A A bit of intuition from Newtonian gravity

The influence of D-branes’ backreaction upon gravitational interactions between open strings

Abstract

We argue that gravitational interactions between open strings ending on D3-branes are largely shaped by the D3-branes’ backreaction. To this end we consider classical open strings coupled to general relativity in Poincaré backgrounds. We compute the linear gravitational backreaction of a static string extending up to the Poincaré horizon, and deduce the potential energy between two such strings. If spacetime is non-compact, we find that the gravitational potential energy between parallel open strings is independent of the string’s inertial masses and goes like at large distance . If the space transverse to the D3-branes is suitably compactified, a collective mode of the graviton propagates usual four-dimensional gravity. In that case the backreaction of the D3-branes induces a correction to the Newtonian potential energy that violates the equivalence principle. The observed enhancement of the gravitational attraction at arbitrary large distances is specific to string theory; there is no similar effect for point-particles.

Theoretische Natuurkunde, Vrije Universiteit Brussel and

The International Solvay Institutes,

Pleinlaan 2, B-1050 Brussels, Belgium

raphael.benichou@vub.ac.be

1 Introduction

D-branes [1] have played an important role in the recent developments of string theory. D-branes can be described using either an open or a closed string language. In the open string sector D-branes are boundary conditions for the worldsheet conformal field theory. In the closed string sector D-branes are sources for the closed string fields, and consequently they are associated with solutions to the equations of motion of supergravity. In a regime in which the bulk closed strings decouple from the open strings living on the D-branes, these two points of view give equivalent descriptions of the same objects. This observation lead to the formulation of the AdS/CFT correspondence [2][3]. On the other hand when the open and closed strings sectors are coupled together we have to take into account both aspects of the D-branes physics.

D-branes are also useful objects for model-building in string theory. Generically at low energy the theory living on D-branes is a gauge theory. One can build D-branes configurations such that the resulting low-energy effective theory resembles the Standard Model of particles physics, or a supersymmetric extension thereof. This gauge theory is coupled to gravity which is mediated by closed strings living in the bulk. These models also offer an interesting perspective to address cosmological questions such as inflation.

Let us consider a configuration of D-branes arranged on a manifold . In some regime it may be legitimate to treat the D-branes as probes. Then the gravitational sector of the low-energy effective theory is (a supersymmetric extension of) general relativity living on , and coupled to the gauge theory. When the D-branes’ backreaction is taken into account this picture has to be modified. Indeed gravitons being elementary excitations of the bulk geometry, they are sensitive to the D-branes’ backreaction. The goal of this paper is to investigate how the gravitational sector of the low-energy effective theory is affected by the D-branes’ backreaction.

Let us introduce a motivation for this work. It is a question of importance to identify observable features of string theory that would not be reproduced by a theory of point particles. From the point of view of a distant observer, when a string approaches a gravitational horizon it becomes tensionless and its proper size grows. Thus in the neighborhood of horizons strings and point-particles should behave quite differently. D-branes are heavy objects that create a horizon. Open strings attached to D-branes stretch up to the D-branes horizon. Consequently their proper length is typically infinite. So we may expect that interactions mediated by bulk degrees of freedom have different manifestations for open strings and for point particles, even at macroscopic distances. The results we will derive indicate that it is indeed the case.

In this paper we focus on open strings attached to D3-branes spanning four-dimensional Minkowski spacetime. The space transverse to the D3-branes may be compact or non-compact; we will deal with both cases. D3-branes create a coordinate horizon. The classical gravity solution for D3-branes does not contain any curvature singularity and can be continued beyond the horizon [4]. Here we adopt the view that spacetime finishes at the horizon, where the D3-branes sit. Any closed string that crosses the horizon is turned into a set of open strings attached to the D3-branes (see e.g. [5] for an early development of this picture in the context of black holes thermodynamics). Open strings attached to the D3-branes extend up to the horizon, and their proper length is infinite (see section 3.1 for explicit computations).

The main result of this paper is the computation of the classical gravitational potential energy between two open strings attached to a stack of D3-branes. We use an effective field theory to describe gravity in the bulk. The open strings are described by the Nambu-Goto action and couple to gravity via their stress-energy tensor. As long as the open strings are not heavily excited their worldsheets are localized close to the D3-branes. We assume that the near-horizon geometry of the stack of D3-branes is the product of the Poincaré patch of with a compact manifold:

(1.1)

where is the radius. The D3-branes live at , where the metric develops a horizon. The prototypal example is the case of D3-branes located on a smooth ten-dimensional manifold. The compact space is then a five-sphere of radius , which takes the value:

(1.2)

where is the string coupling, is the string length squared and is the number of D3-branes. We assume that the radius is large with respect to the string scale . In this regime we can trust the field theory description of gravity. It is convenient to work with the coordinate , so that the metric of the Poincaré patch of is conformally flat1:

(1.3)

The horizon and the stack of D3-branes are now located at .

In order to simplify the computations we will first discard the compact factor in the metric (1.1) and work in five dimensions. In section 2 we describe linearized gravity in Poincaré . In section 3 we compute the linear gravitational backreaction of an open string that stretches up to the Poincaré horizon. In section 4 we evaluate the gravitational potential energy between two such strings. We show that this potential energy is independent of the open strings’ inertial masses and goes at large distance like , where is the distance between the two strings as measured by a Minkowski observer. We propose the following interpretation for this result: the dominant contribution to the gravitational interaction comes from the parts of the open strings located very close to the horizon, in a region where the geodesic distance between the strings is small and the effect of curvature is negligible. Accordingly the open strings interact like infinite strings in five-dimensional flat space.

In section 5 we discuss the influence of the compact factor in the metric (1.1). In section 6 we generalize our results to compact spacetimes. Eventually section 7 contains a summary and a discussion of the results, as well as directions for future research.

2 Linearized gravity in Poincaré

2.1 Linearized Einstein’s equations

The Poincaré metric (1.3) satisfies Einstein’s equations in the vacuum with appropriate negative cosmological constant. In this section we study the linearized Einstein’s equations for a small perturbation of the metric (1.3). We follow the analysis of [6] (see also [7]). We work with the action:

(2.1)

where is the five-dimensional Newton’s constant, is the Ricci scalar, is the cosmological constant and is the Lagrangian describing the matter sources.

Let us introduce a small perturbation in the metric (1.3). We can always choose a coordinate system such that the perturbed metric takes the form :

(2.2)

We treat the metric perturbation as a small fluctuation. Linearized Einstein’s equations read :

(2.3)
(2.4)
(2.5)

where the indices are lowered and raised with the -dimensional Minkowski metric , , and and are respectively the linearized part of the Ricci scalar and Einstein’s tensor built with the -dimensional metric .

We will now show that we can essentially reduce the linearized Einstein’s equations (2.3) to a scalar wave equation in . First let us take the trace of the last line in (2.3) and combine it with the first line to eliminate the Ricci scalar. We obtain:

(2.6)

This equation is easily integrated to get , which can be set to zero outside the sources with a reparametrization of the form:

(2.7)

Such a reparametrization preserves the form of the metric (2.2). Next let us define as:

(2.8)

The second line in (2.3) is then:

(2.9)

This equation is in turn straightforwardly integrated to obtain , which can also be set to zero outside the sources with a reparametrization of the form:

(2.10)

Eventually the last line in (2.3) reads :

(2.11)

where is the anti-de Sitter Laplacian. The right-hand side is known explicitly in terms of the matter stress-energy tensor thanks to equations (2.6) and (2.9), together with the gauge choice that both and vanish away from the sources.

2.2 Green functions

In this section we construct a Green function that allows for the integration of equation (2.11) (see for instance [6][7][8][20] for similar discussions). Let us introduce the functions defined as:

(2.12)

The virtue of this field redefinition is that it transforms the wave equation into a Schrödinger-like equation:

(2.13)

where is the following linear differential operator:

(2.14)

So equation (2.11) can be rewritten as:

(2.15)

Let us introduce the Green function for the operator :

(2.16)

This Green function can be constructed from a complete set of normalized eigenfunctions (see e.g. [9]). The eigenfunctions of the operator can be written in terms of Bessel functions of the first and second kinds. They read:

(2.17)

The functions (2.17) are delta-function normalizable as goes to infinity, i.e. close to the horizon. In this region they can be approximated by trigonometric functions:

(2.18)

Near the AdS boundary at the functions are not normalizable2. We deduce that the Green function is:

(2.19)

To study time-independent configurations it is convenient to introduce the time-integrated Green function:

(2.20)

where we introduced . In a Newtonian regime, the time-integrated Green-function is proportional to the potential energy between two static point-like sources. This potential energy can be understood from a four-dimensional perspective as a sum of contributions from the Kaluza-Klein modes of the bulk graviton field. Let us briefly discuss the asymptotic -dependence of the time-integrated Green-function, for . At large distances, i.e. for , we have:

(2.21)

This is the same behavior as for the potential energy associated with a massless interaction in ten-dimensional flat spacetime. On the other-hand, at small distances () we have:

(2.22)

This is the same behavior as for a massless interaction in five-dimensional flat spacetime. This is to be expected since at small geodesic distances with respect to the radius, the effect of the curvature becomes negligible.

3 Gravitational backreaction of open strings

In this section we consider a static open string ending on a stack of D3-branes. We will compute the linear backreaction of this string on spacetime geometry using the results of the previous section. Similar computations can be found in [6][7].

3.1 Nambu-Goto open string in Poincaré

We consider a classical string living in the Poincaré patch of . The string is described by the Nambu-Goto action :

(3.1)

We want to study open strings attached to D3-branes, so we will consider solutions of the Nambu-Goto equations of motion that extend from up to the Poincaré horizon at . We can think of such a string as being stretched between a large stack of D3-branes at , and a parallel probe D3-brane located at .

For simplicity we study static configurations only. It is straightforward to check that the following string satisfies the Nambu-Goto equations of motion:

(3.2)

with running over the interval . This string is represented in Figure 1. On this solution the Nambu-Goto action evaluates to:

(3.3)

where is the inertial mass of the string:

(3.4)

We define the length of the string (3.2) as the proper length of the (one-dimensional) intersection between the worldsheet and a spacelike hypersurface. A natural slicing of spacetime is given by the hypersurfaces normal to the timelike Killing vector of the D3-branes background: these are the constant- hypersurfaces. With this natural definition the length of the string is infinite:

(3.5)

The effective string tension is redshifted towards the horizon, which renders the mass of the string finite even if its length is infinite.

Figure 1: The classical string defined in (3.2). It extends from up to a stack of D3-branes located at .

If both ends of a static string are attached to the stack of D3-branes at , the Nambu-Goto equations of motion require that the entire worldsheet be located on the horizon. It is reasonable to expect qualitative agreement between the gravitational behavior of the string defined in (3.2), and an oscillating string of energy with both endpoints attached to the stack of D3-branes.

The stress-energy tensor

The Nambu-Goto action can be written as the integral over spacetime of a Lagrangian density:

(3.6)

The string stress-energy tensor is evaluated as:

(3.7)

For the open string defined in (3.2) with , the non-zero components of the stress-energy tensor are:

(3.8)

Since spacetime admits a timelike Killing vector , we can construct a covariantly constant mass-energy current . The mass of the string can be recovered as the flux of this current through a spacelike hypersurface.

3.2 Computation of the linear backreaction

We have now gathered all the ingredients needed to compute the linear gravitational backreaction of the string defined in (3.2). First we compute the various terms appearing on the right-hand side of equation (2.15). From equation (2.6) we deduce :

(3.9)

and from equation (2.9) we deduce :

(3.10)

The integration constants were chosen such that away from the string.

We can now integrate equation (2.15) to obtain :

(3.11)
(3.12)
(3.13)

Let us compute explicitly :

(3.14)
(3.15)
(3.16)

From the first line to the second we integrated by parts the derivatives with respect to the Minkowski coordinates, and used equation (2.16). From the second line to the third we integrated by parts the derivatives. Next we perform the integral over the Minkowski coordinates:

(3.17)
(3.18)
(3.19)

From the first line to the second we used expression (2.20) for the time-integrated Green function. We introduced which is the euclidean distance away from the string. To proceed further we have to evaluate an integral over the Bessel function . A primitive of this function can be written in terms of the following generalized hypergeometric function:

(3.20)

At large negative argument this function behaves like:

(3.21)

The integral relevant for our computation is:

(3.22)

The two terms in the previous equation give two contributions to the computation (3.17) of , that we denote respectively and . The first one can be explicitly evaluated as:

(3.23)

From now on we focus on the region of spacetime where , which will be the most interesting one for our purposes. In this domain, which is denoted as region 1 in Figure 2, simplifies to:

(3.24)

Notice that this result depends only on the large-argument behavior of the eigenfunctions of the operator , given in equation (2.18). In other words it relies only on the way the graviton wave-functions behave close to the stack of D3-branes. It is not sensitive to the geometry far away from the stack of D3-branes.

Now let us consider the second contribution to , namely . We can evaluate this contribution thanks to a small-argument expansion of the generalized hypergeometric function, keeping only the first term in the infinite sum (3.20). We find that gives a small correction to the previous result:

(3.25)

Let us briefly discuss this intermediate result. The first contribution is the result associated to an infinite string. The second term can be understood as a correction due to the fact that the string does not extend up to the boundary at , but ends up at . The previous equation states that this correction is irrelevant in the region , and the string behaves like an infinite string. This is actually expected: in this region the geodesic distance away from the string is small (even if is large), so the string appears as an infinite string in flat spacetime. The flat-space analysis of appendix A then predicts that the gravitational backreaction goes like . This is in agreement with the computation we just performed. Let us also mention that a point-like object would produce a different backreaction, which would go like or depending on whether the geodesic distance away from this object is respectively smaller or larger than the radius (see equations (2.21) and (2.22)).

From equations (3.24), (2.12), (2.8) and using the fact that is traceless, it is straightforward to deduce the linear backreaction of the string in the region :

(3.26)

The other components of the metric perturbation can be computed along the same lines, or deduced from using symmetry and the gauge choices we made. Let us stress again that this result is independent of the geometry far away from the D3-branes.

Linear perturbation theory breaks down as the metric perturbation becomes of order one. Actually the curve defined by gives an estimation of the location of the horizon created by the string. Notice that the intensity of the gravitational field at a given increases with : the redshift of the effective string tension towards the horizon is overwhelmed by the decrease of the geodesic distance away from the string.

Figure 2: Backreaction of an open string in . The string lies at and extends up to . In region 1 the decay of the gravitational field away from the string is slow.

In the other regions of spacetime the linear backreaction typically comes with a larger (negative) power of . For instance in the domain where (which is denoted as region 2 in Figure 2) equation (3.23) gives:

(3.27)

As previously gives a small correction:

(3.28)

These results follow from a small-argument expansion of the Bessel functions. Thus they depend on the geometry far away from the stack of D3-branes.

We will not try to evaluate the metric perturbation outside the domain where since it will not be needed for the computation we have in mind, namely the evaluation of the gravitational attraction between two open strings. Moreover the result would be sensitive to the details of the geometry far away from the D3-branes.

3.3 Deformation of the stack of D3-branes

Before we added the open string, the stack of D3-branes was located on the Poincaré horizon at . The string induces a deformation of the stack of D3-branes, which can be computed using the DBI action. This analysis was performed in [10] for a single D3-brane. The solution to the DBI equations of motion corresponding to a semi-infinite fundamental string ending on a single D3-brane reads:

(3.29)

where is the euclidean distance away from the string along the D3-brane, and is the radial coordinate away from the D3-brane (cf equation (1.1) and Figure 3). We identified with the excited scalar field living on the D3-brane worldvolume, in agreement with the analysis of [11] [12] where it is shown that the backreaction of D3-branes essentially does not modify the flat-space result derived in [10].

A generalization of this solution for a fundamental string ending on a stack of D3-branes can be obtained by assuming that the D3-branes are all equally deformed. Then the stack of D3-branes behaves effectively like a single D3-brane for which the tension has been multiplied by . It follows that the solution we are looking for is the same as , up to a normalization. We can fix this normalization for instance by demanding that the energy of the solution be equal to the string tension times the length of the string. We find:

(3.30)

The assumption that all D3-branes are deformed in the same way means that the open string endpoint sources the diagonal in the gauge theory living on the D3-branes. There are other solutions to the DBI equations of motions for which this assumption is not satisfied. We will consider only the solution (3.30), which will be sufficient for our purposes. Let us also point out that the radius of the string horizon estimated from equation (3.26) by setting matches with the DBI radius (3.30), up to a numerical factor. This suggests that the horizon created by the string can be identified with the position of the (deformed) stack of D3-branes. A similar interpretation can be found in [13].

In terms of the coordinate, we obtain that the deformed stack of D3-branes is located along the curve given by:

(3.31)

where is a dimensionless parameter. Using equation (1.2) we can estimate in terms of and :

(3.32)

where we discarded a numerical factor that is not relevant for our purposes. We notice that as soon as the string coupling is small, the parameter is large.

4 Gravitational interactions between open strings

Let us now consider two static open strings that are attached to the same stack of D3-branes. In this section we will estimate the gravitational potential energy between these two strings, by evaluating the Nambu-Goto action for one string in the gravitational field created by the other one. We do not have the ambition to provide an exact computation, thus we will not keep track of the numerical factors.

Let us mention at that point that we are doing a computation in a non-supersymmetric theory. The static configuration of fundamental strings and D3-branes that we are considering is typically a BPS configuration once embedded into superstring theory. In that case no net force is expected between the strings. This means that the gravitational interaction is exactly canceled by other forces, presumably associated to the Kalb-Ramond and dilaton fields. Nevertheless the gravitational force we evaluate has a physical manifestation in non-supersymmetric circumstances.

4.1 Evaluation of the gravitational potential energy

The strings are parametrized as in (3.2). We denote by the euclidean distance between the two strings, as measured by an observer living on the D3-branes. We are interested in the macroscopic behavior of the gravitational interaction, hence we assume that the distance is large (in particular is much larger than ). Up to first order in the metric perturbation, the Nambu-Goto action for a probe string placed in the gravitational field created by a backreacting string (cf Figure 3) reads:

(4.1)

The integral over extends from the probe string endpoint at up to the point where the probe string touches the stack of D3-branes at , that we evaluated in equation (3.31).

Figure 3: Evaluation of the gravitational attraction induced by the backreacting string upon the probe string. The gravitational potential energy is dominated by a contribution coming from region 1, localized near the endpoint of the probe string.

We are interested in the large distance behavior of the gravitational potential energy between the strings. In the evaluation of the Nambu-Goto action (4.1) the integral over is dominated by the region where the perturbation of the metric is of order :

(4.2)

The second term in the previous expression gives the gravitational potential energy between the two static strings. The integral appearing in this term is the proper length of the piece of the probe string contained in the region . It turns out to be independent of :

(4.3)

where is the dimensionless parameter introduced in (3.31). We deduce an estimation for the gravitational potential energy:

(4.4)

4.2 Interpretation

Let us now discuss the interpretation of the result (4.4) we just obtained. First let us stress that the dominant contribution to the gravitational interaction between the strings is localized near the endpoints of the strings, as can be seen in Figure 3. Accordingly the potential (4.4) does not depend on the inertial masses of the strings involved. Indeed the inertial mass of the probe string in Figure 3 is almost entirely located in region 2, that provides a subleading contribution to the gravitational potential.

The second important point is that the potential energy goes like at arbitrary large despite the fact that gravity lives in more than four dimensions. This behavior is very different from what we would obtain for point-particles. Indeed the Newtonian potential between two point-particles goes like in (cf equation (2.21)), and it goes like in five-dimensional flat space. Notice that this result is also qualitatively different from the gravitational potential energy between strings of finite energy in flat spacetime (see appendix A): as soon as the distance between such strings is much larger than their size, they would behave like point-particles. The potential (4.4) has the same large- behavior as for point particles in four-dimensional flat spacetime, or equivalently as for infinite parallel strings in five-dimensional flat spacetime (cf Appendix A).

It is interesting to compare our result (4.4) with the gravitational potential (A.3) for a point mass close to a one-dimensional extended object. The open strings potential energy (4.4) matches with the Newtonian potential between an infinite string of tension and a mass in five-dimensional flat space. This mass is the product between the probe string tension and the proper length (4.3).

We can now provide an interpretation for the result (4.4), which is illustrated in Figure 4. Even if the two strings are widely separated, the geodesic distance between points located near the Poincaré horizon is very small. At distances that are much smaller than the radius the influence of the curvature is negligible and spacetime is essentially flat. Close to the horizon the strings appear as being infinitely extended along the radial direction, and thus interact like infinite strings in flat space. Notice that the redshift of the strings’ tension near the horizon is compensated by the decrease of the geodesic distance between the strings.

The usual expectation when dealing with extended objects of finite energy is that a point-particle approximation is reliable as soon as the distance away from the objects is large enough. For open strings attached to backreacting D3-branes, our results show that this is not the case3. Indeed the classical open strings we are considering have a finite energy, which can be arbitrarily small. But from the point of view of gravity they behave very differently than point-particles. This holds even if the distance separating the strings (as measured by an observer living on the branes) is arbitrarily large. The reason is essentially that the backreaction of the D3-branes stretches the open strings up to an infinite length, and decreases the geodesic distance between the strings as we approach the D3-branes’ horizon. Thus we identified large-distance manifestation of the microscopic differences between point-particles and strings. We will give further comments on this point in section 7.

Figure 4: The D3-branes’ backreaction creates an infinite throat. In region 1 the geodesic distance between the open strings is small compared to the curvature radius, and the open strings interact like infinite strings in flat space. This region provide the leading contribution to the potential energy (4.4).

5 Higher dimensional spacetimes

Let us now discuss the influence of the compact transverse space that typically appears in the spacetime metric (1.1) in the neighborhood of D3-branes. Since our goal is once again to evaluate the gravitational energy potential between two open strings, we will repeat the steps we followed previously in a new set-up. To avoid too much repetition we will focus on the differences with the previous analysis.

The discussion of section 4.2 already gives some clues about what the result will be. The enhancement of the gravitational potential energy between open strings (4.4) is due to the fact that the geodesic distance between the strings is small near the horizon. If the strings are widely separated along an additional transverse compact manifold, we can expect that the dominant gravitational interaction localized near the string endpoints will fade away. On the other hand if the strings are close enough along the transverse compact manifold, we should recover a stringy enhancement of the gravitational attraction.

We will now make this argument more precise. For the sake of simplicity we assume that the compact space is a circle of radius , but the generalization to higher-dimensional compact spaces is rather straightforward (the result will be given in section 7.1). Let us also mention that is the near-horizon geometry of D3-branes in non-critical six-dimensional backgrounds [14].

Green function

We want to construct a Green function that allows for the integration of the wave equation in . We perform the same field redefinition as in (2.12). We obtain:

(5.1)

The linear differential operator is a six-dimensional generalization of the operator introduced in (2.14):

(5.2)

where the angular variable parametrizes the circle. The delta-function normalizable eigenfunctions of the operator are:

(5.3)

with

(5.4)

The Green function is then:

(5.5)

and the time-integrated Green function reads:

(5.6)

The behavior of this potential in the region depends on the large-argument behavior of the Bessel functions, which is given by:

(5.7)

Linear backreaction of an open string

We consider a classical string extending up to the Poincaré horizon, defined by:

(5.8)

We compute the linear gravitational backreaction of this string. We obtain for (cf equation (3.17)):

(5.9)
(5.10)

The integral of the Bessel function can be estimated as:

(5.11)

where the ellipses contain a term which depends on . This term gives a subleading correction to the linear backreaction in the region . In this particular region we obtain:

(5.12)

The integral over can be evaluated in terms of the Gaussian hypergeometric function:

(5.13)

We can expand this function for :

(5.14)

We deduce for :

(5.15)

At large , behaves like . We recognize4 in the previous equation the Fourier decomposition of a function localized around , with a width of order and a height of order . We approximate this function as:

(5.17)

We observe that the backreaction does not spread in the transverse compact space. It is localized close to the string at . In the region where and (region 1 in Figure 5) we deduce for :

(5.18)

The behavior of the backreaction was expected: it is the same as for an infinite string in flat six-dimensional spacetime. The growth of the gravitational field with (at fixed ) is faster than in the five-dimensional case. This is a consequence of the localization of the backreaction along the transverse compact space, which increases with .

In the other regions of spacetime the linear backreaction comes with a larger negative power of .

Figure 5: Backreaction of an open string in . In region 1 the decay of the gravitational field away from the string is slow. The different equations define the surfaces that surround this region.

The gravitational potential energy between two open strings

The gravitational potential energy between two open strings follows from the previous analysis. First let us consider two strings that are close one to the other in the compact transverse space. Concretely we assume that the separation between the two strings along the is much smaller than the radius . It means that the two strings extend in the same direction away from the D3-branes. This is always the case in particular for two strings stretching between the same stacks of D3-branes (cf Figure 6). From the previous discussion we deduce that the gravitational potential energy is dominated by a contribution coming from region 1 in Figure 5:

(5.19)

where is defined as:

(5.20)

The upper bound in the integral in (5.19) is chosen such that the integration finishes either when the probe string touches the stack of D3-branes at , or when the condition fails to be satisfied at . We obtain:

(5.21)

In that case we recover the stringy enhancement of the gravitational attraction (4.4) first obtained in a five-dimensional set-up.

On the other hand if the open strings are broadly separated along the compact transverse space (i.e. for ), or equivalently if they extend in very different directions away from the D3-branes, there is no strong enhancement of the gravitational potential energy due to the extended nature of the strings.

Figure 6: The gravitational potential energy between two parallel strings (e.g. A and B) is given in (5.21). There is no similar enhancement of the potential energy between strings that extend away from the branes in very different directions (e.g. A and C).

6 Compactification of the transverse space

In the previous sections we studied the gravitational interaction between open strings in non-compact spacetimes. Now we will generalize our computations to the case where the space transverse to the stack of D3-branes is compactified. To this end we will work in the Randall-Sundrum II set-up [15]: the Poincaré patch is truncated to the region , and the fields satisfy Neumann boundary conditions along the wall at . Indeed this set-up can be used as a toy-model for D3-branes localized on a Calabi-Yau compactification of string theory [16]. For our purposes, the main difference with the previous cases is that the graviton admits a normalizable collective mode that propagates usual four-dimensional gravity throughout spacetime.

Green function

Linearized Einstein’s equations (2.3) are unchanged. To integrate them we have to construct the Green function associated to the linear differential operator introduced in equation (2.14). The Neumann boundary conditions at are satisfied by a linear combination of the eigenfunctions (2.17):