Large D gravity and charged membrane dynamics with nonzero cosmological constant

In this paper, we have found a class of dynamical charged ‘black-hole’ solutions to Einstein-Maxwell system with a non-zero cosmological constant in a large number of spacetime dimensions. We have solved up to the first sub-leading order using large D scheme where the inverse of the number of dimensions serves as the perturbation parameter. The system is dual to a dynamical membrane with a charge and a velocity field, living on it. The dual membrane has to be embedded in a background geometry that itself, satisfies the pure gravity equation in presence of a cosmological constant. Pure AdS / dS are particular examples of such background. We have also obtained the membrane equations governing the dynamics of charged membrane. The consistency of our membrane equations is checked by calculating the quasi-normal modes with different Einstein-Maxwell System in AdS/dS.


Introduction
It is now well-known that black hole solutions simplify a lot in a large number of space-time dimensions (denoted as D), the key reason being that the 'blackening factor' of the black hole/ brane metric reduces to its asymptotic form exponentially fast in space as we take D to infinity. The sole effect of the black hole, then, confines within an infinitesimally thin region (referred to as 'membrane region') around its event horizon. Also, it turns out that there is a large O(D) gap in the spectrum of linearized fluctuations around these large-D black holes. The slowly varying modes, which are finite in number, are decoupled from the fast modes in the sense, that they also decay exponentially outside the same membrane region [1]. Considering all these facts together, it is natural to expect a nonlinear completion for these decoupled Quasi-Normal Modes, leading to new dynamical black hole solutions of Einstein equations.
These solutions are always perturbative and could be constructed only in a large number of dimensions as an expansion in 1 D . Nevertheless, they are useful for several purposes. Firstly, they generate a new class of dynamical black hole solutions of Einstein equations which are very difficult to solve otherwise (even numerically).

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Secondly, through these constructions, we could see a duality between the dynamical horizons of the black hole/brane metric and a co-dimension one dynamical membrane, embedded in the asymptotic background geometry. This membrane-gravity duality allows us to analyse the complicated dynamics of the black holes from a different angle, which might turn out to be useful in future for some realistic calculation.
In this paper, our goal is to extend the 'background covariant' technique of [35] to Einstein-Maxwell system in presence of cosmological constant. For this case, the dual system would be a codimension-one dynamical charged membrane, embedded in the asymptotic dS / AdS metric.
The motivation for our work is two-fold. The first is, of course, to see how the whole technique of background-covariantization works for Einstein-Maxwell system, which, in terms of complexity, is just at the next level, compared to the pure gravity system. Indeed we have noticed that unlike the uncharged case, only naive covariantization of the flat spaceequations of the charged membrane (as derived in [9]) will not give the correct duality and we need to add a term proportional to the background curvature even at the first subleading order in O 1 D expansion. This is indeed one of the interesting observations in our paper. The second piece of motivation is as follows. We know that in asymptotically AdS geometry there exist another set of perturbative solutions to Einstein-Maxwell system. These are black holes/branes constructed in a derivative expansion and are dual to dynamical charged fluid living at the boundary of AdS. Recent works can be found in [18,34,38]. At this point, it is natural to ask whether there exists any overlap regime for these two types of perturbations, and if so, whether the two metrics agree. In the best possible scenario, the outcome of this comparison could be a duality between the dynamics of a charged fluid and charged membrane in a large number of dimensions, where gravity does not have much role to play. Our construction in this paper is one necessary step towards such duality.
The outline of this paper is described as following. We start with the Einstein-Maxwell-Hilbert action in section 2. We write the Einstein-Maxwell equations in a simplified form and sketch the general solutions of metric and gauge field using the large D perturbative technique. The next section is devoted mainly to guess the initial ansatz for the metric and gauge field. We consider some educated guess to reach the starting point of the perturbation. Section 4 backs up the choice of our ansatz and describe the conditions imposed in leading order for the ansatz to satisfy the Einstein-Maxwell equations. We state the scaling laws of different tensors in powers of D. In section 5, we have covered, in details, the strategy to solve the equations. We have mentioned the subsidiary conditions on the auxiliary functions and the gauge choice to construct the most general structure of metric and gauge field corrections. Our analysis does not require any coordinate dependency of the background. Instead, the sub-leading order corrections are parametrized by some smooth function, a one form and charge field. The subsidiary conditions help to fix these functions in the background. One of the advantages of the large D technique is that it reduces the dimensionality of the problem by one, by integrating the radial direction. In few specific cases, like ours, it reduces the coupled non-linear PDEs (involving two variables) to ODEs. In section 5, this simplification is discussed. To solve the ODE s we need the boundary conditions. We have devoted a subsection in section 5 for this purpose. Section 6 JHEP12(2018)034 deals with the strategy we mentioned in section 5 to solve the first subleading corrections. The next section summarise the results of this paper. It include the solutions of the Einstein-Maxwell equations along with the membrane equations governing the dynamics of membrane. In section 8, we prove the consistency of our membrane equations. We calculate the light quasi-normal modes of few known solutions of Einstein-Maxwell system using our membrane equations. Section 9 wraps up this paper with a conclusion. The explicit details of the calculations are provided in the appendices.

Set up
In this section, we shall describe our basic set-up for this problem.
We start by looking at the Einstein-Hilbert-Maxwell action in presence of cosmological constant Λ.
Here G is the determinant of the space-time metric G AB , the corresponding Ricci scalar is denoted as R and F AB is the field strength tensor for the U(1) gauge field A B . Following [35] we scale to define the cosmological constant Λ as: Varying the Einstein-Maxwell action, we get the following Einstein and Maxwell equation.
In the Maxwell equation ∇ is the covariant derivative with respect to G AB . The first equation (2.3) could be simplified a bit. Note that, by contracting this equation with G AB , we get a relation among R, F AB F AB and Λ. Substituting this relation back into the expression of E AB we find the simplified version of the Einstein equation, which we shall use for our further computation. So the final set of differential equations that we are going to solve in this paper using the 'large D perturbation technique', are the following.
As mentioned before, our objective is to solve the equations perturbatively where the perturbation parameter is 1 D . Schematically the solution will take the following form.

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whereḠ AB is some exact solution of pure Einstein equation in presence of cosmological constant (i.e., the first equation of (2.4) with gauge field set to zero). In this paper, the precise goal of our computation would be to determine G (1) AB and A (1) M given some arbitrarȳ G AB , satisfying the above constraint.

The leading ansatz
Our goal is to determine the first subleading correction to the metric and the gauge field. But, as it is true in any perturbative calculation, we must know the leading solution before we could determine any subleading term. On the other hand, the leading ansatz could never be derived, since typically there is no unique answer to this. In some sense, we have to start with an educated guess for G Now from [35], we know the form of the ansatz for arbitraryḠ AB but without the gauge field. It is given by almost the same ansatz one has in asymptotically flat space, except that the explicit appearance of the flat space Minkowski metric η AB has been changed tō G AB . Also from [9], we know the form of the leading ansatz in asymptotically flat space in presence of gauge field.
Here ψ and Q are any smooth functions and O A dX A is a one-form which is null with respect to η AB . Taking the cue from the uncharged case, one very natural guess for the leading ansatz in Einstein-Maxwell system in the arbitrary asymptotic background would be to simply replace the explicit appearance of η AB toḠ AB , i.e., and now O A is null with respect toḠ AB . 1 Note that the ψ = 1 hypersurface is null with respect to the metric [G AB ] leading . Also at large D, as one goes finitely away from this hypersurface, the metric either blows up (if ψ < 1) or reduces to its asymptotic formḠ AB . However, we could easily see that the metric is non-trivial and finite only within a region of thickness of order O 1 D in the following way. 1 We could intuitively understand why such simple replacement works for the leading ansatz. As described in the introduction, all these geometries will necessarily possess an event horizon and the non-trivial gravity effects of the black holes will be confined within a thin region of the thickness of order O 1 D around this event horizon, which we shall refer to as 'membrane region'. Because of this infinitesimal thickness, inside the membrane region, the details of the asymptotic geometry become irrelevant at the very leading order, and the same ansatz works as long as we replace asymptotic background as required. See [9] for a more detailed discussion on this point.
Clearly, our ansatz does have the required form with a non-trivial membrane region. Also, the singular part of the space-time is shielded by the membrane at ψ = 1, which we shall identify with the event horizon of the space-time. Now we have to check whether this ansatz satisfy the relevant equations at leading order.

How the ansatz solves (2.4) at leading order
In this section, we shall see that the above ansatz indeed solves (2.4) at leading order provided O A and ψ satisfy certain conditions on the (ψ = 1) hypersurface. But before getting into the details of the equations, we shall first describe our 1 D expansion in a little more details.

Scaling with D
Note that the first equation in (2.4) is a set of D(D+1) 2 equations and the second equation is a set of D equations for total D(D+1) 2 metric components and D gauge field components. Hence, as we increase D (or decrease our perturbation parameter 1 D ), both the number of equations and the number of unknowns increase and no perturbation technique can work in such a situation. This has been discussed in detail in [8][9][10] and [35]. We shall follow their strategy to have a meaningful 1 D expansion. We shall assume that the metric G AB and the gauge field A M are dynamical only along some fixed finite number of dimensions p + 1 and the rest of the (D − p − 1) dimensions are protected by some symmetry. In terms of equation, we mean the following.
where, {a, b} = {0, 1, . . . p}.G ab (x a ) is a dynamical and finite p + 1 dimensional metric and dΩ 2 is the line element which takes care of the infinite (D − p − 1) symmetric space.
f (x a ) is an arbitrary constants of the (p + 1) dynamical coordinates. However, as in [10,35], for our computation, we do not need to use any detail of this decomposition. The sole effect of this symmetry would be to impose some scaling rules (with D) on different derivative structures. Below we are simply stating these rules and would request the reader to go through the discussion in [10,35] for their justification.
• For a generic tensor •Ḡ AB is such that all components of Riemann tensor, evaluated onḠ AB is of order O(1), which further implies that JHEP12(2018)034

Conditions imposed due to leading ansatz
Now we shall simply substitute our leading ansatz (3.2) in (2.4) and shall compute the leading piece, keeping in mind the scaling rules we have mentioned above. The details of the computation are all presented in appendix C. Here we shall only present the final result of this computation. As mentioned before, the leading pieces in both E AB and E A turn out to be of order O(D 2 ). Up to corrections of order O(D), they have the following form In equation (4.2) (and from now on throughout the paper) all raising, lowering and contraction of indices have been done usingḠ AB . Note according to our scaling rules, both (∇ · O) and (K = ∇ · n) are of order O(D) since they are divergences of order O(1) vectors n A and O A . From (4.2) we could easily see that the leading order piece will vanish, or our ansatz will solve the equation at the very leading order (O(D 2 )) provided Note that we have imposed the conditions only at (ψ = 1). If we are finitely away from ψ = 1, the two equations will anyway vanish, because of the ψ −D factor in f andf . To have a non-trivial effect on the equations, we can deviate from this hypersurface only in a power series in 1 D and therefore the effect of such deviation will always be suppressed. We do have to take care of it in our subleading calculation, but at this order, it does not matter.
For convenience, we shall define a unit time-like vector field u A which is orthogonal to n A and defined as u A = n A − O A . In terms of u A the second equation of (4.3) reads as Note that in the language of our 'D scaling rules', the O(1) vector u A is a special case since its divergence is also of O(1) instead of being O(D).
In summary, the final form of our leading ansatz is given by (3.2), where ψ and O A satisfy the conditions given in (4.3).

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5 Subleading corrections: the strategy Once we have our leading ansatz, we are ready to go for the subleading corrections. In this section, first, we shall describe the strategy very briefly, along with the conventions and the choices we shall be using for the solution. See [35] for elaborate discussions on these points. Then we shall implement this strategy for our particular case to get the final set of coupled ODEs.

Brief description of the algorithm
In a nutshell, the algorithm to determine the first subleading correction is as follows.
We have constructed an initial metric and gauge field ansatz G M and have seen that G

Subsidiary conditions
Following [35] we shall write the final answer for G (1) AB in terms of ψ, Q, O A and their derivatives. The advantage of presenting the answer this way is that we never need to choose any specific coordinate system for the background, and therefore our solution will have explicit background covariance.
However, such a final answer does not make sense unless ψ, Q and O A are some known functions of the background. Note that the conditions (4.3) (which has to be satisfied only at ψ = 1 hypersurface) are not enough to determine these two functions and the one-form everywhere in the space-time. Therefore there is a huge ambiguity in fixing these functions everywhere. For convenience, we shall fix this ambiguity, by imposing some conditions (which, following [8,9,35], we shall refer to as 'subsidiary conditions') on ψ, Q and O A JHEP12(2018)034 externally. Our choice of subsidiary conditions (these have to be satisfied everywhere in the background space-time) would be the following.
Note that this choice of subsidiary conditions are consistent with (4.3), and also maintain 'the background covariance', in the sense that to specify them we do not need to choose any coordinate system. Being differential equations, the subsidiary conditions will fix these functions up to some boundary conditions. We shall specify the boundary conditions on ψ = 1 hypersurface. In other words, given the shape of the ψ = 1 hypersurface, and Q and O A on this hypersurface, (5.2) will determine them everywhere in space, (i.e. at all ψ = 1). 3 It will turn out that these boundary values ( i.e., the Q and projected O A fields on the hypersurface and its extrinsic curvature) are not completely free and the Einstein-Maxwell system could be solved only if they together satisfy some integrability conditions. These are the equations that govern the dynamics of the dual charged membrane and one of the key results of this paper.

Gauge choice
In this section, we shall specify a choice of gauge 4 and shall parametrize the metric and gauge field correction accordingly. To fix the general coordinate invariance we need D conditions on the metric and the U(1) gauge fixing will require one condition on the gauge field. Our choice would be as follows The most general form of the metric and the gauge field correction consistent with (5.3) is where P AB is the projector perpendicular n A and u A AB satisfy the following conditions [32] for an explicit construction of ψ in terms of the extrinsic curvature of the ψ = 1 hypersurface in large-D approximation. 4 It is important to distinguish the subsidiary conditions from coordinate and U(1) gauge covariance of the Einstein-Maxwell system. Here we have chosen to express our final answer in terms of some auxiliary functions ψ, Q and OA. The purpose of the subsidiary conditions is to fix or define these functions. We have defined these auxiliary functions in a way so that we do not need to fix any particular coordinate system for our analysis. On the other hand, our gauge choice does fix the coordinate system upto some possible residual gauge invariance.

Reducing the PDE to ODE
Naively if we compute the leading contribution of (5.4) to (2.4), we shall get linear partial differential operators on the unknown functions (G (s 1 ) , A (s) etc.) appearing in G (1) AB and A (1) M . However, the key simplification arises as follows.
Suppose we choose the ψ = constant surfaces to foliate the space-time. It turns out that at large D, part of the metric and gauge field vary parametrically fast in the direction of increasing ψ, compared to the directions along the constant ψ hypersurfaces. In other words, each component of the metric and the gauge field correction could be decomposed as a product of 'fast-varying' and 'slowly-varying' pieces. The derivatives of the 'slowlyvarying' pieces are parametrically suppressed. As a consequence, if we were to evaluate (2.4) on the metric and gauge field correction at a given order of O 1 D k , (assuming the system of equations have been solved up to O 1 D k−1 ) 'slowly-varying' pieces should be treated as constants. This reduces the complicated PDE of Einstein-Maxwell system to a set of in-homogeneous ODEs along the fast varying direction (namely the direction of increasing ψ) and the 1 D expansion takes the form of an effective derivative expansion along these [ψ = constant] hypersurfaces.
From the above discussion it is clear that the first step in determining the subleading corrections would be to decompose the metric and gauge field functions as products of slowly and fast varying pieces as we have done below.
Here ζ = D(ψ − 1). Clearly ζ dependent parts are the fast varying pieces whose derivatives will have explicit factors of D. Each of these fast varying functions are multiplied by 'slowlyvarying' scalar, vector and tensor structures. It turns out that at a given order there are only a finite number of slowly varying structures that could appear. Scalar structures are denoted by S (i) ; vector structures, perpendicular to both u A and n A are denoted by V AB denotes the traceless tensor structure, perpendicular to both n A and u A . The total number of such 'slowly varying' scalar, vector and tensor structures at order O(D) are denoted by N S , N V and N T respectively. A similar decomposition in terms of these scalar, vector and tensor structures is also possible for the sources S AB and S M (see the next section for more details).
Once we substitute these decomposed sources, i.e the metric and the gauge field in the schematic equation (5.1), its different component reduce to second order inhomogeneous ODEs for the unknown functions S v (ζ) and T (i) (ζ). As explained before, these are ODE s (as opposed to PDE s) simply because the derivatives of the slowly varying structures do not contribute at this order. Each structure could be treated as independent constant, thus decoupling the equations in different superscript '(i)' sectors. Of course, along with this, due to the symmetry of the equations, the scalar,

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vector and the tensor sector will decouple as well in the usual way. These two types of decoupling of the resultant ODEs lead to a vast simplification. With an appropriate choice of boundary conditions and a set of constraints on our scalar and vector data (which turns out to be the equation that governs the dynamics of the dual charged membrane -one of the main results of our paper), we could integrate them.
In a nutshell, this is how we determine the subleading corrections to the metric and the gauge field.

Boundary conditions
As explained above, the relevant equations in the end would be a set of second order ODEs for each of the functions S Generically we need two boundary conditions for each of them to fix the integration constants. We shall impose them at ζ → ∞ and ζ → 0.
The fact that asymptotically the full space-time metric G AB should reduce to the backgroundḠ AB fixes the boundary condition at ζ → ∞ end. It simply says all the functions should vanish as ζ goes to ∞. It turns out that S (i) 1 (ζ) could be unambiguously determined using this single condition. For the rest of the functions, we need another condition at ζ → 0 end of the space-time. Now, note that our final solution is parametrized by a dual system of a dynamical charged membrane, embedded in the asymptotic geometry, with a velocity field u A living on it. We need to unambiguously define these parameters in terms of our full space-time geometry and the gauge field containing the black hole. These definitions fix many of the integration constants.
In the metric sector, we shall follow [39] to fix our choices of parameter. We shall choose the ψ = 1 hypersurface to be the horizon of the space-time and u A to be the null generator of the horizon to all orders in 1 D expansion. As explained in [10] and [39], this implies the following boundary conditions. lim ζ→0 S (i) Once the membrane is defined, we shall have the all order definition of the parameter Q through the following equation.
This definition fixes how a (i) For rest of the two functions T (i) (ζ) and a (i) v (ζ), the integration constants at ζ = 0 ends are fixed by demanding that the solution has to be finite on the horizon.

JHEP12(2018)034 6 First subleading correction: explicit solution
In this section, we shall implement the strategy outlined in the previous section to determine the first subleading correction to the metric and the gauge field. We shall refer to the appendices B, C and D for some of the details of the calculation.

Classification of structures
As explained in the previous section the first step would be to classify the slowly-varying structures that can appear at the first subleading order. In table 1 we shall give a list of such structures that will appear in the final answer. Here Q A ≡ ∇ A Q Q . Apart from the list of scalars mentioned in table 1, we could also have some scalar terms proportional to the background curvature. However, we are not giving a separate name to such terms and shall be writing them explicitly whenever they occur.

Source
Now we shall write the explicit expression for the source S AB and S M at first subleading order, which turns out to be of order O(D). We shall decompose the sources into different components.
Now the different components could be further decomposed into different scalar vector and tensor structures as appeared in the table 1.
Here,R uu = u AR AB u B andR AB = (D − 1)λḠ AB is the Ricci tensor w.r.t. the background metricḠ AB .

Scalar
Vector Tensor Table 1. List of membrane data.

Homogeneous part
As mentioned before, the schematic form of the equations at order O(D) is where H AB and H M consist of the linear differential operators acting on the unknown functions appearing in equation (5.4) and S AB and S M are the sources that do not depend on G (1) M . Now once we have decomposed the functions into slow and fast pieces (see equation (5.5)), the homogeneous part -H AB and H M reduce to ordinary linear differential operator on the fast varying functions (the functions that depend on ζ). Below we are presenting the final form of H AB and H M . See appendix D for the detailed derivation.
For convenience, we shall first decompose H AB and H M in the following way. where, And for the second equation in (2.4),

Decoupling the ODE s
The set of coupled ODE s mentioned in the previous subsection could easily be decoupled, by taking appropriate projection. In this subsection we shall present the decoupled ODEs and their solutions in the form of integrals.

Trace-less tensor sector in the metric correction
Consider the following combination, This combination reduces to the decoupled ODE s for T (i) (ζ) s. Now from the table 1 we could see that there exists only one tensor structure at this order, or in other words, JHEP12(2018)034 according to the notation of equation (5.5), N T = 1. To unclutter the notation in this case, let us denote T (i) (ζ), simply by T (ζ) without the subscript. The relevant equation turns out to be the following. ( After imposing the boundary conditions (finite at ζ = 0 hypersurface and vanishing as ζ → ∞) this equation could be integrated.
For Q → 0 limit, the traceless tensor sector correction vanishes, which is consistent with [35].

Vector sector
Consider the following three combinations These three equations give a set of coupled ODE s for the unknown functions V (i) and a (i) v . Note that, by construction, these equations are decoupled for each independent vector structure V (i) A appearing in the source V (1) A and A A , i.e., the superscript i s are not mixed. However, for a given i, we have to do some work to decouple V (i) and a (i) v . Moreover, we have three equations for two unknown functions, leading to the following consistency constraint.
From the theory of 'constraint equations' in any gauge theory, it follows that if we satisfy such constraints along one constant ζ hypersurface, they will be satisfied everywhere, provided we solve the dynamical equations (in this case the equations involving two derivatives w.r.t. ζ and therefore determining the 'ζ evolutions' of the unknown functions) correctly [40]. Here we shall impose these constraints on ζ = 0 or ψ = 1 hypersurface, which will lead to the constraint equations on our membrane data.

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Note that the l.h.s. of equation (6.21) vanishes at (f = 1) (or equivalently ψ = 1) hypersurface which further implies that the combination of vector structures V (1) A , which generically should be of O(1), 5 turns out to be of order O( 1 D ) because of equation (6.21).
This is clearly one of the 'integrability condition' for our set of ODEs. This membrane equation is one of the key results of our paper. This is one of those equations that govern the coupled dynamics of the membrane's shape, its charge and the velocity field. Here, ∇ is the covariant derivative w.r.t. to the intrinsic membrane coordinates denoted by the Greek indices µ, ν.
P ν µ = projector perpendicular to u µ = δ ν µ + u ν u µ K µν is the extrinsic curvature of the membrane in terms of the membrane coordinates and K is the trace of the extrinsic curvature.
As we have explained, this membrane equation ensures that (6.21) is satisfied only at the hypersurface ψ = 1. However, for the consistency of the set of ODE s we need (6.21) to be satisfies at every value of ψ. This is an internal consistency test of our systems of equations. It is a consequence of the theory of 'constraint equations' in any theory with gauge invariance. For our case we could easily verify it and the proof is given in appendix E.
Next, we proceed to decouple and integrate. We shall use the second and the third equation for this purpose. Let us first introduce some new notation to denote the decomposition of the sources in our basis of vector structure as given in table 1 Here the ζ dependence of f andf are given as

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Above notation would help us to decouple different 'i' sectors in (6.20). The second equation of (6.20) will take the following form. 6 Substituting (6.25) in the third equation of (6.20) we get the decoupled equation for a Equation (6.26) could be easily integrated and therefore (6.25) could be easily integrated.

Scalar sector
For every i (i.e., for every independent scalar structure in table 1) we have unknown functions in the scalar sector, namely S Below we are first quoting the equations relevant in the scalar sector.
For Q → 0 limit, the second term of second equation of (6.20) gives back the vector constraint on the membrane data for the uncharged case and it is also suppressed by O( 1 D ) outside ψ = 1. Then the vector sector correction in the metric becomes zero. For more discussion on this, see [35].

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The consistency of this set of 5 equations demands Note that on ψ = 1 hypersurface the above two constraints (6.30) 7 reduces to the following These two are the two scalar constraints on the membrane dynamics. We shall refer to them as scalar and charge membrane equations. Equation (6.22) and the two equations in (6.31) are the key results of our paper. Note that in (6.22) and in the first equation of (6.31), if we just replace the covariant derivatives ∇ with partial derivatives ∂, they reduce to two of the constraint equations derived in [8]. In other words, these two equations are the simplest possible covariantization of their 'flat-space' counterpart. However, the same could not be said about the second equation in (6.31) as it includes a term proportional to the Ricci tensor with both indices projected along the u direction. The charge conservation equation does depend on the background curvature ( hence, the cosmological constant) in a non-trivial way. Note also, that without 7 Actually there is one more constraint. Note that if we take projected trace of HAB, then it is of order O(D), whereas the naive D counting suggests that it should be of order O(D 2 ) since HAB itself is of order O(D). This imposes a serious constraint on the source at order O(D), namely From equation (6.2) we could see that this is indeed true.

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the charge there is no such non-trivial curvature dependence in the membrane equations (see [35]. Our equations, in Q → 0 limit, very simply reduce to that of [35]).
We are yet to solve for the scalar sector. However, before going into the solution of the scalar sector, we want to check that the equations (6.30) are satisfied for all ψ, which, in abstract terms, is a consequence of gauge and coordinate invariance of our theory. It is a similar kind of consistency check, as we did for the vector sector. For easier understanding, we have explicitly showed the verification of the equations (6.30) in appendix E.
Once the consistency is ensured we could proceed to decouple and integrate the equations. We shall start with the solution of S s . We shall use the third and the second equations of (6.28) and (6.29). Note that in the second equation of (6.29), once we substitute the solution for S , which again has a very simple homogeneous piece and therefore could be integrated easily.
Before carrying out these steps let us introduce few notations to unclutter our equations. As in vector sector, we shall decompose the sources in the third and the second equations of (6.28) and (6.29) in terms of different scalar structures listed in table 1.

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In this new notations it is easier to decouple the different 'i' sectors, and the third and the second equation of (6.28) and (6.29) take the following form.
Integrating the second equation of (6.34) we get Substituting the solution the first equation of (6.34) we get the solution for S

Final results
In an expansion in the inverse power of dimension, we found a class of dynamical 'black hole' solutions to Einstein-Maxwell equation in presence of cosmological constant. Our algorithm, in principle, works to all order. We have calculated the explicit solution upto the first subleading order.
In this section, we shall summarize our final result. For convenience, we shall repeat some of the definition and conventions, described in the previous sections, again here.
•Ḡ AB is any solution of Einstein equation in presence of cosmological constant and vanishing electromagnetic field.
• ψ is a smooth function such that ψ = 1 is the horizon of the full space-time and ψ −D is a harmonic function with respect to the background metricḠ AB . 8 In the Q → 0 limit, the s (i) metric vanishes (as there is no metric vector sector correction in the uncharged limit) except for s (3) metric . Using equation (6.31) and the same logic in the vector sector, the r.h.s. of the first equation in (6.33) is suppressed in this order and hence there is no scalar sector correction in the uncharged limit. Thus, this result is consistent with [35].

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• O A is a null geodesic vector field in the background satisfying • Q is another smooth function satisfyinḡ The first subleading correction G AB and A M are parametrized in the following way.
The scalar, vector and tensor structures S (i) , V where v gauge are given in (6.23) and (6.33). In the Q → 0 limit, the tensor, vector and scalar sector corrections in the metric vanish, which is consistent with [35].

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The dual membrane is defined in terms of a smooth function ψ, a smooth one form O, a velocity field u A and charge field Q everywhere in the backgroundḠ AB . The "integrable condition" or the membrane equations governing the dynamics of the membrane are: ∇ is the covariant derivative w.r.t. the induced metric on the membrane. All the quantities used in (7.8) are w.r.t. the coordinates intrinsic to the membrane, denoted by the Greek indices. All the lowering and raising has been done w.r.t. the induced metric. The greek indices can take (D − 1) values. The final results of our paper are (7.7), and (7.8).

Quasi normal mode calculations
In this section, we will be showing the consistency of our membrane equations. Membrane equations are well-posed initial value problem. In principle, it gives the complete dynamical description of any black-hole system. Hence, we will be deriving the light quasi-normal modes for few known solutions of Einstein-Maxwell equation using our membrane equations and will be comparing them with the pre-obtained results calculated purely from the gravitational analysis.

QNM for charged black brane
We will be looking at the light quasi-normal modes of the charged black-brane solution in AdS/dS background. For simplicity, we will be starting with AdS charged black brane in Poincare Patch. The black brane geometry is given as, The background metric in Poincare patch is, The horizon of the charged black brane is at r 0 = 1 [35] and on the horizon, the velocity field is u 0 = −1. We consider a small perturbation on this exact solution as,

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where is the linearization parameter, a represents the (D − 2) linear x a coordinates. The coordinates along the membrane (t, a) are denoted by the Greek indices µ. The induced metric on the membrane up to linear order in is, We follow a convention that, ∇ µ → Covariant derivative with respect to induced metric g ind µν ∂ a → Covariant derivative with respect to the metric of the coordinates (a, b) which is δ ab Using normalization condition of the velocity field g ind µν u µ u ν = −1, in linear order we get, The projector P µν on the membrane and perpendicular to u µ direction are, The structure of the vector membrane equation is written similar to [35], where K µν , K are the extrinsic curvature and trace of the extrinsic curvature w.r.t. the membrane coordinates. K µν is the pull back of the extrinsic curvature K M N (w.r.t. the full spacetime coordinates) [35].

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The relevant quantities that we need to calculate to determine the QNM frequencies are, (For detailed calculation the readers are requested to go through [35]) We are summarizing here the algorithm to calculate (8.11) in the way similar to [35]. The vector membrane equation (8.6) can be written as: The translation symmetry in the background along the x a directions are broken by the small fluctuations that we have considered. So, E b ∼ O( ). Also, P t t = 0, P b t ∼ O( ). Hence E tot t ∼ O( 2 ) which is considered to be zero at linear order in .
Again, for E tot a , we consider only O( 0 ) terms for E t as P t a ∼ O( ). Using the above mentioned quantities in (8.10) and the decomposition,

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where ∂ 2 = ∂ a ∂ a . This is a vector equation with δ ab metric. Taking the divergence ∂ a E tot a with respect to x a and using the relation derived from scalar equation (8.9), we get, We use the fact that ∂ 2 δr is of O(D), and consider only the terms which are of O(D) in (8.14). Thus we get the scalar frequencies, The scalar frequencies in the uncharged case is which match with the results in [1,13,17,33,35]. The general solution for shape function with momentum k a is, Now let, For Charge quasi-normal mode case, we have calculated the quantities that we need. They are mentioned below,

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We take the expansion of charge fluctuation as, where, we have used the subsidiary condition n. ∇δQ = 0. Using these quantities and the expansion in the charge membrane equation (the second equation in (7.8)) and equating the coefficients of e −iω q t we get, Also equating the scalar modes e −iω s ± t we get,

QNM for Reissner-Nordstorm blackhole in global AdS
To calculate the light quasi normal modes of the AdS Reissner-Nordström solution, we consider the global AdS/dS as the background metric, where L is the AdS or dS radius and the value of σ is 0 for flat space, +1 for AdS, −1 for dS. The horizon of RN black hole is at r 0 = 1 and on the horizon, the velocity field is u 0 = − 1 − σ L 2 1 2 . We will consider a small perturbation on this exact solution as follows, where a represents the (D − 2) angular θ a coordinates and µ denotes coordinates along the membrane (t, θ a ). The induced metric on the membrane upto linear order in is, The effective equations of charged black brane in [18] involves the scaling in O 1 √ D . In [33,35], the effective equations of uncharged black brane has been reproduced from covariant uncharged membrane equations. We have also considered a scaled co-ordinate transformation and a particular choice of scaling to reproduce the QNM's of charged black brane which is consistent with our results and also with [18]. The details are in F.

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We will follow a convention that, ∇ µ → Covariant derivative with respect to induced metric, ∇ a → Covariant derivative with respect toΩ ab (The metric along (D-2) dimensional unit sphere) Using the normalization condition g ind µν u µ u ν = −1, in linear order we get, The projector on the membrane perpendicular to u µ is, (The detailed calculation of the below mentioned quantities are available in [35]) Let us consider the structure of the membrane equation in the limit Q → 0, which is the membrane equation for the uncharged case in [35].
where K µν is the pull back of extrinsic curvature K AB along the membrane and K is the trace of extrinsic curvature K µν . In our case (where Q = 0) the vector membrane equation is: Hence, the extra terms appearing in our case proportional to Q are, Hence, our equation can be represented as,

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The components that would be relevant for the linearized membrane equation are taken from [35], Using equations (8.32) the linearized vector membrane equation in the angular directions evaluates to and, This is a vector equation in sphere coordinates. If we take divergence, ∇ a with respect to Ω ab then we get, Now we will consider the scalar membrane equation, For convenience we can decompose δu a as δu a = v a + ∇ a Φ, where ∇.v = 0. Then, ∇.u = 0 implies, Now decomposing the perturbations in spherical harmonic modes, we get the equation in leading order in O(D) to solve the scalar QNM frequencies.

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which matches with the results of [20]. Their results are obtained from an entirely different approach and the matching shows the consistency of our analysis. Now we calculate the vector QNM frequencies. In (8.34) we have already solved for δr and Φ. So considering the divergence less vector part we get, Then using (8.38) we get, In the limit Q → 0 the result matches with [6,35]. The vector fequencies for the charged case are calculated in [20]. In that paper, m is zero for non rotating black hole. Hence it matches with our result. 10 For Charged field quasi-normal mode case, the quantities that we need up to relevant order are, Using these quantities in the charge membrane equation and comparing the coefficients of e −iω q l t , we obtain, The charge frequency also matches with [20] completely. The matching of the quasi-normal modes with the pre-calculated results is a good consistency check of our analysis. 10 In [20] there are two vector modes for Reissner-Nordstorm blackhole, where l the scalar mode frequency can be written in terms of the vector mode frequency lv as l = lv ± 1. So, the vector frequency in terms of lv isωv = −i lv −1 . Readers can go through [6,20] for more details on this.

JHEP12(2018)034 9 Conclusions and future directions
To summarise in a nutshell, the objective of this paper is to solve the Einstein-Maxwell equations in presence of cosmological constant using the large D perturbative expansion up to the first sub-leading order. The inverse of the space-time dimension serves as the perturbation parameter. We have solved for the first subleading correction of the metric and the gauge field. We do not need to choose any specific coordinate system and it holds for both asymptotic AdS and dS geometry (or more precisely to any smooth geometry that satisfies pure Einstein equation in presence of cosmological constant).
The membrane-gravity duality suggests that there is a one to one correspondence with the gravity solutions and the dynamical membrane which is defined by its shape and a velocity field on it. In our case, as we are dealing with a U(1) gauge field coupled to gravity, the dual membrane also has a charge density field on it. The membrane dynamics is captured entirely in terms of the scalar, vector and charge membrane equations. We have obtained these dual membrane equations and have seen that they emerge nicely from the Einstein-Maxwell system.
It turns out that the charge membrane equation contains a curvature dependent term even in the first subleading correction. It is one of the most interesting results in this paper. It implies that unlike the uncharged case, the naive covariantization of the flat space membrane equations will not work for charged membrane. In the last part of our paper, we have shown a consistency check of our membrane equations by computing the light quasi-normal modes of few known solutions of the Einstein-Maxwell equations both in Poincare patch and global AdS. Our QNM results match with the already calculated QNM frequencies, done from a purely gravitational approach in the literature.
A very natural extension of this work would be to extend the calculation to the next order in 1 D expansion. From our experience of the uncharged case, it seems that it is the order where we expect the entropy production.
It would also be interesting to classify the stationary black hole / brane type geometries using the time independent solutions of our membrane equation which is done in [41]. The obvious next step would be to explore the near-stationary geometries (which are dual charged fluid dynamics in our case) and compare them with the large-D limits of hydrodynamics coupled with a conserved charge.

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Raising the indices w.r.t. full metric g AM , We are using a subsidiary condition here, Hence from (A.4a) Hence,

JHEP12(2018)034 B Computation of the equation of motion
Christoffel connections.
Also, by using the null vector condition, we can easily see that, Here all the derivatives ∇ are w.r.t. the background metric η AB . For our convenience we are writing Ricci tensor. Now we will compute the Ricci tensor R AB from the Christopher connections.

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where,R we will use the above relations to compute each term separately in both of the equations of motion in (2.4).

C.1 Notations and identities
Identities.
Note that in the last identity, the first line vanishes because of the subsidiary conditions.
Here the highlighted term are the ones that could contribute at order O(D 2 ). Final form of the equation.

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Adding (C.8) and (C.9) Adding (C.11), (C.12) and (C.10) with L 2 we find Here we have used the subsidiary condition n · Q = 0. It is ok to use the subsidiary condition for order O(D) terms as n · Q does not appear at order O(D 2 ). In E AB , the form of order O(D 2 ) piece now,

C.2.2 Calculation of O(D) piece
We shall write S AB at order O(1) as To compute the different component we have to do the following projections.
Now we shall decompose and simplify different terms into scalar vector and tensor structures that would be requited for O(D) computation. (1) (C.20)
Now, the way we are organizing our computation, the O(D) pieces in the source, will have two different types of contribution: the first one coming from those terms in (C.3), (C.4) and (C.5) which naively looks like O(D 2 ), but an order O(D) piece is also hidden in it, once we cancel the leading piece. This is true only for S 1 and S 2 . We shall refer to such terms as 'Type A' contribution. The rest of the order O(D) will be called 'Type-B' contribution.
Total contribution to S 1 from (C.10) and (C.14) Here I have used all subsidiary conditions.
Total contribution to S 2 from (C.10)

C.3 The final form of the source S AB
We shall write the answer in terms of∇ · u ≡ Π AB ∇ A u B . We shall use the following identities to simplify the answer Here∇ 2 u C and∇ 2 K denote the followinĝ

C.4 Source from the Maxwell equation S N
We have already shown in appendix A that we can do the raising of ∇ M F M N entirely w.r.t. η AB and the covariant derivative is also w.r.t. the background η AB .

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Form of the source C.4.1 More identities and notation We shall first simplify the sum of the first three terms. These are the terms which can naively contribute at order O(D 2 ) and once they are canceled, will contribute 'type-A'
The final form of the source S N trace-less tensor sector (D.2a) Here, AB are the 'slow' data given on the membrane. Any derivative of the these data are O(1) except the divergences, i.e. ∇.
The metric with the first sub-leading correction.
And the inverse metric: The christoffel connections are calculated w.r.t. g AB .
AB ) Ricci tensor. Now Ricci tensor w.r.t. full metric g AB ,

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can be written in the form, AB (with respect to only ansatz metric where notations are Calculation of homogeneous part from the gauge field in E AB . Now we are calculating the homogeneous contribution from the sub-leading order gauge field corrections 1 D a M added to the initial ansatz A 2f O M . The gauge field with the first sub-leading correction looks as, where, We define the field strength w.r.t. g So, the total homogeneous part of the tensor equation of motion is, Homogeneous part due to S (i) Then,

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Then, So, Here we will calculate homogeneous part due to the metric correction, where h AB is raised with respect to η AB . Now field strength tensor raised with respect to g AB is, Here F M N is raised with respect to η AB .

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The equation of motion for gauge field is, Component of this homogeneous part in the direction normal to the membrane surface is, For metric correction of the form, trace-less tensor sector (D.48)

D.2.2 Gauge field equation homogeneous part due to gauge field correction
Now we calculate the homogeneous part due to gauge field correction.
(both of the terms are defined before) (D.50)

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Since, for h AB = 0, Γ C AC =Γ C AC , covariant derivative with respect to total metric will become covariant derivative with respect to background.
Here, ∇ M is the covariant derivative w.r.t.
Now we want to compute the covariant derivative of the anti symmetric tensor F M N w.r.t. g AB . Then following the previous formula, (D.55)

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So, the homogeneous part Using the following identity for the divergence of the source in the tensor sector source.
After substituting the solution for T and the identity above, the consistency equation takes the following form The r.h.s. of (E.2) is proportional to the membrane equation and therefore is of the order of O 1 D both on and away from the membrane.

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E.2 Verification of the consistency constraints (6.30) We will be showing the following constraint holds.
The third equation in (6.20) can be written as, Integrating we get,f Also, we will be using the boundary condition, Using the above result in the second equation of (6.30) and also v (i) gauge from (6.23), we obtain Here we will need the following identities Next, we will be doing a Taylor expansion around ψ = 1 of the second term and third term in the second equation of (6.30), also using the fact (ψ − 1) = ζ D ,

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Hence, adding (E.5) and (E.7), As the r.h.s. is proportional to the scalar membrane equation, C guage is O 1 D both on and away from the membrane. Now we will show the consistency of the other constraint.

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The source terms are evaluated as, (E.14) We have put the membrane equation Adding (E.13) with (E.14) we arrive at, As the r.h.s. is proportional to the scalar membrane equation, C metric is O 1 D both on and away from the membrane.

F QNM for the scaled membrane equation
The scalar membrane equation is,∇ .u = 0 (F.1) ∂ a δu a + (D − 2)∂ t δr = 0 (F.2) Following the line of [35] and from (F.1), it is obvious that the temporal and spatial frequencies are related by a factor of 1 √ D . Without loss of generality we assume that the temporal frequencies are of O(1) and the spatial frequencies are related by equation (F.1). 11 Now we will scale the coordinates as, x a → y a = √ Dx a . So the metric becomes, ds 2 = −r 2 dt 2 + dr 2 r 2 + r 2 dy a dy a D + dx i dx i (F.3) where a, b indices take p number of values and i, j indices take 1 to D − p − 2. We can consider fluctuations only in the p number of x a directions. Now we can write,

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In this new scaled coordinate we can write the membrane equation as, The total contribution from (F.7) is found to be, 12 The contribution from (F.1) is, The charge equation can be written as, These are the scaled non linear version of the membrane equation. Now we have to calculate the QNM for this system by linearizing with respect to the equilibrium system. Such an equilibrium system will be, where, k a V a = 0. Upto the linear order in , Then, upto the linear order in ,

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Now using, (F.16), (F.17) and (F.18) along with the simplification in (F.15) we have calculated the scalar mode QNM for different modes, These results match exactly with our earlier results upto the scaling k → k √ D (which is naturally expected) and are also consistent with [18]. This analysis shows that even after a scaled co-ordinate transformation and a special kind of scaling of O 1 D , it is possible to reach the charged black brane equations (upto the linear order) in [18] from our covariant membrane equations by a suitable field redefinition.
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