Branes are embedded surfaces in a given background (bulk) spacetime. Assuming
a warped bulk, we investigate, in analogy with the case for geodesics, the
notion of {\em focusing} of families of such embedded, extremal 3--branes in a
five dimensional background . The essential tool behind our analysis, is the
well-known generalised Raychaudhuri equations for surface congruences. In
particular, we find explicit solutions of these equations, which seem to show
that families of 3--branes can focus along lower dimensional submanifolds
depending on where the initial expansions are specified. We conclude with
comments on the results obtained and possibilities about future work along
similar lines.Comment: 11 pages, one figur

Kar, Sayan

English

arXiv

IACS Institutional Repository

IndHinJ /V/V.S 80(9),S7.V877 (2U()6)Focusing of branes in warped backgroundsSayiin KarDlikaiHuciiI oI tMiv^ics and Ccnlre ior Tlieorclical Sliulics, Indian Insiitiilc nl ’Icchnoingy, Kluiragpui-721 302, WcM Bengul, IndiaI',-mail sayan(f«Us mkgp crncl inV lisliatl - lirancs aie em bedded surlutcs in a given backgimmd (hulk) spaceiime Assuming a war|icd hulk, we invcsdgaic m iiniilogy .Mil, ilh L.JSI Uii geodLsus ihc notion ol i/ miii- oI laniilics ol siith embedded, exiieinal Vbranc'. m a live dimensional baekpround The I ■.I'lii.il Ion) behiiul mir analysis, is (he well-know n gencialised HayLliaudliuii cqualions for suilaLe Lonpiiicnccs In pariunla i we find ,s|>luii snhiimiis ol iliLse cqualions, w h ith  seem lo show  ihai l.im ilies of 3-braiies ean locus along low er dim ensional subm iinilolds ,1, ii, luliii " on wlieie llie iiiilial expansiuiis aie spetilied  We u in clu d t with eoiiimcnls on ihe resulls obtained iiiul possibililies aboul luiurc volt .iloiiL' similai linesK iss so rd s  k .iv i haul hill I equaim iis, em bedded su rla ie s  w aiped  spaLciim es I'U S N ils  . (M 20 -q 02 40 -kI. I n l io d u L l io nI Ik nniioii ol geodesic tocLisiiig diid ihe conditions under II Ldii ocLiii, lollow rmrn the well-known analysis die Kaschaudluin equation lor the expansion 0 (see !l, Ui loi (Liiihcr details) Let us brielly recall this in Didei lo set the background for further liisaissiDii ihc Raychaudhun equation (for limelikc L^i'iJisKs, assuming shear and rotation to be zero) isdo I .- f -0^  ---- d A  3(I)ii^ ie IS die tangent vector to the geodesic One di'ivts lioin ( 1 ) dial if £) < 0 tor some A = and if > 0, 0 ~ oo at a finite value of A (locusmg)ni.iy also convert the above equation into a second >rilei lorm by using 0  = 3{F'/r) In the second order form, looks loi ihe existence of zeros in F at limic A, in "i‘lei U) establish the focusing theorem The criterion for cxisiencc' ol zeros follows from the theory of ordinary ^^ '"-'^ ■ntial equations of the form F"(\) + H{a)F(\) = 0, "'‘I IS the same as mentioned at the beginning of this i’>iragrdph Through the use of the Einstein equialion in I 'convergence condition, > 0 one amves at theso called sliong energy condition and its variations (e.g. weak, null, dominant energy conditions) Physically, (ocusmg implies the rather simple fact that m an attractive giavilational field, the woildlmes ol te.st particles lend to converge and ultimately meet Focusing is therefore a pre­condition on the existence of spacetime singularities (31, though It can be completely benign, i.e. just a singularity in the congruence, without there being an actual spacetime singularity at the local point.Geodesies aie one dimensionnl surfaces m a given background Their generalisation to higher dimensional embedded surfaces is provided through the so-callsd minimal .surfaces. Such surlaces arise as the solutions of the vaiiational problem for the area action (Nambu-Golo). It IS therefore quite likely that the geodesic deviation (Jacobi) equation and the Raychaudhun equations will also have generalisations for the case of such minimal surfaces. This was achieved by CapoviHu and Guven in (41 Focusing in such a scenario (i e a congruence of surlaces as opposed to a congruence of curves [5]) is the mam topic ol discussion in this article We lestnct ourselves to simple examples In particular, we assume a five dimensional warped background in which we have a '‘our dimensional, limclike embedded surface This is the currently fashionable braneworld picture (6J, Wc shall enumerate the©  2 0 0 6  I A C S874 Silvan Katvarious ciitcria K'L|uiictl lo oblaiii icsulls foi Ihc focusing of such a congruLTKc ol tiinclikc hypcrsurlaccs in a given warped backgroLindIn ScLlion 2, we review ihe generalised Raychaudhun equaiions for surf.iee eongruentes In Seclion 3, we specialise lo 3-branes in a w.iipcd five dimensional backgiouiul and obiain solulions of ihc generalised ei|ualions binally in Seclion 4, wc piovide some remarks and LOiiclude willi commcnls on luluie directions2. The generalised Rayehuudhuri (Capovilla-(juven) ei|ualioiisIn Ihe case ol a family ol geodesic cuives the quantities lhat dial acidise a flow aie the expansion {()), ihc rotation (w,^) and Ihe sliear (rr,j) 'fbe expansion measures ihc rule of change of ihe cross-seclional aiea ol a bundle of geodesies If il goes lo negative mnnily wilhin a finilc value ol Ihe affine paramcler, we have focusing Shearing coiresponds to a change of sliape (ciicle lo ellipse, say) ol the cmss-scclional area and ihc lolalion implies a twist boi families ol surlaces these ciuanlilies aie icpiaced by a geneialiscd expansion a generalised shear T,„, and a geneialiseci lolalion riie index a corresponds lo Ihecocjidmaics ol the embedded sui lacc I bus, we have an expansion, a loiaiion and a shear alone each mdcpendeni clneclion on the surlace ll is obvious lhal these quaiUilies are nol mdependeni - they aic uiupled lo each othei and lhat is whal makes iheii evolution qiiilc dilficull lo study We shall assume sheai and rotation lo be zcro-this is possible undei eeitain conditions and wc take il lhal they aic salisfiecl 'riiiis, we h.ive only one cquaiion foi the geneialiseci expansion given by (41V"d 1 — Ofr' 1 (M- )' -  0," ^  - I) “(Ary, -  -fOne can convert the above equation to a second orcltr form by using and y = (N -  D )lnF This yieljj^finallyv ,,v "F + (M ’) ; r  = o (4,which IS a variable mass wave equation on the embedded surface Solving this with apprcipriatc initial conditio,i., will therefore provide the criteria for focusingIn an earlier papci |51, we had arrived at the criiencm for focusing of strings (two dimensional timelike world sheets, l-biane,s) in an arbitrary background It turned om lhal, using a theorem in the theory of partial differeniul equations, the existence ol zeros in F seem to be guaraiileed if the condition> 0 ,  (5, ^where 'K is the Ricci scalar of the worldshceiFor the case of 3-branes in a waqicd background of five dimensions, N -  D -  I and ihe quantity (M‘) Uirm oul to beIf the worldbiane hypeisurface is Hat, then the quantile further simplified and we have the cquaiion lor itii' expansion taking Ihc formV V"/ '  ^ A E^ “^E \F  = 0 |7t(2)w'heie N and I) are the dimensions ol the background and ihe embedded suifuce and(3)In the above expression, E^ l constitutes the tangent veeloi basis chosen such that («, I? runfiom I lo /)) aie the normals, with (/,/ lun from I lo iV -  D) Also = 0 K“'’‘ arc theN -  I) cxliinsic cuivalines (one along each normal dircciion) and is the Riemann tensor of thebackgioLind spaeciime Figure I provides a piclorial representation of ihc embedding in the A/ = 3, D = 2 caseFor background Finslem spaces with it is cas\lo sec that the nbc^ ive equation takes the simple formv„v"/-’ + 4 / \ r - oWe shall now tiy to analyse the abovemcntioncd equations and look for ways to understand and quantify ihe noiion of focusing of surfaces3. Explicit solutions and the meaning of focusingRecall lhal the background five dimcnsftinal line element ol a warped brancworld model [6J is given as(ts  ^ -  '^(r]^,f,(h''dx^) +dcT^.Here, ct is usually referred to as the extra dimension and the CT = constant slice is a four dimensional timelike hypcrsurlace which is the 3-brane and represents our four dimensional woildAssuming the tnvial embedding ■ f = fi, JT = A',, y = y,, z = Zi, CT =G^ y (constant) with (t^,x^,y^,z^) being the coordinates on the 3-brane, find lhat the tangents and normals take the simple formsFocusing ofbranes in warped backgrounds 875j f  '^ ‘^•’(1,0,0,0,0),2/'" -> '(0, 1. 0 , 0, 0),J- ’(0,0,1,0,0),‘' ‘‘^ "’(0,0,0,1,0),--(0,0,0,0,1) (11) lhL‘ inJiKcd metric on the brane is scaled Minkowski, leI Mil;: (he Ricci tensor components lor the background iiiH- e le m e n t ,  it is easy to write down the generalisedK.iVL.'ijLidhuii (Capovilla-Guven) equations We havedr+ V' F -4 /-"(c r ,)F :^ 0 , ( 12)$URrAC£j;i3HGPUENgE F jxal P9IPIfiLDSiNCALONC; a l in e FOCUSING TO A POINTHere, il IS worth noting that the non-trivial curvature of the live dimensional background makes its appearance only through the above (o -  k relation The geometric stability criteria discussed in [7], in addition, implies y '(n ‘(,)>0 The above solution foi F would lead to the following expressions lor the tour expansions f/J, Wc have0^  = (U tan {k.x ~ (01 + 0„),V 7) = , 7) +00)w'lriL' uc have swiiched back to as the,11 ludinaics on the 3-brane (instead of the (t|.V|.V|,r|) iKiilinm ci eailier) Normally, one docs not solve a Ray- Juadhiiii equalion m order to arrive at a (ocu.sing theorem. I l l , iiMial melhod is to look for zeros of F at finite values I du- pauirielers (heie /, v.v.r) or analyse the first order (I'Miinii However, the above equation, though a partiali . luniliont Ql / .i• ijMire 1 Till uip ii^ -ii fijiuic shows Ihc embedding ol surfaces for the -il l I'l ,1 iwn climcnsK)n,il one in a 3D background The other figuics 'I'limisii,iii‘ a suifuce congruence a focal curve and a focal poinl 'III' liL'iKcs aiL- all tiualiinlivc and do noi pertain lo any of llu* equalionsdillcrc-niial equation, is simple enough and we may solve " directly to get further insight. But, to solve the above '^dh:ili()M, we do need boundary conditions The central n, where do we impose those conditions l-et >^rsi tiy a simple .solution of the form .- Acos(^c.jr-aJM-0o) , (13)A and are constants. It is easy lo see that k and be related via a ‘dispersion relation’ of the form :(14)(15)(16)One immediately notices a couple of imporiani facts from the above(i) Assuming an initial condition defined on the2-biane given by the equation k . x - ( O i “ 0(inm« one can sec that there is a possibility ot a divergence to negative infinity (starting with a finite negative value ot 0,) in 0, as one approaches the 2-brane surtace k.x -  tur -1 0„ -  { n i l )  fl[, ^,, however, is [positive initially and divci^es lo positive infinity at k.x ~ u)t + 0„ -  { n il)(ii) Both 0, and cannot siiniiltaneously go tonegative mliimy because il one assumes a initially negative expansion (or one, the other has to be positive and vice veisaThe above discussion can also be carried out assuming the .solution for F as T = A cos (fc.jc + cut + 0,j). This will correspond to a time reversed situation The general results are similar with reversal of signs at appropriate places Alternatively, assuming a sinusoidal form or a linear combination of sines and cosines does not yield anything newer In either case, the point lo note is that if wc provide initial conditions on a 2-brane surface then the locusmg is on the 2-brancIf focusing IS defined lo be the approach lo negative infinity (at a finite value ol the coordinates that define the surface) of all the expansion vector components then we do not have focusing in this ca.se. In other words, giving initial values on a 2-brane does not yield a focusing in the sen.se delmed above However, one can also argue that being a vector field on the embedded surface (here, the 3-branc) the divergence of any one component would correspond lo a singularity in the surface congruence Thus, one may extend the usual notion of a focal point to include focal cur\>es and focal surfaces These features arc illu,slratcd in Figure 1 for the A/ = 3, D = 2 ca.seAnother worthwhile question to ask is ; can we have solutions such that (i) none of the expansions divergeH76 Sayan Karanywhcie, (ii) all the expansions diverge (to negative inrinity) somewhere The answer to (i) is straighttoiward Assume a solution ol the formF -  A cosh (k.x ~ (ot +0^,) (17)rhis, when substituted in the equation lor f \  will give a “dispelSion relation’ of the lormcu = -  4 /‘"(rr„)f'^^'""’. (I«)Here, wc require /"(cr,, )<() (which goes against the ciiterion of geometric stability discussed in (7)) m order to have anything mcamnglu) Furthermore, we need 4|/"(CT„ )| > A  ^ With these assumptions (if at all permissible) one can show that the expansions are all proportional to tanh (Ar.jc and, theiefore, theyare all finite everywhereThe answer to the second issue (n) is somewhat more involved 1o sec this, let us assume a solution m the lormF = TU)X(x) (lO)fhis will yield oidmary differential equations of the lormThese turn out to beCT = --yJoF + ruf tan |^tu“ +ai|" \ j ,  0^  = -(0^ lanrUn v,6 = yfio] +0)] ta n h |,j(v f^ a jv j.tanl^fu*" + 4 / "(fr„)f'^^‘"‘'T j ^(2^):Oh(M)(2);,7’ + (o>' +4J = ( 20)(21)(22)If (u^  = k- > 0 and /"(<r„) > 0. then both the solutions can be oscillatory and, typically, we have .where and are appropriate separation consiaius It IS easy to sec that m the above expressions for 6[^ ont (anno/ have all of them going to negative infinity even i! we assume all ol them to be initially negative In faci, oiiu of the components (here ft) never diverges Thus, proviilin;; a initially negative expansion vector field (at a given po iiiD  can lead to a focusing at a point though all the expansion coefficients will not diverge thereThus, the important lesson we have learnt is that ihn notion of locusing ol surfaces is quite different (as ii should be!) from the case ol geodesic curves The mLiin difference arises because ol the fact that there arc several coordinates (parameters) that define a surface as opfioscJ to the single parameter that is required to define a cLir\c The expansion, is thus a vector field defined on the surface and the divergence of its components at some value or set of values would indicate locusing Tlu' notion of focusing along a special submanifold U''-m otan[av+ (\)|. (23) illustrated above) depends on where (le. on whitli submanifold) we impo.se the initial condition We have(24) afso noted that all components of the vector field niiiv not diverge to negative infinityU)^  ^ - (25) In the above example, we assumed that the inducedF'rom the above, it is apparent that if wc impose an initially negative expansion at some t = r„ and on k.x + 0, = cl wc can have all expansions going to negative infinity at finite values of t and r.y.z The focal surface, m this case would be spacelikc and two dimensional One can go further and assume solutions of the form F ^ n t ) X ( x ) n v , z )  (26)or,/• =r(0A:(,v)y(y)Z(z). (27)In the first of these, one finds a focal curve (one dimensional) and the final one we have a focal point (zero dimensional) Let us evaluate the expansions for the latter.metric on the brane is fiat If this is iiot the case, then there are bound to be further complications Foi instance, one can think of two scenarios • (a) an induced, sialie sphciically symmetric metric on the brane, (b) an indued cosmological metric In both these cases, the equation for the quantity F would change drastically through the wave operator defined on the curved submanifold and uho through the intrinsic curvature of the induced metric general point made in this article about the role of w h e re  wc impose initial conditions and the possibility of focusing along all possible lower dimensional submanifolds h o w e v e r  remains unaltered, though explicit solutions will surely he very differentFinally, it is u.scful to try and see what can be saidFocii\inf> ofbranes in warped hackgroimds 877loLiising by looking al the lirst ordei equation .11 I he beginning ol the paper. Al a general level., assumes “ > 0 then we have\ /i * ----- ^ ~ N - n  " (32)\l,,ie s|veiriL.illy, (or the waqxid background with a trivially iMiKikled 3-brane (the context we have been discussing I, .^i (he lust ordci equation gives .(33)liK siand.ml analysis for geodesic locusing can be utilised In, ihe alH)\e c(|uation provided we assume that the RHS1,1 iiic above eqii.ilion is ncf^auvc II such is the case, in,,I uc L.in say (hat will go to negative infinity within ,, iimiIl \akie ol the parametei t We cannot, howevei,s.i, ,iiivihing concicle about the behaviour o( the other 'i<n - \ \.:) exiepl that they should be such that the lull i.'llS I'l ihe ab o v e  equation is negative4 ( rau hiding remarksLIS now conclude with a summary of the lesults■i^ naincil.\i III a waipecl background, a lamily of embedded huncs witlr an initial expansion vector field, can loLus along lower dimensional .submanilolds llimclike Ol spacc-hke) depending on where ( / e on winch submanifold) the initial expansions arc speed lediHi ii) II the initial expansion vector components arc '^ivcn on a timelikc 2-hrane then the surface Longruciice will have a future expansion vector w'lih some (hut not all) of its components diverging to negative infinity (ii) On the other hand, a s|iecilication of the initial expansion on a two dimensional .section of a spacehke hypersurface can give rise to a future expansion vector with allcomponents diverging to negative infinity, (in) Finally, specifying initial conditions on a curve (one dimensional) or a point (zero dimensional) will lead, once again, to a future expansion vector with some of its components diverging to negative infinity All of the above could be termed as analogs of the usual geodesic focusing though the situation 111 (ii) IS maikcdly diffeient from the one.s in (i) and (m). One is templed to con)ecturc (based on the example discussed) that for a D dimensional timchke. embedded hypersurface, all components of the expansion vector can diveigc on a 0  -  2 dimensional submanifold(C) As mentioned before, the above issues change quanlilalivciy il we have curvature on the brane or we have a non-trivial embedding oi if we look at F2uclidean .signatuie induced metrics Higher codimension bianes/surfaces involve furihci complications due to the presence of more than one normal vector A detailed treatment of all the above cases mentioned is a topic of future investigation and will be communicated in due courseHdercnii'sI 11 R M W.ild CJciii'Kil Krldin itv (Incli.i Ovcrscu'. Press) (2006)(2 1 h Poisson/\ We/onv/s/ 'l Inolkil fhf Maihrinaiic\ oj Himk Hole M filuniu  I (Carrihnclpe ( ’jmbiulge Umvcrsily Picss) (2004)|3 | .S W ll.iwking iind G P R Hllis Hic .Sc ale Stnutnrr ofSpaie- hniL (Cambridge rjiiibndpe University Press) (1973)14| R Capovilla and J Guven Hev 1)52 1072 (1995),L / j / in s 7  (uoiii Fh\\ 2H 271 ( 199H). It Carlei Contemp Math 20.1 207 (1997). .S Kar /*/iv> Kee l).54 6408 (1996) ilml D53 2071 (1996), 1)52 2036 (199S) S Kai Generahwd RayilumtUuin l\(ptalum\joi S tnn^\ and Memhuwe\ Wrillcn version of inviled lalk al IAGRG18 (1996) (IM.St, Chennai Rcporl no 117)151 SKarP/iv* Rev 1)55 7921(1997)|6 | L Randall and R Sundrum /'/iv» Rev Lett 83 3370 (1999), L Riiiidall and R Sundrum Phy.\ Re\ l,eli 83 4690 (1999) [71 S Pal and S Kar Llai\ Quant (jiav  23 2571 (2006)

Focusing of Branes in Warped Backgrounds

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/7595/1/Focusing%20of%20Branes%20in%20Warped%20Backgrounds_Sayan%20Kar.pdf

Focusing of branes in warped backgrounds

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