The Petrov-like boundary condition at finite cutoff surface in Gravity/Fluid duality
aa r X i v : . [ g r- q c ] A ug The Petrov-like boundary condition at finite cutoff surface inGravity/Fluid duality
Yi Ling , , , ∗ Chao Niu , † Yu Tian , , ‡ Xiao-Ning Wu , , , § and Wei Zhang , ¶ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China Center for Relativistic Astrophysics and High Energy Physics,Department of Physics, Nanchang University, 330031, China School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China Institute of Mathematics, Academy of Mathematics and System Science,Chinese Academy of Sciences, Beijing 100190, China Hua Loo-Keng Key Laboratory of Mathematics, CAS, Beijing 100190, China
Abstract
Previously it has been shown that imposing a Petrov-like boundary condition on a hypersurfacemay reduce the Einstein equation to the incompressible Navier-Stokes equation, but all thesecorrespondences are established in the near-horizon limit. In this paper, we demonstrate thatthis strategy can be extended to an arbitrary finite cutoff surface which is spatially flat, and theNavier-Stokes equation is obtained by employing a non-relativistic long-wavelength limit. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] . INTRODUCTION It has been known that the excitations of a black hole horizon dissipate very much likethose of a fluid since the 70’s of last century[1–4]. From then on, the gravity/fluid duality hasbeen heavily investigated and lots of important progress have been made[5–42]. Remarkably,recent progress in AdS/CFT correspondence has shed more insightful light on this duality.The method of hydrodynamical expansion of the metric was initially proposed to study thedual fluid living on the boundary of spacetime, in which the regularity condition is imposedon the horizon and a long-wavelength expansion is needed[24]. Later, an alternative waywas proposed to reduce the Einstein equation to the Navier-stokes equation by imposinga Petrov-like boundary condition on the cutoff surface[28]. The key idea of this strategyis to consider the perturbations of the extrinsic curvature of the cutoff surface directly,rather than those of the metric. As a result, the Brown-York stress tensor is treated asthe fundamental variable which due to the holographic dictionary can be identified with thestress-energy tensor of a fluid living on the cutoff surface[24, 41]. In another word, we mayextract the hydrodynamical behavior of gravity directly, needless to solve the perturbationequation for the explicit form of the perturbed metric. In literature the advantages of thisstrategy have been continuously disclosed. It has been successfully applied to a spacetimewith a spatially curved cutoff surface[31], or a spacetime with a cosmological constant aswell as matter fields[32, 37]. In particular, our recent investigation indicates that it can beapplicable for a very general spacetime which is only required to contain a weakly isolatedhorizon without rotation[39]. Nevertheless, comparing with the conventional hydrodynami-cal expansion method, the method of imposing Petrov-like boundary condition contains anobvious weak point, which sticks to the near-horizon limit. Namely, in this approach wealways take the non-relativistical limit with the near-horizon limit simultaneously. Whilerecently our understandings on the gravity/fluid duality have been significantly pushed for-ward by investigating the hydrodynamical behavior of gravity at finite cutoff surface basedon the Wilsonian approach or the renormalization group point of view[14, 20, 21]. One keyobservation in this approach is that any interacting quantum field theory at finite temper-ature should be described by hydrodynamics when viewed at sufficiently long length scales.Since the radial coordinate r of the AdS bulk corresponds to the energy scale of the boundaryfield theory, the near-horizon limit only captures the low-frequency limit of linear response2f the boundary theory fluid. Therefore, if one intends to move away from the low-frequencylimit, he needs consider fluid membrane at a hypersurface with constant-radius and finitedistance from the horizon. A flow equation for the radius-dependent response function hasbeen derived, for instance in [14], which can be viewed as a renormalization group flow tolink the gravity/fluid duality near horizon and that at infinity. In this paper we intendto establish such duality at finite cutoff surface with the Petrov-like boundary conditionmethod. Previously in original Ref.[24] it has been pointed out that in Rindler spacetimethe perturbed metric in bulk obtained by hydrodynamical expansion may be subject to thePetrov-type condition at finite cutoff surface(also see the similar check in [38]). However,would any perturbation constrained by the Petrov-like boundary condition at finite cutoffsurface lead to Navier-Stokes equation? Moreover, can this observation be extended to moregeneral spacetime background? We intend to investigate these issues in this paper. Wewill demonstrate through explicit models that once the near-horizon limit is replaced bythe long-wavelength limit, the incompressible Navier-Stokes equation can still be derived bydirectly imposing Petrov-like boundary condition on the finite cutoff surface such that thegravity/fluid duality can be established . Of course in this extension we will only focus onthe cutoff surface which is spatially flat since the long-wavelength limit is introduced.To keep this paper in a concise version, we will just present our main results in the mainbody, but leave all the detailed calculation in the appendix. II. PETROV-LIKE BOUNDARY CONDITION ON THE FINITE CUTOFF SUR-FACE FOR RINDLER SPACETIME
The framework of imposing Petrov-like boundary condition on the cutoff surface has beenintroduced in previous literature, and we refer to Refs.[28, 31] for details. Here we just repeatits basic definition and setting. The Petrov-like boundary condition on a hypersurface Σ c is The need of a long-wavelength limit in finite cutoff case can be understood from the viewpoint of holog-raphy. The radius of the cutoff corresponds to the energy scale of a dual field theory on the boundary. Innear-horizon case, the energy scale of dual theory approaches to zero, which means any perturbation ofthe dual theory are low energy modes which can be described by hydrodynamics. However, in finite cutoffcase, the energy scale of dual theory is not low enough such that not all perturbations have contributionsto the hydrodynamical degrees of freedom. In this sense we need the long-wavelength limit to pick outthose low energy perturbation which corresponds to the degree of freedom of hydrodynamics. C ( ℓ ) i ( ℓ ) j ≡ ℓ µ m iν ℓ α m j β C µναβ = 0 , (1)where C is the Weyl tensor and the Newman-Penrose-like vector fields satisfy the relations ℓ = k = 0 , ( k, ℓ ) = 1 , ( k, m i ) = ( ℓ, m i ) = 0 , ( m i , m j ) = δ ij . (2)As the simplest example we firstly demonstrate how to derive the Navier-Stokes equation atfinite cutoff surface in Rindler spacetime. A p + 2-dimensional metric is ds p +2 = − rdt + 2 dtdr + δ ij dx i dx j , i, j = 1 , ...p. (3)Setting r = r c , then we obtain an embedded hypersurface Σ c and the induced metric h ab onΣ c reads as ds p +1 = − r c dt + δ ij dx i dx j ≡ − ( dx ) + δ ij dx i dx j . (4)In coordinate system ( t, x i ), one can easily check that the non-vanishing component of theextrinsic curvature K ab is K tt = −√ r c /
2. In order to extract the dynamical behavior of thegeometry in the long-wavelength limit as well as the non-relativistical limit simultaneously,we introduce a parameter λ by rescaling the time coordinate with x = λ τ and the spacecoordinates with x i = √ λ x I such that ds p +1 = − λ dτ + 1 λ δ IJ dx I dx J . (5)Note that, precisely speaking, here the wavelength is long compared to the local temperature,which is the natural characteristic scale in the theory (see, e.g. [12, 16]). Obviously the non-relativistic limit and long-wavelength limit can be implemented by taking λ → t ab ≡ Kh ab − K ab . Incoordinate system ( τ, x I ), we expand the components of Brown-York tensor in powers of λ as t τ τ = 0 + λt τ τ (1) + . . .t τ I = 0 + λt τ I (1) + . . .t I J = 12 √ r c δ I J + λt I J (1) + . . .t = p √ r c + λt (1) + . . . , (6)4here t is the trace of Brown-York tensor. In terms of the Brown-York tensor, the Hamil-tonian constraint can be written as ( t τ τ ) − λ h IJ t τ I t τ J + t I J t J I − t p = 0 . (7)Directly taking the perturbation expansion, we find the leading order is trivially satisfied bythe background while the sub-leading order with λ reads as t τ τ (1) = − √ r c δ IJ t τ I (1) t τ J (1) . (8)Now we turn to the Petrov-like boundary condition. In terms of the Brown-York tensor in( τ, x I ), this condition becomes λt τ τ t I J + 2 λ h IK t τ K t τ J − λ t I J,τ − λt I K t K J − h IK t τ ( K,J ) + λδ I J [ tp ( tp − t τ τ )+2 λ∂ τ tp ] = 0 . (9)Similarly, one finds the leading order of the expansion is automatically satisfied by thebackground, while the sub-leading order with λ gives t I J (1) = 2 √ r c δ IK t τ K (1) t τ J (1) − √ r c δ IK ∂ J t τ K (1) − √ r c δ IK ∂ K t τ J (1) + δ I J t (1) p . (10)Until now we have obtained the sub-leading order of the Hamiltonian constraint and thePetrov-like boundary condition. The next step is plugging these results into the momentumconstraint. Its time component and space component will be identified as the incompress-ible condition and the Navier-Stokes equation respectively. Such technical steps have beenused in Refs.[28, 31, 32] and [37]. Hence, substituting all these results into the momentumconstraint ∂ a t ab = 0 , (11)and identifying t τ I (1) = 12 √ r c v I , t (1) = p √ r c e P , (12)as the velocity and the pressure fields of the dual fluid, we obtain the incompressible conditionand the Navier-Stokes equation on a finite cutoff surface as ∂ I v I = 0 , (13) ∂ τ v I + δ JK v K ∂ J v I − √ r c δ JK ∂ J ∂ K v I + ∂ I e P = 0 . (14) The original definitions about the Hamiltonian constraint and the momentum constraint on the cutoffsurface can be found in [28]. Moreover, their specific forms for the models in our current paper havepreviously been presented in [28], [32] and [37] respectively.
5e find that the kinematic viscosity ν c is cutoff dependent with ν c = √ r c . First of all,we remark that we have obtained the same results as those obtained by hydrodynamicalexpansion of the metric in [24] . Secondly, we find that the previous results obtained forthe cutoff surface in near-horizon limit in [28] can be treated as a special case of our currentwork. As a matter of fact, transforming the coordinate system ( τ, x I ) into ( τ, x i ) which isapplied in [28] we find ∂ τ v i + v j ∂ j v i − ∂ j ∂ j v i + ∂ i P = 0 , (15)where v i is defined as dx i /dτ correspondingly. Therefore, in near-horizon limit one obtainsthe standard incompressible Navier-Stokes equation with unit shear viscosity. III. PETROV-LIKE BOUNDARY CONDITION ON THE FINITE CUTOFF SUR-FACE FOR A BLACK BRANE BACKGROUND
Next we will treat the Petrov-like boundary condition on the finite cutoff surface fordifferent backgrounds in a parallel way. The general framework for Petrov-like boundarycondition in this context is presented in [32]. Firstly we consider a black brane backgroundwith a metric as ds p +2 = − f ( r ) dt + 2 dtdr + r ˜ δ ij d ˜ x i d ˜ x j , i, j = 1 , ...p, (16) f ( r ) = r (1 − r p +1 h r p +1 ) , Λ = − p ( p + 1)2 . Where r h is the position of the horizon. Setting r = r c , we have the embedded hypersurfaceΣ c and its metric reads as ds p +1 = − f ( r c ) dt + r c ˜ δ ij d ˜ x i d ˜ x j ≡ − ( dx ) + δ ij dx i dx j . (17)It is obvious that this is a intrinsically flat embedding, so that p +1 e R ij = p e R ij = 0 . (18) Transforming the coordinate system from ( τ, x I ) to ( t, x i ), we easily find the kinematic viscosity in ( t, x i )is r c as derived in [24]. Moreover, we point out that the Navier-Stokes equation has the same form (orhas the same kinematic viscosity ) in ( τ, x I ) and ( x , x i ) coordinate systems, which can be easily provedby dimension analysis and can be viewed as an alternative representation of the scaling symmetry of NSequation presented in [24]. Thus we will always identify our derived equations in ( τ, x I ) system to thosein ( x , x i ) system as well. ds p +1 = − λ dτ + 1 λ δ IJ dx I dx J . (19)Straightforwardly, we take the perturbation expansion for Brown-York stress tensor andsubstitute it into the Petrov-like boundary condition and constraint equations step by step.We find in this case the sub-leading order of the Hamiltonian constraint becomes t τ τ (1) = 2 √ f r c − r c ∂ r c f + 2 f δ MN t τ M (1) t τ N (1) + 2 f − r c ∂ r c f + 2 f t (1) . (20)While from the Petrov-like boundary condition we have t I J (1) = 2 √ f r c r c ∂ r c f + ( p − f δ IK t τ K (1) t τ J (1) − √ f r c r c ∂ r c f + ( p − f δ IK t τ ( K,J )(1) − fr c ∂ r c f + ( p − f δ I J t τ τ (1) + r c ∂ r c f + pfp [ r c ∂ r c f + ( p − f ] δ I J t (1) . (21)If we identify t τ I (1) = r c ∂ r c f + ( p − f √ f r c v I , (22) e P = fr c ∂ r c f − f δ MN v M v N + 2 √ f r c ∂ r c fp [ r c ∂ r c f + ( p − f ]( r c ∂ r c f − f ) t (1) , (23)then the momentum constraint leads to the incompressible condition and the Navier-Stokesequation as ∂ I v I = 0 , (24) ∂ τ v I + δ JK v K ∂ J v I − ν c δ JK ∂ J ∂ K v I + ∂ I e P = 0 . (25)Where ν c = √ fr c r c ∂ rc f +( p − f is the viscosity of the dual fluid. As argued in previous section, theform of Navier-Stokes equation will not change when one transforms the coordinate systemto ( x , x i ) ∂ i v i = 0 , (26) ∂ v i + v j ∂ j v i − ν c ∂ j ∂ j v i + ∂ i P = 0 , (27)with the same viscosity ν c = √ fr c r c ∂ rc f +( p − f .Specially, when r c → r h , ν c = 0; r c → ∞ , ν c = 1 p . x , x i ) into ( τ, x i ), we find that theprevious results obtained for the cutoff surface in near-horizon limit in [32] can be treated asa special case of our current work. Secondly, comparing our results with the previous resultspresented in [27], the viscosity of both results are equal to zero when the hypersurface movesto horizon. However, when the hypersurface moves to infinity, our viscosity approaches to p ,in contrast to the results in [27] in which the viscosity tends to divergence. Such a differencemay be understood from the fact that the Petrov-like boundary condition we employed inthis paper is different from the boundary conditions in [27] and [14], which is the Dirichletboundary conditions plus a regular condition on the horizon. Different boundary conditionsimply that the dual field theory that we obtained may be different from that obtainedthrough the hydrodynamical expansion of the metric. It is well-known in holography thatdifferent boundary conditions lead to different realizations of holographic duals (for instance,see [36] and references therein for alternative realization of Kerr/CFT correspondence). Inthis sense, we think we have presented a new way to establish the gravity/fluid duality atfinite cutoff in a black brane background. Finally, we remark that it should be interestingto investigate the ratio of shear viscosity to entropy density at finite cutoff in our formalismin future, which has been found with some universal behavior in literature[14]. IV. PETROV-LIKE BOUNDARY CONDITION ON THE FINITE CUTOFF SUR-FACE FOR A BACKGROUND WITH MATTER
The last model is on the gravity/fluid duality in spacetime with matter fields. The generalframework for Petrov-like boundary condition in this context is presented in [37]. Here weconsider a 4-dimensional magnetic black brane, which is a solution to the Einstein equationcoupled to the electromagnetic field with a metric as ds = − f ( r ) dt + 2 dtdr + r ˜ δ ij d ˜ x i d ˜ x j , i, j = 1 , , (28) f ( r ) = r − µr + Q m r , Λ = − . Here µ is the mass parameter and Q m is the magnetic charge. The electromagnetic fieldstrength is given by F = √ Q m d ˜ x ∧ d ˜ x . (29)8fter a straightforward but tedious calculation (the relevant detailed calculation is presentedin the appendix A, B and C), one obtains the sub-leading order of the Hamiltonian constraintfrom the expansion as t τ τ (1) = 2 √ f r c − r c ∂ r c f + 2 f δ MN t τ M (1) t τ N (1) + 2 f − r c ∂ r c f + 2 f t (1) . (30)While from the Petrov-like boundary condition, the sub-leading order of the expansion readsas t I J (1) = 2 √ f∂ r c f δ IK t τ K (1) t τ J (1) − √ f∂ r c f δ IK t τ ( K,J )(1) − fr c ∂ r c f δ I J t τ τ (1) + r c ∂ r c f + 2 f r c ∂ r c f δ I J t (1) . (31)In the presence of matter fields, we note the momentum constraint becomes ∂ a t ab = T µb n µ , (32)where T µb is energy-momentum tensor of the matter field. Similarly, if we identify t τ I (1) = ∂ r c f √ f v I , (33) e P = fr c ∂ r c f − f δ MN v M v N + √ f r c r c ∂ r c f − f t (1) , (34) e J J = − √ f∂ r c f F nJ (1) , (35)then from the momentum constraint we obtain the incompressible condition and the stan-dard incompressible magnetofluid equation as ∂ I v I = 0 , (36) ∂ τ v I + δ JK v K ∂ J v I − ν c δ JK ∂ J ∂ K v I + ∂ I e P = f I . (37)Where ν c = √ f∂ rc f is the viscosity of the dual fluid and f I = e J J F I J as an external force termappears on the right hand side of the equation due to the coupling of the background andthe perturbations of the electromagnetic field.As argued in previous section, the Navier-Stokes equation has the same form in coordinatesystems ( τ, x I ) and ( x , x i ). Thus, we have the incompressible condition and the standardincompressible magnetofluid equation in ( x , x i ) as ∂ i v i = 0 , (38) ∂ v i + v j ∂ j v i − ν c ∂ j ∂ j v i + ∂ i P = f i , (39)9ith the same viscosity ν c = √ f∂ rc f . Specially, its asymptotic behavior is r c → r h , ν c = 0; r c → ∞ , ν c = 12 . V. SUMMARY AND DISCUSSIONS
By explicit construction we have extended the Petrov-like boundary condition to the finitecutoff surface and derived the incompressible Navier-Stokes equation in the long-wavelengthlimit. In each model, we have computed the value of shear viscosity and discussed itsasymptotical behavior when the position of cutoff approaches to horizon or infinity. Ingeneral the kinematic viscosity is cutoff dependent and such a dependence asks for furtherunderstanding from the side of holographic renormalization group flow. In special case whenthe cutoff surface approaches to the horizon, our results go back to the previous ones withoutemploying a long-wavelength limit, implying a deep analogy between the near-horizon limitand the long-wavelength limit.This work, as well as previous works imposing the Petrov-like boundary condition inthe near-horizon limit, only involves the electromagnetic field as the most simple matterfield in the bulk (see, however, [40] for the perfect fluid case as a step further). Moregeneral matter fields may lead to further problems, such as the anisotropy caused by theaxion field [29], which is rather interesting. It is also challenging to extend this frameworkto a finite cutoff surface which may be spatially curved. When the spatial part of thehypersurface is compact, the long-wavelength limit seems not applicable. As emphasized in[19], taking the long-wavelength limit is essential to reduce the partial differential equationto ordinary differential equation. However, if the section of cutoff surface is compact, thenthe wavelength should have an upper bound such that the long-wavelength limit can notexist globally. Nevertheless, for some special non-flat cutoff surface the long-wavelengthlimit maybe exist. We leave these issues for further investigation in future.
Acknowledgments
Wei Zhang is very grateful to Cheng-Yong Zhang for useful discussion and help. Thiswork is supported by the Natural Science Foundation of China under Grant Nos.11175245,101075206, 11275208, 11178002. Y.Ling also acknowledges the support from Jiangxi youngscientists (JingGang Star) program and 555 talent project of Jiangxi Province.
Appendix:A. The Petrov-like boundary condition in the last model
From now on, we will present the detailed calculation of the last model with respect tothe gravity/fluid duality in spacetime with matter fields following the general frameworkpresented in [37]. We have the embedded hypersurface Σ c and its metric reads as ds p +1 = − f ( r c ) dt + r c ˜ δ ij d ˜ x i d ˜ x j ≡ − ( dx ) + δ ij dx i dx j = − λ dτ + 1 λ δ IJ dx I dx J . Similarly as we fix the induced metric h ab on the cutoff surface, we also fix F ab | Σ c , whichcould be regarded as the Dirichlet-like boundary condition. Then we have F τI | r c = 0 .F nb , F ab and F ab could be written in terms of F µν on Σ c as F nτ | r c = F nτ h ττ , F nI | r c = F nJ h IJ ,F τ I | r c = F τJ h IJ = 0 , F IJ | r c = F KL h KI h LJ . Then, the perturbation of electromagnetic field should take the following form F nτ = 0 + λF nτ (1) ,F nI = 0 + λF nI (1) . Now we will give the detailed calculation from the Petrov-like boundary condition to equation(31). Firstly we remark that in the presence of matter fields, the Weyl tensor can beexpressed in terms of the intrinsic curvature and extrinsic curvature as well as the energy-momentum tensor through Eqs.(3)-(6) in [37]. Moreover, since the extrinsic curvature isrelated to the Brown-York stress tensor, we can finally rewrite the Petrov-like boundary11ondition in terms of Brown-York stress tensor as λt τ τ t I J + 2 λ h IK t τ K t τ J − λ t I J,τ − λt I K t K J − h IK t τ ( K,J ) + λδ I J [ tp ( tp − t τ τ ) + 2 λ∂ τ tp ]+ λ p ( T δβ n β n δ + 2Λ + T + λ T ττ − λT δτ n δ ) δ I J − λT I J = 0 . The energy-momentum tensor of electromagnetic field takes the form T µν = 14 g µν F ρσ F ρσ − F µρ F νρ . We have T δβ n β n δ = T nn = 14 F ρσ F ρσ − F nρ F nρ ,T = p − F ρσ F ρσ ,λ T ττ = − F ρσ F ρσ − λ F τρ F τ ρ , − λT δτ n δ = − λT nτ = 2 λF nρ F τ ρ , − T I J = − δ I J F ρσ F ρσ + F Iρ F Jρ . The Petrov-like boundary condition further becomes λt τ τ t I J + 2 λ h IK t τ K t τ J − λ t I J,τ − λt I K t K J − h IK t τ ( K,J ) + λδ I J [ tp ( tp − t τ τ ) + 2 λ∂ τ tp ]+ λ p ( − F ρσ F ρσ − F nρ F nρ − λ F τρ F τ ρ + 2 λF nρ F τ ρ + 2Λ) δ I J + λF Iρ F Jρ = 0 . Moreover − F ρσ F ρσ = − F nτ F nτ h ττ − F nI F nJ h IJ − F IJ F KL h KI h LJ , − F nρ F nρ = − F nτ F nτ h ττ − F nI F nJ h IJ , − λ F τρ F τ ρ = − λ F nτ F nτ , λF nρ F τ ρ = 2 λ ( F nτ F τ τ + F nI F τ I ) = 0 ,F Iρ F Jρ = F nJ F nL h IL + F JK F LM h LI h MK . So, the Petrov-like boundary condition reads as λt τ τ t I J + 2 λ h IK t τ K t τ J − λ t I J,τ − λt I K t K J − h IK t τ ( K,J ) + λδ I J [ tp ( tp − t τ τ ) + 2 λ∂ τ tp ]+ λ p [ − F nτ F nτ h ττ − λ F nτ F nτ − F nI F nJ h IJ − F IJ F KL h KI h LJ + 2Λ] δ I J + λF nJ F nL h IL + λF JK F LM h LI h MK = 0 . Our calculation is applicable for a general spacetime with matter fields, thus we keep p as general untilwe get back to the last model with a magnetic black brane, in which p is set to 2. p = 2 into above equation, we get the Petrov-like boundary condition λt τ τ t I J + 2 λ h IK t τ K t τ J − λ t I J,τ − λt I K t K J − h IK t τ ( K,J ) + λδ I J [ t t − t τ τ ) + λ∂ τ t ]+ λδ I J [ − F nτ F nτ h ττ − λ F nτ F nτ − F nI F nJ h IJ − F IJ F KL h KI h LJ + Λ]+ λF nJ F nL h IL + λF JK F LM h LI h MK = 0 . Taking the perturbation expansion for Brown-York stress tensor and electromagnetic field,we find the leading order of the expansion is automatically satisfied by the background whilethe sub-leading order with λ reads as t I J (1) = 2 √ f∂ r c f δ IK t τ K (1) t τ J (1) − √ f∂ r c f δ IK t τ ( K,J )(1) − fr c ∂ r c f δ I J t τ τ (1) + r c ∂ r c f + 2 f r c ∂ r c f δ I J t (1) . B. The Hamiltonian constraint in the last model
Here we give the detailed calculation from the Hamiltonian constraint to equation (30).The Hamiltonian constraint is p +1 R + K ab K ab − K = 2Λ + 2 T µν n µ n ν , a, b = 0 , . . . p, µ, ν = 0 , . . . p + 1 . In terms of t ab = Kh ab − K ab in coordinate system ( τ, x I ), we get( t τ τ ) − λ h IJ t τ I t τ J + t I J t J I − t p − − T µν n µ n ν = 0 . Considering the last term on the left-hand side of the above equation − T µν n µ n ν = − T nn = F nτ F nτ h ττ + F nI F nJ h IJ − F IJ F KL h KI h LJ , then the Hamiltonian constraint becomes( t τ τ ) − λ h IJ t τ I t τ J + t I J t J I − t p −
2Λ + F nτ F nτ h ττ + F nI F nJ h IJ − F IJ F KL h KI h LJ = 0 . Now, considering the perturbation of the electromagnetic field and meanwhile taking theperturbation expansion for Brown-York stress tensor, we find the leading order of the ex-pansion is automatically satisfied by the background while the sub-leading order with λ reads as t τ τ (1) = 2 √ f r c − r c ∂ r c f + 2 f δ MN t τ M (1) t τ N (1) + 2 f − r c ∂ r c f + 2 f t (1) . . The momentum constraint in the last model Following discussion is about the momentum constraint ∂ a t ab = T µb n µ . The time component of the equation is ∂ a t aτ = T µτ n µ . Because ∂ a t aτ = ∂ τ t τ τ + ∂ I t I τ = λ∂ τ t τ τ (1) − λ ∂ I t τ I (1) + . . . ,T µτ n µ = T nτ = 0 , then at leading order it gives rise to ∂ I t τ I (1) = 0 . The space component of the equation is ∂ a t aI = T µI n µ . Similarly, since ∂ a t aI = ∂ τ t τ I + ∂ J t J I = λ∂ τ t τ I (1) + λ∂ J t J I (1) ,T µI n µ = T nI = − (0 + λF nJ (1) ) F I J = − λF nJ (1) F I J , then at leading order we have ∂ τ t τ I (1) + ∂ J t J I (1) = − F nJ (1) F I J . [1] S. W. Hawking and J. B. Hartle, Commun. Math. Phys. , 283 (1972).
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