Stress Tensor for Large-$D$ membrane at subleading orders

In this note, we have extended the result of \citep{radiation} to calculate the membrane stress tensor up to ${\cal O}\left(\frac{1}{D}\right)$ localized on the codimension-one membrane world volume propagating in asymptotically at/AdS/dS space-time. We have shown that the subleading order membrane equation follows from the conservation equation of this stress tensor.


Introduction
It has recently been shown that in a large number of space-time dimensions(D) there is one-to-one corresondence [2][3][4][5] between dynamics of a black hole and a co-dimension one membrane propagating in asymptotic spacetime of the black hole. 1 In [1] the author has constructed a stress tensor up to O(1) defined on the membrane world volume propagating in asymptotically flat space-time and demonstrated that the leading order membrane equation follows from the conservation equation of the stress tensor. In this note, we have extended the result of [1] to calculate the stress tensor up to O 1 D defined on the membrane world volume but propagating in asymptotically flat/dS/AdS space-time and have demonstrated that the subleading order membrane equations [5] are just the conservation equation of this stress tensor.
The computation of [1] essentially captures the leading non-trivial part of the dual membrane stress tensor. But for several reasons, it seems that determining the stress tensor at next order is going to be quite useful.
For example, in [34] the authors have written an improved stress tensor that would work even for finite D at least for stationary metrics. However, in the same paper, they have also reported that their finite D improvement, once compared against the stress tensor for gravity systems with a dual hydrodynamic description in derivative expansion, does not match up to the relevant orders. Needless to say, in their construction the leading and the first subleading terms of the membrane stress tensor played a very important role and the subsubleading terms of the stress tensor will add very useful new data to the whole programme of finite D improvement.

Final Result:
In this section, we shall write our final result -the expression of the membrane stress tensor up to corrections of order O 1 D 2 . Conservation of this stress tensor would result in the membrane equation derived in [5] -the equation that governs the dynamics of the membrane.
In our case the membrane, which is a codimension-1 hypersurface, is embedded in AdS/ dS space. More precisely, the metric of the embedding space satisfies the following equation Where, dimension(D) dependence of Λ is parametrized as follows The membrane is characterized by its shape (encoded in its extrinsic curvature K µν ) and a velocity field (u µ ), unit normalized with respect to the induced metric of the membrane. The membrane stress tensor, that we report below, is a symmetric two-indexed tensor, constructed out of this velocity field, extrinsic curvature and its derivatives. For convenience, we shall decompose the stress tensor in the following way 8πT µν = S 1 u µ u ν + S 2 g (ind) µν + V µ u ν + V ν u µ + W µν (1.1) Where, Here, g (ind) µν is the induced metric on the membrane,∇ µ is the covariant derivative with respect to g (ind) µν . Membrane velocity (u µ ) can also be viewed as a vector field(u A ) in the full background space-time. u µ is related to u A through the following equation Where, X A are the coordinates in the full space-time and y µ are the coordinates on the membrane world volume. The extrinsic curvature of the membrane K µν is defined as follows Here, n A is the normal to the membrane and Π AB is the projector orthogonal to the membrane defined as Π AB = g AB − n A n B .

Strategy:
The two key principles that fix this stress tensor are the following • Conservation of the stress tensor should reproduce the membrane equation up to the relevant order.
• This stress tensor should be the source of the gravitational radiation, generated from the massive fluctuating membrane.
In fact, it is the second principle that finally determines the algorithm to be used to derive the stress tensor. The algorithm is such that the first principle is automatically ensured and we have used it in the end as a consistency check for our long calculation. Below, we shall just write down the steps to be used so that the final construction is consistent with the second principle. However, we shall not write the justification for any of these steps as they are explained in detail in [1] and explanation is completely independent of the order in terms of (1/D) expansion.
• Step-1: Codimension one membrane is given by a single scalar equation ψ = 1.
Define ψ > 1 region as ' outside of the membrane' and ψ < 1 as ' inside of the membrane'. 'Outside region' is the one that extends towards asymptotic infinity and contains the gravitational radiation. • Step-2: Next, we would like to write a space-time metric for both outside and inside region, with the following properties.
1. The metric would solve Einstein equation (in presence of cosmological constant) linearized around pure AdS/dS metric.
2. The metric would fall off as ψ −D in the outside region and would be regular in the inside region.
3. The metric should be continuous across the membrane though its first normal derivative need not be.
It turns out that in 1 D expansion, the above two conditions uniquely fix the metric on both sides, in terms of the induced metric on the membrane, which we read off from the large-D metric determined in [5].
• Step-3: Once we have determined the metric on both sides, the discontinuity of its normal derivative across the membrane is also fixed unambiguously. The conserved stress tensor associated with the membrane is computed from this discontinuity. More precisely, it is the difference between the two Brown York stress tensors on the membrane evaluated with respect to the inside and outside metric.
Here, 8πT (in) AB and, 8πT = 0. So, T AB can equally well be regarded as a tensor T µν that lives on the membrane world volume.
Calculationally, this is very lengthy. In the main text, we have just written the final results, most of the lengthy derivations are in the appendices. The organization of this note is as follows: In section 2 we have linearized the Large -D solution known up to subleading order and have changed the gauge and subsidiary condition (as discussed just below eq.(2.1)). In section 3 we have constructed a linearized solution of Einstein's equation in the inside region of the membrane. In section 4 we have calculated the membrane stress tensor and in the section 5 we have shown that the subleading order membrane equation follows from the conservation of this stress tensor.
2 Linearized Solution : Outside(ψ > 1) In this section, we shall work out the metric in the outside region. However, what we are finally interested in is just the difference between Brown York stress tensor across the membrane. To compute it, we need to know the metric only very near the membrane. The large D solution as described in [5] already determined the metric in this near membrane region even at non-linear order. For our purpose, we shall simply read off the 'outside metric' from [5]. In fact, we have to pick out only the part that is enough to solve the linearized equations. In other words, we need only that part of the metric which could be recast as G (out) In the first subsection, we have described the large-D solution and read off the piece needed.
The main calculation of this section involves a change of gauge and 'subsidiary conditions' (conventions that fix how the basic fields would evolve away from the membrane, see [3] for more details). In the next two subsections, we have described the new set of conventions, that are more useful for our purpose and performed the required changes on the metric, read off in the first subsection. Needless to say, all steps are worked out in an expansion in 1 D .

Large-D Metric upto sub-subleading order : Linearized
In this subsection we will just quote the solution of Einstein's equation up to second subleading order in 1 D expansion as derived in [5] and we will linearize the solution in ψ −D . The solution is given by Here, g AB is the background metric and Where, x 0 y e y e y − 1 dy And, Here,R ABCD is the Riemann tensor of the background metric g AB and∇ is defined through the following equation -for a generic n-index tensor W A 1 A 2 ···An We want the sub-subleading order metric in linearized order in ψ −D . So, we need to calculate the above integration (2.4) in linearized order in ψ −D . The answers are the following. See A for details.
Using (2.7), we can write the full metric G AB as We will get

Change of Gauge Condition
Large-D solution [5] has been derived in the gauge condition O A h AB = 0. But, it turns out that in the calculation of the stress tensor it is more convenient to use the gauge condition n A h AB = 0. In this subsection, we will implement this gauge transformation.
We do the following infinitesimal coordinate transformation Under the above coordinate transformation, metric transforms as follows Now, using (2.9), we get We choose the coordinate transformation in a way such that n A M ′ AB = 0. Now using the expansion (2.15) Now, using the following decomposition Using (2.10), (B.34) and (B.35) we can finally write M ′ AB as Where,

Change of Subsidiary Condition
AB -we have used in the calculation of the stress tensor. Because, we have imposed the condition AB in a power series expansion in (ψ − 1) and will determine different coefficients by satisfying Acting on the above equation by Π C A Π C ′ B (n.∇) and them equating the coefficient of (ψ − 1) 0 we get Equating the coefficient of (ψ − 1) we get The final form form of h Where, AB on the surface ψ = 1 becomes The final form form of h So, finally, we have brought the large-D solution in the following form G (out) In this section, we shall construct the 'inside solution' i.e, the metric for region ψ < 1. As we have mentioned before, we want this metric to be regular throughout the 'inside region' in order to make sure that the membrane is the sole source of the gravitational radiation in this system. Note that the solution presented in [5] continued to be a solution even when ψ < 1. However, this solution diverges at the location of the black hole, the point where ψ approaches zero and also it does not have any discontinuity across the event horizon -the location of the membrane. Therefore, unlike the 'outside solution' we have to construct the inside solution from scratch maintaining the regularity and the fact that on the membrane it reduces to the same induced metric as the one read off from the 'outside solution'.
We shall write the inside metric in the following form AB satisfies the gauge condition n Ah (m) AB = 0. At linearized order, Christoffel symbol for (3.1) is given by Where,Γ A BC is Christoffel symbol of g AB and ∇ A is covariant derivative with respect to g AB . Now, Ricci tensor is given by Where,R AB is Ricci tensor for g AB .
Einstein equation in the inside region Projecting the above equation perpendicular to n A and n B we get Using the following decomposition forh We can solve forh AB by solving (3.7) order by order in 1 D expansion. The final form form ofh The final form form ofh The final form form ofh Adding (3.9) and (3.13) we get

Stress Tensor
In this section, we will derive the expression for membrane stress tensor. The membrane stress tensor is given by the discontinuity of the Brown-York stress tensor across the membrane. 2

Outside(ψ > 1) Stress Tensor
The outside stress tensor is given by Where, p (out) and,∇ is covariant derivative with respect to G AB and K (out) are the followings. See C for details.
Putting the expression for K

Inside(ψ < 1) Stress Tensor
The inside stress tensor is given by and,∇ is covariant derivative with respect to G Putting the expression for K (in) AB and K (in) from (4.10) in (4.6) and using the fact thath

Membrane Stress Tensor
Membrane stress tensor is given by We can simplify the calculation of stress tensor by using a trick. We define Then from (4.12) we can very easily see that Let's call this proportionality factor ∆. With this notation membrane stress tensor becomes Now, from the condition K AB T AB = 0 we get Where, So, the full stress tensor becomes Where, S 1 , V A , W AB are given respectively by (4.17), (4.18), (4.19) and S 2 is given by

Conservation of the Membrane Stress Tensor
The final expression of membrane stress tensor (1. Here, we want to make some comments about how we have done the large-D calculation in Mathematica. We choose the following background metric which is pure AdS metric written in a slightly different coordinates than usual Poincare patch coordinates r → log r will give usual Poincare patch metric . Here, 'a' runs over some finite p dimension and µ runs over large D − p − 2 dimension. ψ and u A are only functions of (t, r, x a ) but does not depend on x µ . We can effectively do our calculation in finite p + 2 dimension. We will calculate the contribution that will come from the large D − p − 2 dimension by hand and will accordingly take into account. For example, if we want to calculate∇ B∇ B u A (where A, B runs over full D dimension), the first thing to note is that it has non zero component only along 'a' direction and it is given bŷ Similarly, we can calculate all the quantities appearing in the expression of the stress tensor.

Conclusions
In this note, we have calculated the membrane stress tensor up to order O 1 D and showed that the conservation of this stress tensor gives the subleading order membrane equation.
Very briefly, our procedure is as follows : given the large-D solution outside the membrane -linearize the solution -search for a regular solution inside the membrane region with the condition that the induced metric is continuous on both sides of the membrane -construct the Brown York stress tensor for inside and outside regionthe difference of the Brown York stress tensor across the membrane is the membrane stress tensor.
As it turns out, the computation leading to the stress tensor at subsubleading orders is extremely tedious, though the final result is relatively compact and simple (presented in section 1.1). Still one might wonder what is the point of taking up such a calculation. The key motivation we have already mentioned in the introduction. It is about the finite D completion of membrane stress tensor [34]. Let us elaborate a little more on that.
Algebraically, the large D expansion technique and membrane-gravity duality are very similar to that of derivative expansion and fluid-gravity duality. It turns out that in fluid-gravity duality it is possible to write an expression for the fluid stress tensor that works exactly for the complicated stationary solutions like rotating black holes. But that exact expression is nothing but a truncation of the fluid stress tensor at second order in derivative expansion. Now in [34] the authors have proposed a finite D completion for large-D stress tensor, which does not quite work and the mismatch arises again at second order in terms of derivative expansion. All these facts and the experience with fluid gravity correspondence naturally lead to the hope that a second order calculation in terms of ( 1 D ) expansion would help in the final goal set by [34].
Though this second order membrane stress tensor is just a small step towards this final goal. We think, the following would be the next few steps, which might help to construct a finite D completion of the membrane stress tensor (if it exists), by generating more data • A detailed matching with the hydrodynamic stress tensor dual to the same gravity system in the regime of overlap for these two perturbation techniques ( namely 1 D expansion and derivative expansion (see [35,36])). Now after computing the membrane stress tensor, we could extend this matching to include the effect of the gravitational radiation as well.
• Recasting known rotating black hole solutions in arbitrary D, in the language of large D expansion, capturing few terms that could contribute in a stationary situation, to all orders.
• Finally, evaluating the second order membrane stress tensor on the rotating black holes, hoping some novel pattern or truncation would emerge out of this exercise, that will tell us in general how stationarity is encoded in this large-D expansion technique.
We find all of the above projects are interesting, themselves. They will teach us a lot about how perturbation works in gravity and how they could be used to have analytic control over the otherwise difficult to handle dynamics of gravitating systems. We leave all these for future work.
for collaboration at the initial stage. I would also like to thank Yogesh Dandekar and Suman Kundu for many illuminating discussions over the course of the project. I would like to acknowledge the hospitality of ICTS Bangalore, TIFR Mumbai and IISER Bhopal while this work was in progress. Finally, I would like to acknowledge my gratitude to the people of India for their steady and generous support to the research in basic sciences.
A Calculation of integrals (2.4) at linear order We just want e −R term of the integration. Expand in e −R we get. Now, x 0 y e y e y − 1 dy = π 2 6 (A.7)

Substituting (A.6) and (A.7) in (A.4) we get the final expression
The f 1 (R) integration is very straightforward Now, we want to calculate the following integration We can expand the integrand in e −x and then can do the integration term by term. Doing the integration term by term, we get So, finally the second line of f 2 (R) becomes Now we will calculate the first line of f 2 (R) Using eq (A.8) we get Now we need to do the following integration Now, we will calculate the following integration. Expanding the integrand in e −x and doing the integration term by term we get So, finally the first line of f 2 (R) becomes B Some Details of Linearized Calculation Comparing coefficient of (ψ − 1) 1 we get Comparing coefficient of (ψ − 1) 2 we get This implies we want ξ A to be correct up to order O 1 A to be correct up to order O 1 D 2 and ξ (2) A to be correct up to order O 1 D . Now, using the following expansion Now, we will calculate ξ Adding (B.9) and (B.8) we get Finally we get, Adding (B.1) and (B.2) we get, 14) Now, we will calculate ξ Now writing the expression for L BA we finally get Now, we will simplify (B.19). First, we will simplify the first square bracketed terms.
Using, (B.20) and it's symmetric part the first square bracketed terms become Now, we will simplify the second square bracketed term of (B.19) Finally, we will try to simplify the third square bracketed term of (2.14) Using, (B.24) and it's symmetric part the third square bracketed terms become Now we will calculate different terms in (2.20). First we will calculate ξ Next, we will calculate ξ Now, we need to calculate different terms of (B.28) Using (B.29) in (B.28) we get Adding (B.27) and (B.30) we get the expression of ξ Next, we will calculate ξ Using the identity (D.1) and (D.2) we can write the above equation as First we will write t AB and v A in a convenient way. From (2.5), t AB can be written as Where, From (2.5), v A can be written as Where, Where, Using the following two identitŷ we get, To simplify the above expression we will use the following identity. We will not give the derivations of these identities. The derivations are quite straightforward Using (B.49) and (B.50) we can write C (0) Where, Using 2nd identity of (B.44) we can write the above equation as To simplify the above expression we will use the following identity To prove the above identity we have used subsidiary condition P A B (O · ∇)O A = 0 and the second order membrane equation( 2.17 in [5] ). Using (B.57) we get Now, we will simplify the above equation Collecting the coefficient of (ψ − 1) 0 of (B.60) Using (3.8), the leading order(O(D)) terms of (B.65) In the last equation, we have used the fact thath (0) can nowhere be O(D). Taking trace of (B.66) From, subleading order(O(1)) of (B.65) Taking trace, Collecting coefficients of (ψ − 1) of (B.60) at order(O(D)) Taking trace, We want to calculate the above expression on ψ = 1. But to calculateh ) After a bit of simplification divergence of the above equation becomes In the derivation of the above equation we have used the following identities In the last line we have used the divergence of leading order membrane equation So, finally we get h (1,1) Now, we will calculatẽ h (1,1) We want to calculate the above expression on ψ = 1. But to calculateh (1,1) CC ′ | part-2 on ψ = 1 we need the (ψ − 1) and (ψ − 1) 2 dependent terms of h Using (B.88) we can write Term-3 as CC ′ + u C B (2) Where,

B
(2) Here, We will use the following identities to simplify (B.96). we are just stating the identities without proof, proofs are quite straightforward.

B
(1) CC ′ +u C B (1) In the derivation of (B.102) we have used the following identity CC ′ as given in (3.9).

Calculation ofh
(2) CC ′ From (B.74), the non-vanishing terms ofh (2) CC ′ are the following h (2) For the calculation ofh (2) CC ′ we need (ψ − 1) dependent terms ofh Using the identity, (n · ∇)u A Π E B (n · ∇)u E − u A (n · ∇)n B n E (n · ∇)u E + u A Π E B (n · ∇)(n · ∇)u E + (n · ∇)u B Π E A (n · ∇)u E − u B (n · ∇)n A n E (n · ∇)u E Inverse of (C.1) at linear order is Using, the gauge condition n A h AB = 0, we get Here, Γ E CC ′ is Christoffel symbol with respect to g AB and δΓ E CC ′ is defined as Here, ∇ C is covarint derivative with respect to g AB K (out)  Now, from (4.8) (C.14) Where,Γ Here, Γ E CC ′ is Christoffel symbol with respect to g AB and δΓ E CC ′ is defined as Here, ∇ C is covarint derivative with respect to g AB Now,  Stress of extrinsic curvature is given by (C.20)

D Important Identities
In this appendix we will mention the identities we have used in this note. The identities have been calculated on ψ = 1 hypersurface. We are not giving the derivations simply due to the fact that the derivations are very lengthy but nevertheless the derivations are quite straightforward. Identity-1:

Background metric g AB
Induced metric on the membrane as embedded in g AB g (ind) µν Full non-linear metric outside the membrane G AB as read off from [5] Linearized metric outside the membrane G (out) Projector on the membrane as embedded in g AB Π AB = g AB − n A n B Projector perpendicular to both the normal of the P AB = g AB − n A n B + u A u B membrane as embedded in g AB and the velocity Projector on the membrane as embedded in G Extrinsic curvature of the membrane K AB when embedded in g AB