Interaction of Cosmic Strings with Gravitational Waves: A New Class of Exact Solutions

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    Interaction of Cosmic Strings with Gravitational Waves: A New Class of Exact Solutions

    A. Economou and D. TsoubelisDivision of Theoretical Physics, Department of Physics, University of Ioannina,

    P.o. Box 1186, GR 45110 Ioannina, Greece(Received 5 August 1988)

    A new three-parameter family of radiative, cylindrically symmetric solutions of the Einstein vacuumequations is presented. Its members represent analytic models of a cylindrical pulse of gravitational ra-diation reflecting oA' a straight-line cosmic string.

    PACS numbers: 04.30.+x, 04.20.Jb, 98.80.Cq

    Among the nontrivial topological structures that grandunified theories with spontaneous symmetry breakingpredict to have formed in the very early Universe, cosmicstrings are supposed to be the ones with the highest prob-ability of surviving up to the present cosmological era. 'This conclusion has led to an extensive effort to findcharacteristic effects which would provide the basis forverification of the existence of cosmic strings observa-tionally.

    The first important result in this direction was ob-tained by Vilenkin. 2 On the basis of an approximatesolution of the Einstein field equations, he concluded thata straight-line cosmic string of vanishing thickness willgive rise to the effect of lensing or multiple image pro-duction, even though no gravitational field appears in thestring s vicinity. This is due to the fact that, in the pres-ence of a cosmic string of the above type, the space-timemanifold remains flat but assumes a conical structurewhereby its angular spread around the string becomesless than 2tt rad.

    Subsequent studies based on exact solutions of theEinstein equations corresponding to finite cross-sectioncylinders as models of open cosmic strings have givenample support to Vilenkin's main result. Only the rela-tion D 8ttlt between the angular defect D of the space-time manifold in which a straight-line cosmic string isembedded and the latter's linear mass density lt has beenquestioned so far. Most of the above studies, however,have been based on static models of the string's space-time background. But this background is subject to thedisturbances produced by gravitational waves emitted byother objects in the string's vicinity. On the other hand,the "false-vacuum" region represented by a cosmicstring must not be expected to be homogeneous from thevery beginning of the string's formation. Thus, a relaxa-tion process can be envisaged whereby, as the stringevolves to its "lower-energy configuration" (probably,represented by one of the static models mentionedabove) gravitational waves are emitted. Therefore, thedevelopment of a more realistic picture of the gravita-tional efIects associated with cosmic strings requires theconstruction of the appropriate time-dependent models.

    In this Letter we present an analytic model of a

    Vilenkin-type cosmic string interacting with cylindricalgravitational waves carrying both of the degrees of free-dom that are possible for these kinds of waves. Morespecifically, the model represents a pulse of gravitationalradiation reflecting off a straight-line cosmic stringwhich occupies the axis of a cylindrically symmetricspace-time. Its construction is based on a new class ofexact radiative solutions of the Einstein vacuum equa-tions which depends on three parameters (a, P, and y,below), two of which are free. Analytic models ofstraight-line cosmic strings interacting with gravitationalwaves have also been presented by Xanthopoulos, Gar-riga and Verdaguer, and Economou and Tsoubelis.What distinguishes the model presented below from theones just mentioned is a set of features of which the fol-lowing are worth pointing out. As noted above, the grav-itational waves involved in the present model have bothof the degrees of freedom which cylindrical waves cancarry. These degrees of freedom are referred to as the +and x polarization modes, respectively, and when bothare present it is not possible to find a coordinate systemin which the space-time metric is globally diagonal. As aresult, the corresponding Einstein equations are so hardto integrate that no analytic solution of this type wasknown as late as in 1985, when Piran, Safier, and Starkpublished a thorough numerical analysis of the behaviorof such solutions. Things are much simpler when onlythe + mode is present. In this case the metric can berendered diagonal [let X 0 in Eq. (5), below] and thecorresponding solutions are known as Einstein-Rosen'waves. It is these kinds of waves that appear in theGarriga-Verdaguer models mentioned above.

    Within the category of the nondiagonal solutions, onthe other hand, the models presented below have the ad-vantage of being the first ones to have the following,physically important, characteristics. First, the globalparameter a which measures the strength of the cosmicstring is uncoupled from the parameters expressing thecharacteristics of the gravitational wave. Second, theparameter P which represents the strength of the gravita-tional wave is freely variable within a range which in-cludes the value P =0 at which the wave is switched off.One of the consequences of the above characteristics is to

    2046 1988 The American Physical Society


    render the physical interpretation of the present solution unambiguous and, thereby, to answer some questions regardingthe physical meaning of similar solutions obtained by Xanthopoulos6 and the present authors recently in the mannersuggested in Ref. 8.

    The new class of solutions described above can be derived most conveniently by the application of a limiting pro-cedure to the line element

    ds2 c2X[x +1) 'dx +(y 1) 'dy ]+(X/Y)(x + l)(y l)dg +(Y/X)(dz cod&) (i)where

    X (1 px) +(l qy), Y p (x +1)+q (y 1),

    q2 l2 p2 (3)As shown in Ref. 8, the above line element represents

    a cylindrically symmetric, Petrov type-D solution of theEinstein vacuum equations. The cylindrical symmetry ofthe metric can be made explicit by the coordinate trans-formation

    t -xy, p-[(x'+ l)(y' I)] 't', (4)which brings the line element given by Eqs. (1) and (2)into the Jordan-Ehlers-Kompaneets canonical form"

    ds f( dt +dp )+e 2 p dy+e"(d.-&dy)', (5)


    f-c'(x'+y') 'X, e'~- Y/X, g-co,and x,y are expressed in terms of t and p according toEq. (4). When f 1 and ~ co 0, Eq. (5) gives theline element of Minkowski space-time in cylindricalcoordinates. This makes clear the meaning of the vari-ous coordinate systems that appear in our analysis.

    Now, let (a,P, y) be the set of parameters defined bythe equation

    (c,l,p) (a/yq, Pq, yq), (7)and consider the limit q ~ of Eqs. (3) and (6). If Eq.(2) is taken into account, it is easy to verify that the re-sulting expressions read

    P2+ y2 ~ 1 (8)

    f-(a/y)'(x'+y') 'A, e'~-8/A,X-(2P/yB)[y (y 1)(x +1)(1 P)(y 1)], (9)A=y x +(y P), 8=y (x +1)+(y 1),

    to-(2/p Y) [lp'(y 1)(x'+1)+q(1+ l' lq px)(y' 1)] .

    The range of the coordinates appearing in Eq. (4) isgiven by x 6 R, y E [l,~), p 6 [0,2tr), and z 6 IR, whilec, p, l, and q are real parameters, the last three of whichsatisfy the condition

    ds' a ( dt +dp )+p dP +dz'. (io)Thus, by choosing P 0 and t a t & 1 we end up with aflat space-time characterized by an angular defect D &,where

    D. -2~(I tat -').Next, let PNO and consider the region near the sym-

    metry axis p Q. In this region p tt t. Thus,x=t p t/2(t +1), y= 1+p t/2(t'+1), (12)

    which, together with Eq. (9), implies thatds =a F[ dt +dp +(p/a) dp ]+F 'dz, (13)


    F-[(P-'+ y't']/y'(t'+ i) (14)Therefore, for t t t 1, i.e., for very early and late times,the metric in the axis region approaches the one given byEq. (10).

    Let us, now, turn our attention to the asymptotic re-gion p 1, tt t. Here,

    x = t/p+ t (t 1)/2p, y =p+ (1 t )/2p, (15)which implies that

    ds = (a/y) [ dt +dp +(y/a) p dP ]+ [dz 2(P/y) (P 1)dp] (16)

    Therefore, the angular defect measured at spacelikeinfinity is given by

    D) 2x(1 I y/a t ) .

    trespectively. Equations (4), (5), (8), and (9) give thenew three-parameter class of solutions announced above.The corresponding metric retains the Petrov type-Dcharacter of the one we started with and, provided

    t P t ~1, it is regular everywhere except on the symmetryaxis where, in general, it is quasiregular. ' In equivalentterms, the axis region of a typical member of the newclass of space-time models constructed above has a coni-cal structure which reveals itself in the angular defect tobe calculated shortly. On the basis of the results quotedin the introduction, we attribute this behavior to thepresence of a straight-line cosmic string occupying thesymmetry axis.

    In order to unveil the physical meaning of the newsolution let us first set P 0 to obtain



    This is larger than the angular defect measured near theaxis by an amount BD, where


    b'D=D& D( =2~(I I yl )/lal &0, (is) ac aceau av

    x +12C ln (a/y) 1 P 2 +JJ 2 (20)

    unless P =0, in which case I y I =1 and bD vanishes. Wewill now show that the enhancement of the angular de-fect expressed by Eq. (IS) is due to a pulse of cylindricalwaves which, at t = ~, approaches the axis along thenull direction t p, gets reflected off the axis region att 0, and, eventually, recedes to the asymptotic regionp ee along the outgoing null direction t p.

    In order to prove the above claim, let us consider thequantity

    C(t,p) =& ln(fe ), (i9)which is referred to as Thorne's C energy. ' For cylin-drically symmetric systems, it provides a well definedmeasure of the energy per unit length in the z directionconfined within a region of coordinate radius p. FromEq. (9), it follows that in our case

    p'(u v) 1 1(2 p')~ p'(1+uv)] i+u''1+v' (23)

    Equations (22) and (23) imply that on the symmetryaxis, where u =v, the C energy is constant and equal toC(, where

    C( =In l al . (24)

    c& - I a/y I, (2S)

    Comparing this with Eq. (11) we conclude that a non-vanishing angular defect near the axis and, therefore, thepresence of a cosmic string, is equivalent to the C energyin this region being positive definite.

    Let it, now, be noted that the C energy is constant andequal to C( in the regions I, where v 1, and I+,where u 1, as well. Similarly the C energy is constantand equal to C&, where

    Introducing the null coordinates u and v, where

    u=t p, v=t+p,we find that

    2C ln(a/y) 1 1+ 1+uv2 E

    E= [(1+u')(1+v')]' ', (22)

    in the asymptotic region III, where u 1, v1. Butthis ceases to be the case in region II which has the sym-metry axis and regions I and III as its boundaries. Infact, in subregion II where u &0, i.e., in a world tubecentered on the radial null direction v =0, there is an en-ergy flux towards the axis, while in II+ where v &0there is an energy flux away from the axis of symmetry.In order to determine the shape of the ingoing pulse,consider a null direction along which v =const and useEq. (23) to find that

    lim ac . ac=0, limu -~ au u -~ av 2(i+v2)[(2 P )(1+v )1 +P v] (26)

    Equation (26) shows clearly that at t = ~ there is indeed a flux of gravitational radiation towards p 0 which isconcentrated around the null direction t = p. Similarly, along u =const

    lim CU ~ au

    R aclim ~p2(1+u 2) [(2p )(1+ 2) ii2 p2u] ' au (27)Thus, an outgoing pulse appears as t ~ which is iden-tical in form with the ingoing one at t = ee.

    The above analysis shows clearly that an angular de-fect appears in the wake of the pulse of gravitationalwaves incident on the string occupying the axis. As a re-sult, after the pulse has crossed the cylindrical surfacep =pp 1 inwards, the angular defect measured atp) pp is found to have increased by bD. This increasewhich, according to Eqs. (1S), (24), and (25), is givenby

    cord with the intuitively expected result that the angulardefect produced by a dynamic, cylindrically symmetricsystem will increase (decrease) as a result of the system'sabsorption (emission) of gravitational radiation.

    We are indebted to Professor Basilis C. Xanthopoulosfor a critical reading of the manuscript and suggestionsfor its improvement.

    8D =2' [exp( C ()exp( C & )], (2s)lasts for as long as the pulse is confined in the regionp & pp, i.e., for Bt = 2pp. Obviously this process is in ac-

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