Dilatonic black holes in superstring gravity

We solve the dilaton field equation in the background of a spherically symmetric black hole in type II superstring theory with $\alpha'^3$ corrections in arbitrary $d$ spacetime dimensions. We then apply this result to obtain a spherically symmetric black hole solution with dilatonic charge and $\alpha'^3$ corrections in superstring theory compactified on a torus. For this black hole we obtain its free energy, entropy, temperature, specific heat and mass.


Introduction
A frequently considered effect of string theory is the result of corrections in the inverse string tension (α ′ ) in the form of higher-derivative terms in the effective action. Curvature-squared corrections (first order in α ′ ) to spherically symmetric d−dimensional black holes in string theory were first obtained in [1]. Higher order corrections, quartic in the Riemann tensor (third order in α ′ ) to the same black holes were obtained in [2].
Article [3] deals with the effects of string compactification on a torus from 10 (or 26) to arbitrary d dimensions on spherically symmetric black holes with corrections of order α ′ . In this article, we study the same effect on the same type of black holes but with α ′3 corrections, as those studied in [2]. In such compactification, one must pass from the string to the Einstein frame, by a conformal transformation involving the dilaton field. Therefore we must determine the solution to the dilaton in the background of a spherically symmetric black hole.
The article is organized as follows: in section 2, we will solve the dilaton field equation, in the background of a spherically symmetric black hole in d dimensions, in the presence of curvature corrections of order α ′3 . In section 3 we revise the α ′3 −corrected noncompactified solution of [2] and we compute some of its thermodynamical properties (free energy, entropy, temperature, specific heat and mass); a few of which had not been previously computed. In section 4 we present the calculations leading to the α ′3 −corrected d−dimensional solution compactified on a torus. Finally we derive the same thermodynamical properties as before, and we compare the results of the two solutions. 2 The dilaton in the background of a d−dimensional black hole with α ′3 corrections The most general static, spherically symmetric metric in d spacetime dimensions can be written in spherical coordinates as (1) f, g are arbitrary functions of the radius r; d Ω 2 i is the element of solid angle in the (d − 2)−sphere. For pure Einstein-Hilbert gravity in vacuum, the solution to the Einstein equations is [4] R H being the horizon radius. This is the d−dimensional extension of Schwarzschild's solution.
We are interested in extending this solution in the presence of a dilaton, but considering stringtheoretical α ′ corrections. The effective action we are thus considering, in the Einstein frame, is Here, Y (R) is a scalar polynomial in the Riemann tensor representing the leading higher derivative string corrections to the metric tensor field, and λ ′ is, up to a numerical factor, the suitable power of the inverse string tension α ′ for Y (R). The dilaton field is φ, and w is the conformal weight of Y (R), with the convention that w (g µν ) = +1 and w (g µν ) = −1. L matter contains terms, up to the same order in α ′ , including other fields than the metric and the dilaton, depending on the type of superstring theory we are considering. In this article we are only considering gravitational α ′ corrections: we can therefore consistently settle L matter to zero. From (3), and defining T µν = δY (R) δg µν , the equations of motion for the dilaton and metric fields are In both equations above we have eliminated certain terms involving powers of φ (namely, we have omitted the factor e 4 d−2 (1+w)φ , taking it equal to 1) , since those terms would only contribute at higher orders in our perturbative parameter λ ′ .
In this article we are focusing in particular in curvature corrections of order α ′3 , which are present in general in string theories, and in particular are the leading corrections in type II superstring effective actions [5]. In this case, λ ′ = ζ (3) 16 α ′3 , ζ(s) being the Riemann zeta-function, and We are interested in computing the first λ ′ corrections to φ and g µν , using (4) and (5), taking (1) with f (r) = g(r) as the λ ′ = 0 metric and working perturbatively in λ ′ . In particular we take the λ ′ = 0 metric in order to compute Y (R), since this term is already multiplied by λ ′ , obtaining We also have Putting everything together, replacing f (r) by (2) and defining we get for (4) We simply integrate this equation, obtaining The integration constant Σ, as will become clear below, is the dilatonic charge. Integrating again, defining the incomplete Euler beta function as B(x; a, b) = x 0 t a−1 (1 − t) b−1 dt, and further defining from now on z = ζ(3) At the horizon one only has a coordinate (but not curvature) singularity. From (4), this means that also φ(r) and φ ′ (r) must be nonsingular at R H . From (10) we see that, in order to avoid φ ′ becoming infinite at r = R H , Σ must take a precise value, given by Equation (11) with Σ given by (12) is the solution for the dilaton in the background of a spherically symmetric black hole with α ′3 corrections in d dimensions. This dilaton solution acts as secondary hair, since it does not introduce any new physical parameter besides the ones of the black hole. While integrating (10), we chose the integration constant so that, at asymptotic infinity, the dilaton vanishes. For large r, φ is approximately given by 1 This solution may also be expressed in terms of the hypergeometric function 2 F 1 , since the relation Taking the first term in the series from the logarithm in (13), φ(r) ≈ Σ r d−3 , one can verify that Σ has the meaning of a dilatonic charge.
At the horizon, φ is indeed regular and given by 2 From (10) and (12), the derivative of the dilaton field is given by Since B d > 0 for d > 4, we see by inspection that φ ′ (r) is a strictly positive function for r > R H ; we conclude that, outside the horizon, φ grows between φ (R H ) given by (14) and 0, its value at infinity. Comparing to the result for the dilaton in the same background but with α ′ corrections obtained in [3], one can find the same leading term − Σ , obtained in the same way (an integration constant after the first integration of (4)), but with a different value of the dilaton charge Σ. It was also found a dependence on the incomplete beta function, but with a different argument: The α ′ corrections considered in [3] were of the same form as (3) 3 The α ′3 −corrected spherically symmetric black hole After having obtained the α ′3 −corrected dilaton, it would be interesting to obtain the α ′3 −corrections to the black hole solution to which it couples. A metric like that would be of the form (1), with The factor 1 − RH r d−3 represents the α ′ = 0 Tangherlini solution (2); the functions f c (r), g c (r), to be determined, encode the z corrections.
That solution for the lagrangian we are considering was obtained in [2], in a system of coordinates such that the horizon radius R H is fixed and has no α ′ corrections. The result is of the form (15), 2 The digamma function is given by ψ(z) = Γ ′ (z)/Γ(z), Γ(z) being the usual Γ function. For positive n, one defines ψ (n) (z) = d n ψ(z)/d z n . This definition can be extended for other values of n by fractional calculus analytic continuation. These are meromorphic functions of z with no branch cut discontinuities.
γ is Euler's constant, defined by γ = limn→∞ n k=1 In the same article, the α ′3 −corrected black hole mass was obtained as The temperature of a black hole of the form (15) is obtained, to first order in z, from In our case, the result is given by We have checked that, for every relevant value of d, E d > 0 and F d > 0. Therefore, we conclude that the α ′3 corrections to the mass and temperature are negative.
The black hole entropy for this solution wasn't studied in [2], but it can be obtained using Wald's formula [7] where H is the black hole horizon, with area A H = R d−2 H Ω d−2 and metric h ij induced by the spacetime metric g µν . For the metric (1), the nonzero components of the binormal ε µν to H are ε tr = −ε rt = − g f . From (3) one also needs This way, taking only nonzero components, one gets from (1) 8πG The λ ′ −corrected term must be evaluated at order λ ′ = 0. At this order φ = 0, f = g and 2

RH r
3d−6 . Therefore one gets for the entropy One can obtain the black hole free energy through the relation F = M − T S : We have checked that, for every relevant value of d, G d > 0. Therefore, we conclude that the α ′3 corrections to the entropy and free energy are positive. The black hole specific heat is given by C = T ∂ S ∂ T . Since for the expressions we obtained only the horizon radius is a variable, we may fully express T and S as functions of R H and vice-versa (including the contribution from z = λ ′ R 6 H ). We then have In our case, we get We see that in the presence of α ′3 corrections this black hole keeps being thermodinamically unstable. When one talks about a black hole in string theory in d dimensions, the original D-dimensional spacetime must have been compactified on some (D − d)-dimensional manifold, with internal coordinates y m and internal metric g mn (y). When passing from the string to the Einstein frame, one needs a transformation under which If one takes this as a conformal transformation of the entire D−dimensional metric (rather than just on the d−dimensional black hole part, as it was done in [2]), it involves the total dilaton field Φ, including the Kaluza-Klein part depending on the internal coordinates y m (rather than just the d−dimensional part φ as we have been considering). This way the size of the compact space becomes spatially varying, being governed by a function h. Since, for the cases we have been considering, the dilaton field depends only on the radial coordinate r, the same is to be expected for the function h. The complete line element is then the sum of the d−dimensional black hole (1) and the compact space: This metric is a solution of the metric field equation (5) for the whole spacetime, in D dimensions, which we write as From (29), the compact space and the black hole cannot be decoupled in general: the respective curvatures appear combined on the terms depending on Y (R) and T µν . In order to avoid this problem, we take the internal space to be a flat torus, with vanishing internal curvature. Also since, for the cases we have been considering, the dilaton field depends only on the radial coordinate r, the same is to be expected for the function h. At order λ ′ = 0, φ = 0 and the conformal transformation (27) is just the identity. This means one should then have g mn (y) = δ mn , h(r) = 1 + 2λ ′ ρ(r).
One must now determine the function ρ(r). By contracting (29) with the D−dimensional metric, one finds the D−dimensional Ricci scalar R D = D µ,ν=1 R µν g µν : But if one rather contracts (29) with just the d−dimensional part of the metric (28), i.e. the black hole metric (1), one obtains R d = d µ,ν=1 R µν g µν : Equations (31) and (32) were obtained from the field equation (29). But one can take directly the D−dimensional metric (28), with the specifications (30), and compute its corresponding Ricci tensor R µν . One can then contract it with the whole metric, obtaining the Ricci scalar R D , or just with the d−dimensional black hole part, obtaining R d . Proceeding this way, one verifies that Combining (31), (32) and (33), we conclude that ρ must satisfy the equation Y (R) and T ρ ρ should be evaluated with the λ ′ = 0 metric (1), with f = g given by (2). Y (R) is given by (6) or (7), before or after replacing the metric. The explicit expression for T µν can be found in article [6]; we choose not to repeat it here, since it is very long and all that we need is the result for T ρ ρ .
After integrating (34) and requiring ρ to be finite at the horizon (a similar procedure to the one taken to obtain (10)), we are left with Equation (35) can be integrated, to finally obtain with φ(r) given by (11). We now proceed to determine the influence of the internal compact space (the torus) on the d−dimensional black hole geometry. There are two nontrivial components of the field equation (29), corresponding to R tt and R rr [1,2]. We use these two equations in order to obtain the two unknown functions µ(r), ε(r) in (15). For ε(r) we obtain the same result as (16), while µ(r) is now given by Metric (1), with f (r), g(r) of the form (15), ε(r) given by (16) and µ(r) given by (38), corresponds to the d−dimensional black hole solution in the presence of a (D − d)− dimensional compact torus we have been looking for.

Thermodynamical properties
In this section, we compute several thermodynamical quantities for the black hole solution we have just found. In each case we compare the result to the corresponding one of the noncompactified solution obtained in [2], since the parameters are the same. This way we can evaluate the physical effects introduced by the toroidal compactification.
The entropy of this black hole solution can be obtained by Wald's formula (22). It is clear from this formula that the λ ′ -correction to the entropy depends only on the λ ′ = 0 part of the metric. Since this part of the metric is the same for the cases we considered, the result for the entropy does not change: it is given by (23).
The free energy of a black hole solution is obtained from the euclideanized action (3), to which one adds a surface term consisting of an integral (on the boundary) of the trace of the second fundamental form, subtracted by the same trace for the boundary embedded on flat space, to render the total surface contribution finite. This surface term also includes contributions for the higher-derivative terms, but these contributions do not affect this calculation [1]. Because we chose a system of coordinates such that R H does not get α ′ corrections, there are no implicit α ′ corrections: all the λ ′ -correction terms in the euclidean action are explicit and should be evaluated using the λ ′ = 0 part of the metric. This means that, just as it happened for the entropy, the result for the free energy for our solution is the same as that for the noncompactified solution obtained in [2], given by (24).
The black hole temperature is given by (20). Using (16) and (38), but also (14), (35), (37), we obtain with The black hole inertial mass is given by , while the gravitational mass is given by We have checked that, for every relevant value of d, ρ(R H ) − R H ρ ′ (R H ) > 0. Therefore, we conclude that also for this solution the α ′3 corrections to the mass and temperature are negative. The magnitude of these corrections is larger than the one of the noncompactified solution of [2].
The black hole specific heat is given by (25). In our case, we get We see that in the presence of α ′3 corrections this black hole keeps being thermodinamically unstable.

Conclusions
In this work, we derived the spherically symmetric solution to a dilaton in the presence of a black hole in string theory with α ′3 gravitational corrections in d spacetime dimensions. We then obtained a spherically symmetric black hole solution with α ′3 corrections from compactified string theory in d dimensions, and we computed its free energy, entropy, temperature, specific heat and mass. We compared the magnitude of the α ′ corrections to these quantities to the ones corresponding to the noncompactified solution obtained in [2], in order to estimate the effects of string compactification. The α ′3 corrections to the free energy and entropy are positive; the magnitude of the corrections is the same in both solutions. The α ′3 corrections to the mass and the temperature, on the other hand, are negative for both solutions; in both cases, their effect is strengthened for the compactified solution we obtained, when compared to that of the noncompactified one. Both solutions are thermodinamically unstable, like the classical Tangherlini black hole. Overall we conclude that the α ′3 corrections do not qualitatively change the thermodynamical properties of these black holes.
In a future work we plan to study some other features of the black hole solution we obtained, like its stability under perturbations of the metric and quasinormal modes.