Holographic dual of a boost-invariant plasma with chemical potential

We construct a gravity dual of a boost-invariant flow of an N=4 SU(N) supersymmetric Yang-Mills gauge theory plasma with chemical potential. We present both a first-order corrected late-time solution in Eddington-Finkelstein coordinates and a zeroth-order solution in parametric form in Fefferman-Graham coordinates. The resulting background takes the form of a time-dependent AdS Reissner-Nordstroem-type black hole whose horizons move into the bulk of the AdS space. The solution correctly reproduces the energy and charge density as well as the viscosity of the plasma previously computed in the literature.


Introduction
In the recent years, the application of the AdS/CFT correspondence [1] to the quark-gluon plasma (QGP) has become a very active research area. One line of research within such holographic studies was initiated by Janik and Peschanski [2] who established a timedependent gravity dual of the boost-invariant flow of an N = 4 plasma. This geometry has mainly been studied in the regime of large proper time, when the system is near equilibrium and approaches the hydrodynamic regime (see however [3,4]). In [5]- [12] higher-order corrections to this late-time background were constructed and found to be equivalent to a gradient expansion of hydrodynamics, see [13] for a review.
An important aspect of the plasma which has not yet received much attention in a time-dependent gravity background is the effects of chemical potentials, even though an asymptotic boost-invariant geometry (without corrections) dual to an N = 4 plasma with U(1) R-charge is known for quite some time [14]. Also the transport coefficients of plasmas with U(1) currents have already been holographically computed in [15]- [19] (up to second order). Such currents are generated, for instance, shortly after the collision of two heavy ions, when the two sheets of color glass condensates have passed through each other and longitudinal color electric and magnetic flux tubes are produced between the sheets [20]. This gives rise to a large topological charge density F µν aF a µν , which in turn leads to an imbalance of the number of quarks with left-and right-handed chirality and chemical potentials µ R and µ L . In addition to the usual baryon chemical potential µ = µ R +µ L 2 , one may therefore also consider a chiral chemical potential µ 5 = µ R −µ L 2 which mimics the effect of an imbalanced chirality.
In this paper we will construct a modification of the Janik-Peschanski background, which will additionally include a time-dependent U(1) gauge field. The bulk theory will be five-dimensional Einstein-Maxwell gravity with a negative cosmological constant and a Chern-Simons term. As in the case without chemical potential, it appears to be difficult to find an analytic solution for all times and we will restrict to solving the equations of motion at late times. As a further simplification, we seek for a solution in which only the time-component of the U(1) gauge field dual to the chemical potential is non-vanishing (the spatial components are set to zero). Asymptotically, at large proper time τ , we may expand the late-time geometry in powers of τ −2/3 . Employing both Eddington-Finkelstein and Fefferman-Graham coordinates we present the late-time solution up to first order (in τ −2/3 ). The resulting background will essentially take the form of a time-dependent AdS 5 Reissner-Nordström solution whose inner and outer horizon move into the bulk of the AdS space. This background can be extended to a full type IIB supergravity solution (by taking the product with an S 5 ) and is dual to a strongly-coupled N = 4 SU(N) supersymmetric-Yang-Mills plasma with a non-vanishing chemical potential.

Late-time background in Eddington-Finkelstein coordinates
In this section we are interested in finding a late-time gravity dual of an expanding N = 4 viscous plasma with non-vanishing chemical potential. The relevant five-dimensional Einstein-Maxwell-Chern-Simons action is given by where α, β, ... denote the 5D bulk coordinates. The cosmological constant is Λ = −6 and the Chern-Simons parameter is fixed as κ = −1/(2 √ 3). Also, 1/(16πG 5 ) = N 2 c /(8π 2 ) for an N = 4 plasma [15]. The corresponding equations of motion are given by the combined system of Einstein-Maxwell equations, and covariant Maxwell equations (with Chern-Simons-term), F αβ is the field strength of the U(1) gauge field A α we wish to introduce in the background.

Boosted black brane solution
Our starting point for the construction of a time-dependent solution is the static AdS 5 Reissner-Nordström (RN) black-hole solution [21]. Using ingoing Eddington-Finkelstein coordinates, we may write the RN metric and gauge field as with mass m and charge q.
Here v is a time-like coordinate (not to be mixed up with the scaling variable v introduced below), x are the spatial coordinates on the boundary, and r parameterizes the holographic direction. The location of the outer horizon r + = r + (m, q) is given by the largest real positive root of V (r + ) = r 6 + − mr 2 + + q 2 = 0. A charged black hole is dual to a fluid at finite temperature T and chemical potential µ. Both the Hawking temperature and the chemical potential are given in terms of r + by [21] (2.6) These relations can be inverted to give m and q as functions of T and µ [15], with r + = πT 2 (γ + 1) , γ = 1 + 8µ 2 3π 2 T 2 . (2.8) Following [9,10], we now consider the corresponding five-dimensional boosted charged black brane solution given by where u µ is the boost velocity along x µ (µ = 0, 1, 2, 3), and m = m(µ, T ) and q = q(µ, T ) as given by (2.7). From this solution we may deduce a time-dependent solution by choosing the frame u µ = (1, 0, 0, 0) and introducing an Eddington-Finkelstein proper time-like coordinateτ and rapidity-like coordinate y. We also substitute the asymptotic late-time behaviour of T [22] and µ, into the explicit expressions for m and q. Here we assumed µ ∝ T , as one would expect for a perfect fluid, such that the quotient µ/T =μ 0 /Λ = const. is independent of time.
This leads to the following metric 1 with coefficients For q = 0 (orμ 0 = 0), this metric reduces to the uncharged (zeroth-order) late-time solution in Eddington-Finkelstein coordinates found in [10,11,12] Note that the size of the outer (and inner) horizon r + (r − ) decreases with time.

Zeroth-order solution and first-order correction
The boosted metric (2.11) is not an exact solution of the Einstein-Maxwell equations. It is a good approximation of the boost-invariant solution at largeτ though. At smallerτ , it receives subleading corrections corresponding to higher-order gradient corrections to the energy-momentum tensor and U(1) current, which will be discussed in section 2.3. These corrections to the metric (2.11) can be found by choosing the following metric ansatz for the time-dependent solution: 2 ds 2 = −r 2 e a(τ ,r) dτ 2 + 2dτ dr + (1 + rτ ) 2 e b(τ ,r) dy 2 + r 2 e c(τ ,r) dx 2 ⊥ , A = d(τ , r)dτ . (2.14) As in the case without chemical potential, we may introduce the scaling variable v = rτ 1/3 and expand the metric coefficients in powers ofτ −2/3 , Similarly, for the coefficient of the gauge field we choose Note that the gauge field has an overall factorτ −1/3 . The existence of a late-time scaling variable v will be shown in section 3.2.
1 There is an additional 1 in the factor (1 + rτ ) 2 in front of dy 2 which is not expected from the boosted solution (2.11). This is to ensure an asymptotic AdS space in the limit Λ → 0, see [11] for details. 2 For this particular ansatz, the Maxwell equation reduces to 1 √ −g ∂ β ( √ −gF βα ) = 0. The Chern-Simons term is absent, since only F rτ and Fτ r are non-vanishing.
The system of Einstein-Maxwell equations (2.2) and Maxwell equations (2.3) can then be solved order by order inτ −2/3 . At zeroth-order inτ −2/3 , we find the coefficients where we defined the time-independent variables with m(τ ) and q(τ ) as in (2.12). In the same way, we also define the variable from the (outer) horizon r + as given by (2.13). A(v), ..., D(v) are in agreement with the metric (2.11) deduced from the boosted black brane.
At first order inτ −2/3 , we find the coefficients where ζ i are the solutions of The resulting expression for b 1 (v) is real, even though we need to consider all six roots of (2.23) including the imaginary ones. Explicit expressions for these roots can be found in appendix A. Note that one of the six roots of this equation corresponds to the outer horizon ζ + . In Reissner-Nordström solutions there is always an upper bound on the chargeq 0 , at which the discriminant of the equation (2.23) vanishes, For larger values ofq 0 , there would be a naked singularity at the origin. Remarkably, this bound is satisfied for any value of the quotientμ 0 /Λ and saturated in the limitμ 0 /Λ → ∞, as can be seen by substituting (2.20) with (2.12) into the bound (2.24). In other words, there is no bound on the chemical potential. Nevertheless, let us assume thatμ 0 ≪ Λ in order to avoid potential stability problems [23], which arise when the black hole is close to extremality.
We still need to fix the integration constants C 1,2,3 . C 1 can be found by requiring regularity of the first-order solution (2.22) at the outer horizon, i.e. C 1 should be a function of the positive root ζ + . More precisely, by choosing , which are singular at v = ζ + . The metric then still contains singularities but they are hidden behind the outer horizon. The constant C 3 is fixed by the requirement that the metric reduces to a pure AdS space in the limit Λ → 0. This simply sets C 3 to zero, There is one remaining integration constant C 2 which can not be fixed at first order. Note that, in general, at each order k there is one integration constant which can only be fixed by regularity at order k + 1 [10], C 2 in our case. Nevertheless, we may guess the correct value for C 2 by comparing with the uncharged solution [10,11,12], in which As for C 1 , it seems natural to replace the horizon ζ H of the uncharged solution by the outer horizon ζ + of the charged solution such that Later in section 2.3 we will justify this value again. It will turn out to correctly reproduce the expected transport coefficients. We have checked that forq 0 = 0 (or, equivalently,μ 0 = 0) the metric reduces to the first-order corrected uncharged solution found in [10,11,12]. Moreover, for the Kretschmann scalar we find (2.28) which is only singular at v = 0. In the limitq 0 → 0, we have m 0 → π 4 Λ 4 and R 2 µνρσ reduces to the corresponding expression in the uncharged case, see [11].
We have thus constructed a natural extension of the first-order corrected boostinvariant plasma geometry of [10,11,12] to the corresponding one with non-trivial U(1) gauge field.

Transport coefficients from the background
In the hydrodynamic approximation, the energy-momentum and U(1) current are given by where the first terms on the right hand side correspond to a perfect fluid with chemical potential. Since the velocity field u µ , energy density ε and charge density ρ vary slowly with the spacetime coordinates, the energy-momentum tensor and current receive higherorder gradient corrections given by (up to first order) where η, σ and ξ denote the viscosity, conductivity and vorticity coefficient, respectively. The corrections satisfy u ν Υ ν = 0 and u ν Π µν = 0. The transport coefficients of the fluid entering these corrections were holographically computed in [15,16,17] (up to second order) by slowly varying u µ , q and m in the boosted solution (2.9) with the space-time coordinates x µ . In this way the hydrodynamic equations are obtained from AdS/CFT without constructing an explicit solution.
In the following we will compute the first-order corrections directly from our timedependent solution using holographic renormalization techniques [24]. Recently, a rigorous holographic renormalization of the Einstein-Maxwell-Chern-Simons theory, including the full back-reaction of the gauge field, has been performed in [25]. The energy-momentum tensor can be obtained from where γ µν is the induce metric on a constant-r hypersurface, which regularizes the boundary. K µν is the extrinsic curvature of on this hypersurface, K the corresponding scalar K = K µν γ µν and G µν the boundary Einstein tensor with respect to the metric γ µν . Substituting our explicit first-order solution into (2.31), we find the time-dependent energy density 3 and b 0 = bτ −1/3 as in (2.20), see appendix B for more details on the computation. The first term in ε(τ ) is the zeroth-order energy density and is in agreement with that in [15], see Eq. (20a) therein. The second term is the first-order correction and formally agrees with that in the uncharged case [10,11,12] but now with a more general shear viscosity η 0 = η 0 (μ 0 , Λ). 4 This correction is also in exact agreement with the first-order gradient correction to the energy-momentum tensor computed in [15]. There [15], the viscosity was found to be
Similarly, the expectation value of the R-charge current can be computed from ρ is the r −n coefficient of the large-r expansion of the gauge field A ρ . Since the spatial components of the gauge field are zero, the second term proportional to κ is absent in our case. Substituting the solution for the gauge field into (2.35), we read off the U(1) charge density withq 0 = qτ as in (2.20). Recalling √ 3q/2 = µr 2 + , we find agreement with the zerothorder charge density in [15], see Eq. (20b) therein. The asymptotic 1/τ behaviour of the charge density was also found in [14]. There are no first-order corrections to the charge density in our case.
More generally, for gauge fields with vanishing spatial components, there are no higherorder gradient corrections. This follows directly from the relation u ν Υ ν = 0. The corrections Υ ν are orthogonal to u ν and cannot come from the near boundary expansion of a gauge field proportional to u ν .

Late-time solution in Fefferman-Graham coordinates
In this section we seek for a time-dependent solution of the Einstein-Maxwell equations (2.2) and (2.3) in Fefferman-Graham coordinates.

General ansatz and near-boundary behaviour
In Fefferman-Graham coordinates, we choose the same metric ansatz as in the uncharged case [2] given by Of course, the warp factors a(τ, z), b(τ, z) and c(τ, z) will be modified due to the effects from the back-reaction of the gauge field. As before, we set the spatial components of the gauge field to zero and assume a non-vanishing time-component, Let us first study the general behaviour of the solution near the boundary at z = 0. Following [3], we choose the small-z expansions a(τ, z) = −ε(τ )z 4 + a 6 (τ )z 6 + a 8 (τ )z 8 + ... , b(τ, z) = b 4 (τ )z 4 + b 6 (τ )z 6 + b 8 (τ )z 8 + ... , c(τ, z) = c 4 (τ )z 4 + c 6 (τ )z 6 + c 8 (τ )z 8 + ... Here the lowest coefficients are determined by the energy and charge density, respectively. For instance, solving the Einstein-Maxwell equations to lowest order in z, we obtain as in [3]. There is no back-reaction of the gauge field on the geometry at this order (ρ(τ ) does neither appear in b 4 (τ ) nor c 4 (τ )). Likewise, the metric does not enter the Maxwell equations at this order. However, other than the energy density ε(τ ), which can be freely chosen (at least at early times), the charge density ρ(τ ) is uniquely fixed by the z-component of the Maxwell equations, which is solved by q 0 = const. Any dependence on the warp factors has dropped out in the Maxwell equations such that ρ(τ ) is independent of ε(τ ). The result (3.7) for the charge density holds for all times τ > 0. Remarkably, the charge density diverges at τ = 0. 5 Solving the system of equations (2.2) and (2.3) order by order, we find the solution up to order z 8 , Generic solutions of viscous fluid dynamics are not expected to be regular in the infinite past (see footnote 4 in [15] in this context): The volume element on the boundary at constant proper time scales linearly with τ . Integrating the charge density (∝ 1/τ ) over this volume element (∝ τ ) yields a constant total charge. Thus, even though the charge density is divergent, the total charge is regular, even at τ = 0, ensuring the validity of the hydrodynamic approximation. and similar expressions for b(τ, z) and c(τ, z). These expressions for the warp factors generalise the corresponding ones for q 0 = 0 found in [3]. They describe the all-time near boundary behaviour of the background as a function of the energy and charge density.

Late-time ansatz for the background
A full analytical all-time solution is difficult to find, even in the uncharged case (q 0 = 0). It is however possible to find a late-time solution. The general late-time behaviour of the energy and charge densities can be found as follows (For the energy density the derivation is very similar to that in [2,5]). In the local rest frame the energy-momentum tensor is diagonal with elements T τ τ , T yy and T xx = T x 2 x 2 = T x 3 x 3 and the current has only a timecomponent J τ while J y = J x 2 = J x 3 = 0. Moreover, we assume that these components depend only on τ .
Using proper time and rapidity coordinates in flat Minkowski spacetime, defined by x 0 = τ cosh y and x 1 = τ sinh y, the tracelessness condition T ν ν = 0, energy-momentum conservation T µν ;ν = 0 and charge conservation J ν ;ν = 0 have the form Here we assumed that the anomaly in the U(1) current is absent, which is true for our simple ansatz of the gauge field. Comparing with the zeroth-order energy-momentum tensor and current given in (2.29), in the frame u ν = (1, 0, 0, 0) we obtain We observe that the asymptotic charge density (3.13) is in exact agreement with the expression (3.7) for the charge density, which is valid for all times. In other words, the late time charge density (3.13) does not receive any higher-order gradient corrections, in agreement with our findings in the previous section. Substituting the asymptotic behaviour (3.13) into the general solution (3.8) and expanding the resulting expressions for large τ , we get (ε 0 = 1) a(τ, z) = − z 4 τ 4/3 + 2 + 30q 0 2 τ 4/3 27τ 10/3 z 6 + 10 − 27q 0 2 τ 4/3 − 54τ 8/3 972τ 16/3 z 8 + · · · , (3.14) b(τ, z) = (3.17) We find that the dominant terms at large τ scale as a n (τ )z n ∼ z n τ n/3 , d n (τ )z n ∼ 1 τ 1/3 z n τ n/3 , (3.18) and similarly b n (τ )z n and c n (τ )z n . As in [2], it is therefore useful to introduce the scaling This suggests the following ansatz at late times, will turn the equation of motions into a system of nonlinear ordinary differential equations for the coefficients a i , ..., d i (i ≥ 0). In principle, this system can then be solved order by order in τ −2/3 .

Zeroth-order solution
In the following we restrict to give an exact solution for the zeroth-order coefficients a 0 (v), ..., d 0 (v). The non-vanishing components of the Einstein-Maxwell equations are At zeroth order, the z-and τ -components of the Maxwell equation both lead to the same equation, The other components are zero. These equations can be simplified a lot. Note that only four out of the five plus one equations are independent. We also find from a linear combination of the τ τ -, zz-and zτ -components of the Einstein-Maxwell equations that b 0 (v) = c 0 (v). Next, the Maxwell equation (3.22) can be solved for d 0 (v), where S 4 is some integration constant which will be fixed below. Substituting this back into the Einstein equations, the two remaining independent equations are given by the τ τ -and zz-components. The first one (τ τ ) is an equation while the second one (zz), can be used to find a 0 (v) as soon as a solution for b 0 (v) is known. Our primary goal will be to solve (3.24) for b 0 (v). a 0 (v) and d 0 (v) can then easily be obtained from (3.25) and (3.23). Later, in order to fix some integration constants, we will need the asymptotic solution close to the boundary which can be expanded in powers of v as (here we present it up to O(v 10 )) (3.26) It shows us that the solution exists and is uniquely fixed by parameters ε 0 and q 0 . Comparing the expression (3.23) with the boundary behaviour (3.26), we may immediately fix the integration constant S 4 as We now solve (3.24) for b 0 (v). By setting b 0 (v) = log(β(v)) , (3.28) we simplify this equation to the form which turns out to be the modified Emden-Fowler equation [27]. Its solution can be written in the parametric form Here S 1 , S 2 , S 3 and p + are some integration constants. One can in principle absorb p + in S 3 but we separate them for the moment. There are two useful expressions for β(v) and β ′ (v), From (3.26), we get the near-boundary conditions Substituting (3.35) into (3.34), we extract the expected asymptotics for β ′ (v) as a function of p, Generically, at large p, (3.33) is approximated by which fixes S 1 as Remarkably both constants S 1 and S 2 do not depend on q 0 . Let us now relate S 3 and p + by setting . v + will be fixed by the requirement that the outer horizon of the geometry is located at v = v + . Formally, the horizon v + is defined as the largest zero of the denominator on the right hand side of (3.25), This can be rewritten in terms of p + . Using (3.33), (3.32) and the expressions for S 1 and S 2 , we get the condition The last step is to fix v + in (3.31). This can be done by substituting the p + solution (3.43) (and all the constants S 1,2,3 ) back into (3.31) and expand v(p) for large p. In this way we determine the constant on the right hand side of (3.36) as a function of v + . Since this constant must be one, we get v + (ε 0 , q 0 ) = exp 44) 7 In order to extract the roots correctly we use the following standard convention: where U(p) is defined as For q 0 = 0, the integral can be performed analytically and v + reduces to the well-known result for the horizon [2], For general q 0 , this integral can in principle be written as a lengthy expression of elliptic integrals of the first and third kind, F (φ, k) and Π(n; φ|m), respectively, which we will not do here. Instead, in Fig. 1 we show the dependence of v + on the charge q 0 for some particular choices of ε 0 . We note that for each ε 0 there is some maximal allowed value of the charge at which the black hole becomes extremal. This value can be found from the condition that the discriminant ∆ of (3.42) vanishes, In Fig. 2 we present some plots of the exact solution and compare them with the power expansions (3.26). For the particular choice ε 0 = 1 and q 0 = 0.6 q extr. 0 the difference between both curves is clearly visible. The function a 0 (v) by definition has a singularity on the horizon, as can be seen in Fig.2(a) dimensions. Near the horizon it approaches some finite constant value µ 0 related to the chemical potential as which confirms the scaling behaviour (2.10).
In summary, the zeroth-order solution b 0 (v) is given by with v + and U(p) given by (3.44) and (3.45), respectively. a 0 (v) and d 0 (v) are obtained by substituting b 0 (v) in (3.25) and (3.23).

Fefferman-Graham vs. Eddington-Finkelstein coordinates
The zeroth-order solution in Fefferman-Graham (FG) coordinates can be related to that in Eddington-Finkelstein (EF) coordinates by the coordinate transformatioñ Transforming the Eddington-Finkelstein metric (2.14)-(2.18) and comparing the result with the power expansion (3.26), we find Comparing also the bound (3.48) with that in Eddington-Finkelstein coordinates given by (2.24), we find some relation between ε 0 and m 0 using (3.52): For q 0 = 0 this relation can be easily checked by our solution with the general form of the metric in [2,5,12]. In this case, we find that our solution (3.50) reduces to and similarly a 0 (v), such that We also note that the transformation (3.51) relates the outer horizons, ζ + in EF coordinates and p + in FG coordinates, as ζ + = √ p + . The chemical potential µ 0 can therefore be written as a function of ε 0 and q 0 . Using (2.12) and (3.52), we find This dependence is shown in Fig. 3 for some particular values of ε 0 . Using this expression with the definition (3.42) for p + and substituting there the maximal value for q 0 (3.48) we find the following bound for the chemical potential: µ 0 (ε 0 , q 0 ) ≤ µ extr. This is not in contradiction with our earlier statement that the disappearance of the horzion does not impose a bound onμ 0 /Λ. Note that if we identify ε 0 with m 0 as in (3.53), then ε 0 explicitly depends onμ 0 = µ 0 and therefore (3.57) is not a bound onμ 0 /Λ.

Conclusions
We constructed a natural extension of the late-time boost-invariant background found in [2] (and [5]- [12]) to a background dual to an expanding N = 4 plasma with chemical potential. The solution we found depends on two parameters, the chemical potentialμ 0 and temperature scale Λ, which are encoded in the mass parameter m 0 and chargeq 0 of a time-dependent AdS Reissner-Nordström-like solution. In Eddington-Finkelstein coordinates the first-order solution is given by the expansion (2.14)-(2.18), with the zeroth-order and first-order coefficients given by (2.19) and (2.22), respectively. We showed that the viscosity of the boundary theory computed from the time-dependent solution is in agreement with that in [15]. We also constructed a zeroth-order solution in Fefferman-Graham coordinates, which we presented in parametric form, see the general ansatz (3.1) and (3.2) with (3.50). FG coordinates may be the preferred choice, when strings [28] or branes [29,30] are embedded into the geometry. Finally, we found the coordinate transformation which maps the zeroth-order solution in FG coordinates to that in EF coordinates. We argued in several ways that the charge density behaves like τ −1 at all times. Unlike the energy density, it can not be chosen freely at early times. This is basically because the charge density behaves like τ −1 at large τ , see e.g. (2.36) or (3.13), and higher-order corrections are absent. It also follows directly from the equations of motion, see (3.7) which holds for all times. Naive extrapolation to early times shows a singularity in the gauge field at τ = 0. However, this does not signal a breakdown of the hydrodynamic approximation since the total charge is constant at all times and therefore regular even at τ = 0 (see footnote 5 on p. 10).
A possible application of the background, when appropriately modified and extended, could be the chiral magnetic effect (CME) [31]. The CME states that, in the presence of a magnetic field and non-zero chiral chemical potential, an electromagnetic current of the type J ∝ µ 5 B is generated in the plasma. The CME is a non-equilibrium process and requires the introduction of gauge fields with (time-dependent) spatial components. For instance, for an electric field E, one needs to introduce the spatial component A 3 = A 3 (τ, r) with boundary condition A 3 → τ E at r → ∞. The time-dependence of the gauge field reflects non-equilibrium physics and requires the back-reaction on the geometry, unless one keeps it infinitesimal [32] (see also [33,34] for an AdS/CFT approach to the CME). In this case one would also obtain higher-order corrections to the charge density due the effects of the Chern-Simons term, which are absent in our solution. An attempt to include an E field in the dual of an electrified plasma (without chemical potential) has been made in [17].
Finally, it would be interesting to find a numerical solution of our backgroundà la Chesler and Yaffe [4] which would hold beyond the hydrodynamic regime.
where we put (4) to covariant derivatives associated with the induced metric, while the derivatives on the right-hand side are defined with respect to the 5-metric. We also define a scalar K = K µν g µν = K µν γ µν , which is used in the Gibbons-Hawking-York part of (2.31).