hepth/0608137
Studies Of The OverRotating BMPV Solution
Lisa Dyson^{*}^{*}*
Department of Physics University of California Berkeley, CA 94720 USA
Theoretical Physics Group, LBNL, Berkeley, CA 94720 USA
We study unphysical features of the BMPV black hole and how each can be resolved using the enhançon mechanism. We begin by reviewing how the enhançon mechanism resolves a class of repulson singularities which arise in the BMPV geometry when D–branes are wrapped on K3. In the process, we show that the interior of an enhançon shell can be a time machine due to nonvanishing rotation. We link the resolution of the time machine to the recently proposed resolution of the BMPV naked singularity / “overrotating” geometry through the expansion of strings in the presence of RR flux. We extend the analysis to include a general class of BMPV black hole configurations, showing that any attempt to “overrotate” a causally sound BMPV black hole will be thwarted by the resolution mechanism. We study how it may be possible to lower the entropy of a black hole due to the nonzero rotation. This process is prevented from occurring through the creation of a family of resolving shells. The second law of thermodynamics is thereby enforced in the rotating geometry  even when there is no risk of creating a naked singularity or closed timelike curves.
Contents
1 Introduction
1.1 Motivation
The possibility of traveling back in time^{1}^{1}1Throughout this paper, when we refer to time travel, we are referring to travel back in time. was introduced when Cornelius Lanczos and later W. J. van Stockum discovered a solution to Einstien’s Equations which is generated by rotating pressureless matter [1, 2]. In this solution, timelike loops that closed on themselves could be formed. This showed that within the context of Einstein’s Equations, it is possible for a trajectory to begin at one point in time and end at that same point. Later, Kurt Gödel studied similar solutions, but with nonzero cosmological constant  rotating dust in Antide Sitter space [3]. These solutions sparked a series of discussions within the scientific community which led to the discovery of other causality violating solutions to Einstein’s Equations. Numerous arguments were also put forth as to why these solutions are physically impossible. See for example [4] and references therein for a review.
Recently, there has been increased interest in time travel in string theory following the discovery by Gauntlett [5] of supersymmetric realizations of the Gödel universe. In [6], Boyda applied Bousso’s prescription of holographic screens [7] to the supersymmetric Gödel universe. They found that for an inertial observer, a causally safe region is carved out by holographic screens. This led to a proposal that holography could play a key role in protecting chronology. Numerous papers have continued the study of closed timelike curves in string theory.
Timetravel is not the only undesirable feature of solutions to General Relativity. There are numerous solutions which have regions of infinite curvature. The physical implications of singular solutions led Penrose to make a ’cosmic censorship’ conjecture [8]. According to his conjecture, if a singularity were created, it would not be causally connected to distant observers. Instead, a black hole would be created. Specifically, in the case of physically reasonable matter undergoing gravitational collapse, the singular regions that are created are contained in black holes. This is the weak cosmic censorship conjecture. The strong cosmic censorship conjecture asserts that timelike singularities never occur. In that case, even an observer inside the horizon of a black hole never ’sees’ the singularity [9].
As a quantum theory of gravity, we can ask what string theory has to say about geometries with naked singularities. As it turns out, there are string theory solutions which naively appear to have singular regions unshielded by horizons. However, upon closer inspection, it has been found that stringy physics comes into play to resolve these unphysical regions [10, 11, 12]. One of the resolutions [12, 13] has been dubbed the ’enhançon’ mechanism since along with the geometric resolution comes an enhanced gauge symmetry. In that case, singularities appear in the naive geometry due to induced negative D(p4) charge when Dp–branes are wrapped on K3 (p4). The singularity occurs when the volume of K3 shrinks to zero, but [12, 13] showed that the geometry receives corrections when the volume takes on a stringy value. In that case, the region where the naked singularity appears in the naive geometry can never by created. The singularity is “excised” (see [14] and reference therein for a review).
In [15], an attempt was made to combine the concept of singularity resolution with chronology protection. It was proposed that the enhançon mechanism is a tool that string theory also employs to correct causally unsound geometries. The specific example of the BMPV black hole and the corresponding dual geometries was discussed. In the BMPV black hole, when the rotation parameter exceeds a certain bound, a time machine is created. The enhançon mechanism was applied to this geometry and it was argued that the time machine is never created. Instead, the matter that would create the time machine is restricted from traveling beyond a chronology protection radius outside of the wouldbe chronology horizon. In this proposal, since the matter that is responsible for creating the closed timelike curves in the naive geometry never reaches the region where they would exist, the time machine can never be formed. The corrected geometry is free of causal violations and chronology is preserved.
Another unphysical feature of geometries can occur when a black hole is present. Since black holes obey the laws of thermodynamics, an adiabatic process that would lower the entropy of a black hole would be unphysical. Indeed, in [16, 17], it was shown that the second law of thermodynamics could be violated for five dimensional black holes when the compact space is K3. This can happen due to the negative Dbrane charge that is induced as discussed above. Specifically, the entropy of the five dimensional black hole made out of D1 branes, D5–branes and units of momentum running along the effective string is . When the D5–branes are wrapped on the K3, curvature couplings induces units of D1–brane charge giving an entropy of . Thus, if we adiabatically drop D5–brane probes into the black hole with charge , the change in entropy is which, for can be negative. Thus the second law of thermodynamics would be violated. In [16, 17], it was shown that probes which could naively lower the entropy of a black hole were prohibited from entering the black hole by the enhançon mechanism.
In the case of the BMPV black hole with units of angular momentum, a similar violation can occur. The entropy is . Assuming that the rotation is carried by one of the charge types, [18], we can consider adding charge (, ). The corresponding change in the entropy is . If the angular momentum satisfies the bound [19], the second law of thermodynamics may be violated when . Notice that the bound appears in a different phase of the geometry (i.e. not the black hole phase) [20], one in which angular momentum is carried by the D1 and D5–branes. This implies that there is a region of parameter space, , where it may be possible to violate the second law of thermodynamics. This can happen independent of the appearance of closed timelike curves (although closed timelike curves may be present).^{2}^{2}2Note also that is the region of parameter space where the entropy of the black hole is derived in the CFT.
1.2 Plan Of Paper
In this work, we study the three above mentioned unphysical features of geometries  naked singularities, closed timelike curves and violations of the second law of thermodynamics. We show how their resolution can be linked through the enhançon mechanism. We begin by reviewing singularity resolution via the enhançon mechanism in the context of the five dimensional black hole. In the process, we find that the interior of a class of enhançon geometries are time machines. In a dual picture, the time machine is made out of fundamental strings coupled to RR flux proportional to the angular momentum. After applying the lessons that we learned from the enhançon to this geometry, following the proposal of [15] we argue that a shell emerges outside of the chronology horizon beyond which the fundamental strings cannot travel. In this case, the causality violating region can never be created and chronology is preserved.
Turning D–brane charge back on, we generalize the chronology protection result for the BMPV black hole [15], allowing for configurations with varying charge. We show that if we begin with a causally sound BMPV black hole and attempt to add matter that would create a time machine, a family of chronology protection shells appear outside of the horizon beyond which the potentially causality violating matter cannot travel. Since the matter does not travel beyond this radius, the time machine is never created and chronology is protected.^{3}^{3}3We note here that our result naturally extends to the region behind the horizon (when in our coordinates). However, we will restrict our discussion to the observable region outside of the black hole.
While studying the generalized BMPV geometry, we consider the charge configurations that can be constructed which, when dropped into the black hole, can decrease its entropy and thus violate the second law of thermodynamics. We show that, not only does the chronology protection mechanism save us from causality violations, but it also serves as an enforcer of the second law of thermodynamics. Probes that would lead to violations of the second law are prohibited from passing through the horizon by the chronology protection proposal. We go on to show that even in the causally sound regime, it may be possible to decrease the entropy of the black hole by dropping an appropriately constructed probe into the horizon. We apply the same analysis used in the chronology protection discussion to show that these probes are restricted from entering the black hole as well. This provides a generalization to the results of [16, 17] where violations of the second law resulting from wrapping Dbranes on K3 were resolved.
The outline of the paper is as follows: In section 2, we review the BMPV black hole. We discuss its fivedimensional form, Dbrane configuration and CFT dual. In section 3, we review the enhançon mechanism. We focus on the the supergravity analysis of the geometry as in [13]. In section 4 we discuss chronology protection. First, we consider the interior of the enhançon geometry in the limit of vanishing D–brane charge in section 4.1. We study the time machine that results and show how chronology is resolved (section 4.2) through the expansion of fundamental strings in the presence of RR flux (section 4.3). In section 4.4, we turn on D–brane charge to reproduce the chronology protection result for the BMPV black hole studied in [15]. The naked singularity that results when is also resolved. We then generalize the chronology protection result to include geometries with causally sound BMPV black hole interiors in section 4.5. In section 5 we study the second law of thermodynamics. We show how probes may be constructed which, when dropped into the black hole, can lower its entropy, violating the second law of thermodynamics. We show how the resolution chronology protection mechanism kicks in at just the right location to prohibit this from happening in section 5.1. We go on to show in section 5.2 that probes which would violate the second law of thermodynamics can be constructed even when no causality violations would result. These probes are also prohibited from entering the black hole, generalizing the results of [16, 17].
2 The Geometry
2.1 Five Dimensional Black Hole
Let us begin by presenting the five dimensional rotating black hole solution [21, 22, 23, 24, 25]. We will consider a black hole with three charges, call them , and . The metric for this geometry is
(2.1) 
where the are harmonic functions associated with the charges, . In addition to the metric, this supergravity solution has the following moduli and gauge fields under which the black hole is charged,
(2.2) 
(2.3) 
(2.4) 
(2.5) 
with .
In order to have a rotating black hole that preserves supersymmetry, the causalityviolating ergoregion that is usually associated with Kerrlike solutions must be absent. In order for the ergoregion to be absent, the horizon of the black hole must be static. By analyzing the above metric in the near horizon limit, one finds
(2.6) 
So the horizon is not rotating. The nonzero rotation parameter does, however, have a nontrivial effect on the horizon. Instead of the standard spherical horizon geometry, the rotation introduces a squashing parameter [23]. The horizon is a squashed sphere. Since the squashing parameter depends on the angular momentum, , one can show that a sensible description of the horizon can break down if becomes too large.
The BMPV geometry has closed timelike curves. Closed timelike curves exist for all values of . However, if is small enough, all closed timelike curves are hidden behind the horizon. When exceeds a critical value, closed timelike curves appear outside of the horizon leading to observable causality violations. The overrotating black hole geometry was studied in detail in [24].
The causality constraint on can be seen explicitly by looking at the angular components of the metric. The chronology horizon, , is the location where vanishes. Closed timelike curves appear when is negative. This occurs if
(2.7) 
Since the horizon is located at the origin in these coordinates, an observer outside of the horizon will see closed timelike curves if
(2.8) 
2.2 Ten Dimensional D–Brane Geometry
The tendimensional supergravity solution describes D1 and D5–branes with momentum running along the effective D–string wrapped on an [25]. The D5–branes are additionally wrapped on or K3. In the case, of K3 naked singularities of repulson type can appear. We discuss this in detail in section 3. The metric, RR field and the dilaton are
(2.10)  
(2.11) 
(2.12) 
is or K3 and the charges , and are given by
(2.13) 
where we have D1–branes, D5–branes, units of right moving momentum, and the angular momentum is quantized in terms of integers . is the asymptotic volume of or K3 and is the radius of the .
This is a IIB supergravity solution with four supercharges. Closed timelike curves are an integral part of this geometry. They can be constructed by considering the curves with tangent vectors [25]
(2.14) 
where we define and is the Hopf fiber. Since is a compact direction, as is required to preserve spacetime supersymmetry and necessary to link this geometry to the black hole [26], for certain values of and curves of this type can be closed. A quick calculation of the proper length of these curves show that they can be timelike in the region . The causality violations are more explicit in the Tdual geometry [15] and as was shown in [27], closed timelike curves are invariant under Tduality.
We can also consider the gauge theory that describes this configuration [28, 21]. If we take the size of the to be much larger than the size of the , the effective description is the 1+1 dimensional field theory living on the world volume of the D1–branes. One can show that the causality bound coincides with a unitarity bound in the dual field theory [21, 25]
3 Review Of Singularity Resolution
As discussed in the previous section, in order to create a black hole we could wrap the D5–branes on a four torus or on K3. In this section, we will review some properties of D5–branes wrapped on K3. Wrapping Dp–branes on K3 induces negative D(p4)–brane charge. This in turn leads to naked singularities of repulson type [29]. In [12, 13] it was found that string theory resolves repulson singularities by using the enchançon mechanism. The enhançon of the D1 D5–brane system wrapped on K3 was studied in detail in [16, 13]. Their results were generalized to include nonzero rotation in [17]. Studying how the enhançon resolves naked singularities will give us insight into how string theory repairs geometries that violate causality [15]. We will study the results of [16, 17] in detail here. Interestingly, we will find that while the enhançon has the desired effect of resolving naked singularities, it can create geometries with closed timelike curves where none existed in the naive geometry. We will use this fact to explore how chronology can be resolved below.
Consider a geometry with D5–branes wrapped on K3 and D1–branes wrapped on the . Wrapping the D5–branes on K3 induces negative D1–brane charge equal to . The supergravity solution of this configuration is as in equation (2.10), but with charges and .
Plugging these charges into the metric, it can be seen that a singularity appears when . This occurs at a radius
(3.15) 
Since the horizon is located at the radius in these coordinates, the singularity is outside of the horizon for , and this geometry has the unphysical feature of having a naked singularity that is causally connected to observers outside of the black hole. In order to see how the enhançon repairs the naked singularity, we can follow the standard procedure of building the geometry by adiabatically bringing in the objects that create the geometry from asymptotically far away. A probe calculation will show us if this is a consistent thing to do. In [12, 13] an equivalence was shown between the worldsheet and supergravity analysis of probes that create the geometry. The authors found that massless modes appear at a radius larger than the repulson radius. They argued that the naive geometry with a repulson singularity is never formed. Instead, the objects that would have created the singularity cannot travel beyond a special radius, dubbed the enhançon radius, were new massless modes appear and geometry gets corrected. The resulting true configuration is free of all naked singularities. We will review the resolution using the the supergravity techniques discussed in [13, 16, 17].
In an effort to construct the repulson singularity, we must determine if it is possible to construct a geometry made up of D5–branes wrapped on K3 and D1–branes wrapped on as above. We will see if this is possible by beginning with a geometry with D5–branes and D1–branes and attempting to bring in D5–branes and D1–branes from asymptotically far away. As in [13, 16, 17], we can study the behavior of the this configuration in the supergravity picture by adiabatically collapsing a shell of D–brane charge from asymptotically far away and determining if this is a consistent thing to do. In order to do this, we patch together two geometries at a radius which is the location of the additional D5–branes and D1–branes and consider what happens when we let . The geometry for both regions is of the same form as (2.10), but we will need to make the following substitutions: The exterior geometry has harmonic functions and but with . For the interior region, replace and with the harmonic functions
(3.16) 
Where
(3.17) 
The metric is smooth across the incision radius. Any discontinuity that appears in its derivative should be interpreted as a function source of stressenergy. Applying the standard Israel geometry matching techniques [30], we find the stress tensor for the shell is with:
(3.18) 
The indices , denote the and directions; , denote the directions; , denote the angular directions along the junction threesphere; and sets the Newton constant.
From the stressenergy tensor, we find that the tension of the shell is
(3.19) 
As in the prototypical D6–brane case, if we consider only D5–brane probes (), the tension of the probe vanishes before the repulson singularity is reached. This happens at a radius
(3.20) 
where and is the asymptotic volume of K3. This radius is precisely the location where the coordinate volume of K3, as measured in Einstein frame takes on the stringy value, . For , the tension of the D5–brane probe is positive. At , the D5–brane probe is tensionless. Beyond the D5–brane probe would have negative tension. Since this would be unphysical, the D5–brane probe cannot travel beyond . Since D5–branes cannot travel beyond and the enhançon radius is greater than the repulson singularity radius, , the repulson geometry, which depends on D5–branes being able to travel into the interior of , cannot be created. Instead a shell of D5–branes forms at and, in the limit of vanishig and , the interior geometry is just flat space. In the case of nonvanishing and , it was argued in [16, 17] that these charges decouple from the D5–branes at the enhançon radius and are free to travel to the origin. Thus, the geometry has D5branes sitting at and a nontrival geometry created by and charge in the interior of the enhançon shell. The shell is also the location where massless modes appear and the gauge symmetry is enhanced. Note also that the stressenergy in the transverse directions, , vanishes, so there are no transverse force acting on the shell and it is consistent to bring it in from asymptotically far away.
For a shell made up of D1–branes, , , the tension is always positive. This implies that there is no obstruction to bringing in arbitrary D1–brane charge. This makes sense since the repulson singularity is caused by the wrapping of the D5–branes on K3. Likewise, the tension of the shell is always positive when the number of D1–branes equals or is greater than the number of D5–branes, . This would imply that we can bring in D5–brane charge only as long as the D5–branes are appropriately “dressed” with positive D1–brane charge. For all values of , a repulson singularity exists in the naive geometry and an enhançon shell appears in just the right location to repair the naive geometry.
4 Resolving Time Machines
We have seen how string theory employs the enhançon mechanism to resolve a class of singular geometries. This was a pleasing discovery because it provided an important example of how string theory resolves physically unsound geometries as a fundamental theory of quantum gravity should. In an effort to determine if string theory has a way of resolving causality violations, the enhançon construction was applied to an “overrotating” BMPV black hole in [15]. It was argued that the above analysis has an analogue in chronology violating geometries. We motivate this result further here by zooming in on the interior of a type of enhançon geometry that was constructed in the previous section. We will discover that closed timelike curves exist for all values of the charges and .
4.1 Causality Violations Inside The Enhançon
To begin, notice, as discussed in [16, 17], the enhançon only depends on D1 and D5–brane charges. Since the momentum charges and do not play a role in creating the repulson singularity, it is natural that they do not play a role in its resolution. In [16, 17], it was argued that the string momentum modes decouple from the D–branes that make up the enhançon shell and are free to travel to the origin. This is an interesting result because it allows for the existence of closed timelike curves in the interior of the enhançon shell, even when there were none in the original naive geometry.
Let us consider the limiting case where we only have D5–branes wrapped on K3 and no additional D1–brane charge (e.g. the harmonic functions are constant in the interior, in equation (3.16)). We find that closed timelike curves exist for all values of the charges and in this limiting case. We discover that, even though the enhançon mechanism saves us from the embarrassing situation of being able to construct a singular geometry unshielded by a horizon, it plays a key role in creating an equally troubling geometry.
To see this explicitly, let us zoom in on the interior of the enhançon. The metric is given by
(4.21)  
This metric is supported by the RR potential
(4.22) 
and a constant dilaton,
(4.23) 
For simplicity, we will rescale this geometry to absorb the constants . Since the rescaled geometry is a IIB solution in its own right [25], we will leave out the added complication of the enhançon shell for the moment and consider the solution for all values of . The rescaled geometry is (using the same coordinate names although it is understood that the new coordinates have been rescaled appropriately)
(4.24) 
The RR potential is
(4.25) 
and the dilaton is constant.
Closed timelike curves are an integral part of this geometry. The chronology horizon is the positive real solution to the equation
(4.26) 
This equation has a nonzero and positive solution for all values of . It follows that is always positive and closed timelike curves exist for all radii .
What can we do about these chronology violations? We can begin by applying what we learned from the supergravity discussion in section 3 to this geometry following the proposal in [15]. Before we do, let us recall what this geometry is made of by performing a Tduality along the direction. In the Tdual picture, the momentum modes become winding modes. The resulting geometry is constructed out of fundamental strings supported by RR flux proportional to the angular momentum parameter . The full solution has the following fields:
(4.27) 
(4.28) 
where we have replace the harmonic function with its dual representing fundamental string charge . Closed timelike curves are invariant under Tduality [27], but one can easily perform a quick computation on the angular directions of this geometry to confirm that the causality horizon is at the same location as in equation (4.26). Thus we find that the fundamental string geometry has causality violations for all values of in the region where .
4.2 Chronology Protection Sphere
We will now return to the supergravity analysis of the geometry to study how the causality violations might be repaired. Beginning with flat space, one can consider the thought experiment of creating the geometry by adiabatically bringing in charge from asymptotically far away as in section 3. In the limit of vanishing , the BPS objects that make up the geometry are fundamental strings^{4}^{4}4We will study the IIA geometry in section 4.3, but will use continue to use the dual language here. With angular momentum turned on, the fundamental strings must be supported by additional RR flux. As we bring in the strings coupled to the RR potentials from infinity, the discontinuity in the derivative of the metric that results represents the function source of stressenergy. We can determine the tension of the shell of charge by deriving the stressenergy tensor as we did for the enhançon in equation (3.18). We do this by matching the geometry in (4.24) with flat space in the interior.
In order to match internal flat space with the string geometry, we must “twist” our coordinates to ensure the metric is smooth across the boundary . Defining , the angular coordinate of the Hopf fiber in the interior is twisted as follows:
(4.29) 
This implies that must satisfy the quantization condition:
(4.30) 
for some integer since is compact. We must also rescale as follows
Before calculating the stress energy tensor, let us write the metric in a more general form:
(4.32) 
where we have the following expressions for and outside of the shell, , and inside the shell, :
(4.33) 
It is clear that the metric is continuous across the boundary . The stress tensor for the shell is
(4.34)  
where . Happily, the transverse components of the stressenergy tensor vanish, indicating that there are no transverse forces acting on the shell that would prevent us from constructing it. From this, we can determine the energy density associated with the shell. The momentum vector is given by
(4.35) 
The energy density associate with the timelike killing vector is given by . From this, we find that the tension of our function source is
(4.36) 
where we have defined .
Asymptotically, the energy density is of the form of a shell of dual fundamental strings as expected,
(4.37) 
Locally, the energy density decreases by . This effect is crucial for repairing the causal sickness of the geometry and is due to the coupling to the RR flux. The tension of the shell vanishes at a critical radius
(4.38) 
This is also the radius where our coordinate transformation simplifies: , . For radii greater than , the tension of the shell is positive. For radii less than , the tension of the shell is negative.
How do we interpret this critical radius? In the spirit of the enhançon discussion in section 3 and proposal of [15] we argue that is a critical radius beyond which our fundamental strings coupled to RR flux cannot travel. At this radius, nontrivial physics comes into play correcting the naive geometry. If the fundamental strings travel beyond , unphysical negative energy states would be present. Since , the matter that constructs our geometry is never able to travel into the region where closed timelike curves would be created. The causally sick region is never created and our usual notion of chronology is preserved. The critical radius is the chronology protection radius as proposed in [15] in the limit of vanishing D–brane charge.
We can also rewrite the stressenergy tensor by using the quantization condition due to the geometric “twist” (4.29):
(4.39) 
The tension then is given by:
(4.40) 
Vanishing tension implies
(4.41) 
for some . The angular momentum is maximal, , when . In terms of the charge quantization given in (2.13), we have:
(4.42) 
From this, we find that yields the following results:
(4.43) 
4.3 Tdual IIA Shell
In the previous section, we showed that the tension of the shell of charges that make up our time machine geometry vanishes at a critical radius. We argued that this radius serves as a minimum value in moduli space beyond which the fundamental strings supported by RR flux cannot travel. It is interesting to see what happens to our geometry in a Tdual configuration. If we perform a Tduality along the direction, the metric given in equation (4.27) can be written as:
(4.44) 
with and with and as defined in (4.33) for the inner and outer regions and with Tdual charge . The Tdual geometry has F1–strings coupled to 1form and 3form RR potentials (proportional to ) for radii , while the geometric twist of flat space in IIB gives rise to flux branes in IIA in the interior as was noted in [19]. We will derive the stress energy tensor in the IIA picture using language similar to [19].
We can rescale the geometry to absorb the factor at the location of the shell. This can be done quite simply with the following identifications:
If we define the functions
we can rewrite in the outer and inner regions as:
where we have defined . With the metric written in this form, the stressenergy tensor is
A quick calculation of the tension of the shell gives
(4.45) 
Requiring that the tension of the shell is nonnegative gives us the chronology protection condition, or equivalently:
(4.46) 
4.4 Resolving The BMPV Time Machine
Let us return to the BMPV black hole. From the enhançon discussion in section 3, we saw that momentum modes decouple from Dbrane charge when attempting to build the singular repulson geometry [16, 17]. We studied the geometry created by the momentum modes separately above in the language of the dual fundamental strings coupled to nontrivial RRflux. We saw that closed timelike curves exist for all values of the charges and . We further saw how chronology can be protected. Following [15], we argued that the fundamental strings expand in the presence of the RR flux to form a wall at a chronology protection radius. In that case, the strings cannot travel beyond this radius and are never able to create the causality violating geometry. Through the expansion of the objects that create the geometry to just the right radius, our usual notion of chronology can be preserved.
In the full BMPV geometry the condition for closed timelike curves to exist is softened and in fact come along with another unphysical feature, a naked singularity [24, 23]. These unphysical features are present only when . In [15], an attempt was made to construct the BMPV geometry by bringing in the charges that make up the geometry from asymptotically far away. There was no obstruction to bringing in the D1 and D5–branes extended along the z direction, but, in the spirit of our discussion of the fundamental strings in the previous sections, it was proposed that the and charges could not travel beyond a chronology protection radius which was the critical radius where the tension of the shell of fundamental strings supported by RR flux vanishes. To get a supergravity description, we match the IIB geometry (2.10) to an interior geometry with only D1 and D5–brane charge,
(4.47) 
where is the metric on a three sphere. In order to match the coordinates in the interior to the exterior geometry the metric must be continuous across the boundary, the interior coordinates are related to the exterior coordinates via a similar geometric twist as in (4.29)
(4.48) 
with and is the scaled radius of the shell once D–brane charge is turned on. We must also perform the coordinate transformation for :
As we saw when we just had and charge turned on, in order to ensure that our internal geometry is just flat space with D1 and D5–branes, we require that
(4.49) 
for some integer since is compact. Also notice that and when , similar to equation (LABEL:coordtz). As can be anticipated, this happens precisely at the proposed chronology protection radius.
One can attempt to build the geometry with an adiabatically collapsing shell of fundamental strings coupled to RRflux. The shell has tension of the same form as in (4.36), but with the rescaled radius.
(4.50) 
The tension of this shell is positive for , zero when and negative when . As discussed in the case of the enhançon and the time machine above, we conclude that the matter that makes up the shell cannot travel beyond the location where . If we attempt to adiabatically bring in the shell of charge beyond the critical radius , the shell has negative tension. In the presence of D1 and D5–brane charge, the value of the radius of the proposed resolving sphere is given by
(4.51) 
Since the chronology protection radius is always greater than the chronology horizon, the resulting geometry is free of all causal inconsistencies. Also, the naked singularity that appears in the “overrotating” geometry due to the destabilization of the horizon (the naive area at is imaginary) is resolved since the interior geometry now only has D1 and D5–branes.
We can rewrite the tension of the shell in terms of the charges using the quantization condition due to the geometric “twist” (4.48):
(4.52)  
Vanishing tension implies
(4.53) 
for some integer . The bound on the angular momentum, agrees with the microscopic bound discussed in [19] when the angular momentum is carried by one type of charge component.
To summarize, the argument is that when the angular momentum parameter exceeds the three charge bound but has not exceeded the single charge bound (which can occur when ), the supergravity description may actually give the correct asymptotic description of a domain wall configuration in which the angular momentum is carried by a single charge, . In that case, beyond a certain region in moduli space, the supergravity solution must be corrected. In fact, the supergravity solution is signaling that the geometry must be corrected by yielding unphysical behavior such as a naked singularity and closed timelike curves. In [19] (see also [31]) a similar domain wall argument was made. In that case, the rotation is carried by two charges, one of which is . A solution was constructed which had an “overrotating” BMPV exterior and a Gödel space interior. The domain wall linking the two geometries was a supertube. The resulting solution was free of causality violations and naked singularities.
4.5 General BMPV Black Hole Interior
We can ask what happens if we begin with a causally sound geometry and add charge to the system that would create closed timelike curves. We find that there is a natural generalization to the chronology protection mechanism which prohibits causality violating probes from falling into the black hole. For each probe, we argue that a wall emerges outside of the black hole at just the right location to prohibit the flow of causality violating charge. In this way, chronology is protected for general BMPV charge configurations.
To show this explicitly, let us begin by considering the limiting case of the causally safe BMPV black hole with angular momentum satisfying . We will add additional charge and to the system. Closed timelike curves will exist in the new geometry if or, to first order,
(4.54) 
Consider the following generalized metric
(4.55)  
Let us begin with an interior geometry with
(4.56) 
and consider bringing in additional charge, and , adiabatically from asymptotically far away. The metric for the exterior geometry is represented by equation (4.55) with
(4.57) 
We will paste these geometries together at a shell of radius . The function source that results represents fundamental strings with charges and (supported by RR flux). In order to ensure that the metric is continuous across the boundary at a radius , we must perform the following geometric “twist”:
(4.58) 
and (leading order) coordinate transformation
(4.59) 
where and the quantization condition becomes
(4.60) 
with rescaled radius . Once we have performed the coordinate transformation to match the two geometries, the interior geometry can be expressed in the same form as equation (4.55), but with
(4.61) 
The metric is now continuous across the surface .
The discontinuity in the derivative of the metric will give us the stressenergy of the function source of fundamental strings coupled to RR flux. We find that the tension of the shell takes on the following form
(4.62) 
To leading order, this expression reduces to
(4.63) 
where is the location of the shell. The form of the tension tells us that the shell contains local nontrivial charge, and as expected. Asymptotically, the tension is of the form of a shell of fundamental strings, . Locally, we get a correction to the energy density proportional to the angular momentum parameter. Also, the transverse components of the energy momentum tensor vanish, where , so there are no forces acting on our shell in the transverse directions prohibiting its construction.
Consider the following function:
The tension is a positive multiple of . Studying the solutions to will tell us how the tension of the shell behaves. First, define
(4.65) 

For , the tension of the shell is positive for all real radii . This can be seen clearly by the fact that is always positive for real .

For , the tension of the shell is positive when and vanishes at the origin.

For , the tension of the shell is positive for radii greater than a critical value, . The tension vanishes at and is negative for . This can be clearly seen by noticing that has a unique real solution, . increases monotonically when and decreases monotonically when .
How can we interpret this results? If we wish to construct a time machine we can begin with the limiting causally sound geometry with charges satisfying and add charges and . If we attempt to bring in this shell of charges from asymptotically far away, if