Asymptotically AdS Solutions of Five Dimensional Gravity-Dilaton Theory

We search for Asymptotically AdS solutions of the background metric in which dilaton back reacts to gravity in five-dimensional gravity-dilaton theory. The five-dimensional gravity-dilaton theory generally appears in the context of the low energy effective action of closed string theory in the Einstein frame. In particular, we consider dilaton which are minimally coupled to gravity in which the potential for dilation is taken to be simple and contain only one exponential term parametrized by a constant a. We solve analytically and show for a simple constant dilaton potential it appears there are no black hole solutions if we turn on the dilaton contribution. On the other hand, the exponential dilaton potential has black hole solutions but they are not in general Asymptotically AdS. We argue that there are some possible Asymptotically AdS black hole solutions in the range of

produced at RHIC and LHC [9]. Then, we generalize the dilaton potential to an exponential form that still inherits the cosmological constant term of the previous case. We will show that there exist some black hole solutions in this case.

From String to Einstein Frame
Consider the following action in the string frame 1 We can go from the string frame to the Einstein frame using a Weyl transformation g E µν = e − 4 D−2 Φ g µν , such that, in the Einstein frame, the action (1) becomes Assume that we can find all classical solution of the fields except for gravity and dilaton. Plugging back those solutions into the action, we can write the effective action in terms of gravity and dilaton as 2 In our case, we will just consider the potential which gives asymptotic AdS solution to the metric. Therefore we take the potential to be of the form V = 12 b 2 + O(Φ), where b is meant to be the AdS radius.

BPS-like Forms
The method that we are going to use here is based on the one used by Miller et al [3]. It is basically using BPS-like equations which are in first order differential equations to solve equations of motion which normally take the form of second order differential equations. Despite the fact that BPS-like equations are simpler to solve, it is not an easy job to get the BPS-like form from the action. One has to able to write all terms in the action as a sum of squares plus a total derivative then one can trust their BPS-like solutions to satisfy the equations of motion. Now, we are going to show how do these BPS-like forms satisfy the equations of motion. Lets, consider a theory in D-dimensions with N fields ϕ i and assume that we can write its action into BPS-like forms plus a total derivative terms, where A i are the BPS-like forms, B is the total derivative terms, and ⃗ ϕ ≡ (ϕ 1 , . . . , ϕ N ). Neglecting the boundary terms, we can find the equations of motions by taking a variation of the action above to zero, Here, we can see the simplest solution of these BPS-like forms that satisfy the equations of motion is when ⃗ A[ ⃗ ϕ, ⃗ ∂ϕ] = 0 and these are our BPS-like equations that we are going to use in many of our calculations. 1 Here, we only consider fields (gravity, dilaton, and U (1) gauge field) that are needed in our calculation. 2 We have scaled the dilaton Φ → √ D−2 8 Φ and assumed that the potential term is just a function of dilaton.

Constant Dilaton Potential
As a first attempt, we take the dilaton potential to be a constant in order to make sure that near the boundary (infinity) the resulting metric is Asymptotically AdS. Let us consider the action that is given by the effective five-dimensional action of gravity-dilaton, with cosmological constant term, We write the following general ansatz for the metric and dilaton in Poincare patch coordinates of AdS that preserve less symmetry(three-dimensional rotation) than four-dimensional Lorentz invariance, with u is the fifth dimension. Substituting these ansatz back to the action (6), we obtain an effective (1 + 1)-dimensional action, after factoring out an overall three-dimensional volume, with the effective Lagrangian is then given by ) .
(9) Note that there is no n ′ 1 -quadratic terms in the Lagrangian, so we can eliminate it by a field redefinition, or equivalently taking a coordinate transformation of u. In order to remove the overall exponential factor in the first integral of the right hand side effective action (9), we choose n 1 = 2n 3 . It turns out to be much better if we would rather substitute n 3 = 1 2 n 1 and get a nicer Lagrangian, Here the degree of freedom is n 1 instead of n 3 . From now on, we will neglect the boundary terms.

Non-Black Hole Solutions
For non-black hole solutions, we make an identification n 1 = 3n 4 to preserve four-dimensional Lorentz invariance and so the metric becomes It is clear that this metric, parametrized by one field n 4 , does not have characteristic of black hole solutions.

Trivial Dilaton Solution
For trivial dilaton solution, we may write the effective Lagrangian in BPS-like forms as follows with BPS-like equations are given by where the solution for dilaton is just a constant. In order to get an AdS solution, we make another identification such that e 2n 4 = b 2 r 2 and the first BPS-like equation of (13) becomes So, the solution for the metric is just the usual AdS metric . (15)

Non-trivial Dilaton Solution
The non-trivial dilaton solution can be obtained by setting the solution of dilaton as a linear function of u. This is true because from the effective Lagrangian (10) we can derive the equation of motion of dilaton that is a linear function of coordinate u such that the first derivative of dilaton is a constant C, Φ ′ = C. To get the BPS-like forms, we can not just substitute this constant into the effective Lagrangian 3 since this will not satisfy the original second order differential equations of motion. What we should do is writing the effective Lagrangian in BPS-like forms while keeping one of its BPS-equations to be the same as the equation of motion of dilaton. The satisfying effective Lagrangian is given by Here we assume the last term in the first integral can be written as an additional contribution to the total derivative term 4 . The BPS-like equations are given by Now, we choose an identification such that the metric becomes asymptoticaly AdS and the choice is e 4n 4 du = b r dr. The first BPS-like equation of (17) has solution where c 1 is an integration constant. The fact that we have two solutions of the first order differential equation might be a reminiscence of second order differential equation of motion.
The explicit form of u in term of r is 3 One may think that there is no difference by doing so but actually it will give at least a different sign in the Lagrangian. 4 In more detail, one can confirm with a paper by Miller et al [3] in the section where they discuss about nonextremal BPS solutions.
C , we are celebrating the Csaki-Recee solution [4]. In this solution, we can see clearly how dilaton changes the metric solution and indeed if we turn-off the dilaton by setting C = 0, we get back the usual AdS solution.

Black Hole Solutions
For black hole solutions, we need to use some general form of a asymptotic AdS black hole solution as the following [5] with f (r) and g(r) are regular function and f (0) = g(0) = 1 and also g(r) is zero at some finite r > 0. So, the metric ansatz (7) can be rewritten to mimic the general black hole from as below Here we can identify e n 1 −n 4 = b 2 r 2 f (r) and e 2n 1 du 2 = e n 1 −3n 4 dr 2 , with e 3n 4 −n 1 = g(r).

Trivial Dilaton Solution
The effective Lagrangian (10) can be written in the BPS-like forms The BPS-like equations are Define p = n 1 + n 4 and q = n 1 − n 4 , so that the first two BPS-like equations of (24) are now The first BPS-like equation of (24) can be solved as with C is an integration constant. Now, let us identify e q = b 2 r 2 so that the metric can be written in terms of functions p and q as follows ds 2 = −e p−q dt 2 + e q dx 2 3 + e 2q+p du 2 = − ( 1 , the metric becomes This metric has a horizon at r 4 h = − b 2 C , with C < 0 and this is the usual AdS-Schwarzschild(black hole) solution after rescaling the time coordinate.

Non-trivial dilaton solution
We can get another solution by redefining 8p = 3n 1 − n 4 and 8q = n 1 − 3n 4 and writing the Lagrangian (10), in BPS-like form, as the following where A, B and C are real constant and they are constrained by C 2 = A 2 + B 2 . The BPS-like equations are given by The metric can be written in terms of p and q, Now, we have to find an identification that gives us asymptotic AdS metric with an event horizon. The easiest way to do it is by identifying e 8p = f (r). So from the first BPS-like equation of (30), we obtain As a simple case take f (r) = b 8 r 8 which give solutions Here we have fixed all constants to give us asymptotic AdS solution (15) near r → 0. Curvature scalar in terms of u coordinate, using the BPS-like equations, is given by Here, we can see that naked singularity happens when f (r) = 0, or in our example is at r → ∞, and so at r 8 = − 24b 6 C 2 we have a horizon, with C 2 < 0 if we define 0 ≤ r < ∞. If we turn off the dilaton contribution, A = 0, we will get back the negative curvature scalar of AdS metric 5 . We can also see that the first term in the curvature scalar is coming from the matter contribution and in this case is the dilaton. This solution in some sence is not a good one because it gives us imaginary number in C if we expect the horizon to be at some finite positive number of r.
The difficulty of finding a correct solutions for metric and dilaton in terms of real functions could arise as the fact that in the corresponding non-black hole solution we obtain a naked singularity at some finite r. In general, we will show that in the black hole solutions even though there is a singularity at some finite r, this singularity turns out to be a naked singularity instead of a horizon. The ansatz for a black hole (31) can be written as From general form of asymptotic AdS black hole (21), if there is a horizon at some finite r = r H then we would expect that e −8q(r H ) = 0 or u = ±∞, from the second BPS-like solution(30). The first BPS-like equation (30) will give us and this function is infinite at u = ±∞. So, as we can see from (34) at r = r H , the curvature scalar will blow up and we can conclude that this singularity is a naked singularity. A more detail study on the non-existence of black hole solutions in this theory can be read in [13].

Exponential Dilaton Potential
In general, we can add more dilaton potential terms into the action (6). Dilaton potential usually comes in terms of exponential form. Here, we are going to consider the action with one exponential form, where a is a constant real number.

Non-Black Hole Solutions
Using the non-black hole ansatz (11), the effective Lagrangian becomes  to dilaton potential such that a ̸ = ± 4 √ 6 . Substituting those fields into the action, we obtain a simple action in BPS-like form where C is a constant. The BPS-like equations are and the solutions in terms of u coordinate are given by where c 1 and c 2 are constant. So, the solution of the original fields (dilaton and metric) is given by One can check that this solution has no naked singularity in the curvature scalar with restriction a > 4 √ 6 and C > 0 or a < − 4 √ 6 and C < 0. This is because the solution has a nice finite secant hyperbolic function, in coordinate u, unlike sinus hyperbolic function of previous Csaki-Reece solution. This could also be a sign the existence of black hole solutions.

Black Hole Solutions
Defining 8p = 3n 1 − n 4 and 8q = n 1 − 3n 4 and using the black hole ansatz (21) and (22), we can rewrite the effective Lagrangian to be Using another fields definition, l = 8p + aΦ and m = m 1 p + m 2 Φ, the effective Lagrangian written in BPS-like form now becomes The solution to (46) in terms of u coordinate is given by (48) Let us compute the area of horizon, which can be read off from the metric, In order to get a finite horizon area, at u = ∞, we impose the last exponential term equal to one by setting B = − a √ 6 4 C and C = 8A, with C, a > 0. Unfortunately, we will show later on, this solution does not give an AdS metric at u = 0 unless a = 0. For the moment, we would expect that the area is infinity at u = 0 where in this case we fix the constant e −Cbc 1 = −1 and a 2 < 8 3 hence the horizon area becomes where f (r) is a regular function at the horizon and equal to one at r = 0. So the entropy density and temperature for this solution are given by where A > 0. Now, let compute curvature scalar for this solution. We can write p in terms of q, with c 3 = 0 for simplification, and its derivatives From formula (34), we can write curvature scalar, while at u = 0 one can show the curvature scalar vanishes for a ̸ = 0 means that the solution is a black hole but not asymptotically AdS. If a = 0, the curvature scalar will be finite both at u = 0 and at u → ∞ which gives us back AdS-Schwarzchild. To find the exact form of this black hole, we need to use identification (50) such that we can write u in terms of r, The metric is given by and solution for dilaton is We can also see clearly from dilaton profile that dilaton is infinity near r = 0 which is not expected for asymptotically AdS solution unless a = 0. Recall that the calculation is done for c 3 = 0. If c 3 ̸ = 0 the curvature scalar will not be zero and, so there might be AdS black hole solutions with some appropriate fixed remaining constants.

Conclusion
We have shown how to construct BPS-like equations for five dimensional gravity-dilaton system in which the dilaton potential is an exponential function parametrized by a constant a. For non-black hole solutions, we obtained the range value of a is |a| > 4/ √ 6. It turned out that there are possible black hole solutions for a in the range |a| < 4/ √ 6. This in accordance with the black hole solution of the constant dilaton potential, namely AdS-Schwarzschild black hole, in which a = 0. We argue there might be some AdS black hole solutions if one of the constants in the solutions is non-zero, or to be precise c 3 ̸ = 0.