5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

1/9

CERN-TH.4973/88February 1988

CERN-TH.4973/88

BEAM-LIKE GRAVITATIONAL WAVES AND THEIR GEODESICS

V. Ferrari and P. PendenzaInternational Center for Relativistic Atrophysics-ICRADipartimento di Fisica "c. Marconi", Dni.vesi.ta di Roma

and

G. VenezianoCERN -- Geneva

A B S T RAe TExact solutions for the gravitational wave pro-duced by an impulsive, massless beam of arbitraryenergy profile are constructed in any number ofspace-time dimensions. Geodesics can be explicitlycomputed and, for homogeneous, axisymmetric,finite-size beams, they exactly focus at the loca-tion of the curvature singularity found in theinfinite shell collision.

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

2/9

1. Introduction. In 1971 Aichelburg and Sex1(AS) [1] found anexact solution of the classical Einstein equations, describing theplane fronted, impulsive wave generated by a massless particle ofgiven energy E (pz=p=E).

Recent1y, Dray and '1 Hooft, using new techniques, were ableto rederive (2) the AS metric and to extend it [3] to the case of aplanar shel1 of null matter with constant energy densHy along Uswavefront. They also constructed [3) the exact solutioncorresponding to the head-on col1ision of two such waves, exibltingsingularities analogous to those found earlier [4,5] in the collisionof sourceless plane waves.

In this note we shan first extend, almost trivially, the workof Dray and 't Hooft to the case of waves produced by a masslessbeam of arbitrary energy distribution on the wavefront (hyper)plane. Having in mind further studies of string-theoreticcorrections, we shall also extend our calculations to any number ofspace-time dimensions.

Then, as a preliminary to the study of the fully-fledged wavecollision problem, we consider the shape of null geodesics for thegeneral axisymmetric case. We are able to find explicit solutionsfor the geodesics and to discuss some of their amusing properties.In particular, we find that the wave associated with a beam offinite size and constant energy density p acts as a perfectconverging lens for massless (null) incident test particles(geOdesics) wHh a focal distance given by:

tF = -zF = (0/2-1) (BuG p r' (1)where GIs Newton's constant and c=1 throughout this paper.

This expression, which agrees completely w ah 6 recentlow energy dens1ty result by one of us [61. also cotnctaes. for anyD*)' with the location of the curvature singularity found inrefs.[4,5,3L re1nforcing the beUef [7] that focusing ts a crucial1ngredlent In the generation of such stngulertttes.

*)The results of ref. 3 for the location of the singularitiescan be extended to ertntreru D1na straightforward way.

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

3/9

2

2. Beam-shaped waves. Following Dray and 't Hooft [2] welook for a metric of the form:

ds2 = -dudv -tf{x,y)8(u) du2 +dx2 -tdi (2)corresponding to an impulsive weva m o v t n q along the posittve zaxis. Here uet-z, y=t+z and the weva-tront is u=O. For theD-dimensional case we genera1ize the ensatz (2) as:

ds2 = -dudv +f(K)8{u) du2+d~ (2')where K 1s the set of (D-2) transverse coord1nates.

Fol1ow1ng eq etn r er.lz l, one ttnds the only non ventsmnqcomponent of the Ricci tensor to be:

Rw = -1/2 L lf 8(u) (3)

where ~ is the d'Alembertian in (0-2) dimensions. Einstein'seouetions then require:Llf 8(u) = -161T G T uu = -16n G p(K) 8(u) (4)

where we neva used the known expression of the stress-energytensor for an impulsive null source haying energy density p ( K > onthe weve-tront. Eq.(4) is immedh:~tely solved for arbitrary p(Kl,since one knows the Green's functions of the problem:

I J . lo g r = 2 ' T T cP)(x,y)b. r 4 - D 1(4-0) = ~2 s I D - 2)(K> (5)

in which Qd = 2n-d/2/r(d/2) is the solid angle in d-dimensions and,2 = 'Il.

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

4/9

3

For th e tw o p erticu ler ceses of 0 single p ortic le (A S) end ofon tnnn tte, hom ogeneous snell (O ray-'t H oott) w e obtain ,respectively:f = -16rr G p, -4-D I ~2 (4-0) (60)

p(K> = p = constont: f = -Brr G p r 2 1(0-2) (6b)F rom now on we sha ll con cen tra te. on the aX lsym m etr ic

case, the z axis b ein g iden tified w ith the beam axis. S in ce pW :;p er) im p lies fO i);:: fer ), it is stra ightforw ard to check that:

fer) ;:: -16TT 6 E(r ) ,.-3 -D1~2 -----} - 86 E (r)/r (7 )1)->4

w here E(r) ts the energy located on the w avefron t w ithin a distancer from the axis. The Q uan tity r(r)/r w m con trol the m agn itude ofthe deflection angles for geodesics. as w e shall now dtscuss,3. Null G eodesics. C on sider a genertc test p ar tic le ofzero m ass travelling. in the far p ast. In the direction op p osite tothat of the w ave and at d istance b (say in the x=x1 d ir ec tion ) fr omthe beam axis. T he corresp ond ing geodesic will ob viou sly H e in th ex-z p lane . In ord er to determ in e it, w e n ote tha t the on lyn on -van ishin g com p onen ts of the C hristoffel sym bols are:rwv ;::-t(r) d /du 8(u )r v = r y = -r(r) 8(u)ru Irr r:; -I'(r) 8(u)/2uu (8)

w here the p rtm a denotes derivative with resp ect to r . The geodesiceq uations are easl1y solved to give (d iscon tinuous, b roken ) straightlin es p a ram etr iz ed by :

x = b + f'(b) u e(u)/2Y :; f'(b)2 u 9(u)/4 + f(b) 9(u) (9)

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

5/9

4

with 9(u) the usual step functlon. Note that et u=O,the pert tcls(the geodesic) suffers a shift [2] in spece find in time given by:

Ilt = ca = llv/2 = f(b)/2 (10)

ThIs shift, whIch Is tn s te n te n eo us due to the impulsivenature of the wave, w1l1 Play an important role In improving thefocusing process .. as utscussec below. It Is amusing to notice thatllt and ca of eQ. (10), after use of eQ.(6a) end insertion into theQuantum plane wave phase px/n. yield precisely the phase shiftfound In ref.[B} (see also ref. [9J), when Quantum strtnq and higherorder classical effects are neglected.

After the Impact with the wave, the test particle hits thez -e xts (x= O ) at:

u = uF = -2bl reb) ----) ~/(4G E(b))D - - > 4

Y = Y F = -bf'(b)/2 + f(b) ----> 4GE(b) + f(b) ( 11)0 - > 4which are immediately convertible into a zF and a tF , Thus a ring ofnun geodesICs lying at a distance b from the 6x1S, converges, aftera time tF to the point ZF' but, in general, z F and a tF depend upon b .For example, in the AS metric in D=4,

tF = 2Gp + tf'/8G E(b) - 4Gpln b/boZF = 2Gp - t J 2 / 8 G E(b) - 4Gpln b/bo ( 12)

where bo is an irrelevant scale [2]. Depending on whether the ratiob/4GE(b) is larger or smaller than I, the particle is either"refracted" or "reflected" by the waye. It is captured by it onlywhen this ratio goes to zero, this being presumably the equ tva len tof a particle's capture by a black hole.

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

6/9

5

Cenwe make both zF end a tF independent of b? This looksa priori hard since one has to solve two conditions, (see eqs.( lOtin terms of the slngle function feb) . Surprisingly. however, andthanks to the shift term feb) . there is 8 solution,

feb) :; const. b (t 3)

E(r) r > - " 2 ( 14)This sotunon precisely corresponds to a beam of constant

energy densHy. Inserting the expression (6b), valid in this case,qtves:

(15)

i.e. the result anticipated in eq.(1).If the beam has constant energy density oyer a radius R

and vanishing (as in ref.[6J) or just decreestnq density outside,perfect focusing at the position (15) still occurs for all geodesicshaying b

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

7/9

6

D efin in g t9 82 = bl uF w e con rew rite eq, (16b) es:1/% + 11 uF = - f(b)/2b

( 17)

w hich agrees w ith ref.(2 ) in the A S case and looks p rec isely as theequ a tion of con juga te p oin ts of a th in an astigm atic len se of focu s Fp rovided thet, aga in :

- f(b )/2 b = cans t = F ( 18)i.e . tha t w e have an hom ogen eou s d isc ! Thus, even for n on p a ra llelin c id en t p ar tic les, an hom ogen eou s, axisym m etr ic , f in ite siz e b eamacts a s a p er fec t c onver qt n 9 In s.

4 . O u tlook. W ha t a re the tm plicetton s of ou r resu lts? W ecan see a few both in term s of p rac tic a l eppttcettons an d in term sof con cep tu al, even fun dam en ta l q uestion s.a ) B y goin g aw ay from the (tran sverse) p lan e sym m etr icsitu a tion (ln fin ite shell) d iscu ssed so fa r in the litera tu re, ou rresu lts cou ld be of astrop hy sica l in terest, sin ce they cou ld ap p ly tothe case of reeltst tcel lg p rodu ced gravita tion al w aves. F ocusin g, insom e lucky c ircum stan ce, cou ld in c rease the flux of gravita tion a len ergy im p in gin g on ea r th, im p roving chan ces of detec tion .

b ) The sin gu la r ity en coun tered in the p lan e sym metr icsitu a tion is often a ttr ibu ted to su ch a sym m etry (see e.g . T ip ler ,ref.[S )). O u r resu lts w ou ld n a ively im ply tha t an in fin ite energyden sity is form ed even in the coutston of fin ite-fron ted w aves,p rovid ed on e of them , a t lea st, is hom ogen eou s (con stan t p ). Theon ly d ifferen ce w ou ld be tha t the singu la r ity is a t a foca l p oin tra ther than on a foca l p lan e.

c ) The rem a rk just m ade b rin gs u p the Q uestion of howfa ithfu lly the geodesic s sim ula te the ac tu a l colltston's propert ies .The q eodestce, b y defin ition , do not in terac t w ith each other . In theac tua l co lltstn n , how ever , the graviton s in terac t w ith each othera fter bein g deflec ted . S uch in terac tion s a re sm all as long as thesp ac ia l sep ara tion of the graviton s exceed s the gravita tion a l rad iu sassoc ia ted w ith their in va r ian t en ergy , b u t b ecom e stron g andn on -Iin eer a fterw a rd s. A s d iscu ssed in ref. [61. on e can on ly a rguetha t a s m uch an en ergy gets focu sed in 6S sm all a region as to

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

8/9

7

imply, at least, the formation of a collapsed object, if not of asingularity. We plen to study soon the full collision problem,possibly in somelimit, in order to clarify this issue.d) Suppose that further work will establish that asingularity does form. It would be then extremely important [6) tostudy whether or not string corrections con ovoid it. In thisconnection we con only make, for the time being, a simple-mindedobservation. Wehave seen(eqs.9, 16) that exact focusing dependsona subtle cancellation between the v-shirt term proportional to fand the deflection term controlled by 1'. The v-shift is related tothe Quantumphaseshift and it was shown in refs. [8] thet stringeffects modify the phaseshift to toke into account the finite sizeA s of the string. Unless a miraculous cencelletion occurs, therewill be a Quantum-string uncertainty of O(Ag)on Y F implying that,an infinite energy density situation is avoided. This point toodeserves.of course, further study.

Oneof us (G.V.)wishes to thank D.Amott andH.Ciofolonifor discussions.

5/13/2018 V. Ferrari and P. Pendenza- Beam-Like Gravitational Waves and their Geodesics

9/9

6

REFERENCES

1 . P.C.Aichelburg and R.U.Se: