Dual geometries for a set of 3-charge microstates
Introduction
In the traditional picture of a black hole, infalling matter settles into a central singularity while Hawking radiation emerges at the horizon. Due to the large separation between the horizon and the singularity the radiation is insensitive to the detailed state of the matter that made the hole, and we get information loss [1].
Some computations in string theory suggest that the black hole interior is quite different; instead of ‘empty space with a central singularity’ we have a ‘fuzzball’ with state information distributed throughout the interior of the horizon. It was shown in [2] that due to the phenomenon of ‘fractionation’ the effective excitations of a D1–D5–P bound state are very light, and in fact extend to a distance of order the horizon radius.
In [3], [4] the 2-charge extremal D1–D5 system was studied. The ‘naive’ geometry of D1 and D5 branes is pictured in Fig. 1(a); we have flat space at and a singularity at . But the CFT analysis implies that the Ramond (R) ground state of the D1–D5 system is highly degenerate, with entropy . In [3] the geometries dual to these states were constructed. It was found that the naive metric did not arise from any of the microstates; instead all states yielded geometries that were ‘capped’ smoothly before reaching .1 No individual geometry has a horizon or singularity but if we draw a surface to bound the area where these geometries differ significantly from the naive geometry then from the area A of this surface we find The radius of this surface is times the Planck length or the string length (the dilaton is bounded, so ). Thus we see that the D1–D5 bound state ‘swells’ up to a radius that increases with the charges, and which is such that the bounding surface constructed above bears a Bekenstein type relation to the count of states.
If we have three charges—D1, D5 and momentum P—then the ‘naive’ geometry is an extreme Reissner–Nordström type black hole. This geometry has a horizon at , and continues to a region which contains a singularity (Fig. 1(c)). The area of the horizon gives and this exactly equals the microscopic entropy obtained from a count of D1–D5–P ground states [6]. But based on the results above we are led to ask if the individual states are described by geometries that ‘cap off’ before reaching as in Fig. 1(d). For three charges the radius of the ‘throat’ asymptotes to a constant as we go down the throat, so the area A obtained at the dashed line in Fig. 1(d) will give (1.2). Thus the nontrivial question in this case is whether the geometries dual to 3-charge microstates are like Fig. 1(c) (with a horizon and singularity inside the horizon) or whether some effects destroy this naive expectation before we reach . Note that in the 3-charge case (unlike the 2-charge case) we do not expect the generic state to be well-described by a classical geometry; quantum fluctuations can be large. But there would still be special cases that are in fact well described by a classical metric, and we can gain insight by constructing these explicitly.
In [7] a perturbation was constructed on an extremal 2-charge D1–D5 state that added one unit of P charge. The equation for linear perturbations was solved to give a regular, normalizable excitation in the limits of small r and large r, and the solutions were shown to agree to several orders in the region of overlap. This indicated that at least this particular 3-charge state was smoothly ‘capped’ as in Fig. 1(d), and did not have a horizon or singularity like Fig. 1(c).
In the present paper we obtain exact geometries dual to a set of D1–D5–P microstates. These geometries will again turn out to be capped as in Fig. 1(d). The microstates are not generic 3-charge states; in particular they have a significant amount of rotation. But the construction does support the general conjecture that all configurations must suffer modifications before reaching and forming a horizon.
While we were finishing this work the paper [8] appeared, which also constructed similar metrics by an interesting though different method based on [9]. If we set the D1 and D5 charges equal in our solution () then the dilaton vanishes, and we obtain solutions that look (locally) like the solutions in [8]. There does appear to be a difference however in the way the final parameters are set in the solution, so that the values of the conserved quantities like angular momenta in [8] appear to be different from the ones that we have. We comment briefly on these issues near the end of our paper.
Section snippets
The D1–D5 CFT
We take IIB string theory compactified to . Let y be the coordinate along with The is described by 4 coordinates , and the noncompact space is spanned by . We wrap D1 branes on , and D5 branes on . Let . The bound state of these branes is described by a -dimensional sigma model, with base space and target space a deformation of the orbifold (the symmetric product of N copies of ). The CFT has
Constructing the gravity duals
In [3] the 2-charge D1–D5 solutions were found by dualizing to the FP system, which has a fundamental string (F) wrapped on carrying momentum (P) along . Metrics for the vibrating string were constructed, and dualized back to get D1–D5 geometries. The general geometry was thus parametrized by the vibration profile of the F string. But a 1-parameter subfamily of these D1–D5 geometries had been found earlier [14], [15], by looking at extremal limits of the general axially symmetric
Conserved charges
We want to take an extremal limit of the above solution. We take this limit while keeping the conserved charges fixed to the values that describe the states defined in Section 2. Thus the solution should describe D1 branes, D5 branes, units of momentum, and angular momenta The volume of the is V and the length of the is . The 10D Newton's constant is . If we dimensionally reduce along then we get the
Regularity of the metric
A sufficient condition for the metric to be regular is that the coefficients of both the metric and the inverse metric be twice differentiable functions of the coordinates. In turn, the inverse metric is well defined if the metric is smooth and the determinant is nonvanishing. For the metric (4.21)
The function f does vanish on a hypersurface, but the combination hf takes a finite value when The function f appears explicitly in the metric only in the
Comparison with the solution in [8]
The metric in [8] was expressed somewhat implicitly, but with some algebraic manipulation it can be brought to the form where We have denoted by what in [8] has been called a, to distinguish it from the
Discussion
We have constructed extremal D1–D5–P solutions dual to a special subset of CFT states. The solutions were smooth, with no singularity or horizon. The solutions thus look like Fig. 1(d) rather than Fig. 1(c). These results support the general conjecture [7] that individual microstates of a black hole do not look like the naive picture of a black hole—‘empty space inside the horizon with a central singularity’. Rather, the horizon arises only as an effective construct when we coarse-graining over
Acknowledgments
S.G. was supported by an INFN fellowship. The work of S.D.M. was supported in part by DOE grant DE-FG02-91ER-40690. We thank Oleg Lunin and Yogesh Srivastava for helpful discussions.
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