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Passage of radiation through wormholes Andrey Doroshkevich,1 Jakob Hansen,2 Igor Novikov,1,3,4 and Alexander Shatskiy1 1Astro Space Center, Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia 2Waseda University, Department of Physics, Tokyo, Japan 3Niels Bohr Institute, Copenhagen, Denmark 4Kurchatov Institute, Moscow, Russia (Dated: January 23, 2009) We investigate numerically the process of the passage of a radiation pulse through a wormhole and the subsequent evolution of the wormhole that is caused by the gravitational action of this 9 pulse. The initial static wormhole is modeled by the spherically symmetrical Armendariz-Picon 0 solution with zero mass. The radiation pulses are modeled by spherically symmetrical shells of 0 self-gravitating massless scalar fields. Wedemonstrate that thecompact signal propagates through 2 the wormhole and investigate the dynamics of the fields in this process for both cases: collapse of thewormhole into theblack hole and for the expandingwormhole. n a PACSnumbers: 04.70.Bw,04.20.Dw J 3 2 I. INTRODUCTION exotic matter” (see [9]). Another type of exotic matter is a ”scalar exotic matter” in the form of a scalar field ] c One of the most interesting features of the theory of with a negative energy density (see [18]). Finally, it can q be in the form of a mixed ”magnetic-negative dust ex- general relativity is the possible existence of spacetimes - otic matter” type in which it is a mixture of an ordered r with wormholes [1, 2, 3, 4, 5, 6]. The wormholes are g magneticfieldanddust(matterwithzeropressure)with topological tunnels which connect different asymptoti- [ negative matter density (see [19]). The physical proper- cally flat regions of our Universe or different universes 2 in the model of Multiverse [7]. ties of different types of wormholes are different (see for v Recently the hypothesis that some known astrophysi- example [19]). The possible consequences of the passage 2 of radiation through a wormhole with a ”scalar exotic calobjects(e.g. the activenucleiofsomegalaxies)could 0 matter” was considered, using an analytical approach, beentrancestowormholeswasconsideredbyKardashev, 7 by Doroshkevich,Kardashev,D. Novikovand I.Novikov 0 NovikovandShatskiy[8,9]andbyShatskiy[10]. Ifsome in [20], where a toy model was used to accept some very . of these wormholes are traversable, then radiation and 2 artificial hypotheses about the final state of the worm- information, at least in principle, can pass between two 1 hole. The results obtained in [20] allowed the authors to regions of the Universe or from one universe to another 8 reachsomeimportantconclusions,but ofcoursewithout 0 universe in the model of the Multiverse. Furthermore, numerical computations, the real dynamics of the pro- : if a wormhole exists, in principle it is possible to trans- v cessesofradiationpropagationthroughawormholecould formitintoatimemachine(space-timewithclosedtime- i X like curves) [11, 12]. Therefore, it is very important for not be considered. Other important analytical analyses oftheexistenceandevolutionofthewormholesandtheir r both the theory and the development of the hypothesis a possible transformation into black holes (BH) was seen about the real existence of the wormholesto analyze the in[21,22,23,24,25,26,27,28]. Otherimportantresults physics of the passage of radiation through a wormhole. are also seen in the papers [29, 30, 31]. According to the wormhole models of general relativity, in the absence of any special matter, the throat of the The new period of the investigation of the wormholes wormhole pinches off so quickly that it cannot be tra- beganfromthe seminalpapers[32,33],wherethe worm- versed even by a test signal moving with the velocity of hole dynamics were analyzed using numerical simula- light [13]. In order to prevent the shrinking of a worm- tions. Paper [32] is devoted mainly to the evolution of hole and to make it traversable,it is necessary to thread aninitialstaticwormholemaintainedbythescalarexotic its throat with so-called ”exotic matter”, which is mat- matterunderthegravitationalactionofthescalarradia- ter that violates the averaged null energy condition (see tionpulsepassingthewormholefromoneasymptotically [5, 14, 15]). Many questions associated with the exis- flat region to other one. In paper [33] the authors ana- tence of wormholes and with exotic matter remain un- lyze the nonlinear evolution of the wormhole perturbed solved, e.g. its stability against various processes, and by the scalar field. different views havebeen expressedin the literature (see The goal of this paper is to continue the line of the for example [5, 13, 16, 17]). We will not discuss this investigation of these two works. We focus on the dy- here. Different types of wormholes may exist depend- namics of the scalar fields during the nonlinear stages of ing on the type of exotic matter in their throats (see the wormholeevolutionunder the gravitationalactionof [9, 18, 19]). For example, it could be a ”magnetic ex- theingoingpulseofthescalarradiation(bothexoticand otic matter” in which the main component is a strong, usual). These dynamics can explain some results of the ordered, magnetic field plus a small amount of a ”true papers [32, 33]. 2 This paper is organized as follows: space, but the wormhole (two asymptotically flat three- In sectionII we presentthe initial model of the worm- dimensional regions joined by a three-dimensional tun- hole. In section III the field equations are written out. nel) nevertheless exists due to the distribution (2) of the The initial value problem is discussed in section IV. In exotic scalar field. Of course there are not any apparent section V we present the numerical solution of the field andeventhorizonsinsuchanobjectandtestsignalscan equationwhichdescribethepassageofthesignalthrough pass through the tunnel in both directions. the wormhole and its subsequent evolution with the for- The initial motivations ofour choiceofthe MT worm- mation of a BH. In section VI we analyze the case of hole as an initial static wormhole were the following. a strong signal. The passage of the signal of the exotic (1)Thecorrespondingsolutions(1)-(2)oftheEinstein scalar field is considered in section VII. Finally we sum- equations are very simple and elegant. marize our conclusions in section VIII. Mathematical (2) It was declared in [18] that the MT wormhole is details andthe details ofthe numericalcode are givenin stable. appendices A-C. However the analytical [34] and numerical [33] analysis (see also Appendix A) demonstrates that the solu- tions (1)-(2) are unstable against small spherical pertur- II. THE MODEL bations. Our numerical experiments confirms this conclusion. In the light of this fact, the analysis of the possibility We study, numerically, the evolution of a simple of the passage of the signals through the MT wormhole spherical model of a wormhole maintained by an exotic and their distortions have a special interest. scalar field and perturbed by an in-falling massless, self- gravitating scalar field. This initially in-falling scalar field is a pulse in the form of a sphericallayer with some III. FIELD EQUATIONS width and amplitude. It imitates a pulse of radiation directed into the wormhole and propagating with veloc- For the numerical analysis we will use the double null ity of light. We do not impose that the in-falling scalar coordinates. The general line element in these coordi- field is weak. We will consider two cases: when this in- nates can be written as: falling field has a positive energy density (imitating real radiation), and when it has a negative energy density ds2 = 2e2σ(u,v)dudv+r2(u,v)dΩ2, (3) (imitating exotic radiation). As an initial static worm- − where σ(u,v) and r(u,v) are functions of the null coor- hole, let us consider the special case of the Armendariz- dinates u and v (in- and out-going respectively). The PiconsolutionoftheEinsteinequationsforthespherical non-zero components of the Einstein tensor are: static state with the effective mass equal to zero [6, 18]. Another name of this wormhole is Moris-Thorne’s (MT) 4r,uσ,u 2r,uu G = − (4a) wormhole. The corresponding line element for such a uu r wormhole can be written as follows: 4r σ 2r ,v ,v ,vv G = − (4b) vv r e2σ+2r r +2rr ds2(MT) =−dτ2+dR2+r2dΩ2, r2(MT) =Q2+R2, (1) Guv = ,ur2,v ,uv (4c) G = 2e−2σr(r +rσ ) (4d) where R is running from to + , Q is a charac- θθ − ,uv ,uv teristic of the strength of t−he∞exotic∞supporting field Ψ Gϕϕ = 2e−2σrsin2θ(r,uv+rσ,uv) (4e) − and dΩ2 = dθ2 + sin2θdϕ2 is the line element on the Theenergy-momentumtensorcanbewrittenasasum unit two-sphere. The generalequations will be givenbe- of contributions from the exotic scalar field Ψ and the low. Herewegivethenon-zerocomponentsoftheenergy- ordinary scalar field Φ with the positive energy density: momentum tensor of the Ψ-field for the solution (1): T =TΨ +TΦ : µν µν µν Tτ(MT) = Q2 (2a) Ψ2 0 0 0 τ 8πr4 ,u 1 0 Ψ2 0 0 TRR(MT) = −8Qπr24 (2b) TµΨν = −4π  00 00,v r2e−2σ0Ψ,uΨ,v r2sin2θe−02σΨ Ψ  Tθ(MT) = Tϕ(MT) = Q2 (2c)  ,u(5),v θ ϕ 8πr4 Thenarrowestpartofthewormhole(1)isatR=0. This Φ2,u 0 0 0 tmhertoraictc(o1r)reisspgoinvdenstionr[2=0]Q. .HTehreepwheyesmicaplhaansaizleystihsaotftfhoer TµΦν = +4π1 00 Φ02,v r2e−2σ0Φ,uΦ,v 00  themetric(1),inthisreferenceframe,thereisnogravita-  0 0 0 r2sin2θe−2σΦ,uΦ,v tional accelerationat any point in the three-dimensional  (6) 3 The u u. v v, u v and θ θ components of the moment there is a rather narrow spherical layer of an − − − − Einstein equations respectively are (with c=1, G=1): in-falling scalar field. There is not any radiation coming to the wormhole from the side of the other entrance. As mentioned,wespecifytheinitialvaluesontwoinitialseg- r 2r σ r (Ψ )2+r (Φ )2 =0 (7) ments, namely u = constant and v = constant. In the ,uu ,u ,u ,u ,u − − r 2r σ r (Ψ )2+r (Φ )2 =0 (8) next section V we consider the case where the in-falling ,vv ,v ,v ,v ,v − − layer consists of the Φ-field. Later in the section VII we r r e2σ r + ,u ,v + =0 (9) consider the case of the in-falling layer consisting of the ,uv r 2r Ψ-field. r,vr,u e2σ For the case of the layer from the Φ-field, the initial σ Ψ Ψ +Φ Φ =0 (10) ,uv − r2 − 2r2 − ,u ,v ,u ,v condition could be specified as follows: First, let us consider the static MT solution [18]: The scalar fields satisfy the Gordon-Klein equation µ Ψ = 0 and µ Φ = 0, which in the metric (3) Q Q R ∇ ∇µ ∇ ∇µ Ψ(MT) = = Ψ(MT) =arctan become: ,R r2 Q2+R2 ⇒ Q (cid:18) (cid:19) (13) 1 Ψ + (r Ψ +r Ψ )=0 (11) In u v coordinates we can write the MT solution as: ,uv ,v ,u ,u ,v r − 1 1 Φ,uv+ (r,vΦ,u+r,uΦ,v)=0 (12) r(u,v)= Q2+ (v u)2 (14) r 4 − r Equations (9)-(12) are evolution equations which are e−2σ(MT) =2 (15) supplemented by the two constraint equations (7) and v u (8). It is noted that none of these equations depends Ψ(MT) =arctan − (16) 2Q directly on the scalar fields Ψ and Φ, but only on their (cid:18) (cid:19) derivatives,i.e. thederivativeofthescalarfieldisaphys- Φ(MT) =0 (17) ical quantity, while the absolute value of the scalar field Equations (14) - (16) completely specifics the MT- itselfisnot. SpecificallywenotetheTΨ = (Ψ )2/(4π) manodmeTnvΨtvum=te−n(sΨor,v()52)/(a4nπd)TcΦom=po(Φnenu)tu2s/(o4f−π)thae,nudenTeΦrgy=- dwoomrmaihnolienicnluud−invgctohoerdininitaiatelssiunrfaalcleosfuou=rccoomnpsutatanttioannadl uu ,u vv v =constant. (Φ )2/(4π)componentsoftheenergy-momentumtensor ,v Now, let us consider the more general case with non- (6), which are part of the constraint equations. Physi- trivial Φ and Ψ scalar fields. In sections V and VI we cally T and T represents the flux of the scalar field uu vv consider the case of a non-zero Φ field and in section throughasurfaceofconstantvandurespectively. These VII we consider the case of a perturbed MT Ψ field on fluxeswillplayanimportantroleinourinterpretationof the initial outgoing u = constant surface. In general we the numerical results in sections V, VI and VII. are free to choose the Φ and Ψ fields (and their derivatives) on the initial surfaces in any way we wish. Our choices for the Φ and Ψ-fields are described in sections IV. INITIAL VALUE PROBLEM V - VII. We are also free to choose r(u,v) on the initial surfaces, this merely expresses the gauge freedom asso- We wish to numerically evolve the unknown func- ciated with the transformation u u˜(u),v v˜(v) (the tions r(u,v), σ(u,v), Φ(u,v) and Ψ(u,v) throughout line element (3) and the equations→(7)-(12) a→re invariant some computational domain. We do this by follow- to such a transformation). We continue to use eq. (14) ing the approach of [35, 36, 37, 38, 39, 40, 41, 42, on the initial surface as our choice of gauge. Hence, the 43] to numerically integrate the four evolution equa- only variable left for us to specify on the initial surfaces tions (9)-(12). These equations form a well-posed ini- is σ. This can easily be found by integrating the con- tial value problem in which we can specify initial val- straintequations eq. (7) and (8), which ensures that the ues of the unknowns on two initial null segments, constraintequationsaresatisfiedonthe initialhypersur- namely an ingoing (v =v0 =constant) and an outgoing faces. Specifically on the outgoing u = u0 hypersurface (u=u0 =constant) segment. We impose the constraint it is found by the integral: equations (7) and (8) on the initial segments. Consis- tency ofthe evolvingfieldswith the constraintequations 1 v r r (Ψ )2+r (Φ )2 ,vv ,v ,v isthenensuredviathecontractedBianchiidentities[37], σ(u0,v)=ln + − dv but we use the constraint equations throughout the do- (cid:18)√2(cid:19) Z 2r,v 0 main of integrationto check the accuracy of the numeri- (18) cal simulation. The initial conditions on the initial ingoing hypersur- Our choice of the initial values corresponds to the fol- face is set equal to the MT solution for all simulations. lowingphysicalsituation;Thereis anMT-wormholeand Hence, by specifying a distribution of the scalar fields Φ atsome distance fromone ofthe entrances,atthe initial andΨonthe initial nullsegments,choosinga gaugeand 4 charge parameter ”Q” we can specify complete initial conditions on the initial null segments. Using a numerical code (described in appendix B), we can then use the evolutionequations,eqs. (9)-(12)to evolvethe unknown functions throughout the computational domain. V. PHYSICS OF THE PASSAGE OF THE Φ-FIELD PULSE THROUGH THE MT- WORMHOLE We investigate the full nonlinear processes arising in the case of propagation of a compact pulse of the scalar field through an MT-wormhole, using results of our numerical simulations. Our numerical code is described in appendix B andtestedinappendix C. Inthis sectionwe analyzethepassageoftheΦ-fieldpulse. InsectionVIIwe investigate the passage of the Ψ-field pulse. Throughout thispaper,theconstantQ(seeeq. (1)),priortoinfluence from scalar pulses, has initial value of Q = 1. The com- putationaldomainforallresultsthroughoutthispaperis v = [8,28] and u =[0,20]. The flux of the scalar Φ-field into the wormhole is specified along initial u=u =0 0 outside the wormhole in the following way: FIG. 1: Static MT-wormhole. Plot shows contour lines of constant r. Thick diagonal line ”a” marks the throat of the v v wormhole (corresponding to the diagonal u=v) with r =1. Φ,v(u0,v)=AΦsin2 π − 1 (19) The thin lines marks lines of constant r increasing outwards v v (cid:18) 2− 1(cid:19) from thethroat of thewormhole with aconstant spacing between lines of ∆r = 0.5 (i.e. line ”b” marks r = 1.5 up to wherev andv marksthebeginningandendoftheingo- 1 2 line ”c” at r=12.). ing scalar pulse, respectively, and AΦ measures the am- plitudeofthepulse. Beforeandafterthepulse,atv <v 1 and v >v the flux through u=u is set equal to zero, 2 0 i.e. Φ (u ,v)=0. The fluxofthe scalarΦ-fieldthrough of the space-time arisesat r =0. It is seen from the fact ,v 0 initialsegmentv =v issetequaltozero: Φ (u,v )=0. that when we come to r=0, the metric coefficient e2σ 0 ,u 0 The expression (19) can readily be integrated to give: in formula (3) tends to infinity (see Fig. 2(b)). Also apparent and event horizons arise. The apparent horizon AΦ v v corresponds to events where world lines r = const are 1 Φ(u ,v)= 2π(v v ) (v v )sin 2π − 0 4π − 1 − 2− 1 v v horizontal or vertical (see [44] page 85). All space-time (cid:18) (cid:18) 2−(210(cid:19))(cid:19) is divided into R- and T-regions(see [13, 45, 46, 47]). In Note that we formulate the initial condition directly theR-regions(lines”a” and”b”),thelinesr= constare for the flux T =(Φ )2/(4π) of the scalar field through time-likeandsignalscangobothtohigherandsmallerr. vv ,v the surface u=u , rather than for Φ itself since T has Inthe T-region(betweenthe twoapparenthorizons)the 0 vv thedirectphysicalmeaning. Alsorememberfromsection lines r = constarespace-likeandasignalcanpropagate IVthatoncethefluxthroughthetwoinitialsurfaceshas to smaller r only. The structure of the apparent horizon been chosen, all other initial conditions are determined depends on the value of the flux of the energy through by the model. In the examples of the results of our com- it. We will analyze it later (see section VI). putations we specify v =9 and v =11. In our compu- Now we consider the evolution of the Φ- and Ψ-fields. 1 2 tations we vary the amplitude AΦ of the pulse. Fig. 3(a) represents the evolution of the flux TΦ of the vv In Fig. 1 is seen the static case of the MT metric (1) scalar Φ-energy into the wormhole. Fig. 3(b) shows the in u v coordinates, without any perturbations of the evolution of the TΦ flux which arises as a result of TΦ − uu vv scalar fields. The throat r =1 corresponds to the diago- being scattered by the space-time curvature. Fig. 4(a) nal u=v. representstheevolutionofthedifferenceoftheTΨ TΨ vv− uu In Fig. 2 one can see a typical example of the evolu- of the fluxes of the Ψ-field in and out of the wormhole. tion of the wormhole due to the action of the in-falling The reasonwhy for the Ψ-fieldwe show the difference of Φ scalar field. Initial values for this Φ-field are v =9, the in and out fluxes rather than the TΨ and TΨ-fluxes 1 vv uu v =11, AΦ =0.01. Fig. 2(a) shows the evolution of the separately,isthefactthatevenforthestaticMT-solution 2 linesr = const. AsisseeninFig.2(a),theevolutioncor- (1), (2), the values of TΨ and TΨ are not equal to zero vv uu respondstocontractionofthe throatdowntor =0, and (butequaltoeachother). Soonlythedifferencebetween contraction of the whole wormhole. The real singularity them is important for the dynamics. Finally Fig. 4(b) - 5 10 8 6 σ) 2 p( x e 4 2 0 0 2 4 6 8 10 12 L (a)Lines ofconstant r,rangingfromr=0tor=5withaspacing (b)Plotofthemetriccoefficient e2σ(L) asafunctionofdistance between linesof∆r=0.25(line”a” marksr=1.25,line”b” parameter L=p(v−v0)2+(u−u0)2 measuringthedistance marksr=5andline”c” marksr=0.75). Thethickdotted lines alongthewormholethroatatu=v (withL=0correspondingto marksthepositionoftheapparenthorizons. Thethickfullydrawn wheretheΦ-pulsefirstreaches thewormholethroat,i.e. linemarksthepositionofthesingularityr=0,hencetheregion v0=u0=9). marked”X” isoutsideofthecomputational domain. FIG. 2: Results of simulation with a non-zero Φ-field modeled after eq. (20) with v =9,v =11 and amplitudeAΦ =0.01. 1 2 4(d) shows the comparisonofall three fluxes atdifferent zero everywhere (no any gravitational forces). Just af- slices: ter the passageof the perturbing compact ingoing signal of the Φ-field, the effective mass become positive by the u=3, 14, 20. mass of the ingoing Φ-shell. In the process of the collapse the fluxes directed in the opposite directions (out In Fig. 3(a) and 3(b) the initial pulse (between ofthe throat)arise(regionsC andDinfig. 4(a)). These 9.0<v <11.0) andsubsequenttails withresonancesare fluxes partlypropagateoutofthe openings ofthe worm- clearly seen. In different regions, TΦ and TΦ are con- uu vv holeandpartlypropagateinside ofthe formingBH.The verted into one another due to curvature scattering and fluxesoftheΨ-fieldgoingoutofthewormholecarryaway resonances. But at later u (see Fig. 4(c), 4(d)) and for negative energy providing origin of the positive mass of v >11 the flux TΨ TΨ of the Ψ-field dominates abso- vv− uu the forming BH. That is emphasized in papers [32, 33]. lutely (in modulus). Thus for these regions this flux of Everything looks like a collapse and an explosion of the the Ψ-field determines the dynamics of the wormhole. Ψ-field. From the Fig. 2-4 the general picture of the evolution of the wormhole under the action of the passage of the compact Φ-field signal looks like the following. At the beginning,aratherweaksignalproducessmallperturba- tionsofthestaticwormhole. Theseperturbationstrigger theevolutionoftheΨ-fieldthatmaintainsthewormhole. The fluxes of the Ψ-field (in- and out-fluxes) determine A very important fact is that the compact signal of the subsequent evolution of the wormhole and lead to the Φ-field propagates through the wormhole from one its collapse. At the beginning of this process the fluxes asymptotically flat region into another asymptotically of the Ψ-field are directed to the throat from both sides flat region before the beginning of the collapse, this is of the wormhole (regions A and B in fig. 4(a)). This especially clear from fig. 3(a). Only a small part of the flux to the throat of the Ψ-field with the negative en- Φ-signalis scatteredback and into the arising BH. After ergy density may lead to the formation of the negative the BH formation the propagation of a signal from one local effective mass. Remember that before the begin- asymptotically flat space into another one is impossible ning of the process the local effective mass was equal to in any direction. 6 (a)ContourlinesoftheingoingΦflux,TvΦv =const.. Thelines”a”, (b)Contour linesoftheoutgoingΦflux,TuΦu=const.. Thelines ”b” and”c” denotes regionsofTvΦv=10−3,TvΦv =10−4 and ”a”,”b” and”c” denotes regionsofTuΦu=3·10−5,TuΦu=3·10−5 TvΦv=3·10−6 respectively. andTuΦu=10−9 respectively. Theletter”d”identifies aregion wheretheoutgoingfluxincreasestowardsther=0singularity,the contour lineclosesttother=0singularitycorresponds to TuΦu=10−4. FIG. 3: Contour lines of the in- and out-going Φ fluxes for the simulation with v =9,v =11 and amplitude AΦ =0.01. In 1 2 1 both plots lines are logaritmically spaced with a factor 102 between the lines. The thick solid line marks the location of the r=0 singularity and theregion ”X” marks the area outside of thecomputational domain. VI. STRONGER Φ-SIGNAL, HORIZONS AND charged BH, see [42]. RATE OF THE COLLAPSE The picture of the evolution of the Ψ-field in the case ofthestrongsignal(seeFig.5(d))isqualitativelysimilar Now let us consider the case of an essentially stronger to the case of weaker signals (Fig. 4(a)). infalling Φ-signal with AΦ =0.5 (see Fig. 5(a)-5(d)). Abouttheeventhorizon. FromFig.2(a)itisclearthat In this case, the collapse of the wormhole arises much events close enough to the singularity r =0 are inside of faster. The essential part of the initial Φ-signal comes the BH and thus inside of the event horizon. Indeed, into the arising BH, this can be seen from figure 5(b). the light signal from these events (vertical and horizon- The picture of the process is very asymmetrical. tal lines) will come to the singularity and cannot go to The asymmetry is clearly seen at the picture of the infinity. At Fig. 2(a) event horizon practically coincide apparent horizon (figure 5(a)). In general, with any with vertical and horizontal asymptotes to the singular- strength of the compact Φ-signal, the apparent horizon ity r = 0. The analogous region inside the BH can be arises at the throat just at the moment of the passage found at the Fig. 5(a). of the Φ-signal (see Fig. 2(a)). At the beginning the dy- Now let us consider the process of the collapse of the namics of the wormhole is rather slow (if the Φ-signal is throat. In Fig. 6(a) one can see this process for different weak enough) and two branches of the apparent horizon amplitudes AΦ of the ingoing signal of the Φ-field. It is are very close from each other. seenthatforsmallAΦ thewormholepracticallydoesnot Whenthefastcollapsebeginsthesebranchesaregoing responds to the Φ-signal during long periods and after in different directions. that collapses very fast. It looks like a very nonlinear In the case of a strong initial pulse of the Φ-field process. (Fig. 5(a)) the shape of the branchof the apparenthori- Fig. 6(b) represents the dependence of the moment of zon from the side of the coming Φ-field signal is quite collapseofthethroatontheamplitudeAΦ oftheingoing different from the one at the opposite side. Φ-signal. Itis seenthatfor smallAΦ the periodbetween The analogous change of the apparent horizon un- the passageofthe signaland the moment ofthe collapse der the action of the incoming signal we observe in the can be very long. 7 0.001 0.0001 1e-05 1e-06 C 1e-07 x u Fl A 1e-08 B 1e-09 1e-10 1e-11 1e-12 10 15 20 25 30 V (a)Contour linesoftheresultingΨ-flux=TvΨv−TuΨu withafactor (b)ComparisonoftheΦandΨfluxesalongthelineu=3. Line of1012 betweenlines. Thelines”e” and”f” denotes ”A” marks|TvΨv−TuΨu|,line”B” marksTvΦv andline”C” marks TvΨv−TuΨu=+10−7 andTvΨv−TuΨu=−10−7 respectively,withthe TuΦu. resultingabsolutefluxincreasingtowardsther=0singularity. Theregionsmarked”A” and”C” marksTvΨv−TuΨu>0,”B” and ”D” marksregionswithTvΨv−TuΨu<0. Thethickdotted line marksthecontour lineTvΨv−TuΨu=0andthefullydrawnthick linemarksther=0singularityasinpreviousplotsaswellasthe region”X” whichmarkstheareaoutsideofthecomputational domain. 1 1 0.01 0.01 A A 0.0001 0.0001 B 1e-06 ux 1e-06 ux C Fl C Fl 1e-08 1e-08 1e-10 B 1e-12 1e-10 1e-14 1e-12 10 15 20 25 30 8 9 10 11 12 13 14 15 V V (c)ComparisonoftheΦandΨfluxes alongthelineu=14. Line (d)ComparisonoftheΦandΨfluxes alongthelineu=20. Line ”A” marks|TvΨv−TuΨu|,line”B” marksTvΦv andline”C” marks ”A” marks|TvΨv−TuΨu|,line”B” marksTvΦv andline”C” marks TuΦu. TuΦu. FIG. 4: Resultsof thesimulation with v =9,v =11 and amplitude AΦ =0.01. 1 2 8 (a)Lines ofconstant r,fromr=0tor=5with∆r=0.2(lines (b)Lines ofconstant TvΦv. Linesarelogarithmicallyspacedwitha ”a” marksr=1.2andline”b” marksr=5). Thickdotted line factor1012 between lines,line”a” marksTvΦv =10−0.5 andline marksthepositionoftheapparenthorizons. ”b” marksTvΦv =10−5. (c)Lines ofconstant TuΦu. Linesarelogarithmicallyspacedwitha (d)LinesofconstantTvΨv−TuΨu,Region”A” and”C” marks factor1012 betweenlines,line”a” marksTvΦv=10−2.5. regionswithTvΨv−TuΨu>0,”B” and”D” marksregionswith TvΨv−TuΨu<0andthethickdottedlinemarksTvΨv−TuΨu=0. FIG.5: Resultsforasimulation withv =9,v =11andamplitudeAΦ=0.5. Forallfigures,thethickfullydrawnlinemarks 1 2 theposition of the singularity r=0, and theregion marked ”X” marks area outside of thecomputational domain. 9 1.2 16 14 1 g 12 0.8 10 r 0.6 L 8 6 0.4 4 0.2 2 a b c d e f 0 0 0 2 4 6 8 10 12 14 16 0.0001 0.001 0.01 0.1 1 L A (a)r asafunctionofthedistanceparameter (b)Distance parameterL=p(v−v0)2+(u−u0)2 with L=p(v−v0)2+(u−u0)2 withv0=u0=9. Thecurves”a”, v0=u0=9measuredalongu=v fromthestartofthepulseto ”b”,”c”,”d”,”e”,”f” and”g” markssimulationswith thepositionofther=0singularityasafunctionofscalarfield AΦ=0.5,0.2,0.1,0.01,0.001,0.00035and0.0001respectively. amplitudeAΦ. FIG.6: Developmentofther=0singularityasafunctionofpulseamplitudeAΦ(withidenticalpulsewidthsofv =9,v =11). 1 2 VII. PASSAGE OF THE Ψ-FIELD sizeofthethroatbecomeslargerandlarger. Thephysical reasonfor this expansionis clear, the additional amount InthissectionwebrieflyconsiderthepassageoftheΨ- of the Ψ-field with the negative energy density creates fieldcompactpulsethroughaMT-wormhole. As wetold gravitational repulsion. This gives the initial push. The at the end of section II, the MT wormhole is unstable expansion at the late v-parameter looks like an almost against small spherical perturbations. Here we consider ”inertial”. not small perturbations in the form of a compact signal Betweentwobranchesoftheapparenthorizonanytest coming from outside. As we will see, the evolution is in particle or radiation can move to greater and greater r agreement with our conclusion about instability of the only. ThisisasocalledexpandingT+ region,see[13,48, MT wormhole. Now we put the Φ-field 0 everywhere 49, 50, 51]. and model the ingoing Ψ-flux as a pert≡urbation to the Fig. 7(b) represent the fluxes of the Ψ-field out of the background Ψ(MT) field supporting the wormhole: throat to greater r. Thecaseoftheexpression(22)withthesignAΨ <0is Ψ(u ,v)=Ψ(MT)(u ,v)+ψ¯(u ,v) (21) 0 0 0 representedinFig.8. NowthereisadeficitoftheΨ-field The form for the perturbation of the Ψ-field into the as compared with the static solution (1), (2). It leads to wormholeis specifiedalmostthe samewayasinthe case collapse of the wormhole. Qualitatively the process of of the Φ-field in the section V. But now instead of the the collapse is the same as it was described in section V. formula (19) we should write down v v ψ¯ (u ,v)=AΨsin2 π − 1 , (22) VIII. CONCLUSIONS ,v 0 v v (cid:18) 2− 1(cid:19) All other remarks about the initial conditions from sec- In this paper we investigated numerically the process tion V are correct here also. But now there is one new of the passage of the radiation pulse through a worm- factorinthiscase. Namely,thispulseofthe Ψ-fieldisan hole and the subsequent evolution of the wormhole as addition to the background of the Ψ-field of the MT so- respond to a gravitationalaction of this signal. It is the lution. This addition can be taken with the sign plus or continuation of the investigations started in the papers minus. This means that AΨ can be positive or negative. [32, 33, 34]. In our work we focus mainly on the evo- The evolution depends critically on this sign. lution of the fields Φ and Ψ. We use the MT solution Fig.7representthecaseoftheexpression(22)withthe [see (1)-(2)] of the Einstein equations as a model of an signAΨ >0. The wormhole expands (Fig. 7(a)) and the initial static wormhole. 10 (a)Lines ofconstant r,fromr=1tor=5with∆r=0.5between (b)Lines ofconstant TvΨv−TuΨu,Region”a” marksregionwith lines(line”a” marksr=1.5andline”b” marksr=11, TvΨv−TuΨu<0”b” marksregionwithTvΨv−TuΨu>0andthethick dottedlinemarksTvΨv−TuΨu=0. FIG. 7: Results for a simulation with v =9,v =11 and AΨ=+0.01 (Φ=0 everywhere). 1 2 (a)Lines ofconstant r,fromr=1tor=10withspacing∆r=0.5 (b)LinesofconstantTvΨv−TuΨu,Regions”A” and”C” marks between lines,lines”a” marksr=1.5,line”b” marksr=10. regionswithTvΨv−TuΨu>0,”B” and”D” marksregionswith Thickdottedlinesmarkspositionsofapparenthorizons. TvΨv−TuΨu<0andthethickdotted linemarksTvΨv−TuΨu=0. FIG. 8: Results for a simulation with v = 9,v = 11 and AΨ¯ = −0.01 (Φ = 0 everywhere). As previously, the thick fully 1 2 drawn line marks theposition of r=0 and theregion marked ”X” indicates theregion beyondour computational domain.