Collisions of Einstein-Conformal Scalar Waves

arXiv:gr-qc/9212012v1 18 Dec 1992Collisions of Einstein-Conformal Scalar WavesC. Klimˇc´ık ∗†P.

Koln´ık ∗‡AbstractA large class of solutions of the Einstein-conformal scalar equationsin D=2+1 and D=3+1 is identified. They describe the collisions ofasymptotic conformal scalar waves and are generated from Einstein-minimally coupled scalar spacetimes via a (generalized) Bekensteintransformation.

Particular emphasis is given to the study of the globalproperties and the singularity structure of the obtained solutions. It isshown, that in the case of the absence of pure gravitational radiationin the initial data, the formation of the final singularity is not onlygeneric, but is even inevitable.PRA-HEP-92/19∗Theory Division of the Nuclear Centre, Charles University, V Holeˇsoviˇck´ach 2, Prague,Czechoslovakia†e-mail: presov @ cspuni12.bitnet‡e-mail: kolnik @ cspuni12.bitnet1

1.IntroductionColliding gravitational waves have attracted a lot of interest in the last twodecades [1]-[10]. Apart from the character of the nonlinearities of the gravi-tational interaction, the interest was probably caused by characteristic cur-vature singularities occuring as the result of the collision of two waves.

Muchwork has been done on the structure of the singularities [6]-[9] in the case ofcollision of either sourceless or various source waves, with the result that thefinal singularity formation is, in fact, generic.A relevant contribution concerning the singularity formation was madeby Hayward [11], who formulated the criterion of “incoming” regularity. Inother words, he proposed to make a clear distinction whether the singular-ity formation occurs for the collision of waves which are initially regular orsingular.

Then the problem reads: Under what conditions may the initiallyregular waves avoid the singularity formation after the collision? For the caseof the purely gravitational (sourceless) waves Hayward himself found that theregular waves generically produce the curvature singularities.

However, therewere also exceptional cases where the singularities were avoided.In D=2+1 the present authors found that after the collisions of regularasymptotic scalar waves the singularity is always formed [12]. We consideredthe minimally coupled scalar field.

A similar conclusion was then obtainedin D=3+1 by Hayward [13], who has shown that if the pure gravitationalradiation is absent in the initial data then the collisions of the minimallycoupled scalar waves always end up in a singularity.In this paper, we wish to study a source field of a different type andfind whether similar conclusions about the inevitability of the singularityformation can be reached.We shall work with the conformal scalar fieldwith the field equation (in the D-dimensional spacetime) [14], [15]∇α∇αφ −D −24(D −1)Rφ = 0,(1.1)following from the actionS =Z ( 12κ −D −28(D −1)φ2)R −12(∇φ)2√−g dDx.Unlike the other massless field equations (i.e. Maxwell, Dirac or Weyl), theminimally coupled massless scalar equation is not conformally invariant.

The2

coupling according to (1.1) cures this “deficiency” and, in any case, it is areasonable alternative for gravitational coupling of the scalar field. The stresstensor for the conformal scalar field is quite different from the ordinary one,and we may therefore test the singularity formation problem in a differentsetting to previously.From a technical point of view, it is not difficult to generate solutions ofthe Einstein-conformal scalar equations from the minimally coupled Einstein-scalar solutions via the generalized Bekenstein transformation [14], [15].

How-ever, the structure of singularities requires independent analysis, since theBekenstein transformation multiplies the original metric by a nontrivial con-formal factor. This operation, in general, may change the asymptotic be-haviour of the Riemann tensor components.

Indeed, consider for instancethe dilaton gravity in D=2+1 with the actionS =Zd3x√−g eφR.There is the following (black hole) solution of the corresponding field equa-tions [16]ds2 = −r −2mrdt2 +rr −2m dr2 + r2dϑ2, φ = log |r sin ϑ|. (1.2)On the other hand under the transformationegαβ = e2φgαβ, eφ = φ,the action S changes into the action of the minimally coupled scalar fieldS =Zd3xq−eg ( eR −2 eφγ eφγ).The metric corresponding to (1.2) becomesds2 = r2 sin2 ϑ−r −2mrdt2 +rr −2m dr2 + r2dϑ2.It is not difficult to demonstrate that the singularity structures of both met-rics differ considerably from each other.In sections 2 and 3 we study the singularity structure of the Einstein-conformal scalar spacetimes in 2+1 and 3+1 dimensions, respectively.

Wearrive again to the same conclusions as for the minimally coupled scalar field,namely, for initially regular waves (without sourceless part) the singularityformation is inevitable.3

2.Conformal scalar waves in D=2+1We start with a reminder of some properties of the spacetimes describingthe collision of line-fronted scalar waves. (It is known that the formation ofsingularities occurs as the result of these processes [12].) In such spacetimes,global coordinates can be introduced such that the metric has the formds2 = −e−eK(u,v)du dv + e−eN(u,v)dx2(2.1)with one space-like coordinate x and a pair of null coordinates u, v. Theform of the metric functions fN, fK follows from the Einstein equations whereon the r.h.s.

the stress tensor for the scalar field eφ stands:eTµν = eφµ eφν −12 egµν egρσ eφρ eφσ. (2.2)Restricting ourselves to the case of the so-called asymptotic waves spacetimes,for which all functions fN, fK, eφ are smooth and fulfil asymptotic conditions(fK, fN, eφ)(u →−∞, v →−∞)=0,(fK, fN, eφ)(u, v →−∞)=(fK, fN, eφ)(u),(fK, fN, eφ)(u →−∞, v)=(fK, fN, eφ)(v),(2.3)we can write down the general solutionfN = −2 ln (1 −f(u) −g(v)) ,eφ=k ln(1 −f −g) + p cosh−1"1 + f −g1 −f −g#+ q cosh−1"1 −f + g1 −f −g#+Z ∞0 [A(ω)J0 (ω(1−f −g)) + B(ω)N0 (ω(1−f −g))] sin (ω(f −g)) dω+Z ∞0 [C(ω)J0 (ω(1−f −g)) + D(ω)N0 (ω(1−f −g))] cos (ω(f −g)) dω,(2.4)subject toZ ∞0 [C(ω)J0(ω) + D(ω)N0(ω)] dω = 0,4

where f(u) and g(v) are functions, k, p and q are real numbers, A(ω), B(ω),C(ω) and D(ω) may be integrable functions or distributions and J0 and N0are zero-order Bessel and Neumann functions.The function fK can be obtained by integration from the relevant Einsteinequations, viz.fNuu −12fN2u + fNu fKu=2κ eφ2u,fNvv −12fN2v + fNv fKv=2κ eφ2v,fKuv=κ eφu eφv. (2.5)The asymptotic conditions (2.3), requiring certain asymptotic behaviour ofthe functions f(u) and g(v), are met by the choicef(u)=[−a(u −us)]1/(1−κp2)for κp2 > 1,f(u)=exp[a(u −us)]for κp2 = 1,andg(v)=[−b(v −vs)]1/(1−κq2)for κq2 > 1,g(v)=exp[b(v −vs)]for κq2 = 1.Now, the plan of the investigation of the properties of these spacetimes wasthe following [12]: firstly we had to exclude the cases when the asymptoticwaves were singular themselves, for the “good physical situation” shouldavoid singularities in the initial data.

Then the formation of final singularitieswas questioned. It turned out that in D=2+1, as the result of the collision ofregular asymptotic scalar waves, the final singularity always appeared [12].Now, what is the situation when the source of colliding waves is not thescalar field, but the conformal scalar field?

The solutions for the selfgravitat-ing conformal scalar field can be easily obtained from the selfgravitating min-imal scalar spacetimes via a generalized Bekenstein transformation [14], [15],linking the D-dimensional scalar field eφ and metric egαβ to the D-dimensionalconformal scalar field φ and metric gαβ as followsφ= κ D −24(D −1)!−1/2tanh κ D −24(D −1)!−1/2eφ,gαβ=cosh κ D −24(D −1)!−1/2eφ4/(D−2)egαβ. (2.6)5

In terms of metric functions fN, fK and analogously defined N and K we have(for D=2+1) the following transformation rulesφ=s8κ tanhrκ8eφ,N=fN −4 ln coshrκ8eφ,K=fK −4 ln coshrκ8eφ. (2.7)We see that (2.3) imply fulfillment of the same asymptotic conditions for thenew metric functions and the new (conformal scalar) field.

It means thatthe metric gαβ also describes the colliding asymptotic waves spacetimes andwe can study the conditions for the regularity of initial data as well as theconsequent creation of singularities after collision1. What is the criterion forthe regularity of the initial data?Consider the (asymptotic) regions where only one of the initial wavesis present while the influence of the other wave vanishes, i.e.

the regions(u →−∞, v ̸= −∞) or (v →−∞, u ̸= −∞). Now the metric is that ofthe single wave there (i.e.

u- or v-independent) and we can introduce theamplitude uh(u)uh(u) = 12e2K( 12N2u −Nuu −NuKu),resp. vh(v)vh(v) = 12e2K( 12N2v −Nvv −NvKv).These are the amplitudes of the single line-wave in the so-called Brinkmanncoordinates.

It was shown in [12] that the boundedness of these amplitudesis the necessary and sufficient condition for the absence of the curvaturesingularities in the asymptotic (incoming) region.Using the transformation rules (2.7) and the Einstein equations for theordinary scalar field (2.5) we can writeuh(u) = 12CH−8e2 eKκ eφ2u(−2 −TH2 +12CH2) +√2κ TH ( eφuu + eφu fKu),1Actually, there is one more branch of the generalized Bekenstein transformation (see[15]) which, however, does not preserve the asymptotic conditions, so we shall not dealwith it anymore.6

where CH=cosh( eφqκ/8), TH=tanh( eφqκ/8) and the other amplitude vh(v)has the analogous form. Now, we need the asymptotic behaviour of the Besseland Neumann functions near zeroJ0(w)∼1 −w24 + · · · ,J′0(w)∼−w2 + · · ·,N0(w)∼(1 −w24 ) ln w + · · · ,N′0(w)∼1w −w2 ln w + · · ·(2.8)From the expressions (2.4) taken in g = 0, f ∼1 it follows thateφu ∼fu"c1 −f + d ln(1 −f) + e + h(1 −f) ln(1 −f) + · · ·#,and then from (2.5) we can obtainfKu = fu" κc21 −f + 2κcd ln(1 −f) + · · ·#,and hencefK = −κc2 ln(1 −f) + bounded.Herec=p −k −Z ∞0 [B(ω) sin(ω) + D(ω) cos(ω)]dω,d=Z ∞0 [ωB(ω) cos(ω) −ωD(ω) sin(ω)]dω,e=p2 +Z ∞0 ω[A(ω) + B(ω) lnω] cos(ω)dω−Z ∞0 ω[C(ω) + D(ω) ln ω] sin(ω)dω,h=12Z ∞0 ω2[B(ω) sin(ω) + D(ω) cos(ω)]dω.

(2.9)Thus if c ̸= 0 the leading term in uh(u) is27f 2u(1 −f)8|c|√κ/8−2κc2−2|c|h−3κ|c| +√2κ(1 + κc2)i.7

It vanishes for |c| = 1/√2κ or |c| = 2/√2κ, but then we cannot suppressthe subleading term proportional to (1 −f)−1/2 or (1 −f)−1 respectively.Hence, we arrived at the first necessary condition for the incoming regularity,viz. c = 0.

In this case the asymptotic behaviour of uh(u) isuh(u) ∼bounded ×h−12κ(3d2 ln2(1 −f) + (6de −d2) ln(1 −f)) + · · ·i,from which the second necessary condition for the incoming regularity canbe extracted: d = 0. It is easy to see that the conditions c = d = 0 are alsosufficient.After a similar analysis for vh(v) we conclude that for the incoming reg-ularity it is necessary and sufficient to fulfilc = d = c′ = d′ = 0,(2.10)where c and d are as above andc′=q −k +Z ∞0 [B(ω) sin(ω) −D(ω) cos(ω)]dω,d′=−Z ∞0 [ωB(ω) cos(ω) + ωD(ω) sin(ω)]dω.Now, we wish to study the region of interaction, i.e.

the region wherethe metric is both u- and v-dependent. There are special points (formingthe so-called caustic) in which the metric functions are singular, namely thepoints where1 −f −g = 0.We have to elucidate what kind of singularity this is, or – in other words – howthe curvature behaves in its vicinity.

In particular, we find the conditionsunder which the scalar curvature R is unbounded while approaching thecaustic2. In terms of metric functions the scalar curvature R readsR = −4eKKuv,which we can rewrite using (2.7) and (2.5)R = −4 e eKCH−4[κ eφu eφv −√2κ eφuvTH −12κ eφu eφvCH−2].2This case is often referred to as the C0 scalar curvature singularity (see [17]).8

In the neighbourhood of the caustic it is convenient to introduce other func-tions t and z instead of f and g, given byt = 1 −f −g,z = f −g. (2.11)Then the caustic is formed by the points t = 0, |z| ̸= 1, the second conditionguaranteeing that we do not deal with the asymptotic caustic studied previ-ously.

Near the caustic the following behaviour of fK and eφ can be derivedfrom (2.4) and (2.8)eφ∼E(z) ln t + · · · ,fK∼−κ E2(z) ln t + · · · ,whereE(z) = k −p −q +Z ∞0 [B(ω) sin(ωz) + D(ω) cos(ωz)]dω. (2.12)So unless E(z) ≡0 the leading term in R isfugv const t−κE2(z)+4|E(z)|√κ/8−2[κE2(z) −√2κ (sign E(z)) E(z)],which vanishes if |E(z)| =q2/κ.

But then the subleading term−4 fugv const t−1is present in R, so R is unbounded. Hence the only case when R may bebounded near the caustic is if E(z) ≡0.

(This, in turn, can only be true forB(ω) = D(ω) = 0 because of the nontrivial dependence of E(z) on z. )However, the condition E = 0 is incompatible with the conditions (2.10)for the asymptotic (incoming) regularity.

We can conclude that after collisionof regular asymptotic conformal scalar waves the scalar curvature singularityat the caustic is always produced. Hence, the conclusion is the same as inthe case of minimal scalar waves.3.Conformal scalar waves in D=3+1In this section, we shall proceed to the collisions of asymptotic conformalscalar waves in D=3+1.

In order to obtain a solution which describes sucha process, we again apply the Bekenstein transformation (2.6). The metricwith the minimally coupled scalar field eφ as the source3 has the form (see3Unfortunately, the exact solutions with scalar sources are known only for the case ofso-called collinear waves, it means egxy = 0 in terms of the metric given below.9

[13])ds2 = −2e−eMdudv + e−eP+eQdx2 + e−eP −eQdy2.The metric functions and eφ, fulfilling the asymptotic conditions analogous to(2.3), can be expressed in the formeP = −ln (1 −f(u) −g(v)) ,eQ=k ln(1 −f −g) + p cosh−1"1 + f −g1 −f −g#+ q cosh−1"1 −f + g1 −f −g#+Z ∞0 [A(ω)J0 (ω(1−f −g)) + B(ω)N0 (ω(1−f −g))] sin (ω(f −g)) dω+Z ∞0 [C(ω)J0 (ω(1−f −g)) + D(ω)N0 (ω(1−f −g))] cos (ω(f −g)) dω,eφ=λ ln(1 −f −g) + π cosh−1"1 + f −g1 −f −g#+ χ cosh−1"1 −f + g1 −f −g#+Z ∞0 [A(ω)J0 (ω(1−f −g)) + B(ω)N0 (ω(1−f −g))] sin (ω(f −g)) dω+Z ∞0 [C(ω)J0 (ω(1−f −g)) + D(ω)N0 (ω(1−f −g))] cos (ω(f −g)) dω,subject to the constraintsZ ∞0 [C(ω)J0(ω) + D(ω)N0(ω)]dω = 0,Z ∞0 [C(ω)J0(ω) + D(ω)N0(ω)]dω = 0.All symbols have analogous meaning as in the D=2+1 case.The last –unexpressed – metric function fM is given by direct integration of the relevantEinstein equations2 ePuu −eP 2u + 2 ePu fMu=eQ2u + 2κ eφ2u,2 ePvv −eP 2v + 2 ePv fMv=eQ2v + 2κ eφ2v,2 fMuv=eQu eQv −ePu ePv + 2κ eφu eφv,(3.1)10

(in our conventions the stress tensor is double of that considered by Hay-ward [13]). In what follows we shall restrict ourselves to the special type offunctions f(u) (and g(v)) [13]f(u)=[−a(u −us)]2/(2−p2−2κπ2)for p2 + 2κπ2 > 2,f(u)=exp[a(u −us)]for p2 + 2κπ2 = 2,ensuring the proper asymptotic behaviour of the metric functions and of thefield in past timelike infinity.

Then the Bekenstein transformation (2.6) yieldsthe new metric functions M, P and Q and new (conformal scalar) field φ:φ=s6κ tanhrκ6eφ,P=eP −2 ln coshrκ6eφ,M=fM −2 ln coshrκ6eφ,Q=eQ. (3.2)We turn to the study of the singularity structure of the new spacetimes (3.2).Near the asymptotic caustic, e.g.

(v = −∞, u = 0), where the metric isv-independent, it is convenient to take a new null coordinate u′ instead of usuch thatdu′ = e−Mdu.Then the following vierbein is orthonormal and parallelly propagated alongthe incomplete geodesics respecting x- and y-symmetry and hitting the asymp-totic caustic:eα(0)=( 1√2, 1√2, 0, 0),eα(1)=( 1√2, −1√2, 0, 0),eα(2)=(0, 0, e12(P −Q)(u′), 0),eα(3)=(0, 0, 0, e12(P +Q)(u′)).11

It is straightforward to compute the only two (mutually independent) nonzerovierbein components of the Riemann curvature tensor asR2020=−18[2Qu′u′ −2Pu′u′ + (Qu′ −Pu′)2],R3030=18[2Qu′u′ + 2Pu′u′ −(Qu′ + Pu′)2].Going back to the original coordinates we haveR2020=−18e2M[2Quu + 2QuMu −2Puu −2PuMu + (Qu −Pu)2],R3030=18e2M[2Quu + 2QuMu + 2Puu + 2PuMu −(Qu + Pu)2].The incoming regularity now requires the boundedness of both components.Using (3.2), we can writeuR−≡R2020 −R3030 = −12Ch−4 e2 eM[ eQuu + eQu fMu −eQu ePu],uR+≡R2020 + R3030 = 14Ch−4 e2 eM[2 ePuu −4rκ6eφuuTh −23κ eφ2uCh+( ePu −2rκ6eφuTh)(2 fMu −ePu −2rκ6eφuTh) −eQ2u],with Ch and Th standing instead of cosh( eφqκ/6) and tanh( eφqκ/6), respec-tively.We have to identify the behaviour of uR± for f ∼1. Taking again intoaccount the asymptotic behaviour of the Bessel and Neumann functions (2.8),we obtain (for f ∼1):eQu∼fu"c1 −f + d ln(1 −f) + e + h(1 −f) ln(1 −f) + · · ·#,eφu∼fu"c∗1 −f + d∗ln(1 −f) + e∗+ h∗(1 −f) ln(1 −f) + · · ·#,where c, d, e, h have the same form as in (2.9) and c∗, d∗, e∗, h∗are given byc∗=π −λ −Z ∞0 [B(ω) sin(ω) + D(ω) cos(ω)]dω,12

d∗=Z ∞0 [ωB(ω) cos(ω) −ωD(ω) sin(ω)]dω,e∗=π2 +Z ∞0 ω[A(ω) + B(ω) ln ω] cos(ω)dω−Z ∞0 ω[C(ω) + D(ω) ln ω] sin(ω)dω,h∗=12Z ∞0 ω2[B(ω) sin(ω) + D(ω) cos(ω)]dω. (3.3)From (3.1) we havefMu = fu"c2 + 2κc∗2 −1211 −f + (cd + 2κc∗d∗) ln(1 −f) + · · ·#,hencefM = −c2 + 2κc∗2 −12ln(1 −f) + bounded.Therefore, the leading singular terms in uR−and uR+ read−8f 2uc2 + 2κc∗2 −12c (1 −f)4√κ/6|c∗|−(c2+2κc∗2−1)−2,4f 2u |c∗|8κ3 |c∗| −2rκ6(c2 + 2κc∗2 + 1)(1 −f)4√κ/6|c∗|−(c2+2κc∗2−1)−2.Both coefficients of proportionality vanish ifc = 0, |c∗| = (2 ± 1)/√6κ,|c| = 12, |c∗| =s38κ,c = c∗= 0,c∗= 0, |c| = 1.In the first two cases there are subleading singular terms, which cannot beeliminated, but in the remaining cases we can exclude them by fitting someother coefficients in the expansions of eφ and eQ.

Hence, the initial data are freeof curvature singularities if c = c∗= d = 0 or if c2 = 1, c∗= d = d∗= h = 0.The analogous conditions have to be satisfied at the other asymptotic caustic.Now, we have to study the components of the curvature tensor at thecaustic 1 −f −g = 0. It is sufficient [13] to consider the scalar curvature Rgiven byR = −eM(PuPv + 2Muv −QuQv),13

and the component Ψ2 of the Weyl spinor in the null spin-frame [13]Ψ2 = 13eM(QuQv −PuPv + Muv).If one of them is unbounded, then there is a (final) curvature singularity atthe caustic. We take the suitable combinations of Ψ2 and R:V1=eMMuv,V2=eM(QuQv −PuPv).Using (3.2) we haveV1=Ch−2e eMfMuv −2rκ6eφuvTh −13κ eφu eφvCh−2,V2=Ch−2e eMeQu eQv −( ePu −2rκ6eφuTh)( ePv −2rκ6eφvTh).We again introduce the functions t and z as in (2.11).The asymptoticbehaviour of eφ and eQ near t = 0 iseφ∼E(z) ln t + F(z)t2,eQ∼E(z) ln t + F(z)t2,whereE(z) = λ −π −χ +Z ∞o [B(ω) sin(ωz) + D(ω) cos(ωz)]dω,E(z) is given as (2.12) and the forms of F(z) and F(z) are not important.Then the leading singular terms of V1, V2 are proportional toE2(z) + 2κE2(z) −1 −4rκ6|E(z)|t2√κ/6|E(z)|+ 12(1−E2(z)−2κE2(z))−2,E2(z) −23κE2(z) −1 + 4rκ6|E(z)|t2√κ/6|E(z)|+ 12(1−E2(z)−2κE2(z))−2,respectively.

Both coefficients of proportionality vanish ifE = 0, |E| = 1,E = 0, |E| =s32κ.14

In the second case, V1 is unbounded due to the subleading term. Hence theonly way to keep both V1 and V2 bounded is to set E(z) ≡0 and |E(z)| ≡1.Therefore both the criterion for the incoming regularity and the necessarycondition for the avoiding of the final singularity are in the case of conformalscalar waves the same as in the case of minimal scalar waves [13].

Hence theconclusions have to be the same, too. In particular, the formation of finalsingularities in the collision of regular asymptotical conformal scalar waves isgeneric.

Moreover, if the pure gravitational radiation is absent in the initialdata, i.e. (−Quu + PuQu −MuQu) = 0, v →−∞,(−Qvv + PvQv −MvQv) = 0, u →−∞,the final singularities are even inevitable.4.Concluding remarksIn 2+1 dimensions, where there is no pure gravitational radiation, the onlygravitational waves are those accompanying light-like matter sources.

For themassless minimally coupled scalar field [12] previously and for the conformalscalar field now, we have shown that collisions of regular waves necessarilyform the final curvature singularities. Since the electrovacua in D=2+1 are(up to some global obstructions of the cohomological origin) equivalent tomassless minimally coupled scalar vacua [18], it is reasonable to conjecturethat collisions of regular general light-like matter waves always end up ina curvature singularity.

Of course, systems with more exotic stress tensorshave to be studied in order to prove this conjecture. From the point of viewof quantum theory it may be worth remarking that the classical phase space,which is to be quantized, does not possess too complicated a structure fromthe geometrical point of view.

Indeed, the spacetimes for all regular initialdata have the same global structure.In 3+1 dimensions the situation is slightly more complicated due to thepresence of the pure (sourceless) gravitational radiation. Indeed, within theframework of pure gravity there are regular initial data the evolution of whichdoes not lead to a curvature singularity at the caustic.

However, if we excludethe sourceless waves in the initial data, we have shown that the collision ofthe regular pure conformal scalar waves inevitably leads to the formation of15

a curvature singularity. A similar conclusion was made for the case of themassless minimally coupled scalar field [13].AcknowledgmentWe are grateful to Sean Hayward for enlightening discussions.16

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