arXiv:hep-th/0306057v2 30 Jun 2003 ∗ noeo hs akrud,fidn t pcrmand spectrum its finding backgrounds, theory worldsheet these string of In- the one quantize curves. in to timelike trying closed well of like particularly stead issues suited non-local probes study with to us provides 7]. 6, objects 5, 4, 3, [2, in the is work realize direction recent theory this Some string conjecture? Does Protection in Chronology backgrounds theory? curves such string Are timelike in gravity. closed allowed of natural of theory is quantum status it a the such freedom, of about degrees wonder extra to timelike, of to addition mainly managed the singularities, by has curvature theory and certain string relativity as resolve Just general physics. both quantum encompasses which theory, theory. field classical quantum in or of curves sense mechanics timelike make quantum closed to mechanics, way with the a spaces is address on there to physics whether ask direction to Another is problem as reason- matter. serving to physical corresponds tensor, equations able momentum Einstein’s in energy like source the would the One that know relativity. general to classical not. the within or open tirely discarded leave be to to have seems timelike with it closed begin having to true, curves backgrounds is whether this of if question develop back- Even not do account, curves them. timelike into closed taken having not is grounds mechanics that quantum namely when Conjecture: sup- Protection to Chronology arguments presented the has port [1] Hawking time- curves. closed having like no by have causality, may violate solutions va- but Other singularities, of theory. regime classical the the of beyond lidity lie generically sin- curvature which have gularities, Some unphysical. seem that solutions lcrncades rke,fil [email protected] fiol, drukker, address: Electronic nsrn hoy h xsec fdnmclextended dynamical of existence the theory, string In string in questions those ask to is all of best Probably en- found be not may questions such to answer The many have Relativity General in equations Einstein’s eateto atcePyis h ezanIsiueo S of Institute Weizmann The Physics, Particle of Department ieiecre,adte hs pcsse iepretyg perfectly like d seem the spaces inside those of volume then out the and made adjust curves, wall timelike can domain o One family a structure. a inside asymptotic G¨odel construct universe the also like We universelook space-time. closed these in Since wrapping everywhere terms. when place kinetic conditions, negative realizin develop by certain modes achieved under is This and theory. spaces, string of solutions not edmntaeta eti uesmercG¨odel-like un supersymmetric certain that demonstrate We .INTRODUCTION. I. ¨ dlsUies naSprueShroud Supertube a in G¨odel’s Universe aa rke,Broe iladJa Sim´on Joan and Fiol Bartomeu Drukker, Nadav h orsodn pctm uv eoe ieiefor timelike becomes > r curve spacetime corresponding the h qain fsprrvt,w ildmntaethat, demonstrate will we supergravity, of equations the elblteetadrcini hc hr sflxby flux is there which and in fluxes, 4-form direction and extra 2-form the RR label as we well as flux NS-NS coordinate time the by t parameterize we which curves. mensions, timelike simplest closed the with is metric this supergravity a since such IIA plane, of single con- type example a of will in solution We rotation fluxes specific with some a supersymmetric. as on well solution centrate as the rotation make of that planes the choose can very [11]. are G¨odel universe and original [10], the Raychaudhuri met- to and dimensional similar Som four by these a studied All of ric 9]. generalizations [2, are in other found solutions Some were metrics [8]. similar of supergravity em- examples dimensional first be five were in will which bedded We G¨odel-like universes, curves. in timelike interested closed having the backgrounds of invalidity sickness the a signaling with action, states solution. effective BPS their find closed will in having we backgrounds of curves subset timelike around specific fluctuations a small For vacuum of them. this dynamics the in BPS analyze branes for to and extended look to to be corresponding will states strategy our interactions, adding osdrmto nteperiodic the in motion consider h ercadflxsare fluxes and metric The esalasm positive and assume supercharges shall we 8 preserves solution parameter one This ds n oa coordinates polar and , huhti em ieapretyvldslto to solution valid perfectly a like seems this Though hsslto 9 a otiilmti ntredi- three in metric nontrivial a has [9] solution This one where spaces, rotating describe solutions these All supergravity many are There precise. more be us Let 2 H 1 = /c htsprue r P ttsi these in states BPS are supertubes that g 3 ine ez tet eoo 60,Israel 76100, Rehovot Street, Herzl 2 cience, r ooeeu,ti ntblt takes instability this homogeneous, are s o tigbackgrounds. string ood − = ec h lsdtmlk curves. timelike closed the hence , uegaiysltoswihlocally which solutions supergravity f mi als hr ilb oclosed no be will there so wall omain − ieiecre,sm world-volume some curves, timelike dt vresltoso uegaiyare supergravity of solutions iverse uetbs u aevr different very have but supertubes, 2 rdr cr + F cr 4 2 2 = ∧ dφ dφ
rdt cr 2 ∗ ∧ + c r y,F , dy ihu oso eeaiy fwe If generality. of loss without dr and ∧ 2 dr + φ ∧ φ r ntepae hr sa is There plane. the in 2 dφ ieto tconstant at direction WIS/06/03-June-DPP dφ 2 ∧ 2 = + y. dy − dy hep-th/0306057 2 rdr cr 2 + X i ∧ =4 9 φ, dφ ( dx i (1) ) y r 2 , . 2 as it stands, it can’t be a valid solution of string theory. This task is simplified by the realization that the solu- One way of realizing this fact is to notice that supertubes tion (1) is a particular case of a family of IIA solutions [12, 13] can exist in the above background. Supertubes discussed in [13, 15]. The authors of [13] prove that are bound states of D0-branes and fundamental strings in general these solutions admit D0 branes, fundamen- having non-vanishing D2-dipole moment. They can be tal strings and supertubes as probes that don’t break realized on the worldvolume of a cylindrical D2-brane any further supersymmetry. Even better, they perform a with electric and magnetic fields turned on, such that probe analysis of these solutions, and show that when the they induce a non-vanishing angular momentum that bal- solution presents closed timelike curves, the worldvolume ances the tension which would naturally tend to collapse theory on the supertube is ill-defined. Our computation the tube. In our discussion, supertubes extend in the y is just a special case of theirs. direction and wrap the angular direction φ at fixed radius The supertube is a cylindrical D2-brane which is ex- r. It is not too surprising that this is the most interest- tended in the y direction as well as the angular direction ing object to probe the above geometry with, since it has φ at fixed radius r about the origin.1 The world-volume the correct rotational symmetry, and couples to all the theory on the supertube is just that of a D2-brane in background fields that were turned on. As will also be curved background, which includes the Dirac-Born-Infeld made clear below, this object preserves the same super- and Wess-Zumino terms symmetries as the G¨odel-like universe itself. Even though −Φ the tension of this supertube is just given by the sum of S = − e − det(G + F) − (C3 + C1 ∧F) , (2) the D0-brane and fundamental string charges, and it can Z p Z have an arbitrary size (by adjusting these charges), we where G is the pullback of the metric and C1 and C3 the will find that for a certain range of charges, some world- pullbacks of the RR potentials. The field strength F volume modes develop a negative kinetic term whenever includes the gauge fields that are turned on in the brane the radius satisfies r ≥ 1/c, so that the supertube wraps and those induced by the NS field a closed timelike curve. This sickness is much like that found by considering a probe brane in the repulson back- F = F − B2 = E dt ∧ dy + B dy ∧ dφ . (3) ground, which led to the enhan¸con mechanism [14]. We interpret it as an instability of the vacuum itself, which E is the electric field and B = B −cr2 the magnetic field, indicates that these supertubes will want to condense. including the term induced by the background. The plan of the rest of the paper is the following. In The equations of motion are solved by E = 1 (which the next section we shall briefly discuss the probe cal- is not critical in the presence of magnetic field) and con- culation of a supertube in our G¨odel universe, and show stant2 B > 0, whereas the radius of the tube is fixed that the world-volume theory may become sick at the to be the angular momentum J, related to the D0 and radius where closed timelike curves appear. In Section fundamental string charge densities q0 = B and qs by 2 3, we describe a family of supergravity solutions, that r = J = q0qs. It is a simple exercise to show that locally look like the G¨odel universe, but have a domain the supertube preserves the same supersymmetries as the wall made out of smeared supertubes separating them background, which is also clear from the analysis of the from a space that does not have closed timelike curves next section, where the metric (1) is shown to belong to asymptotically. The domain wall in these solutions can the family of metrics studied in [13]. be thought of as a regulator, the initial solution (1) aris- After adjusting the electric and magnetic fields as ing in the limit where the domain wall is sent to infinity. above, the Lagrangian density of the D-brane probe is One might think it possible to enlarge the region inside simply the magnetic field L = −B. The supertube ten- the domain-wall to get closed timelike curves, but the sion is just given by the sum of charges it carries, that same problem we encountered in the full G¨odel universe of D0-branes and fundamental strings, as dictated by su- prevents this from happening. persymmetry. Therefore we can seemingly make the tube We conclude with some discussion of the analogy to as small or as large as we wish. the enhan¸con mechanism and how similar problems show To verify that, we can start with a very small super- up for other G¨odel universes of supergravity, invalidating tube and slowly increase the radius. There are a number them too. of ways one could envision this happening. For instance, one could imagine a bath of D0-branes and fundamental strings surrounding the supertube, and allowing the su- II. SUPERTUBES IN A GODEL-LIKE¨ pertube to change its radius by exchanging charges with UNIVERSE.

As mentioned in the introduction, the type IIA Som- 1 Raychaudhuri solution (1) has closed timelike curves. The metric (1) is actually homogeneous. The addition of the supertube breaks translational invariance, but preserves the ro- Thus, one may suspect that an extended probe would tation around the origin. develop some sickness in this background. Our first goal 2 The positivity of the magnetic field is enforced by supersymme- is then to discuss the possible probes in this background. try. 3 the bath, satisfying the BPS condition at all times. Per- Here U and V are harmonic functions in the eight dimen- haps more naturally, one can also consider fluctuations sions spanned by xi, and A is a Maxwell field. Clearly of the radius, in time and/or in the y direction, which the IIA G¨odel like solution (1) is recovered if we take don’t preserve the BPS condition. In this latter case, a U = V = 1 and A = −cr2dφ (where r and φ are polar non-zero potential would also appear. coordinates in the x2, x3 plane). This calculation was done in [13] for the background A very natural question is whether there are more generated by a collection of supertubes, and is trivial to general configurations which are G¨odel near the origin, adapt it to our metric. The result is that the expansion but are asymptotically different, with no closed timelike of the Lagrangian at small velocities is curves.3. If we want to retain translation symmetry in R6 1 2 and rotational symmetry in φ, any source must be L = L0 + M r˙ + ... (4) 2 smeared in those directions, so U and V are harmonic functions on the plane with rotational symmetry. where the dots can contain a potential term if the fluc- We construct the solution by taking U = V = 1 and tuation doesn’t preserve the BPS condition. L0 is the − 2 static Lagrangian density and A = cr dφ near the origin. At a radius R we put the smeared supertube, and for r > R we choose (B − cr2)2 + r2 − c2r4 M = , (5) B Qs r Q0 r U =1+ ln , V =1+ ln , is the mass of the excited mode. This is positive definite 2π R 2π R for r < 1/c, but for larger radii, it becomes negative for A = −cR2dφ . (7) a certain range of charges. How do we interpret this singularity? In similar com- Qs and Q0 are respectively the fundamental string and putations, e.g. when we encounter that the metric in D0-brane charge densities, and are still arbitrary. The moduli space becomes negative we interpret it as mean- choice of A is motivated by requiring continuity of the ing that the effective Lagrangian is not valid, because it is metric through the domain wall. missing some light degrees of freedom. One then corrects This space is identical to G¨odel near the origin, but the effective description by adding those extra degrees of has very different asymptotics. For example, it has finite freedom. An example of this is the enhan¸con mechanism angular momentum, whereas for the original space it di- [14]. It is not exactly clear how to apply this proce- verges. Also, inside the domain wall there are constant dure to the case at hand. Recall that the G¨odel universe magnetic fields for the D0-brane and fundamental string, (1) is homogeneous, so there are closed timelike curves while outside there are mainly electric fluxes (the rota- through every point in space, and the sickness we found tion of space mixes electric and magnetic fields). One is not confined to a bounded region of spacetime. In ad- bad feature is that the dilaton grows as r → ∞. dition we don’t have well defined asymptotic conditions, It is easily verified that the solution satisfies the Is- or well defined global charges that a new solution should rael matching conditions [17], with the supertube stress- reproduce (since it is not clear how to define charge in energy tensor at the junction. Furthermore, the match- G¨odel universes). What we can safely conclude is that ing fixes the D2-brane density in terms of c. A similar the solution (1) is not a good string theory background. calculation was done for the enhan¸con in [18]. There is a very good analog to these solutions in III. A DOMAIN WALL SOLUTION. electromagnetism—a charged solenoid. Outside there is electric flux, with the potential behaving like a log. The current around the solenoid will induce some magnetic In the previous section we argued that the G¨odel-type flux inside, but no electric fields. solution (1) can’t be a correct effective description of a From this construction it is also clear that the super- string theory background. Nevertheless it is fairly easy to tube preserves the same supersymmetries as the G¨odel display supergravity solutions that locally are similar to metric (1). After all, the latter is just the metric induced G¨odel, but are free of closed timelike curves. To do so, we by a large collection of the same supertubes, but this can start by recalling the family of IIA solutions considered also be verified directly by the same calculations as in in [13, 15]. [13, 15, 19]. 2 −1 −1 2 2 −1 1 2 2 ds = − U V / (dt − A) + U V / dy So far we did not specify the radius where the domain 9 wall is located. Clearly when R ≤ 1/c there are no closed 1 2 2 + V / (dxi) timelike curves within the domain wall, but it can also Xi=2 be shown that there are no such curves outside the shell. −1 B2 = − U (dt − A) ∧ dy + dt ∧ dy , (6) −1 C1 = − V (dt − A)+ dt , −1 3 C3 = − U dt ∧ dy ∧ A, The idea of patching another metric outside a finite radius, to construct a solution free of closed timelike curves, was considered Φ −1 2 3 4 e =U / V / . for the original G¨odel metric in [16]. 4

At most, if R = 1/c there are closed null curves on the other solution, separated by a shell formed by the probes shell itself, which might warrant further study. themselves. Of course, an obvious difference is that in If we place the shell at a larger radius there will be the enhan¸con case one keeps the asymptotic form of the causality violating curves. It is therefore natural to think metric and changes the interior, while in our case we do of the full G¨odel solution as the limit when the domain the opposite. wall is taken to infinity and the region outside discarded. This picture may provide a better laboratory for study- In this paper we concentrated on a specific G¨odel uni- ing the problems of the G¨odel universe. In particular, if verse solution of supergravity. Other solutions were stud- the radius is only slightly larger than 1/c, there is only ied in [2, 8, 9] involving rotations in more planes. It is a a thin shell of closed timelike curves. So the sickness is simple check to see that all the solutions in type IIA and not spread over the entire space, and it should be possi- IIB suffer from the same problems we described. One ble to describe a process by which the shell contracts to has to take the same supertube, or in some cases the eliminate the region with closed timelike curves. T-dual objects, and place them so that their circles fol- In any event, it seems like those spaces cannot be cre- low the closed timelike curves, and the rest of the cal- ated. A calculation similar to the one carried out in sec- culation is identical. Therefore all those spaces are not tion II will show that, if one tries to increase the size of good string theory backgrounds. We expect this to be a the region inside the shell, so it exceeds R = 1/c, the general mechanism that eliminates many solutions with same problems will appear, indicating that one cannot closed timelike curves in string theory. increase the radius of the shell further. Acknowledgements

IV. DISCUSSION We are very grateful to Ofer Aharony, Tom Banks, Micha Berkooz, Jacques Distler, Roberto Emparan, One of the most striking uses of D-branes in string Willy Fischler, Surya Ganguli, Shinji Hirano, Eliezer Ra- theory is as probes of spacetime, often exposing the lim- binovici and Kip Thorne for interesting discussions on itations of supergravity, and providing a cure to some of related topics. We are particularly grateful to Roberto its pathological solutions. The enhan¸con mechanism [14] Emparan for pointing out a mistake in an earlier version provides a beautiful example of this. Studying super- of this paper. We would also like to thank the ITP at tubes on the G¨odel-like universe (1) teaches us a similar Stanford university and the organizers of the Stanford- lesson, that the naive supergravity solution is not valid. Weizmann workshop, where this project started. BF As opposed to the enhan¸con case, we find that the space would further like to thank the Santa Cruz Institute for is sick everywhere, and the question of how to “cure” it Particle Physics at UCSC and JS would like to thank the does not seem well posed. Perimeter Institute and ITP at the University of Amster- Nevertheless, the domain wall solutions we discussed in dam for their hospitality. BF is supported through a Eu- section III locally look like G¨odel universes, and present ropean Community Marie Curie Fellowship, and also by certain analogies with the enhan¸con mechanism: in both the Israel-U.S. Binational Science Foundation, the IRF cases, we keep part of the original metric, and replace the Centers of Excellence program, the European RTN net- problematic part (the repulson singularity in one case, work HPRN-CT-2000-00122 and by Minerva. JS is sup- the region with closed timelike curves in the other) by an- ported by the Phil Zacharia Postdoctoral Fellowship.

[1] S. W. Hawking, Phys. Rev. D 46, 603 (1992). 011602 (2001) [arXiv:hep-th/0103030]. [2] E. K. Boyda, S. Ganguli, P. Hoˇrava and U. Varadarajan, [13] R. Emparan, D. Mateos and P. K. Townsend, JHEP Phys. Rev. D 67, 106003 (2003) [arXiv:hep-th/0212087]. 0107, 011 (2001) [arXiv:hep-th/0106012]. [3] L. Dyson, arXiv:hep-th/0302052. [14] C. V. Johnson, A. W. Peet and J. Polchinski, Phys. Rev. [4] L. J¨arv and C. V. Johnson, Phys. Rev. D 67, 066003 D 61, 086001 (2000) [arXiv:hep-th/9911161]. (2003) [arXiv:hep-th/0211097]. [15] J. P. Gauntlett, R. C. Myers and P. K. Townsend, Phys. [5] E. G. Gimon and A. Hashimoto, arXiv:hep-th/0304181. Rev. D 59, 025001 (1999) [arXiv:hep-th/9809065]. [6] C. A. Herdeiro, arXiv:hep-th/0212002. [16] W. B. Bonnor, N. O. Santos and M. A. Mac- [7] R. Biswas, E. Keski-Vakkuri, R. G. Leigh, S. Nowling Callum, Class. Quant. Grav. 15, 357 (1998) and E. Sharpe, arXiv:hep-th/0304241. [arXiv:gr-qc/9711011]. [8] J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and [17] W. Israel, Nuovo Cim. B 44S10 (1966) 1 [Erratum-ibid. H. S. Reall, arXiv:hep-th/0209114. B 48 (1967 NUCIA,B44,1.1966) 463]. [9] T. Harmark and T. Takayanagi, arXiv:hep-th/0301206. [18] C. V. Johnson, R. C. Myers, A. W. Peet and S. F. Ross, [10] M. M. Som and A. K. Raychaudhuri Proc. R. Soc. Lon- Phys. Rev. D 64, 106001 (2001) [arXiv:hep-th/0105077]. don A304, 81 (1968) [19] J. Gomis, T. Mateos, P.J. Silva and A. Van Proeyen, [11] K. G¨odel, Rev. Mod. Phys. 21, 447 (1949). arXiv:hep-th/0304210. [12] D. Mateos and P. K. Townsend, Phys. Rev. Lett. 87,