IACS Institutional Repository

Mohan, M

Maan, A

Jha, P

Control  of Molecular Dynamics in Intense Laser Fields

Indian J. Phys. 80(5), 561-562 (2006)
r Special Publications
♦Commemorative Volumes to mark the ^ lebration of 75th year 
of publication of Indian Journal of Phy^cs : Volumes 1-6
' 'I
1. Frontiers in Materials Physics 
Guest Editor: D Chakravorty 
Price : Rs. 600.00 (10 articles, 247 pi|ges, illustrated)
Date of Publication : 2002
Z  Frontiers in Astrophysics
truest Editor: Sandip K Chakrabarti
Price: Rs. 600.00 (12 articles, 303 pages, illustrated)
Date of Publication : 2002
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Guest Editors : S S Bhattacharyya and S C Mukherjee 
Price : Rs. 600.00 (15 articles, 429 pages, illustrated)
Date of Publication : 2003
4. Frontiers in High Energy Physics
Guest Editors : Amitava Raychaudhuri and Partha Mitra 
Price: Rs. 600.(X) (11 articles, 281 pages, illustrated)
Date of Publication: 2004
5. Frontiers in Condensed Matter Physics
Guest Editors : J K Bhattacharjee and B K Chakrabarti 
Price: Rs. 6 0 0 .0 0 ( 7  articles, 308 Pages, illustrated)
Date of Publication : 2005
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Guest Editors : T  P Singh and Chanchal Das Gupta 
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Guest Editors : J K Bhattacharjee and D S Ray 
Price: Rs. 500.00; US $. 50.00 (289 pages, illustrated)
Date of Publication : December, 2002
HI. 75th Year of Discovery of Raman Effect: A  Special Issue 
of India Journal of Physics [1928-2003]
Guest Editor: S Chakravorti
Price: Rs. 500.00; US $. 50.00 (164 pages, illustrated)
Date of Publication : February, 2003
IV Production of Radioactive Ion Beams [PRORIB-2001 ] 
Proceedings of the DAE-BRNS sponsored International 
Workshop
—A Special Issue of Indian Journal of Physics
Guest Editor: Alok Chakrabarti
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Date of Publication : May, 2002
V  Commemoration Volume on Professor K S Krishnan :
Birth Centenary [1898-1998]
Guest Editors: S Ray and C K Majumdar
Price: Rs. 100.00; US $. 4 0 ,0 0  (332 pages, illustrated)
Date of Publication : December, 1999
VL Physics and Technology of Accelerators : A  Special 
Issue of Indian Journal of Physics on the occasion of 
Golden Jubilee of India's Independence 
Guest Editor: Debashis Basu 
Price: Rs. 100.00; US $. 40.00 (236 pages, illustrated)
Date of Publication: December, 1999
Vn. Recent Trends in Statistical Physics : A Special Issue of 
Indian Journal of Physics 
Guest Editor: J K Bhattacharjee 
Price : Rs. 100.00; US $. 40.00 (140 pages, illustrated) 
Date of Publication : August, 1998
VIII. XI DAE Symposium of High Energy Physics : A  Special 
Issue of Indian Journal of Physics
Guest Editor: Ranabir Dutt and Asim K Ray
Price : Rs. 300.00; US $. 100.00 (395 pages, illustrated)
Date of Publication : February, 1997
IX. Lectures of Theoretical Physics Seminar Circuit 
[1990-92]; A Special Issue of Indian Journal of Physics 
Guest Editors ; C K Majumdar and Partha Ghosh 
Price : Rs. 80.00; US $. 35.00 (100 pages, illustrated) 
Date of Publication: November, 1993
X  Commemoration Volume on Professor Meghnad S a h a . 
Birth Centenary [1893-1993]
Editors : H Banerjee and S P Sen Gupta
Price : Rs. 140.00; US $. 40.00 (240 pages, illustrated)
Date of Publication : December, 1993
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Theoretical Physics Seminar Circuit [1989-90]:
A  Special Issue of Indian Journal of Physics 
Editors ; B K Chaudhuri and S P Sen Gupta 
Price: Rs. 80.00; US $. 30.00 (220 pages, illustrated) 
Date of Publication : March, 1992
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Our Special Publications

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Chatterjee, S

Chandra, P

On the Acoustic Deformation Potentials of some II-IV-V2 Ternary Chalcopyrites and their Mixed Versions

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680 Notes for Contributors
(iO Title
T h e  title shou ld  be b r ie f  and yet con vey  to the in fo rm ed  reader, the particu lar nature o f  the contents o f  the paper.
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T h e  n u m e(s ) o f  the au th o r(s ) and  a d d ic s s (c s ) o i 'th c  m stiiiilc w h ere  the research  w o ik  w a s  don e, sh ou ld  be indicated. T h e  nam e ot the 
author to w h o m  correspondence  is to be  addressed , shou ld  be  underlined  T h e  present a d d re ss (c s ) o f  the a u lh o r (s ), i f  it is d ifferent, m ay  
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International con ven tions in the c h o ice  o f  sy m b o ls , units and notation shou ld  be fo llo w e d .
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and be presented  as fo l lo w s  :
[c ]  U  F an o  and A R P  R ao  Atomic Coliisions a n d  Spectra (N e w  York  : A c a d e m ic ) V o l  1, C h  2, S ec  4, p  25 (1 9 8 6 )
[7 ]  T  A tsum i, T  Isihara, M  K o y a m a  and  M  M a tsu z a w a  Phys. Rev. A 4 2  6391 (1 9 9 0 )
[1 1 ] T  L e -B ru n , M  L a v o lle d  and  P  M o r in  X-ray an d  Inner Shell Processes ( A l P  C o n f.  P roc . 2 1 5 ) eds T  A  C a n so n , M  O  K rause  and  
S M an so n  (N e w  Y o rk  : A l P )  p 846  (1 9 9 0 )
[1 4 ] S B  H ansen , K  B  M a c A d a m  and  L  G  G ra y  J2fh hit. Conf. on Atomic Physics (A n n . A r b o r )  A bs trac ts  p x - 12 (1 9 9 0 )
f 15] H  P au ly  Atomic an d  M olecular Beams Methods eds G  S co le s , D  B ass i, U  B u ck  an d  D  L a in e  (L o n d o n  : O x fo rd  U n ivers ity  P ress ) p 83
(1 9 8 8 )
[19] W Fritsch (private communication) (1988)
[2 1 ] K  B  M jw ^Adam  (to  be p u b lish e d ) (1 9 9 1 )
[2 3 ] T  R o y  PhD Thesis (U n iv e rs ity  o f  C a lcu tta , In d ia ) (1 9 9 2 )
(xi) Footnotes
A s  fa r as p oss ib le , footno tes sh ou ld  be  a v o id e d
(xii) Appendices
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Notes fo r  Contributors 68 J
MPORTANT I n s t r u c t i o n s  t o  t h e  C o n t r i b u t o r s
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IV( *0(8), 851-852 (2006)
Special Publications
♦CommemoriUive Volumes to mark the celebration of 75th year 
of publication of Indian Journal of Physics Volumes 1-6
1 Frontiers m Materials Physics 
Guest Editor DChakravorty
Puce Rs 600 00 (10 articles, 247 pages, illustrated)
Date of Publication 2002
2 Fiontieis in Asliophysics
Guest Editor Sandip K Chakrabarii
Puce Rs 600 00 (12 articles, 303 pages, illustiaied)
Date of Publication ■ 2002
3 Fionliers m Atomic, Molecular & Optical Physics 
Guest Editors S S Bhattacharyya and S C Mukherjec 
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Date of Publication 2003
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Date of Publication 2005
6 Fiontiers in Biophysics
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852 Our Special Publications
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Golden Jubilee ol India's Independence 
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VIII XI DAE Symposium of High Energy Physics AS|y 
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Part, P h ysics
ii iiy hikI a paranielcnzalion orprolon slmuure lunclmn al small \ 
V CiioiiJirnRV N^i 1 Rum'on Gocioi
ulialive ori^ Mii 111 .'tnd revisileti
823-827
82^ -83*5
i l iL  h e a l o l l in c L ir  La n g e n l h a r d  s p h e re  c h a in s  
f i\'aii K u m a ; V e rm a , R a ja l  S r iv a s la v a  im d  K  N  K h iu i i ia
'ritv.:’ i'v propeJlics ol mixed jilyuiie lluondecryslals 
M  M  K i ia n d i ’Ia a k
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i!,‘\ fn) I oiiinbutors
837-839
H4I-84T
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847-830
831-832
I Published hy Ihe Regislriu", Indian Asscx:iatK)n tor the Cullivalioii ot Science, 2A & 2B, Raja Suhodh CliLindra Ma
I K  w t i n n n ' i ' i  . \ / i / . a a  7 o n i n . i i u


English

Kar, Sayan

IndHinJ /V/V.S 80(9),S7.V877 (2U()6)
Focusing of branes in warped backgrounds
Sayiin Kar
DlikaiHuciiI oI tMiv^ics and Ccnlre ior Tlieorclical Sliulics, Indian Insiitiilc nl ’Icchnoingy, Kluiragpui-721 302, WcM Bengul, India
I',-mail sayan(f«Us mkgp crncl in
V 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 lines
K 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 -k
I. I n l io d u L l io n
I 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) is
do 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 the
so 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 pre­
condition 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 S
874 Silvan Kat
various ciitcria K'L|uiictl lo oblaiii icsulls foi Ihc focusing 
of such a congruLTKc ol tiinclikc hypcrsurlaccs in a given 
warped backgroLind
In 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 directions
2. The generalised Rayehuudhuri (Capovilla-(juven) 
ei|ualioiis
In 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 Ihe
cocjidmaics 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 (41
V"d 1 — Ofr' 1 (M- )' -  0,
" ^  - I) “
(Ary, -  -f
One can convert the above equation to a second orcltr 
form by using and y = (N -  D )lnF This yieljj^
finally
v ,,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 focusing
In 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 worldshcei
For the case of 3-branes in a waqicd background of 
five dimensions, N -  D -  I and ihe quantity (M‘) Uirm 
oul to be
If the worldbiane hypeisurface is Hat, then the quantile 
further simplified and we have the cquaiion lor itii' 
expansion taking Ihc form
V 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? run
fiom I lo /)) aie the normals, with (/,
/ lun from I lo iV -  D) Also = 0 K“'’‘ arc the
N -  I) cxliinsic cuivalines (one along each normal 
dircciion) and is the Riemann tensor of the
backgioLind spaeciime Figure I provides a piclorial 
representation of ihc embedding in the A/ = 3, D = 2 
case
For background Finslem spaces with it is cas\
lo sec that the nbc^ ive equation takes the simple form
v„v"/-’ + 4 / \ r - o
We shall now tiy to analyse the abovemcntioncd equations 
and look for ways to understand and quantify ihe noiion 
of focusing of surfaces
3. Explicit solutions and the meaning of focusing
Recall 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 woild
Assuming 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 forms
Focusing ofbranes in warped backgrounds 875
j 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, le
I Mil;: (he Ricci tensor components lor the background 
iiiH- e le m e n t ,  it is easy to write down the generalised
K.iVL.'ijLidhuii (Capovilla-Guven) equations We have
dr
+ V' F -4 /-"(c r ,)F :^ 0 , ( 12)
$URrAC£j;i3HGPUENgE 
F jxal P9IPI
fiLDSiNCALONC; a l in e FOCUSING TO A POINT
Here, 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 
have
0^  = (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 partial
i . 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* equalions
dillcrc-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 the
2-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 to
negative mliimy because il one assumes a initially 
negative expansion (or one, the other has to be 
positive and vice veisa
The 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-branc
If 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.se
Another worthwhile question to ask is ; can we have 
solutions such that (i) none of the expansions diverge
H76 Sayan Kar
anywhcie, (ii) all the expansions diverge (to negative 
inrinity) somewhere The answer to (i) is straighttoiward 
Assume a solution ol the form
F -  A cosh (k.x ~ (ot +0^,) (17)
rhis, when substituted in the equation lor f \  will give a 
“dispelSion relation’ of the lorm
cu = -  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, they
are all finite everywhere
The answer to the second issue (n) is somewhat more 
involved 1o sec this, let us assume a solution m the 
lorm
F = TU)X(x) (lO)
fhis will yield oidmary differential equations of the 
lorm
These turn out to be
CT = --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 there
Thus, 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 infinity
U)^  ^ - (25) In the above example, we assumed that the induced
F'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 different
Finally, it is u.scful to try and see what can be said
Focii\inf> ofbranes in warped hackgroimds 877
loLiising 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 RHS
1,1 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 negative
4 ( rau hiding remarks
LIS 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 led
iHi 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 all
components 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 
course
Hdercnii's
I 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

Banerjee, Narayan

l^ianJ. Phys- « K U ) .  1097-1100 (2006)
Accelerated evpansion of the universe and scalar tensor theory of gravity
N arayan Banerjee
Relativity and Cosm ology Research Centre, Department of Pliysics, Jadavpur University, 
Kollcata-700 032, India
Abstract ; Tlie possible role o f a scalar field, non minimally coupled to gravity, in the search for dark energy is very briefly reviewed 
Some possibilities in the framework o f Brans-Dickc theory are discussed
Keywords : Brans-Dicke iheoiy, scalar field
PACwS N os. ■ 95 36 +x. 95 30 -k. 04 20  -q
1. Introduction
Observations now  have firm ly established that our universe 
IS expanding at an accelerated ra te  The initial indications 
ume from lum inosily-redshift surveys o f som e type-la 
sii|icr-novae f l] ,  w hich show ed that they appear to be 
dimmer than they should have, requm ng  a negative value 
ol ilie dim ensionless deceleration  param eter q. This 
iwameler is defined as q = -{adld^), a being the scale 
'aaor of the universe, a m easure o f the linear dim ension 
111 llie universe This result has later been firm ly supported 
bv oUier h igh  precession data [21- 
This defies all intuition as the dynamics o f the universe 
i'' governed by gravity and gravity  is attractive The 
I'-iiural outcom e o f this observation is indeed an urgent 
!‘carch for some field w hich can give n se  to an accelerated 
(expansion Certainly a host o f different kinds o f  fields 
aipear and cla im  their suitability  as a driver o f this 
aaderalion. As tiKist o f  the m atter present m  the universe 
^ilh the norm al attractive gravitational [nxipaty is invisible 
hence called the dark  matter, this new  tvand o f m atter 
'^ ilh a wrong sign o f gravity is genencally  called the D ark 
A m ongst the dark  eneigy  candidates the first and 
perhaps the m ost talked about one is the rejuvenated 
cosmological constant A. The other exam ples are a scalar 
l^ 'elcl w ith a  positive potential (this form  is genencally  
called the quintessence field), a  C h ^ ly g in  gas, non linear
contribution o f the R icci curvature scalar in E inste in ’s 
equations, and many others. There are excellent review s, 
for exam ple, see [3] In m ost o f the cases the idea is to 
take advantage of the fact (hat unlike ui Newtonian gravity, 
in general relativity  {x-essure also plays a sim ilar role as 
m atter density in  the determ ination o f the gravitational 
intensity. As our universe is hom ogeneous and isotropic 
and hence is correctly described by a Robertson-W alker 
m edic, Einstein equations look like
^ n G p
and
2 - + - = —^ G p ,
(1)
(2)
where p  and p  are the density  and pressure o f  the m atter 
present, G  is the N ew tonian constant o f gravity and k is 
the curvature index having possib le  values am ongst 0  and 
±1. A n overhead dot represents differentiation w ith respect 
to  the cosm ic dm e t. These tw o equations w ill yield
4;rG
(p  + 3p). (3)
Obviously, if  the pressure is sufficiently negative, so 
that p+  3p < 0, one can have  a  negative value o f q. T h e  
dark energy m odels are such that t l^ y  produce an effective
© 2 0 0 6 IA C S
1098 Narayan Banerjee
negative pressure, which drives the accelerated expansion, 
L e  , gives nse to a negative value ol q
Observations further tell us that the accelerated 
expansion is rather a recent phenomenon, having st:irted 
at quite a matured age of the universe when it Uxiked very 
much like today, i .e ., dominated by matter like galaxies and 
dark matter |41 Tins observation indeed came as a relief 
as only a decelerated expansion provides the pertect 
ambience tor the synthesis o f matter in the form of atomic 
nuclei in the radiation dominated era as also for the 
tormation of galaxies in later stages
Tile dark energy candidates all serve the primary 
purpose, i .e ,,  they produce an accelerated expansion dunng 
the matter dominated era. some of them can indeed pnxluce 
the signature llip  ot q  in a recent past but perhaps none 
o f them hits a sound Uieoretical basis regarding why it 
should lie there
In tlie absence o f a final vertlici in favour ol any of the 
dark energy models, scalar tensor theories also stake Uieir 
claims as a possible arena where the answers to this 
challenge can be sought Amongst scalar tensor theories, 
Brans-Dicke theory 15] has been successful to have an 
interest alive for some reasons One important ap^ Teal o f 
this theory had been that it  believed to have general 
relativity as a lim it o f this theory for a very high value ol 
a dimensionless parameter (o This has now been shown 
to be true only in a restricted sense [6], but the theory 
proved iLself to be useful in finding solutions to many a 
cosmological problem For exarrqile, when the inllationary 
nxxlels were all struggling to lind a solution to the graceful 
exit iiroblem, Braiis-Dicke theory came up with the idea of 
an extended inllation 17] Tlie dark energy problem also 
has some solutions in Bnuis-Dicke theory or in some 
nxidification o f tlie same In this article, we shall brie lly  
descnlie some ol the attempts towards tins This is by no 
means exhaustive Only some work w ill be descnbed where 
the present author had been involved in The next section 
descnbes some work in the theory piroper and in the third 
section investigations in some modification ol the theory 
w ill be talked about
2. Acceleration of the universe in Brans<Dickc theory
The lie ld  equations in Brans-Dicke theory along w ith a 
quintessence scalar field y / w ill be
+  k  1 \  ^d(l)
2  ad> (t)'
where is the density o f normal matter and p  aand;
are the density and pressure due to the quintessence 
scalar held ^  and they are given by
P v  = -V '^ + V ( i/ r )
and
(7)
The equations are wnlten m units where 8 ;rG  = 1 V'(^ ) 
is the potential o f the quintessence field  Both the scal,v 
fields 0 and w ill satisfy the respective wave equalior;s 
in view o f which the field equations w ill yield th^  ^
conservation ot matter as
p„ I'leing a constant The pressure o f the normal matter ih 
taken to be zero so as to l:>e consistent w ith the prescni 
matter dominated universe This set o f equations indeed 
gives vanous possibilities [8]. For different choices ol the 
potential V {y/), the model can yield power law soluimiis 
for the scale factor a  as 
a  -  flof"
where n > 1 so that q = - ( n - l ) / n  has a negative value 
For examjile, lo r a spatially flat universe (k  = 0), V{\i/) = 
V q \I/^ yields n  = 4/3, l e . ,  q  = -1 /4 . For this rale ol 
acceleration the model works lo r the entire time span, l e ,
0 < / < oo For a faster rate o f acceleration, however, the 
model works only upto a finite  future, 0 < / < The 
model can produce non-decelerating solution with nonzero 
spatial curvature {k 0) as well, but those are consistenl J 
only w illi q -  0, i . e , d coasting universe
As a bye-product o f tlie model, it is also shown dial 
w itli some choices o f the parameters, A: = 0 is an attractor, 
i.e , the tlatness problem has some solution in this model 
The most attractive feature o f working in Brans-Dicke 
theory is that it  can produce an accelerated expansion for 
the spatially fla t universe even in the absence of any 
qumlessence fie ld  [9]. I f  we put and equal to zero 
in  the fie ld  equations, a power law soluUon for 
factor can be obtained as a = Oof" where
(4) For 0) = -5 /3 , the deceleration parameter q
Consistency o f the equations demand that - 2 <  o) <
= -1/4 
-3/2
Accelerated expansion o f the universe and scalar tensor theory offfravity 1099
jjid ihis range of values for o) yields different rales of 
aLCcleraiion This result is quite interesting as the desired 
,i,LeleniUon can be driven by the Brans-Dicke scalar field, 
,^l„ch IS already there in the purview o f the theory and 
.me does not have to invoke any quintessence field by 
h iJid
bul the problem in these models are two-fold Tliey 
only an acceleration, the relevjuit values o f eo fail to 
,linw a signature flip  in q  in the matter dominated era The 
scloiiJ problem is that tliese accelerated nxidels all need 
,1 vtry low negative value o f o), o f the order oi unity 
WhcrtLis local astronomical observations clearly indicate 
[\u\ this value should be very high [101
} ALii'lcrulion in modified Brans-Dicke theories
Br.ms-Dicke scalar held is non mininuilly coupled to gravity 
,uul has an interference term with the Ricci scalar R  in the 
u'lion as follows
A : l [ ( l > R + l ,„ ^ P g ) d * X ,
_ 2{U + 3,2
3— = p +-------a- ^  4 - (10)
2« + 3.^ 2 
w 4 ’ (11)
p ^ 3 I I p = Q  (12)
An overhead bar obviously denotes quantities in the 
transtormed version and A = lii(i) It can be shown that i f  
we lake e/j to l)e a constant, llien one can have accelerated 
expansions, bul there is no sign llip  in g like tlie mLxlels 
in the previous section Bul i f  the theory is generalized to 
incorporate a varying aj [11|, there is obviously the 
possibility that the universe enters Uie accelerated phase 
only in the recent past |12J The scale laclor a is not a 
simple power lunclion of time, and g is given by
g=^-l + 3 + 1/
(13)
(8)
wIkic IS the lagranguui for any held other than the 
Br.uis-Dicke field It is amply clear dial 0  does not have 
.i!iv direct interaction with matter ol any lorm  Now if the 
ilii'iry IS m otiilied so that 0 \ ya& an iiUeraclion w ith  matter 
i'- Mill, one arrives at niiiny possibilities Two examples 
uill be described in this section in case o f a spatially flat 
iimveise (k  = 0)
It It IS assumed that the universe is filled with a 
piussureless llu id  consistent w ith the jiresent universe but 
itit mailer does not conserve itself and rallier has an 
iiiiuaLlH)n with the Brans-Dicke field, then the conservation 
i^ qualion ol matter becomes
n + 3 U f )  =  Q ,  (9)
where Q  lakes care o f the interaction between mailer and 
ihc scalar field and actually is a measure o f tlie rale at 
'diieh energy is pumped m from one fonn to Ihe other Q  
= 0 obviously indicates an absence of interaction Tlie  
*'dd ecjualions in this case is rather involved Bul w ith a 
unitormal transformation fhe equation system
more tradable like
the matter consca^aUon equation becoiws
where r / is  small dimensionless constant Tlie deceleration 
parameter as given by this equation indeed has a signature 
ll ip  in the recent past The era at which g enters its 
negative phiise is seen to be not cnicially sensitive to the 
value o f (X.
Tile problem ol this model is that the interaction 
between the scalar filed rmd matter is not well undersUxid 
But this indeed opens up a possibility, as this model does 
not require a stringent restndjon on the value of aJ, and 
even high values ol oj  can be quite consistent w ith an 
accelerated exiiansion o f the universe
Another interesting tx)ssibilily is to use a quintessence 
scalar field in Brans-Dicke theory as described in tlie 
previous section bul with an interaction between 0  imd 
the quintessence field ^  which slowly roles down a 
potential V  =  V (\f/)  The interaction is such that the 
elfective potential looks like
= 0 '*V (v /) ,  (14)
SO that the Brans-Dicke field, winch is already there in tlie 
theory, helps universe enter the accelerated phase quite 
late in the history o f the evolution. Here, is a constant. 
This potential IJ replaces V(vr) in eqs (4) and (5) The 
motivation comes from the fo llow ing example It is quite 
well known Uiai an innaUonary expansitxi. i.e., an accelerated 
expansion o f the universe was required in the early phase 
o f the evolution so as to solve certain problems like that 
o f flatness (or line tuning) and horiziin problem amongst 
others The problem o f these inHationary models was that 
of a graceful exit, how the universe can get out o f this 
inflalionary expansion to enter a more sedate decelerated 
expansion. The idea of an extended inflation came out in
noo Narayan Banerjee
in v e s U g a lio n s  in  B r a n s - D ic k e  th e o ry  w h e r e  a  s c a la r  f ie ld ,  
s lo w ly  r o ll in g  d o w n  its  p o ten tia l, a m id  in d ic a te  the  |X )ss ib le  
s o lu t io n  o f  th is  g r a c e fu l e x it  p n ih le m  [7J It w a s  a lso  
s h o w n  fhat a  p o te n tia l  a s in  e q  ( 1 4 )  c o u ld  in d e e d  p r o d u c e  
an in fla t io n  in  th e  ea r ly  s ta g e  fm d tlie  in tla to n  f ie ld  h ad  an  
o s c il la t io n  d u r in g  th e  la ter  p h a s e s  m a r k in g  an  e n d  to  th e  
in f la t io n a r y  s ta g e  [1 3 ]  T h e  r e c e n t  p r o b le m  is  j u s t  th e  
r e v e r s e  o f  that, a  g r a c e fu l en try  s o  to  sa y . w h e r e  th e  
q u in t e s s e n c e  f ie ld  s h o u ld  h a v e  an  o s c i l la t o r y  b e h a v io u r  in  
th e  b e g in n in g  and  g r o w s  d u r in g  th e  la te r  s ta g e s  to  m a rk  
the o n s e t  o l an a c c e le r a te d  e x p a n s io n
It is  s h o w n  llia l in d e e d  th e  B r ju is -D ic k e  l lie o r y  c iu i 
g iv e  r is e  to  th is k in d  o f  a s itu a t io n  W ith  s o m e  c o n d it io n s  
o n  tlie  c o n s ta n ts  o l  the m ix lc l  and  s o m e  arbitriiry c o n sU u ils  
o f  in te g r a t io n , th e  q u in t e s s e n c e  h e ld  y/ c a n  h a v e  tui 
o s c il la to r y  b e h a v io u r  to  start w ith  a n d  g r o w s  la te r  to  d r iv e  
the a c c e le r a te d  p h a se  114] F o r  a n  e x p o n e n t ia l ly  e x p a n d in g  
u n iv e r se , th e  pote iU n il V s h o u ld  b e  an e x p o n e n t ia l  I u n c tio n  
o l {// H o w e v e r , lo r  a p o w e r  la w  e x p m is io n  w ith  n > 1 as  
in  th e  p r e v io u s  s e c t io n ,  th e  lo r m  o f  d ie  fKitentKiJ V  c o u ld  
b e  q u ite  arb itrary  T h is  is  q u ite  an  i i i le r e s lin g  fe a tu r e , as 
th e  lo r m  o f  th e  p o te n tia ls  u s e d  to  d r iv e  th e  a c c e le r a te d  
e x p a n s io n  o f  th e  u n iv e r se ,  as m e n t io n e d  ea r lie r , a re  n o t  
b.'ised o n  a scm nd  b a c k g r o u n d  p h y s ic s ,  h e r e  th is  fea tu r e  
IS g e n e n c ,  q u ite  in d e p e n d e n t  o l the  c h o ic e  o l the  p o te n tia l  
A ls o ,  th e  v a lu e  o f  O) is n o t  n e c e s s a n ly  r e str ic te d  to  a v e r y  
lo w  v a lu e , s o  th e  th eory  can  w o r k  fo r  th e  lo ca l a stro n o m ica l  
r e su lts  as w e l l .
Acknowkdgniciits
I w o u ld  l ik e  to  th an k  P r o f  J a y a n ta  B h a tta c h a iy a  for giving 
m e  th e  o p p o r tu n ity  f o r  p a y in g  th is  s m a ll  in b u ie  to our 
b e lo v e d  te a c h e r  A K R
R eferen ces
11 ] S PerimuUcr e t  a l ,  A j i r p h y s  J  517 565 (1997). A G Rjess n  ai 
A s r r r m .  J  116 1009 (1998)
121 S Brjdle, O LJiav, J P Oslnkcr and P J Sleiliardl .Srirnir 
1532 (2003)
13] V Salmi lAsiro-pti/0403324], T Padnianablian V h y s  R q ,
235 (2003)
14] A G Riess lAslro-ph/0104455)
[5] C 11 Dran.s diiil R 11 Ditke P h y s  R e v  124 925 (1%1)
16] N Hiuierjec and S' Sen P h y s  R e v  D56 1334 (1997), V Ijmoij
I b , d  59 084021 (1999) \
17] C Malliiazlingan and V D John C l a s s  Q u a n tu m  Grnv I 1 2'j 
(1984), IJ t.u and P J Sleinliardl P h y s  R e v  U f i  62 3/(i
18] N Danurjec and D Pavon C la s s  Q u a n tu m  ( J r a v  18 593 (’OOi
19] N Baneijec and D Pavon P h y s  R e v  1)63 043504 (200lj
110] CM Will T h e o r y  a n d  E x p e r i m e n t  m G r a v i t a t i o n a l  /’/juk', 
(3rd edn) (Cambndge Camhndge University Piess) (1993]
[11] K Nordivedl Aj/ropV J  161 1059 (1970)
[12] S Das and N Banerjee G e n  R e l a t i v  G r a v i i 3H 785 (20061
[13] N Banerjee and B Ram P h y s  Lett A  229 83 (1997)
114] N Bancriee and S Das M o d  P h y s  L e t t .  A  (in prcs.s) Igriji; 
0605110]


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Control of Molecular Dynamics in Intense Laser Fields

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