We obtain an exact solution of the supergravity equations of motion in which
the four-dimensional observed universe is one of a number of colliding
D3-branes in a Calabi-Yau background. The collision results in the
ten-dimensional spacetime splitting into disconnected regions, bounded by
curvature singularities. However, near the D3-branes the metric remains static
during and after the collision. We also obtain a general class of solutions
representing $p$-brane collisions in arbitrary dimensions, including one in
which the universe ends with the mutual annihilation of a positive-tension and
negative-tension 3-brane.Comment: RevTex, 4 pages, 1 figure, typos and minor errors correcte

A. Papapetrou

C. N. Pope

D. Z. Freedman

G. W. Gibbons

H. Lü

English

arXiv

Gibbons, G. W.

Lü, H.

Pope, C. N.

Texas A&M University

Physical Review Letters

Brane Worlds in Collision

Texas A&amp;M Repository

arXiv:hep-th/0501117v3  18 Aug 2007DAMTP-2005-3 MIFP-05-01 hep-th/0501117Brane Worlds in CollisionG.W. Gibbons,1 H. Lu¨,2 C.N. Pope2∗1DAMTP, Centre for Mathematical Sciences, Cambridge University Wilberforce Road, Cambridge CB3 OWA, UK2George P. & Cynthia W. Mitchell Institute for Fundamental Physics,Texas A&M University, College Station, TX 77843, USA(Dated: January 14, 2005)We obtain an exact solution of the supergravity equations of motion in which the four-dimensionalobserved universe is one of a number of colliding D3-branes in a Calabi-Yau background. Thecollision results in the ten-dimensional spacetime splitting into disconnected regions, bounded bycurvature singularities. However, near the D3-branes the metric remains static during and afterthe collision. We also obtain a general class of solutions representing p-brane collisions in arbitrarydimensions, including one in which the universe ends with the mutual annihilation of a positive-tension and negative-tension 3-brane.PACS numbers: 11.25.-w, 98.80.Jk, 04.50.+hThe idea that our spacetime is a brane in a higher-dimensional world leads naturally to the suggestion thatthe big bang is the result of a brane collision [1, 2]. Col-liding brane-world scenarios offer an alternative to con-ventional four-dimensional cosmological models, and animportant challenge is to make predictions that distin-guish between intrinsically higher-dimensional effects andconventional lower-dimensional physics. In order to doso, a better understanding of brane-brane interactionsis needed. Studies of D3-D3 and D3-D3 collisions haveso far been restricted to approximations based on ef-fective field theories on the brane, or on perturbativestring-theory techniques [3, 4]. There have been no ex-act treatments based on the supergravity equations ofmotion. The purpose of this letter is to provide an exacttreatment in the case of colliding D3-branes, moving in aCalabi-Yau background. Our technique extends to moregeneral brane-collision processes, which we shall also dis-cuss briefly. It requires an extension to p-branes of ananalysis of black-hole collisions developed in [5, 6], whichmade use of a remarkable time-dependent generalisation[5] of the static Majumdar-Papapetrou metrics [7]. Thecollision of extreme Reissner-Nordstro¨m black holes hadbeen discussed previously using a modulus-space descrip-tion [8, 9]. The importance of the work of [6] was thatat the expense of introducing a cosmological term, thedynamics could be studied exactly.The Kastor-Traschen solution [5] of Einstein-Maxwelltheory with cosmological constant Λ > 0 takes the formds24 = −H−2 dt2 +H2 dy2 , A = H−1 dt ,H = h t+Φ(y) , Φ(y) ≡∑kMk|y − yk| , (1)where h = −√Λ/3. Note that if the masses Mk are setto zero, the solution is de Sitter spacetime, and the ap-parent singularity at t = 0 is a mere coordinate arte-fact. (It corresponds to a null hypersurface separat-ing the maximally-extended de Sitter spacetime into twoantipodally-related parts, with t > 0 and t < 0. In fact,the involution t→ −t is the antipodal map.)When t is negative, the surfaces t = constant areeverywhere spacelike. Near each yk, there is an infi-nite throat which is very similar to the throat of theasymptotically-flat extremal Reissner-Nordstro¨m solu-tion, for which H = 1 + Φ(y). Far from the throats,however, there is a contracting k = 0 de Sitter universeds24 ∼ −dτ2 + a2(τ) dy2, with a(τ) = e−hτ = ht, and τgoing to +∞.When t is positive, however, one cannot go too faraway from the throats before the function H becomesnegative. In fact, the time-dependent locus H = 0 isa spacetime singularity. In [6], a detailed discussion isgiven of the locations of the horizons that form aroundthe black holes. The simplest case to consider would bethat of two black holes, of massesM1 andM2, with M =M1+M2 <14 |h|−1. This represents the head-on collisionof two black holes. As t approaches zero, one has twoouter apparent horizons which ultimately coalesce, anda single time-independent Reissner-Nordstro¨m-de Sitterblack hole is formed at late times [6].After the appearance of [5], Maki and Shiraishi ob-tained a general class of time-dependent multi-particlecharged solutions [10], coupled to a scalar field φ. In gen-eral, the scalar field couples to the cosmological constantin Liouville-like fashion. Some four-dimensional exam-ples were analysed in [11]. In some cases the cosmologicalconstant can be set to zero. The modulus-space metricsfor these multi-particle solutions were given in [12]. Theslow motion of NS-NS 5-branes was discussed in [13]. In[14], a particular case was used to study an expandinggas of D0-branes, allowing a comparison of the modulus-space approximation with the exact solution of [10].In order to discuss brane scattering, we shall lift someof the Maki-Shiraishi solutions to higher dimensions.Their Lagrangian in (N + 1) dimensions isL = R − 4N − 1 (∂φ)2 − e−4aφ/(N−1) F 2 − Λ e4bφ/(N−1) .(2)2We restrict attention to solutions described by Case (II)in [10], for whichds2 = −V −2(N−2)N−2+a2 dt2 + a2(t)V2N−2+a2 dy2 ,A = −√N − 12(N − 2 + a2)1√C (a(t))a2V −1 dt ,e−4aφ/(N−1) = C (a(t))2a2V 2a2/(N−2+a2) , (3)whereV = 1 +∑kµk√C (a(t))N−2+a2 (N − 2) |y − yk|N−2,a(t) =( tt0)1/a2,Na2− 1 = a2 t20 ΛC (N − 1) . (4)By contrast with the Kastor-Traschen case, the pres-ence of the scalar field allows us to take Λ = 0, by choos-ing a2 = N . In the notation of [15], this coupling strengthof the dilaton to the Maxwell field is characterised by theconstant ∆ taking the value 4. In fact ∆ = 4 is preciselythe coupling that one encounters, in any dimension, fora Maxwell field arising from the Kaluza-Klein vector in acircle reduction [15]. It is also the coupling strength forany field strength in maximal supergravity in any dimen-sion [15, 16]. Thus, for example, if N = 9 the vector fieldis the Ramond-Ramond vector of type IIA string theory,which may also be interpreted as arising from M-theoryvia Kaluza-Klein reduction [14].To construct colliding D3-branes we require N = 6,and the Maxwell field strength may be viewed as aris-ing from the 5-form of type IIB supergravity. Liftingthe solution from seven to ten dimensions, we obtaina time-dependent D3-brane solution, in which the onlynon-trivial fields are the metric and the 5-form, given byds210 = H−1/2 (−dt2 + dx2) +H1/2 dy2 ,F(5) = dt ∧ d3x ∧ dH−1 + ∗(dt ∧ d3x ∧ dH−1) ,H = h t+Φ(y) , Φ(y) ≡∑kMk|y − yk|4 , (5)where h is an arbitrary constant (which we shall take tobe negative, to parallel the discussion in [6]). Sendingt → t + 1/h one gets a standard supersymmetric staticD3-brane if h = 0. However, the time-dependent h 6= 0solution is non-supersymmetric.If the non-negative constantsMk, which correspond tothe 3-brane charges, all vanish, then we obtain a vacuumsolution of the Einstein equations, which is a generalisa-tion of the familiar Kasner solution,ds210 = (ht)−1/2 (−dt2 + dx2) + (ht)1/2 dy2 ,= −dτ2 +(3hτ4)−2/3dx2 +(3hτ4)2/3dy2 , (6)where we have defined hτ = 43 (ht)3/4. The singularity att = 0 is, by contrast with the de Sitter case, a true curva-ture singularity, and the Kasner metric cannot be contin-ued from negative to positive t. In the ten-dimensionalmetric the three spatial dimensions are increasing in sizeas t increases towards zero from an initial negative value.From the point of view of four dimensions, the interpre-tation is simplest in the Einstein conformal gauge, whichis achieved by writingds210 = e2αϕ ds24 + e−2αϕ/3 dy2 , (7)where α2 = 3/16. Introducing the Einstein proper-timecoordinate T = 3hτ2/8, the four-dimensional Einstein-frame metric becomesds24 = −dT 2 + (32hT )2/3 dx2 . (8)This is of the form expected for gravity coupled to a mass-less scalar field, which behaves like a perfect fluid with astiff-matter equation of state for which the energy den-sity equals the pressure. In the Einstein conformal gauge,the spatial 3-sections contract as T increases towards zerofrom an initial negative value; T = 0 is a big-crunch sin-gularity. If instead we run time backwards, the solutionrepresents an expanding universe with a big-bang singu-larity. The discussion above emphasises the point thatfour-dimensional physics should be analysed in the Ein-stein conformal frame.Now let us consider the case where the constants Mkare non-zero. Starting with t negative but increasingtowards zero, the solution represents 3-branes moving ina background Kasner universe, in which, measured in theten-dimensional metric, the transverse space contractswhile the D3-brane world volume expands. From thepoint of view of four dimensions, the situation is similarto the pure Kasner case discussed above, as long as onestays well away from all of the D3-branes, because therelevant conformal factor ise−2αϕ = (ht)3/2(1 +1ht∑kMk|y − yk|4)3/2. (9)At any fixed negative t the surfaces of constant timeare smooth, asymptotically flat and non-compact. Neareach 3-brane, y ∼ yk, the 10-metric is static, and well-approximated by the product metric on AdS5 × S5. Be-cause the transverse directions are inhomogeneous, thefour-dimensional interpretation depends upon where oneis in the transverse space.The situation as t goes to zero is more complicatedthan in the pure Kasner case. If t is exactly zero, weobtain a non-singular ten-dimensional configuration thatis not asymptotically flat, which may be thought of asa union of AdS5 × S5 components. An analogous solu-tion has been discussed for AdS2 × S2 in [17, 18, 19], inconnection with anti-de Sitter fragmentation.If the charges Mk are non-zero the solution contin-ues to exist in a neighbourhood of each 3-brane when t3FIG. 1: The level sets of Φ(y) for two mass points: Theuniverse after the collision is confined inside a level set Φ(y) >−ht. As time progresses, the level set ∂Dt shrinks and splitsinto two components, which then shrink around the two D3-branes.becomes positive, by contrast with the Kasner solution.Specifically, the metric isds210 = (−ht)−1/2( Φ(y)(−ht) − 1)−1/2(−dt2 + dx2)+(−ht)1/2( Φ(y)(−ht) − 1)1/2dy2 , (10)where Φ(y) is given in (5). For all positive times, themetric exists inside a domain Dt bounded by the levelset Φ(y) = −ht. Inside the domain Dt, but away fromthe D3-branes themselves, the transverse dimensions ex-pand while the three spatial dimensions contract evenwith respect to the four-dimensional Einstein-frame met-ric, if one stays at fixed y. However, if one moves in thetransverse space in such a way that Φ(y)/(−ht) remainsapproximately constant (and greater than 1), then thecorresponding four-dimensional universe has expandingspatial sections.At small positive t, the domain Dt is a large connectedball containing all the D3-branes. As t increases, the do-main Dt shrinks, and eventually splits into disconnectedpieces which shrink down onto each mass point. In otherwords, the universe splits into disconnected regions con-taining the individual D3-branes. Near each D3-brane,however, the metric is static, and again well approxi-mated by the product metric on AdS5 × S5.Consider, for example, the case of two mass points,which is illustrated in Fig. 1. At negative times the D3-branes are approaching one another in a head-on colli-sion, with the universe contracting about them. The D3-branes never actually meet, but rather scatter off eachother into a universe bounded by spacetime singularities.In the solutions we have discussed so far, the six-dimensional transverse space is flat. In more realisticexamples one might want to consider D3-branes movingin a Calabi-Yau manifold. In fact, the solutions given by(5) may be generalised to the case where the flat trans-verse metric dy2 is replaced by any Ricci-flat 6-metric,and H = ht+Φ(y) where Φ(y) is any harmonic functionon the Ricci-flat 6-manifold. In particular, we can takethe Ricci-flat manifold to be Calabi-Yau.The time-dependent D3-brane solutions that we ob-tained above admit a straightforward generalisation tothe case of an arbitrary p-brane supported by a (p+ 2)-form field strength with a dilaton coupling characterisedby ∆ = 4 in the notation of [15]. The relevant D-dimensional Lagrangian can be expressed asL = R− 12 (∂φ)2 − 12(p+2)! ecφ F 2(p+2) , (11)wherec2 ≡ ∆−2(p+ 1)(D − p− 3)D − 2 = 4−2(p+ 1)(D − p− 3)D − 2 .(12)The time-dependent p-brane solution is given byds2 = H−D−p−3D−2 (−dt2+dx2) +Hp+1D−2 gmn(y) dym dyn ,F(p+2) = dt ∧ dpx ∧ dH−1 ,φ = 12c logH , H = ht+Φ(y) (13)and h is an arbitrary constant. Here gmn(y) dym dynis any Ricci-flat (D − p − 1)-metric, and Φ(y) is anharmonic function in this metric. One might, for ex-ample, take the flat metric gmn(y) dym dyn = dy2 andΦ(y) =∑kMk/|y − yk|D−p−3.These solutions can be derived by lifting the Case(II) time-dependent Maki-Shiraishi solutions [10], settinga2 = N so that ∆ = 4. Alternatively, they can be de-rived by using T-duality symmetries to relate them topp-waves. We begin by noting that the pp-wave metricds2 = 2dx+ dx−+4(2hx+)2Φ(y) (dx+)2+gmn(y) dym dyn(14)is Ricci-flat for any Φ(y) that is harmonic in the Ricci-flat metric gmn(y) dym dyn. It is invariant under theboost given by x+ → λx+, x− → λ−1 x−. We maytherefore perform a Kaluza-Klein reduction on this boostsymmetry by first introducing coordinates (t, z), given byx− = t e12hz and x+ = −(2/h)e−12hz. This givesds2 = −H−1dt2 +H(dz +H−1 dt)2 + gmn(y) dymdyn ,(15)with H = h t + Φ(y). Reducing on the boost Killingvector ∂/∂z gives a time-dependent 0-brane. If one isstarting from D = 11 supergravity, this gives the D0-brane that was discussed in [14], which is the special caseN = 9 = a2 of Maki and Shiraishi.By performing a sequence of T-duality transformationsin the standard way (see, for example, [20]), this imme-diately gives us all the time-dependent p-brane solutionsthat we have presented above. A further extension is toconsider harmonic intersecting p-branes. It is straightfor-ward to see that any one (but only one) of the harmonicfunctions can be generalised from 1+Φ(y) to ht+Φ(y).A further generalisation is to apply a discrete elec-tric/magnetic duality transformation in four dimensions.4Consider, for example, the metric representing a maxi-mally electrically charged Kaluza-Klein black hole:ds24 = −H−1/2 dt2 +H1/2 dy2 ,F = −dt ∧ dH−1 , φ = 12√3 logH , (16)with H = ht+Φ(y). After making the duality transfor-mation we obtain the magnetic solution, with the samemetric and withF = dA = ∗3dΦ , φ = − 12√3 logH (17)where ∗3 denotes the Hodge dual in the flat 3-metric dy2.Lifting to five dimensions, we obtain the Ricci-flat metricds25 = −dt2 +H dy2 +H−1 (dz +A)2 . (18)The last two terms in (18) give a time-dependent se-quence of hyper-Ka¨hler 4-metrics. The interpretation ofthe metric (16) is that it represents an assembly of freely-expanding point particles in a background Friedman uni-verse with a scale factor a(T ) proportional to T 1/3. Inthe slow-motion approximation the motion is free andhence the branes separate linearly with time [21].It would be of great interest to study the stabilityof Horˇava-Witten brane models by finding exact time-dependent solutions; some preliminary results are re-ported in [22]. Here, we present a different collidingbrane-world model, which however captures the essenceof all brane-world instabilities. We take Φ(y) =M |y1| in(5), and reduce on (y2, . . . , y6), obtaining the 5-metricds25 = −(ht+M |y1|)1/3 (−dt2+dx2)+(ht+M |y1|)4/3 dy21 ,(19)which describes a negative-tension time-dependent 3-brane at y1 = 0, supported by a 5-form. If the solution isreflected about y1 = L, and if y1 = 2L is identified withy1 = 0, we obtain an S1/Z2 orientifold with an additionalpositive-tension brane at y1 = L. The proper distancebetween the 3-branes is 6[(ht+ML)5/3−(ht)5/3)]/(5M).This decreases monotonically as t increases to 0. Defin-ing T = 2(ht + ML)3/2/(3h), the Einstein metric in-duced on the positive-tension brane is of the Friedman-Lemaitre form (8), driven by a massless scalar field, i.e.the radion. At t = 0 a singularity forms on the negative-tension brane, and moves towards the positive-tensionbrane, causing the complete annihilation of the universeat t = ML/(−h), i.e. T = 0. This is the way the brane-world ends, not with a whimper but a bang.G.W.G. is grateful to Indrajit Mitra for helpful dis-cussions, and to the George P. & Cynthia W. MitchellInstitute for Fundamental Physics for hospitality duringthe course of this work.∗ Research is supported in part by DOE grant DE-FG03-95ER40917[1] G.R. Dvali and S.H.H. Tye, Brane inflation, Phys. Lett.B450, 72 (1999), hep-ph/9812483.[2] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok,The ekpyrotic universe: Colliding branes and the ori-gin of the hot big bang, Phys. Rev. D64, 123522 (2001),hep-th/0103239.[3] N. Turok, M.J. Perry and P.J. Steinhardt, M theorymodel of a big crunch / big bang transition, Phys. Rev.D70, 106004 (2004), hep-th/0408083.[4] L. McAllister and I. Mitra, Relativistic D-brane scatteringis extremely inelastic, hep-th/0408085.[5] D. Kastor and J.H. Traschen, Cosmological multi-black-hole solutions, Phys. Rev. D47, 5370 (1993),hep-th/9212035.[6] D.R. Brill, G.T. Horowitz, D. Kastor and J.H. Traschen,Testing cosmic censorship with black hole collisions,Phys. Rev. D49, 840 (1994), gr-qc/9307014.[7] S.D. Majumdar, Phys. Rev. 72, 930, (1947); A. Papa-petrou, Proc. Roy. Irish. Ac. Vol. 851, (191), (1947).[8] G.W. Gibbons and P.J. Ruback, The motion of extremeReissner-Nordstrom black holes in the low velocity limit,Phys. Rev. Lett. 57, 1492 (1986).[9] R.C. Ferrell and D.M. Eardley, Slow motion scatteringand coalescence of maximally charged black holes, Phys.Rev. Lett. 59 (1987) 1617.[10] T. Maki and K. Shiraishi, Multi-black hole solutionsin cosmological Einstein-Maxwell dilaton theory, Class.Quant. Grav. 10, 2171 (1993).[11] J.H.Horne and G.T.Horowitz, Cosmic censorship and thedilaton, Phys. Rev. D48, 5457 (1993), hep-th/9307177.[12] K. Shiraishi, Moduli space metric for maximally chargeddilaton black holes, Nucl. Phys. B402, 399 (1993).[13] A.G. Felce and T.M. Samols, Low-energy dynam-ics of string solitons, Phys. Lett. B308, 30 (1993),hep-th/9211118.[14] D.Z. Freedman, G.W. Gibbons and M. Schnabl, Ma-trix cosmology, in “The New Cosmology”, R.E. Allen,D.V. Nanopoulos, C.N. Pope, AIP conference proceed-ings, Vol. 743, 286, (2004), hep-th/0411119,[15] H. Lu¨, C.N. Pope, E. Sezgin and K.S. Stelle, Stain-less super p-branes, Nucl. Phys. B456, 669 (1995),hep-th/9508042.[16] M.J. Duff, R.R. Khuri and J.X. Lu, String solitons, Phys.Rept. 259, 213 (1995), hep-th/9412184.[17] D.R. Brill, Splitting of an extremal Reissner-Nordstromthroat via quantum tunneling, Phys. Rev. D46, 1560(1992), hep-th/9202037.[18] J.M. Maldacena, J. Michelson and A. Strominger,Anti-de Sitter fragmentation, JHEP 9902, 011 (1999),hep-th/9812073.[19] S. Ng and M.J. Perry, Brane splitting via quantum tun-neling, Nucl. Phys. B634, 209 (2002), hep-th/0106008.[20] E. Bergshoeff, C.M. Hull and T. Ortin, Duality in thetype II superstring effective action, Nucl. Phys. B451,547 (1995), hep-th/9504081.[21] P.J. Ruback, The motion of Kaluza-Klein monopoles,Commun. Math. Phys. 107, 93 (1986).[22] W. Chen, Z.W. Chong, G.W. Gibbons, H. Lu¨ andC.N. Pope, Horava-Witten stability: Eppur si muove,hep-th/0502077.

Crossref

https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/179514/document-1.pdf?sequence=1&isAllowed=y

Brane Worlds in Collision

Abstract

Similar works

Full text

Available Versions

Texas A&M University

Texas A&M Repository

Crossref

Brane Worlds in Collision

Abstract

Similar works

Full text

Available Versions

Texas A&M University

Texas A&amp;M Repository

Crossref

Texas A&M Repository