar mode (sound mode)of gravitational perturbations, so they examine the perturbation problem. They found thatthe near-horizon physics shows a special behavior at the special point, namely the specialpoint is characterized by the appearance of an extra regular incoming solution. The fieldequation in general has 2 solutions, the incoming mode and the outgoing mode. We areinterested in the retarded Green’s function, so we choose the incoming mode. However, atthe special point, ◦ Both solutions are regular at the horizon. ◦ One cannot distinguish between the incoming and the outgoing modes.Then, the incoming mode is not uniquely defined, and as a result, the Green’s function isnot unique.They conclude the regularity of the extra solution just by looking at the mode function, butthis definition is ambiguous. Even if a mode function diverges at the horizon, multiplying anappropriate function which vanishes at the horizon can produce a new mode function whichis regular. Since the regularity is a key feature of the pole-skipping, it is worthwhile to studythe regularity more in details. This is a purpose of this paper.The regularity of an arbitrary mode function is not meaningful. From the bulk pointof view, we should require a mode function not to produce a curvature singularity. We use“regularity” in this sense. We reanalyze the scalar mode in the SAdS black hole backgroundusing variables where regularity can be clearly seen. We also compute curvature invariantsfor perturbations.In a perturbation problem, it is often useful to use a “master variable.” But one needs to becareful since it may fail at particular points in momentum space. Instead, it is straightforwardto analyze the special point if one uses the full set of gauge-invariant variables and the fulllinearized Einstein equation. This allows one to analyze the other systems where mastervariables are hard to find.Our results are summarized as follows: ◦ At the special point, two solutions are indeed regular, and the curvature remains finite. ◦ The outgoing mode with ℑ ω > ω -plane. (In this paper, we focus on the upper-half ω -plane.) ◦ The field equation has the regular singularity at the horizon r = 1, but at the spe-cial point, it becomes a regular point in the incoming Eddington-Finkelstein (EF)coordinates. As a result, two regular solutions appear. While this paper and the companion paper [25] are in preparation, there appeared preprints[26, 27] which have some overlap with ours. 3/19 s we see below, the incoming-wave boundary condition is not uniquely defined at the specialpoint. The boundary condition is imposed by deviating from the special point and takingthe δω, δq → δω/δq . Theexistence of two regular solutions reflects the slope dependence.
2. Sound mode
We consider the 4-dimensional pure gravity S = 116 πG Z d x √− g ( R − , Λ = − L , (9)and consider the Schwarzschild-AdS (SAdS ) black hole: ds = r ( − f dt + d~x ) + dr r f , (10) f = 1 − r − . (11)For simplicity, we set the AdS radius L = 1 and the horizon radius r = 1. The Hawkingtemperature is given by 2 πT = 3 / dr ∗ := dr/ ( r f ), anincoming wave behaves like e − iω ( t + r ∗ ) , and an outgoing wave behaves like e − iω ( t − r ∗ ) . Thus,it is convenient to set v = t + r ∗ and work with the incoming Eddington-Finkelstein (EF)coordinates. The metric becomes ds = r ( − f dv + d~x ) + 2 dvdr . (12)In the EF coordinates, an incoming wave behaves like e − iωv , and an outgoing wave behaveslike e − iωv e iωr ∗ ≃ e − iωv ( r − iω/f ′ (1) = e − iωv ( r − iω/ (2 πT ) . (13)The horizon consists of the future horizon and the past horizon. We use the incoming EFcoordinates, so the horizon r = 1 corresponds to the future horizon (for a finite v ). In the pole-skipping, the issue of the incoming and outgoing modes is rather confusing, soit would be helpful to make a few remarks.
Definition of incoming/outgoing mode.
At the special point, ω, q are pure imaginary, andthe incoming (outgoing) nature is somewhat obscure, so let us first start from the definitionof these modes.The incoming and outgoing modes at the future horizon behave like e − iω ( t ± r ∗ ) . This meansthat we first define them for ω ∈ R . In a black hole background, we are also interested in ω ∈ C , and the incoming and outgoing modes are defined by analytic continuation from We use upper-case Latin indices
M, N, . . . for the 4-dimensional bulk spacetime coordinates anduse Greek indices µ, ν, . . . for the 3-dimensional boundary coordinates. The boundary coordinates arewritten as x µ = ( t, x i ) = ( t, x, y, · · · ). Lower-case Latin indices a, b, · · · are used for the 2-dimensionalsubspace ( v, r ). 4/19 ∈ R . This definition seems to work fine in most cases. Then, one would conclude that thedistinction between 2 modes is clear even at the special point. However, as we see below,one cannot distinguish between the incoming and the outgoing modes at the special point. Importance of boundary condition.
To clarify our statement about the pole-skipping, letus make a few incorrect statements. If one said that“there is an extra regular solution at the special point, so one must include it,”this is not really correct. What solution one chooses depends on the boundary condition. Ifone just considers the general solution, one does not impose a boundary condition. Anothermisleading statement is“the outgoing mode is also regular at the special point, so one must include it.”Again, whether one selects the outgoing mode or not depends on the boundary condition.According to the standard AdS/CFT rule, the choice of boundary condition at the horizonreflects the choice of Green’s function. The incoming mode corresponds to the retardedGreen’s function. If one is interested in the retarded Green’s function, one should not includethe outgoing mode. Regularity is just a prerequisite.Therefore, what matters eventually is the incoming-wave boundary condition at the specialpoint. As we discuss below, the incoming-wave boundary condition is not uniquely defined[see Eq. (27)], or the incoming mode is not uniquely determined at the special point. This isthe reason why we have to include both solutions. One solution is normally interpreted asan outgoing mode. We exclude it since we normally compute the retarded Green’s function.It is excluded from the point of view of regularity as well. But at the special point, it is alsoregular. Moreover, it should be taken into account to define the incoming mode.
Advanced Green’s function?.
The outgoing mode is in general prohibited from regu-larity, but it is prohibited at the future horizon H + . Again, according to the standardAdS/CFT rule, the choice of a boundary condition at the horizon simply reflects the choiceof Green’s function. The incoming (outgoing) mode at the future horizon corresponds tothe retarded (advanced) Green’s function. So, one would argue that these modes must betreated symmetrically and the outgoing mode should also be allowed.The asymmetry comes from the incoming EF coordinates. We use the incoming EF coor-dinates, so we impose the boundary condition at the future horizon H + . In order to computethe advanced Green’s function, one should impose a boundary condition at the past hori-zon H − , not at the future horizon H + . In other words, one should use the outgoing EFcoordinates. The outgoing mode is allowed at H − . We consider gravitational perturbations of the form h MN ( r ) e − iωv + iqx . (14)As is well-known, gravitational perturbations are decomposed as scalar mode, vector mode,and tensor mode. (For p = 2, there is no tensor mode though.) The perturbations are decom-posed under the transformation of boundary spatial coordinate x i (see Appendix A for thedetails). For example, the scalar mode transforms as scalar under the transformation. In this aper, we are interested in the energy-density 2-point function. This corresponds to solvingthe scalar mode, which has 7 perturbations. For p = 2, they are given by h vv , h vr , h rr , h vx , h rx , h xx , h yy . (15)The scalar mode has 7 perturbations, but they are redundant due to the diffeomor-phism. Normally, one fixes the gauge h rM = 0, which reduces to 4 perturbations. Then,one constructs gauge-invariant variables which are invariant under the residual gaugetransformation. This is the formalism advocated e.g. , by Kovtun and Starients [21].Instead, we do not fix the gauge and carry out analysis in a fully gauge-invariant manner.We essentially follow the formalism by Kodama and Ishibashi [22] and use the variableswhich are invariant under the full diffeomorphism . In either case, there are 4 variables. Wedenote the gauge-invariant variables as h vv , h vr , h rr , and h L defined by Eqs. (A26).So far, we use only the gauge invariance to reduce degrees of freedom. We now imposethe equations of motion. Then, these 4 gauge-invariant variables are not independent, andthe equations of motion leave us only a single degree of freedom which obeys a second-orderdifferential equation. They are referred as the master field and the master equation. Thelinearized Einstein equation reduces to0 = h rr + 2 r f h vr , (16a)0 = 2 ( ( i w − r ) (cid:0) i w r − q (cid:1) r f + i w − q r ) h vv + (cid:18) w − q + q f (cid:19) r f h rr − i w (cid:26) f (cid:18) w r (cid:19) − (cid:18) q r (cid:19) + f (cid:27) h L , (16b)0 = h ′ vv − (cid:18) − q i w r (cid:19) h vv r − (cid:18) i w q i w f (cid:19) r f h rr − r h L , (16c)0 = h ′ rr + 3 ( − i w r + (cid:18) q + i w rr ( r − i w ) − (cid:19) f − q i w ( r − i w ) (cid:18) f (cid:19) ) h rr rf − q r + 3 i w r r − i w h L r f + 2( q − i w r )3 i w ( r − i w ) h vv r f , (16d)0 = h ′ L − (cid:18) i w r + f (cid:19) f h L r − (cid:18) − q i w r (cid:19) f h vv r − q i w r f h rr , (16e)where ′ = ∂ r and w = ω πT , q = q πT v B = √ q πT . (17)In terms of w and q , the special point is located at ( w ⋆ , q ⋆ ) = ( i, ± i ).There are 2 constraint equations which do not involve r -derivatives and 3 differentialequations which have one r -derivative. The latter 3 are not independent; one is redundantfrom the constraint equation (16b). The constraint equations (16a) and (16b) allow us to In the terminology of Kodama and Ishibashi, the scalar mode here is called “scalar-type”perturbations. 6/19 hoose 2 independent variables. Both obey first-order differential equations, so one gets asecond-order differential equation for a single variable which is the master equation.Thus, there is only a single degree of freedom, but note the choice of a master field is notunique. One can choose any of 4 gauge-invariant variables h vv , h vr , h rr , and h L and linearcombinations as a master field. Among them, a few choices are worth mentioning: ◦ From the boundary point of view, it is natural to choose a master variable which does notinvolve r -derivatives of metric perturbations since one imposes the Dirichlet boundarycondition at infinity. This is the choice, e.g. , by Kovtun and Starinets [21] . ◦ It is often useful to rewrite the master equation in the form of a Schr¨odinger equation.This is the choice of Ref. [15].
We are interested in the regularity of the perturbations, and we eventually show this fromgeometric quantities. Any variable is fine in principle. But it would be better if one couldcheck regularity from the behavior of the mode function. So, it is worthwhile to pause hereand to consider which variable is suitable for that purpose.As mentioned in Introduction, an arbitrary mode function is not very suitable. A divergingmode function can be regular by multiplying an appropriate power of ( r − C ∞ ) at the future horizon H + .Unlike the Schwarzschild coordinates, the incoming EF coordinate system is regular at thefuture horizon. Thus, the metric perturbations must be smooth there as well. Then, theRiemann tensor components of the perturbed spacetime are also smooth. The master variableof Ref. [15] is not appropriate for that purpose. It is not a metric perturbation itself, and itis unclear if all metric perturbations are regular or not.As we see below, gauge-invariant metric perturbations are expanded as a Taylor seriesfor the H + -incoming mode. On the other hand, perturbations are not a Taylor series andcurvature invariants diverge for the H + -outgoing mode. Before we solve the sound mode at the special point, let us solve the problem for a generic( w , q ) as a warmup exercise. The choice of the master variable is not unique, but we areinterested in regularity of all metric perturbations. So, it is better to choose a variable whoseregularity guarantees the regularity of the other variables. The variable h rr is most suitablefor the purpose. The other variables are expressed by h rr and h ′ rr (Appendix B.1), and the h rr regularity at the horizon guarantees the regularity of h vv , h vr , and h L . However, thisfails when i w + q = 0 which includes the special point ( w , q ) = ( i, ± i ), so the special pointmust be examined separately. Note that our gauge-invariant variables implicitly depend on h ′ MN through η a [see Eq. (A26)], soa general linear combination depends on h ′ MN . 7/19 he h rr -equation becomes0 = h ′′ rr + (cid:18) r + f ′ f − i w − rr f (cid:19) h ′ rr + 3 r f (cid:18) − i w r − q r (cid:19) h rr . (18)The equation has regular singularities at r = 0 , ∞ , and at 3 zeros of f which includes r = 1.According to Ref. [15], the near-horizon behavior is important for the pole-skipping, so solvethe equation by a power series expansion around r = 1: h rr = ( r − λ X n =0 a n ( r − n . (19)At the lowest order, one gets the indicial equation and obtains λ = 0 , λ = − i w . (20)The coefficient a n is obtained by a recursion relation. The λ -mode is the incoming modesince h rr ∝ e − iωv , and the λ -mode is the outgoing mode. We show regularity carefully fromcurvature invariants later (Sec. 4). But as discussed in the previous subsection, one may checkregularity from h rr . The λ -mode is regular at the horizon, but the λ -mode is singular atthe horizon in general when ℑ w >
0. This result does not apply to the special point thoughsince the master variable h rr fails there.One needs a slight modification when λ and λ differ by an integer. In such a case, thesmaller root fails to produce the independent solution since the recursion relation breaksdown at some a n . Suppose that the smaller value is λ . Write 2 solutions as h (for λ ) and h (for λ ). Then, the second solution in general takes the form h = h ln( r −
1) + ( r − λ X n =0 b n ( r − n . (21)In any case, the leading behavior near the horizon comes from the second term when ℑ w > λ -mode is again singular in general.
3. Special point
The master variable is a useful technique, but it has some problems: ◦ First, it is often not easy to find a master variable. ◦ Second, the choice of the master variable is not unique, so one needs to find whichvariable is most suitable. This poses a problem particularly for the pole-skipping sincesome master variable like h rr breaks down at the special point so should not be usedthere.In order to analyze the special point, it is straightforward to use the full linearized Einsteinequation (16) instead of a master equation. At the special point, the linearized Einsteinequation becomes0 = 2 r h vv − ( r + r + 1) h rr − r + 1) h L , (22a)0 = h ′ vv − r + 1 h vv + ( r + 2) ( r + r + 1) ( r + 3 r + 2)4 r ( r + 1) h rr , (22b)0 = h ′ rr + 3 1 + r r + r h rr , (22c)0 = h ′ L − r h L − ( r + 2) ( r + r + 1)2 h rr . (22d) or a generic ( w , q ), the linearized Einstein equation has a regular singularity at the horizonat r = 1. However, at the special point, the regular singularity becomes a regular point.Namely,The special point is characterized by the regular singularity at the horizon r = 1becoming a regular point in the incoming EF coordinates.As a result, two independent solutions both become regular.One can obtain the solution explicitly at the special point . In this particular example, h rr is independent from the others and obeys a first-order differential equation, so one cansolve it as h ⋆rr = C R ( r ) , (23a) R ( r ) := (cid:18) r + r + 1 (cid:19) / exp (cid:26) −√ (cid:18) Arctan 2 r + 1 √ − π (cid:19)(cid:27) (23b) ∼ C { − r −
1) + · · · } ( r → . (23c)Then, the rest is solved as h ⋆L = r { C I ( r ) + C } , (23d) h ⋆vv = 32 ( r + 1) " C (cid:26) ( r + r + 1) r ( r + 1) R + I ( r ) (cid:27) + C , (23e) I ( r ) := Z r dr ′ ( r ′ + 2) ( r ′ + r ′ + 1)2 r ′ R ( r ′ ) . (23f)Two solutions are indeed regular at the horizon. The h rr -solution depends only on 1 integra-tion constant C . This is rather unusual, but the full solution still depends on 2 integrationconstants C , C . We consider linear perturbations, so an overall constant is not relevant,but the solution is not unique due to C /C .The above solution is the general solution, and we have not imposed a boundary condi-tion at the horizon. The field equation has two independent solutions. Normally, one is theincoming mode, and the other is the outgoing mode. We usually pick up the incoming modeto compute the retarded Green’s function.At the special point, the regular singularity becomes a regular point. As a result, twosolutions are both regular. Does this mean that there exists a regular outgoing mode? Belowwe argue that the incoming mode is not uniquely determined at the special point. For the right interpretation, move away from the special point: w = w ⋆ + δ w = i + δ w , q = q ⋆ + δ q = ± i + δ q , (24a) h MN = h ⋆MN + δ h MN . (24b)Instead of q , it is convenient to use the combination η := q i w . (25) Actually, one can obtain the general solution not only for the special point but also for i w + q = 0.9/19 way from the special point, the field equations have a regular singularity at r = 1 asusual, and the distinction between the incoming mode and the outgoing mode should beclear. Thus, we move away from the special point and approach the special point δ w , δ q → δ w /δ q how one approaches the special point.The resulting equations are rather lengthy, so we present them in Appendix B.2, but forexample, the δ h ′ L -equation is given by0 = δ h ′ L − (cid:18) r r + r + 1 (cid:19) δ h L r − rr + r + 1 δ h vv − r f δ h rr + 3 i δ w r f " δηi δ w rr + r + 1 h ⋆vv + (cid:26) − δηi δ w r ( r + 1) f ( r + r + 1) (cid:27) h ⋆L . (B2d)It may look complicated, but the structure is simple: the perturbation obeys an inhomoge-neous differential equation, and the source is given by the special point solution h ⋆MN . Weare interested in the retarded Green’s function, so we impose the incoming-wave boundarycondition on the perturbations δ h MN . An incoming wave is written as a Taylor series in theincoming EF coordinates, so the homogeneous part must be expanded as a Taylor series.However, the source term is proportional to ( r − − , so the equation in general producesan outgoing mode. To avoid this, we require that the source term is also written as a Taylorseries.From Eq. (B2d) and Eq. (22a), we obtain conditions for the special point solution: h ⋆L h ⋆vv (cid:12)(cid:12)(cid:12)(cid:12) H + = 23 δηi δ w , h ⋆rr h ⋆vv (cid:12)(cid:12)(cid:12)(cid:12) H + = 227 (cid:18) − δηi δ w (cid:19) . (27)This is the incoming-wave boundary condition for h ⋆MN . The boundary condition does notuniquely determine h ⋆MN and depends on the slope δη/δ w . The exact solution we obtaineddepends on the combination C /C . This reflects the slope dependence. Conversely, givena δη/δ w , one has to choose the combination C /C appropriately. Imposing the boundarycondition on the special point solution (23), we obtain C C = − γ γ − c ⋆ , (28)where γ := δη/ ( iδ w ).We emphasized the gauge-invariant variables h ⋆MN in this paper, but in order to obtainthe Green’s function, it is convenient to use a master variable `a la Kovtun and Starinets[21]. One imposes the Dirichlet boundary condition asymptotically, and the master variabledoes not involve r -derivatives of metric perturbations. The master variable Z G is written interms of our variables h MN as Z G = 1 g xx (cid:18) h vv − g ′ vv g ′ xx h L (cid:19) . (29)Using the special point solution (23), Z G asymptotically behaves as Z ⋆G = C ( c ⋆ + I ∞ ) " r + 32 r + 12 r √ e − π √ c ⋆ + I ∞ ! + O (cid:0) r − (cid:1) . (30)Then, the Green’s function at the special point depends on c ⋆ , and it is not unique there. . Regularity from curvature invariants So far, we have studied regularity at the horizon from the behavior of gauge-invariantvariables. In this section, we use curvature invariants to show regularity further.We consider pure gravity, so R MN ∝ g MN . Then, the curvature invariants R MN R MN and R remain unchanged under perturbations. Thus, we use the Kretschmann scalar R KLMN R KLMN . We thus consider the perturbed Kretschmann scalar δ ( R KLMN R KLMN ) := R KLMN R KLMN − R KLMN R KLMN , (31)where the boldface letters indicate background values. But this quantity is not appropriate.Under the gauge transformation x M → x M + ξ M , a scalar S transforms as S → S − ξ M ∂ M S . (32)As a result, the perturbed Kretschmann scalar itself is not gauge invariant.The gauge-invariant Kretschmann scalar can be constructed in the same manner as gauge-invariant variables (see Appendix A) and is given by δ ( R KLMN R KLMN ) GI := δ ( R KLMN R KLMN ) + η r ∂ r ( R KLMN R KLMN ) . (33)See Eq. (A27) for η r which is constructed by h MN . The explicit form of the gauge-invariantKretschmann scalar can be found in Appendix B.3.First, assume i w + q = 0, and use the master variable h rr . Earlier we found ( λ , λ ) =(0 , − i w ). Using Eq. (B3b), the perturbed Kretschmann scalar becomes δ ( R KLMN R KLMN ) GI ≃ − a ( w + i )( w + 2 i ) , (for λ ) (34)so the λ -mode is indeed regular. On the other hand, δ ( R KLMN R KLMN ) GI ≃ a ( i w − q ) w ( w + i ) ( r − i w , (for λ ) (35)Thus, the λ -mode is singular in general when ℑ w >
0. The λ -mode corresponds to theoutgoing mode. We are interested in the retarded Green’s function, so we are interested inthe incoming mode, but the outgoing mode is prohibited from regularity as well.Note that we consider perturbations of the form h MN ( r ) e − iωv + iqx . Because the backgroundspacetime is not flat, the perturbations contribute to the the perturbed Kretschmann scalarat O ( h MN ). Thus, the above results (as well as expressions in Appendix B.3) should bemultiplied by e − iωv + iqx . The Kretschmann scalar in real space is obtained by taking the realpart.When i w + q = 0, h rr fails, and one cannot use the above result. At the special point( w ⋆ , q ⋆ ), one can use the exact solution (23) and Eq. (B3a). One obtains δ ( R KLMN R KLMN ) GI ≃ C . (36)Thus, the Kretschmann scalar remains regular at the horizon. Again, one should multiplythe result by e − iω ⋆ v + iq ⋆ x . But in this case, there is no need to take the real part. ω ⋆ and q ⋆ arepure imaginary, but this just comes from the fact that we write the perturbation in plane-wave form. Recall that it actually behaves as e λ ( v − x/v B ) . Thus, the perturbations grow intime and so does the Kretschmann scalar. The perturbation breaks down eventually at latetime. But the point is that the perturbation does not produce the diverging Kretschmann calar near the horizon. Note that the integration constant C does not appear. In fact, theKretschmann scalar is independent of C .One can obtain the general solution when i w + q = 0 (footnote 6). When i w = − q = −
1, one can show that the Kretschmann scalar diverges at the horizon. Therefore,Two solutions are regular only at the special point when ℑ w > ω -plane. Remarks.
In this paper, we show regularity using curvature invariants. Actually, it isdifficult to show regularity of perturbations because there are many kinds of spacetimesingularities. There are two common singularities: ◦ s.p. (scalar polynomial) curvature singularity ◦ p.p. (parallelly propagated) curvature singularityFor a s.p. curvature singularity, curvature invariants diverge. For a p.p. curvature singularity,all curvature invariants remain finite , but a tidal force diverges. Such singularities appear,for example, ◦ in the extreme limit of some black p -branes and ◦ in the Lifshitz geometry [23].We have shown that 2 solutions do not produce a s.p. singularity at the special point, butstrictly speaking, we have not shown that they do not produce a p.p singularity.A p.p. singularity is the one where the Riemann tensor components diverge in a p.p. framealong at least one non-spacelike curve. Physically, a radially infalling observer experiences alarge tidal force. For example,(1) Compute the Riemann tensor R abcd in a convenient orthonormal frame, usually in thestatic frame.(2) An infalling observer measures the curvature not in the static frame but in anotherorthonormal frame which is related to the static frame by a local radial boost.This procedure is a little complicated, but for our purpose, one does not need to carry outcomputations explicitly. The argument goes as follows:(i) Suppose that our criterion holds, i.e. , 2 solutions are smooth at the future horizon.Then, the Riemann tensor in the EF frame is regular there.(ii) The magnitude of the Riemann tensor in general becomes large in the boosted frame(for a diagonal metric) [24]. But we use the incoming EF coordinates which is regularat the future horizon. Thus, the boost from the EF frame to the observer frame doesnot have a divergence at the horizon.(iii) Then, the regularity of the Riemann tensor in the EF frame implies the regularity ofthe Riemann tensor in the observer frame. Thus, 2 solutions do not produce a p.p.singularity.Nevertheless, we must stress that proving no p.p. singularity is very difficult. If the Rie-mann tensor diverges only along one curve, the spacetime is p.p. singular. In order to show In the literature, some authors do not impose this condition for a p.p. singularity. Then, s.p.singularities are part of p.p. singularities. 12/19 hat there is no p.p. singularity, one needs to examine the Riemann tensor along all curves,which is impossible in practice. What we can argue is that the perturbed spacetime is unlikelyto have a p.p. singularity: ◦ First, as discussed above, h MN is smooth at the horizon. ◦ Second, it is reasonable to focus on the radially infalling geodesic among all curves.Our interest is whether the outgoing mode is regular or not. The outgoing wave gets aninfinite boost from the incoming wave point of view, so the radially infalling geodesic islikely to give the most strict condition.
5. Discussion
In this paper, we examine the pole-skipping phenomenon using variables where regularity canbe clearly seen. The Kretschmann scalar is also computed to show regularity. It is straight-forward to analyze the special point if one uses the full set of gauge-invariant variables. Thisallows one to analyze the other systems where master variables are hard to find. However,dealing with the full set of equations is in general complicated. One way is to formulate theproblem as an eigenvalue problem [25].We observed that the regular singularity at r = 1 becomes a regular point at the specialpoint in the incoming EF coordinates. As a result, two regular solutions appear. One woulduse this criterion to explore special points in the other systems [25].We show regularity using curvature invariants, but there are cases where one cannot usecurvature invariants to show regularity. A simple example is the vector and the tensor modesof gravitational perturbations. For those modes, the perturbed Kretschmann scalar vanishes.We consider linear perturbations, so the quantities with different transformation propertiesdecouple. The Kretschmann scalar transforms as a scalar, but the these modes transformdifferently. Thus, for those modes, the outgoing mode does not produce a s.p. singularity.However, it is likely that the outgoing mode (for a generic ℑ w >
0) is not smooth at thehorizon. Then, it should produce a p.p singularity. In any case, the vector and tensor modesdo not have special points in the upper-half ω -plane [25], so the outgoing mode is not anissue there. Acknowledgments
We would like to thank Pavel Kovtun and Kengo Maeda for useful discussions. This researchwas supported in part by a Grant-in-Aid for Scientific Research (17K05427) from theMinistry of Education, Culture, Sports, Science and Technology, Japan.
A. Gauge-invariant variables
We consider the background spacetime ds = g ab ( y ) dy a dy b + e Φ δ ij dx i dx j , (A1) here y a = ( v, r ), x i = ( x, y ) , and e Φ = r . The 2-dimensional metric g ab is given by g ab = − r f
11 0 ! , g ab = r f ! . (A2a) A.1. Maxwell field example
Let us start from the Maxwell field A M . We assume that perturbations take the plane-waveform A M ∝ e − iωv + iqx . As we see below, the perturbations are decomposed asscalar: A v , A x , A r , (A3)vector: A y . (A4)It is not difficult to find variables which are invariant under the gauge transformation A M → A M − ∂ M λ . For the scalar mode, A v = A v + ωq A x , (A5) A r = A r − iq A ′ x . (A6) A v and A r are not independent: they are related by the Maxwell equation, and there is onemaster field for the scalar mode.It is not difficult to figure out the gauge-invariant variables for A M : they are just pro-portional to the field strength F MN . But for a systematic analysis, proceed as follows. Wedecompose the perturbations under the transformation of the boundary spatial coordinate x i . The scalar (vector) mode transforms as scalar (vector) under the transformation. TheMaxwell field consists of A M = ( A a , A i ). A i can be decomposed as A i = ∂ i A L + A T i , ∂ i A T i = 0 . (A7)The scalar mode consists of A a ( A v , A r ) and A L ∝ A x , and the vector mode is A T y = A y .For λ ∝ e − iωv + iqx , the gauge transformation δA M = − ∂ M λ (A8)becomes δA a = − ∂ a λ , (A9a) δA x = iqδA L = − iqλ , (A9b) δA T i = 0 . (A9c)Gauge-invariant variables eliminate the gauge parameter λ by combining variables. Thevariables A T i are gauge invariant by themselves. From Eq. (A9c), the gauge parameter λ is For the quantities defined in the p -dimensional subspacetime ( e.g. , h (1) ai below), the index i is raisedand lowered with δ ij . For simplicity, we consider the p -dimensional metric which is proportional to δ ij , but the extention to γ ij ( x ) is easy. Replace ∂ i with D i , the covariant derivative with respect to γ ij . Some expressions must be symmetrized since D i do not commute.14/19 xpressed by the perturbation A L as λ = − δA L . Substituting this into Eq. (A9a) gives δ ( A a − ∂ a A L ) = 0 , (A10)so the gauge-invariant scalar perturbations are given by A a := A a − ∂ a A L . (A11)In terms of components, Eq. (A11) reduces to Eqs. (A5) and (A6). A.2. Gauge-invariant metric perturbations
Now consider metric perturbations h MN = ( h ab , h ai , h ij ). Again, the perturbations aredecomposed as scalar, vector, and tensor mode. h ab gives 3 scalar perturbations. Just asthe Maxwell field example, h ai is decomposed as h ai = ∂ i h a + h (1) ai , ∂ i h (1) ai = 0 , (A12)and h a gives 2 scalar perturbations and h (1) ai gives vector perturbations. (The superscript“(1)” refers to the spin.) In a similar manner, h ij is decomposed as h ij =: h L δ ij + P ij h T + 2 ∂ ( i h (1) T j ) + h (2) T ij , (A13)where ∂ i h (1) T i = 0 , ∂ j h (2) T ij = 0 , h (2)
T ii = 0 , (A14)and P ij is the projection operator given by P ij := ∂ i ∂ j − p δ ij ∂ k . (A15)The first term of h ij is the trace part which is a scalar perturbation. The rest is the tracelesspart which is decomposed as scalar h T , vector h (1) T j , and tensor perturbations h (2) T ij . (For p = 2, there is no tensor mode.) Thus, ◦ The scalar mode consists of 7 perturbations ( h ab , h a , h L , h T ). ◦ The vector mode consists of perturbations ( h (1) ai , h (1) T i ).In components (for p = 2), the scalar mode is h xx = h L − q h T , h yy = h L + 12 q h T , h vx = iqh v , h rx = iqh r , (A16)and the vector mode is h ay = h (1) ay , h xy = iqh (1) T y . (A17)Again consider the gauge transformation δ G x M = ξ M . ( δ G refers to a gauge transforma-tion.) The infinitesimal transformation ξ i is decomposed as ξ i =: ∂ i ξ L + ξ T i , ∂ i ξ T i = 0 . (A18)Only ξ a and ξ L appear for the scalar mode. The scalar mode transforms as δ G h ab = − ∇ ( a ξ b ) , (A19a) δ G h a = − h ξ a + ∂ a ξ L − ξ L (cid:0) ∂ a Φ (cid:1) i , (A19b) δ G h L = − h ξ a ∇ a e Φ + 2 p ∂ k ξ L i , (A19c) δ G h T = − ξ L , (A19d) nd the vector mode transforms as δ G h (1) ai = − h ∂ a ξ T i − ξ T i (cid:0) ∂ a Φ (cid:1) i , (A19e) δ G h (1) T i = − ξ T i , (A19f)where ∇ a is the covariant derivative with respect to g ab .In order to obtain gauge-invariant variables, we again express gauge parameters ξ a , ξ L , and ξ T i by perturbations. For the vector mode, Eq. (A19f) expresses ξ T i by δ G h (1) T i . SubstitutingEq. (A19f) into Eq. (A19e), we obtain gauge-invariant vector perturbations: δ G (cid:16) h (1) ai − ∂ a h (1) T i + 2 h (1) T i ∂ a Φ (cid:17) = 0 → h ai := h (1) ai − e Φ ∂ a (cid:16) e − Φ h (1) T i (cid:17) . (A20)For the scalar mode, Eq. (A19d) expresses ξ L by δ G h T . Substituting Eq. (A19d) intoEq. (A19b), ξ a is expressed by δ G h a and δ G h T : ξ a = δ G η a , (A21a) η a := 12 ∂ a h T − h T ∂ a Φ − h a , (A21b)Substituting ξ a into Eq. (A19a), we obtain δ G (cid:0) h ab + 2 ∇ ( a η b ) (cid:1) = 0 → h ab := h ab + 2 ∇ ( a η b ) . (A22)Similarly, Eq. (A19c) becomes δ G (cid:18) h L + η a ∇ a e Φ − p ∂ i h T (cid:19) = 0 → h L := h L + η a ∇ a e Φ − p ∂ i h T . (A23)Let us write these formulae in components. For p = 2, the gauge-invariant vectorperturbations are h vy = h vy + ωq h xy , (A24) h ry = h ry − r iq (cid:18) h xy r (cid:19) ′ . (A25)The gauge-invariant scalar perturbations are h vv := h vv − i ω η v + g ′ vv ( − g vv η r + η v ) , (A26a) h vr := h vr + η ′ v − (cid:0) i ω + g ′ vv (cid:1) η r , (A26b) h rr := h rr + 2 η ′ r , (A26c) h L = h yy + 2 r ( η v − g vv η r ) . (A26d)From Eq. (A21b), η a becomes η v = − i q (cid:18) h vx + ωq h xx − h yy (cid:19) , (A27a) η r = 1 i q (cid:26) r i q (cid:18) h xx − h yy r (cid:19) ′ − h rx (cid:27) . (A27b) . Some formulae B.1. Expressions of gauge-invariant variables by h rr h vv r i w + q (cid:18) P vv ( r ) f r ddr + 3 Q vv ( r ) (cid:19) h rr , (B1a) P vv := i w (cid:26) w + 1 r − f (cid:18) q + 1 r (cid:19)(cid:27) , (B1b) Q vv := i w (cid:18) w + 1 r (cid:19) (cid:18) − i w r (cid:19) + i w f (cid:26) i w (cid:18) i w − f (cid:19) − w r + 2 i w r (cid:27) + 3 i w ( i w − q ) f (cid:18) − i w r − f (cid:19) + 2 i w + q r f . (B1c) h vr = − r f h rr . (B1d) h L − r = 14 r i w + q (cid:18) P L ( r ) f r ddr + 3 Q L ( r ) (cid:19) h rr , (B1e) P L := ( i w − q ) (cid:18) − i w r (cid:19) − i w (cid:18) − r (cid:19) (cid:18) i w + 1 r (cid:19) + q f , (B1f) Q L := ( i w − q ) (cid:18) − i w r (cid:19) (cid:18) − i w r (cid:19) − i w (cid:18) − r (cid:19) (cid:18) i w + 2 + w r (cid:19) (B1g)+ f (cid:26) i w (cid:18) − r (cid:19) (cid:18) i w + 1 r (cid:19) − ( i w − q ) (cid:18) − i w r (cid:19) − i w + q r (cid:27) . Note that h vv and h L are proportional to ( i w + q ) − , and it diverges at i w + q = 0 whichincludes the special point ( w , q ) = ( i, ± i ). Thus, the master variable h rr fails at the specialpoint, and the special point must be examined separately. B.2. Field equations near the special point r δ h vv − ( r + r + 1) δ h rr − r + 1) δ h L + 27 iδ w r ( r + 1)( r − ( r + r + 1) " − r (3 r + r + r + 1)9( r + 1) (cid:26) δηiδ w + r ( r − r + 1)3 r + r + r + 1 (cid:27) h ⋆vv + (cid:26) r f r + 1) (cid:18) r + r + δηiδ w (2 r + 1)(2 r + 3 r + 1) r ( r + r + 1) (cid:19)(cid:27) h ⋆L , (B2a)0 = δ h ′ vv − (cid:18) − r (cid:19) δ h vv r + 32 (cid:18) − f (cid:19) r f δ h rr − r δ h L + iδ w (cid:26) δηiδ w h ⋆vv r − (cid:18)
12 + δηiδ w f (cid:19) r f h ⋆rr (cid:27) , (B2b)0 = δ h ′ rr + 3 1 + r r + r δ h rr + 27 iδ w ( r + 2 r + 3) r f " − r + 1)( r + r + 1)3( r + 2 r + 3) (cid:26) δηiδ w + r (3 r + r + 4 r + 1)3( r + 1)( r + r + 1) f (cid:27) h ⋆vv + (cid:26) r − r − r + 2 r + 3) (cid:18) δηiδ w r + 6 r + 3 r + 411 r − r − (cid:19) f (cid:27) h ⋆L , (B2c) = δ h ′ L − (cid:18) r r + r + 1 (cid:19) δ h L r − rr + r + 1 δ h vv − r f δ h rr + 3 iδ w r f " δηiδ w rr + r + 1 h ⋆vv + (cid:26) − δηiδ w r ( r + 1) f ( r + r + 1) (cid:27) h ⋆L . (B2d)As discussed in the main text, we require that the source terms are written as Taylor series.Note that the source term of Eq. (B2b) does not contain ( r − − , so the equation gives nocondition. B.3. Gauge-invariant perturbed Kretschmann scalar
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