The Alcubierre Warp Drive: On the Matter of Matter

Transcription

1 The Alcubierre Warp Drive: On he Maer of Maer Brendan McMonigal, Gerain F. Lewis, and Philip O Byrne Sydney Insiue for Asronomy, School of Physics, A28, The Universiy of Sydney, NSW 26, Ausralia (Daed: February 24, 212) The Alcubierre warp drive allows a spaceship o ravel a an arbirarily large global velociy by deforming he spaceime in a bubble around he spaceship. Lile is known abou he ineracions beween massive paricles and he Alcubierre warp drive, or he effecs of an acceleraing or deceleraing warp bubble. We eamine geodesics represenaive of he pahs of null and massive paricles wih a range of iniial velociies from c o c ineracing wih an Alcubierre warp bubble ravelling a a range of globally subluminal and superluminal velociies on boh consan and variable velociy pahs. The key resuls for null paricles mach wha would be epeced of massive es paricles as hey approach ±c. The increase in energy for massive and null paricles is calculaed in erms of v s, he global ship velociy, and v p, he iniial velociy of he paricle wih respec o he res frame of he origin/desinaion of he ship. Paricles wih posiive v p obain eremely high energy and velociy and become ime locked for he duraion of heir ime in he bubble, eperiencing very lile proper ime beween enering and evenually leaving he bubble. When ineracing wih an acceleraing bubble, any paricles wihin he bubble a he ime receive a velociy boos ha increases or decreases he magniude of heir velociy if he paricle is moving owards he fron or rear of he bubble respecively. If he bubble is deceleraing, he opposie effec is observed. Thus Eulerian maer is unaffeced by bubble acceleraions/deceleraions. The magniude of he velociy booss scales wih he magniude of he bubble acceleraion/deceleraion. I. INTRODUCTION The fundamenal limi on he speed of paricles implied by Special Relaiviy has long hough o be a limi o how humans can eplore he cosmos. However, wih he deformaion of spaceime permied by General Relaiviy, globally superluminal movemen is possible. The Alcubierre Warp Drive spaceime [1] allows a ship o ravel beween wo locaions a an arbirarily large velociy as measured by observers on he ship, as well as a he origin and desinaion. Only a small number of papers have eamined he ineracions of ligh wih Alcubierre warp bubbles [2 6], wih deailed analysis only by Clark e al. [7]. Furhermore here has been almos no coverage in he lieraure of he ineracions of massive paricles wih warp bubbles. Pfenning and Ford [8] invesigae hese ineracions only briefly, discussing only he ineracion beween a warp bubble ravelling a consan velociy and Eulerian maer, ha is maeaionary in he res frame of he origin/desinaion of he ship. This paper fills his void, providing a deailed analysis of he ineracions of null and massive paricles wih an Alcubierre warp bubble a boh consan and variable velociy, via he calculaion of represenaive geodesics hrough Alcubierre spaceime. The ouline of he paper is as follows. In secion II we inroduce he Alcubierre spaceime [1] and discuss he main concerns regarding is validiy ha have been proposed. In secion III we ouline he variable velociy monigal@physics.usyd.edu.au gfl@physics.usyd.edu.au; gfl/ poby36@uni.sydney.edu.au pahs used and equaions of moion before presening our analysis for ineracions wih bubbles a consan velociy in III A. We eend his o one way rips in secion III B and rounds rips in III C. In secion IV we summarize and conclude. II. ALCUBIERRE WARP DRIVE The Alcubierre spaceime is asympoically fla wih he ecepion of he walls of a small spherical bubble surrounding supposedly a ship. The general idea behind he Alcubierre warp drive is o choose an arbirary pah and deform spaceime in he immediae viciniy such ha he pah becomes a imelike freefall pah i.e. a geodesic. Since he ship is following a geodesic, he ravellers eperience no inerial effecs. The Alcubierre meric [1] can be described by ds 2 = d 2 + (d v s ()f( )d) 2 + dy 2 + dz 2, (1) where f is he shape funcion which deermines he form of he bubble wall spaceime disorion, normalised o uni value a he cenre of he bubble. In Alcubierre s original paper i was proposed f( ) = anh(σ( + R)) anh(σ( R)), (2) 2anh(σR) where () = ( s ()) 2 + y 2 + z 2 is he disance from he ship, v s () is he global velociy of he ship, and hence he bubble, and can be arbirarily large, and R and σ are arbirary parameers deermining he radius of he warp bubble and he hickness of he bubble wall respecively. For he purposes of his paper, a qualiaively

2 2 similar bu mahemaically more manageable equaion was used, namely f( ) = 1 ( R ) 4 for rs < R and oherwise. The global velociy is ha as measured by Eulerian observers, ha is anyone in fla spaceime and in he res frame of he origin/desinaion of he ship. I is sraighforward o see from he line elemen ha he form of global velociy is v() = d() d. From his definiion of global velociy, i is also sraighforward o see from he line elemen, ha dτ = d for he ravellers on he ship, where τ is proper ime. Thus he proper ime of he Eulerian observers and he ravellers is he same, and hence no ime dilaion is eperienced. Alcubierre produced his soluion by wha is ermed meric engineering, ha is, sipulaing he required spaceime geomery and solving for he necessary energy disribuion. This mehod is problemaic as i can resul in seemingly unphysical soluions. As noed by Alcubierre, he sress energy momenum ensor is negaive for all observers, even when operaing a arbirarily low velociy [9, 1]. This implies a requiremen of negaive energy densiies, which can only be achieved by eoic maer[11] hus violaing he Energy Condiions [12]. This violaion implies abiliy o creae Closed Timelike Curves (CTCs) and heir associaed problems. In addiion o hese issues, he Alcubierre Warp Drive suffers from he Tachyonic Problem as described by Coule [13]. To circumven hese problems, modificaions he meric have been proposed, however any spaceime ha permis apparen superluminal ravel violaes he Weak and Null energy condiions, and by eension opens he way for CTCs and heir associaed problems [14 16]. Barceló e al. [2] sugges ha even if hese problems were viewed simply as engineering issues, here would sill be criical problems due o semiclassical insabiliy. III. THE INFLUENCE ON PARTICLES The following analysis is resriced o he - plane, and hus () simplifies o he signed disance from he ship s (). Firs we look a he ineracions of null and massive paricles wih a bubble a consan velociy, hen wih a bubble on a one way rip, and finally wih a bubble on a round rip. 1. Non-uniform Pahs Two ypes of variable velociy pah are used in his paper. One way rips characerised by a logisic curve of he form in equaion 3 and velociy given by equaion 4, and round rips characerised by gaussian funcions of he form in equaion wih velociy given by equaion 6. s () = b 1 + ep( d a ) + e (3) v s () = d s() d v s () = d s() d = ( s () = b ep = sb ( a ep b ep( d a ) a(1 + ep( d a ))2 (4) ( d a ( d a ) s ) + e () ) s ) ( ) s 1 d a (6) In each pah, he parameers a,b,d, and e respecively se he pah lengh in, he pah lengh in, he midpoin in, and he journey origin in. For he gaussian pah, here is an addiional paramee, which is an even posiive ineger. For large values, he pah more closely resembles a op ha. The geodesic equaions 2. Equaions of Moion d 2 α dλ 2 + d β d γ Γα βγ dλ dλ =, (7) describe freefall pahs hrough spaceime. Timelike geodesics are parameerised by τ whereas null geodesics mus be parameerised by an affine parameer λ. The nonzero Chrisoffel symbols are [17] Γ = Γ = Γ = v 3 s()f 2 ( ) f(), (8) Γ = Γ = Γ = vs()f(r 2 s ) f(), (9) Γ = v s () f(), (1) Γ = v 2 s()f( )(v 2 s()f 2 ( ) 1) f() f( ) v s() v s () f() 3. Paricle Energy (11) The energy of he paricles a he ship and ouside he bubble is calculaed using E = p u obs (12) where u α obs is he 4-velociy of he observer, and pα is he 4-momenum of he paricle. For an Eulerian observer a res wih respec o he (, ) coordinae sysem, he 4- velociy is (1,,, ); ineresingly we can show ha for he raveller a he cenre of he bubble he 4-velociy is also (1,,, )[8, 18]. The redshif is hen calculaed

3 3 using z + 1 = E emied E observed. Thus he relaive increase or decrease in paricle energy is represened by he quaniy b = E observed E emied = 1 z + 1 (13) which we define he blueshif. As his is calculaed from he ime componen of he 4-velociy, i is also represenaive of he ime dilaion for massive paricles when measured a by Eulerian observers or observers on he ship. 4. Horizons A useful quesion o ask is when he global velociy of he paricle is larger han he global velociy of he ship. For null paricles, we sar wih ds 2 = and rearrange o find d d = ±1 + v s()f( ). Subsiuing ino v p v s () and solving for ouhape funcion gives p () s () R v 1 4 s () (14) which defines wo posiions. We define he posiion on he side owards which he bubble is moving o be he fron horizon and he oher o be he rear horizon. Noably, hese posiions are ouside he bubble radius when v s () < 1 and hus he horizons only eis while he ship is moving a superluminal velociy. For massive paricles, his quesion is more difficul. Saring from he normalisaion u u = 1, we find (vs()f 2 2 ( ) 1)u 2 2v s ()f( )u u + u 2 = 1, which using u u = d d becomes v2 s()f 2 ( ) 1 2v s ()f( )v p + vp 2 = 1 (u ) We found ha when a bubble caches up o a 2 paricle wih a nonzero global velociy in he same direcion as he ship, u diverges, which implies 1 (u ). 2 This reurns us o he same equaion as for null paricles, and hus he same equaion for he horizons, bu due o our assumpion ha he paricle is ineracing wih he bubble from he fron, his can only ell us abou he fron horizon. We revisi he rear horizon for massive paricles laer. A. Consan Velociy Bubbles 1. Null Paricles Figure 1 shows ligh rays ineracing wih a warp bubble ravelling a a consan subluminal global velociy of.. When he bubble velociy is subluminal, all forward and backward moving ligh rays pass hrough he bubble, however he forward moving ligh rays ake longer o do so wih respec o boh observers on he ship and Eulerian observers ouside he bubble. The ligh rays ei he bubble wih he same frequency wih which hey ener i, however hey are spaially displaced by a small FIG. 1. The lef panel is a spaceime diagram of a ship which has been ravelling a a consan subluminal velociy of. ineracing wih ligh rays from ouside he bubble. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The righ panel is he same ineracion wih respec o he ship. The abscissa represens he disance wih respec o he ship, and he ordinae represens he coordinae. The blue and red lines are forward and backward ravelling ligh rays respecively, and he magena and black lines represen he bubble walls and ship respecively FIG. 2. As in Figure 1 for a consan superluminal bubble velociy of 4. disance in he direcion of ravel of he ship. The magniude of he displacemen is due o he ime spen in he bubble and he bubble velociy, larger for longer ime spen in he bubble and for largehip velociy. Thus he displacemen for forward ravelling ligh rays is slighly larger han ha of backward ravelling ligh rays. Figure 2 shows he same siuaion for a warp bubble of consan superluminal velociy 4. The backward ravelling ligh rays sill pass hrough he bubble, eiing he bubble wih he same frequency wih which hey ener i. As he bubble caches up o he forward ravelling ligh rays, hey ener he bubble and asympoe owards he posiion given by Equaion 14 corresponding o he fron horizon before reaching he ship. Due o his, he space behind he bubble is virually devoid of forward ravelling ligh rays. 2. Massive Paricles Figure 3 shows he pahs aken by massive paricles wih respec o he ship when ineracing wih a bubble of superluminal velociy. When he iniial paricle velociy is zero, he paricle passes hrough he bubble, which agrees wih he resul presened by Pfenning and Ford [8]. This is he pah which akes he longes coordinae ime

4 FIG. 3. Pahs of paricles ineracing wih a bubble of consan superluminal velociy. The abscissa is he disance wih respec o he ship, and he ordinae is he coordinae. All paricles ener he bubble a =. Going clockwise from he boom lef, he pahs are for backward ravelling ligh, v p = 1 1 o 1 7 in uni powers of 1, v p =, v p = 1 7 o 1 1 in uni powers of 1, and forward ravelling ligh. The bubble and ship are marked as in Figure o reach he ship and subsequenly leave he rear of he bubble. Of he pahs ha leave he bubble, no oher pah spends more coordinae ime in he bubble and hence his pah resuls in maimal spaial displacemen. Paricles wih negaive iniial velociy have a pah beween ha of zero velociy paricles and backward ravelling ligh. The larger he magniude of he negaive velociy, he earlier he pah diverges from ha aken by zero velociy paricles and he more closely he pah approimaes ha aken by backward ravelling ligh. As he magniude of he negaive velociy decreases, he pah aken diverges from ha aken by zero velociy paricles laer, bu even paricles wih a negaive velociy of only 1 7 spend less coordinae ime in he bubble han zero velociy paricles by almos a magniude. Posiive velociy paricles similarly have a pah beween ha of zero velociy paricles and forward ravelling ligh. Posiive velociy paricles never reach he ship, and all asympoe o he same posiion given by Equaion 14 corresponding o he fron horizon. As he magniude of he velociy increases, he pah aken by he paricles approimaes ha aken by forward ravelling ligh, and as he magniude of he velociy decreases, he pah aken follows he zero velociy paricle pah for longer before rapidly diverging and asympoing owards he pah aken by ligh. Figure 4 shows he pahs aken by massive paricles wih respec o he ship when ineracing wih a bubble of subluminal velociy.. Paricles wih non posiive iniial velociy pass hrough he bubble as fouperluminal bubble ineracions, bu ake longer o do so wih he zero velociy pah aking almos a magniude longer han for he c bubble ineracion. Posiive iniial velociy paricle pahs are divided ino pahs which resemble ha aken by forward ravelling ligh and pahs which FIG. 4. Pahs of paricles ineracing wih a bubble consan subluminal velociy.. The abscissa, ordinae, ship and bubble are as in Figure 3. In he upper panel, he pahs are as in Figure 3 wih he eclusion of forward ravelling ligh as i does no inerac wih a subluminal bubble from he fron. In he lower panel, he pahs are for velociies larger han hose of Figure 3 by he magniude of he criical velociy (which in his case is.8) wih he eclusion of backward ravelling ligh as i does no inerac wih a subluminal bubble from behind; hey are disribued he same way. we refer o as slow maer pahs; his division occurs a a velociy we define he criical velociy, v c. Paricles ravelling a he criical velociy are he slowes paricles which pass hrough he bubble from behind. This is similar o how zero velociy paricles represen he slowes pah which passes hrough he bubble from infron. These paricles are he slowes in he sense ha heir velociy magniude is he lowes, and hey ake he larges amoun of coordinae ime o pass hrough he bubble. As he velociy increases above he criical velociy, he pah aken diverges from ha of paricles a he criical velociy earlier, more closely approimaing he pah aken by forward ravelling ligh. For paricles wih iniial posiive velociy lower han he criical velociy, here are wo possibiliies, eiher he velociy is above ha of he bubble and hence he paricle will cach up o he bubble from behind or he velociy is below ha of he bubble and hus he paricle will inerac wih he fron of he bubble firs as he bubble caches up o he paricle. In he firs case, he paricle is ejeced from he bubble back ou he rear wih a reduced bu sill posiive velociy below he ship velociy. For paricles wih iniial velociy closer o he criical

5 Global Velociy.8 FIG.. Slow maer ineracions wih a subluminal bubble of velociy.. The lef panel is a spaceime diagram as in Figure 1 where he ordinae represens he coordinae and he abscissa represens he coordinae. The righ panel shows he same ineracion where he abscissa represens he global velociy. The ineracions shown are for paricles wih velociies from vp =.2 o.48 in seps of.2 and heir corresponding final velociies. velociy, he pah follows ha of he criical velociy paricles for longer, hus geing closer o he ship, before diverging and being ejeced from he bubble. Similarly in he second case, he paricle is ejeced ou he fron of he bubble wih an increased velociy larger han he ship velociy. For paricles wih a smaller iniial velociy, he pah follows ha of he zero velociy paricles for longer, hus geing closer o he ship, before diverging and being ejeced from he bubble. While he paricular resuls presened are for our choice of shape funcion only, we can generalise o any shape funcion consising of epansion behind he ship and conracion infron of he ship. We found ha paricles ruly a res before enering he warp bubble come o res when hey reach a region of fla spaceime again; he resuls presened sugges ha his firs occurs as he paricle leaves he bubble, bu infac i occurs upon reaching he fla spaceime region immediaely surrounding he ship. Our choice of shape funcion resrics his region o a single poin and hus our use of numerical mehods does no allow i o land direcly on his poin, insead he paricle passes he ship as if i had a infiniesimal negaive velociy. We found ha while wihin regions of conracing space, he magniude of he velociy of paricles increases. This means ha paricles wih negaive velociy accelerae hrough he fron half of he bubble, and ha paricles wih posiive velociy accelerae forwards away from he ship. Thus any shape funcion of his form will resul in paricles wih zero iniial velociy coming o res a he ship, separaing he pahs which pass hrough he bubble from hose which do no. Figure shows slow maer pahs for a bubble velociy of.. The closer he iniial velociy is o he bubble velociy, he closer he final velociy is o he bubble velociy. Furhermore, here is a symmery beween he wo ineracions such ha if a paricle has iniial velociy < vp1 < vs, hen he bubble will cach up o he paricle and ejec i wih a new velociy vs < vp2 < vc. If his paricle were o cach up o anoher warp drive ravelling a he same velociy (and hus having he same criical velociy), hen afer enering he bubble i would be ejeced ou he rear wih final velociy vp1 ; his symmery is eviden in Figure. This symmery is no eviden in Figure 4 as he final velociies obained by he slow maer in Figure 4 ineracing from he fron do no correspond o he iniial velociies of he slow maer in Figure 4 ineracing from behind, and vice versa. 3. Blueshifs The differen ineracions ha can occur beween paricles and a warp bubble are summarized in Figure 6. N+ and N represen paricles wih iniial negaive velociy meeing a superluminal and subluminal bubble respecively, and passing hrough i. P+ and P represen paricles wih iniial posiive velociy which are overaken by a superluminal and subluminal bubble respecively, and are subsequenly capured in he fron of he bubble and ejeced from he fron of he bubble respecively. B+ and B represen paricles wih iniial posiive velociy greaer han he ship velociy, which cach up wih a subluminal bubble from behind, and pass hrough he bubble and are released ou he back of he bubble wih a reduced posiive velociy respecively. The curved line separaing B+ and B marks he criical velociy of he bubble. I is imporan o remember ha while he ship velociy is only depiced up o vs = 1., i can be arbirarily large. No ineracions occur along he line vp = vs as he paricles and he bubble never mee. The line vp = is included only in he regions N+ and N, and similarly he line vp = vc is included only in he region B+. Thus P and B are open regions. The lines vp = 1 and vp = 1 correspond o backward and forward ravelling null paricles respecively and can be hough of as being par of he regions hey bound. Thus N and B+ are closed regions. The regions ha conain null pahs, i.e. N+, P+, N and B+, represen all ligh-like pahs. The regions which do no conain null pahs, i.e. P and B represen all slow maer pahs. The following blueshif resuls apply o boh null and massive paricles. The op image in Figure 6 shows he blueshif observed a he ship for all paricle ineracions. Paricles in he P+ are capured in he fron of he bubble and never reach he ship, and similarly paricles in he slow maer regions are ejeced from he bubble wihou ever reaching he ship, hus here is no value for hese regions. Paricles in he B+ region, i.e. caching up o he ship from behind, have a blueshif of < b 1 which is uniy only when he ship velociy is. As he ship velociy increases owards 1, he observed blueshif decreases owards he limiing value of. The blueshif also increases wih he paricle velociy bu he dependence on he ship velociy is much larger due o he resricions on he paricle velociy for reaching he ship. Paricles in he N regions, i.e. meeing he ship head on, have a blueshif of b 1 which uniy when he ship veloc-

6 FIG. 7. The lef panel is a spaceime diagram of ligh ineracing wih a ship ravelling on a superluminal one way journey. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The blue and red lines are forward and backward ravelling ligh rays respecively, and he magena and black lines represen he bubble walls and ship respecively. The righ panel is he same scenario wih respec o he ship, hus he abscissa represens he disance wih respec o he ship. FIG. 6. The upper panel is blueshifs seen by observers on he ship. The lower panel is for blueshifs seen ouside he bubble by observers saionary wih respec o he origin. iy is. In addiion, he blueshif is also uniy whenever he paricle velociy is. When boh he ship velociy and paricle velociy are nonzero, increasing he paricle velociy owards 1 increases he observed blueshif, and similarly increasing he ship velociy increases he observed blueshif. As here is no bound on he ship velociy in he N + region, he blueshif observed can be arbirarily large. All of he above menioned blueshifs are given by b = 1 v s v p (1) The boom image in Figure 6 shows he blueshif observed a he origin or desinaion, or indeed anywhere ouside he bubble for an Eulerian observer. The paricles which reach he ship pass hrough he bubble and ei wih he same velociy and energy wih which hey enered, i.e. he blueshif for he regions N +, N and B + is uniy. The blueshif for he slow maer is more complicaed. In he P, i.e. when a subluminal bubble caches up o paricles, he blueshif increases wihou bound as he ship velociy approaches uniy. As he paricle velociy increases owards he ship velociy, he blueshif drops owards uniy. The blueshifs for he P region are given by b = 1 + v s v p 1 (γ c 1) where γ i = (16) v s 1 v 2 i In he B region, i.e. when slow maer caches up o a subluminal bubble, here are wo bounds o he blueshif. As he paricle velociy approaches he ship velociy from above, he blueshif rises owards uniy, whereas as he he paricle velociy increases oward he criical velociy, he blueshif drops oward. The blueshifs for he B region are given by (1 γ c ) 2 + γpv 2 s 2 b = γ c γpv 2 s 2 + γ p (1 γ c ) 2 ( ) (17) + vs 2 γ 2 p γc 2 Technically he paricles in he P + canno leave he bubble, and so here is no value given for his region. However, he ime componen of he 4-velociy he paricles in his region increases eponenially for he duraion of he ime hey are caugh in he bubble. Hence if he ship were o evelow o below he speed of ligh such ha hey could escape and inerac wih ouside observers, hey would be observed o have eremely large energies. B. Superluminal One Way Trips 1. Null Paricles Figure 7 shows ligh ineracing wih a bubble which reaches a superluminal maimum velociy of 1. As he bubble acceleraes, he backward ravelling ligh rays are released from he bubble wih a spaial separaion lower han ha hey had before enering he bubble. Similarly during deceleraion he backward ravelling ligh rays are

7 7 released from he bubble wih a spaial separaion lower han heir iniial spaial separaion. These increases and decreases in spaial separaion scale wih he magniude of he bubble acceleraion and deceleraion respecively. As firs glance hese effecs imply ha he spaial separaion of he ligh rays should be unchanged around he cenre of he journey, however his does no occur in Figure 7 unil around 3/4 of he way hrough he journey. The reason for his is ha ligh rays ake ime o pass hrough he bubble, and hus would only appear o be unaffeced by acceleraion/deceleraion if he magniude of he effecs of he acceleraion and deceleraion were equal. As he period of acceleraion (i.e. he firs half of he journey) is symmeric o he period of deceleraion (i.e. he second half of he journey) on he one way rips in his chaper, he ligh rays which appear unaffeced are hose which reach he cenre of he bubble a he midpoin of he journey. This occurs for ligh rays which ener he bubble approimaely 1/4 of he way hrough he journey and ei he bubble approimaely 3/4 of he way hrough he journey, as is observed. The effecs on he change in spaial separaion can be eplained in erms of he difference in he velociy of he bubble while he ligh rays are enering and leaving he bubble. Figure 7 shows ha he rae which he backward ravelling ligh rays ener he bubble is he same as he rae wih which hey leave he bubble. Thus he change in spaial separaion is due only o he difference beween he disance ha is covered by he ship beween picking up ligh rays and he disance covered beween dropping off he ligh rays. Figure 7 shows ha as he bubble caches up o forward ravelling ligh rays, hey are capured and asympoe o he fron horizon given by Equaion 14. The dependence on he ship velociy of he posiion of his horizon in he bubble is now visible. Forward ravelling ligh inside he bubble prior o and during he period of bubble acceleraion eiher asympoes oward he fron horizon in he bubble or is dropped ou he rear of he bubble during he journey. If he ligh is behind he rear horizon, given by Equaion 14, hen he ligh will move owards he rear of he bubble and be dropped off upon reaching he bubble edge. If he ligh is infron of he rear horizon, hen i will asympoe owards he fron horizon. As he posiion of he horizons is dependan on he bubble velociy, during periods of large acceleraion, forward ravelling ligh wihin he bubble can be overaken by he rear horizon. As eplained earlier, he horizons only eis fouperluminal bubble velociies, and hus upon he bubble velociy decreasing back o subluminal velociy, all he forward ravelling ligh ha was previously capured in he fron horizon is released a once. Due o he capure of forward ravelling ligh, superluminally ravelling bubbles creae a region behind hemselves where all forward ravelling ligh from ha space has been swep up ino he forward horizon. However, due o he release of forward ravelling ligh from behind he rear horizon, his region of space is no enirely devoid FIG. 8. The lef panel is a spaceime diagram of massive paricles wih iniial velociy -.1 ineracing wih a ship ravelling on a superluminal one way journey. The righ panel is he same as he lef panel for massive paricles wih iniial velociy -.8. In each panel, he abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The magena and black lines represen he bubble walls and ship respecively. of forward ravelling ligh, bu conains a sparse disribuion of highly redshifed forward ravelling ligh. The range of blueshifs observed a he ship are 1 b < 1.9 for backward ravelling ligh. The range of blueshifs observed ouside he bubble are.27 < b < 3.89 for backward ravelling ligh. Blueshifs are no observed for forward ravelling ligh as i does no reach he ship, and does no leave he bubble. 2. Massive paricle ineracions for a superluminal one way rip Figure 8 shows massive paricles wih negaive iniial velociy ineracing wih a warp bubble on a one way rip which reaches a maimum velociy of 1. All he paricles pass hrough he bubble. The rarefacion of he massive paricles creaed during he bubble acceleraion and conracion during he deceleraion mach ha shown for he superluminal bubble ineracing wih ligh rays in Figure 7, however he change in he energy of he massive paricles is now visually apparen via he change in he velociy which can be seen as he gradien when ouside he bubble in Figure 8. Thus we can see ha he massive paricles which are sparsely released behind he bubble during he acceleraion have a reduced velociy and he massive paricles which are released rapidly during deceleraion have an increased velociy. The magniude of he velociy increase and blueshif obained scales wih he magniude of he inial velociy. The final velociy and blueshifs obained increase owards hose of backward ravelling ligh as he iniial velociy approaches 1. As discussed earlier, all massive paricles wih posiive iniial velociy ineracing wih a superluminal bubble from he fron asympoe o he fron horizon and receive an eponenial increase in energy. However, as was menioned wih ligh, forward ravelling massive paricles which are already wihin he bubble during he acceleraion o superluminal velociies eiher coninue forwards

8 Global Velociy and asympoe o he fron horizon or move owards he rear of he bubble and are dropped off depending on wheher hey are infron of or behind a criical poin similar o he rear horizon for ligh. This criical poin is closer o he ship han he rear horizon, as he massive paricles are ravelling slower han ligh and hence find i harder o keep up wih he bubble han ligh does. The criical poin is dependan on boh he velociy of he ship and he velociy of he massive paricles and should be inerpreed as he posiion in he bubble behind which massive paricles wih such a velociy would no make i o he ship, bu would insead be ejeced from he rear of he bubble. Addiionally, he rear horizon only forms upon he ship reaching he speed of ligh, however, as he criical poin applies o massive paricles, i forms (i.e. is inside he bubble) earlier. The velociy of ligh is consan, so if he ship velociy is below he speed of ligh hen he ligh evenually reaches he ship. However while in he rear of he bubble, massive paricles lose velociy and hus he criical poin forms even before he ship reaches he velociy of he massive paricles. We can hink of he criical velociy defined earlier, as he velociy of massive paricles for which he ship velociy generaes a criical poin a he bubble edge, and hus any massive paricles below his velociy aemping o ener he bubble will already be behind he criical poin. FIG. 9. The lef panel is a spaceime diagram of a ship on a superluminal one way rip ineracing wih massive paricles wih iniial velociy.8 caching up o he bubble as he ship acceleraes away o superluminal velociy. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The righ panel is same ineracion showing he global velociy of each paricle and he ship, hus he abscissa represens global velociy. The magena and black lines represen he bubble walls and ship respecively Global Velociy FIG. 1. The lef panel is a spaceime diagram of a ship on a superluminal one way rip ineracing wih massive paricles wih iniial velociy.1. The bubble caches up o he massive paricles as i deceleraes from superluminal velociy. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The righ panel is he same ineracion showing he global velociy of each paricle and he ship, hus he abscissa represens global velociy. The magena and black lines represen he bubble walls and ship respecively. Figure 9 shows massive paricles wih iniial velociy.8 enering a bubble while he bubble is sill ravelling subluminally. The massive paricles ener he bubble over he period of acceleraion, and hus by he ime he las several paricles have enered he bubble, hey are below he criical velociy of he bubble and hus are ejeced from he rear of he bubble before he bubble reaches superluminal velociy. The remainder of he massive paricles wih he ecepion of he firs four paricles o ener he bubble are one by one overaken by he criical poin and hus are also ejeced from he rear of he bubble. The laer he massive paricles are overaken by he criical poin, he greaer he reducion in he velociy of he massive paricles; his reducion can resul in final velociies arbirarily close o zero. Similarly he laer he massive paricles ejeced due o being below he criical velociy ener he bubble, he smaller he reducion in he final velociy of he massive paricles. The combinaion of hese wo effecs resuls in massive paricles being released from he rear of he bubble wih consisenly decreasing velociy, which ranges from he iniial velociy of he massive paricles o arbirarily close o zero. Of he massive paricles which are no overaken by he criical poin, only he firs wo o ener he bubble make i o he fron horizon before he bubble deceleraes below he speed of ligh and he horizon ceases o eis, and hus only he firs wo paricles o ener he bubble obain velociies close o uniy. Figure 1 shows massive paricles wih velociy.1 ineracing wih a bubble deceleraing from a superluminal velociy. This shows how during deceleraion massive paricles are ejeced infron of he bubble a a range of velociies from he iniial velociy of he massive paricles up o arbirarily close o he speed of ligh depending on how long he massive paricles have been in he bubble. Figure 11 shows he pah for massive paricles which are spaially saionary and inside he bubble a he origin of he journey. As wih ligh and massive paricles discussed earlier, he massive paricles from he rear porion of he bubble are dropped off from he bubble wih increasing spaial separaion. As menioned earlier, massive paricles wih zero iniial velociy are unaffeced by he bubble acceleraing or deceleraing, and hus upon being dropped off from he bubble hey reain a zero velociy. Massive paricles in he fron porion of he bubble are compressed inwards owards he ship, and massive paricles immediaely around he ship are largely unaffeced.

9 FIG. 11. The lef panel is a spaceime diagram of massive paricles iniially in he bubble wih zero velociy ineracing wih he ship ravelling on a subluminal one way journey. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The righ panel is he same ineracion wih respec o he ship, hus he abscissa represens he disance wih respec o he ship. The magena and black lines represen he bubble walls and ship respecively FIG. 12. The lef panel is a spaceime diagram of forward and backward ravelling ligh ineracing wih a ship ravelling on a round rip. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The righ panel is he same ineracion wih respec o he ship, hus he abscissa represens he disance wih respec o he ship. The blue and red lines are forward and backward ravelling ligh rays respecively, and he magena and black lines represen he bubble walls and ship respecively. C. Superluminal Round Trips 1. Null Paricles Figure 12 shows ligh ineracing wih a warp bubble on a superluminal round rip. The firs par of he rip is very similar o he one way rip shown in Figure 7, wih a buildup of forward ravelling ligh a a fron horizon and he spreading ou and bunching up of backward ravelling ligh rays during he acceleraion and deceleraion respecively. The second half of he rip is similar o wha a one way rip in he opposie direcion would look like, wih ligh ravelling in he same direcion as he ship being capured and ligh ravelling in he opposie direcion o he ship passing hrough he bubble and being bunched up during deceleraion a he origin of he rip. A new feaure visible is he formaion of a fron horizon during he second half of he journey wih ligh asympoing owards i from boh sides for much longer han for FIG. 13. The lef panel is a spaceime diagram of massive paricles wih zero iniial velociy ineracing wih a ship ravelling on a round rip. The abscissa represens he coordinae and he ordinae represens he coordinae. The aspec raio is 1:1 such ha ligh rays ravel a 4 in fla spaceime. The righ panel is he same ineracion wih respec o he ship, hus he abscissa represens he disance wih respec o he ship. The magena and black lines represen he bubble walls and ship respecively. a normal one way rip, due o he buildup of ligh in he bubble from he firs half of he journey. In boh halves of he rip, when he bubble velociy dropped below he speed of ligh, any ligh ha had been capured was released as a burs. The pah of ligh is mainly included as reference for he massive paricle pahs. 2. Massive paricles Figure 13 shows he same siuaion as for ligh fopaially saionary massive paricles. Figure 13 shows ha he longer he massive paricles are inside he bubble, he closer hey ge o he ship. This is o be epeced, as earlier we showed ha massive paricles wih zero iniial velociy would evenually pass hrough a bubble a consan velociy, however his journey is no long enough o allow his o happen. Before he massive paricles can pass all he way hrough he bubble, he bubble begins he reurn journey. Since he pah is symmeric, and he acceleraion and deceleraion of he bubble has no effec on massive paricles wih zero iniial velociy, as was menioned earlier, he massive paricles leave he bubble a he same spaial posiions as hey enered he bubble wih no change in heir velociy. If he journey had been long enough o allow he massive paricles o pass hrough he bubble, any displacemen incurred due o passing hrough he bubble on he firs par of he journey would be offse by a corresponding displacemen incurred while passing back hrough he bubble on he second par of he journey. Thus any symmeric pah would leave massive paricles wih zero iniial velociy unperurbed. As wih he res of he repor, we epec massive paricles wih nonzero velociy o behave in a similar manner o ligh, more closely approimaing he behaviour of ligh as he iniial velociy of he massive paricles approaches ha of ligh. Figure 14 shows he same siuaion for massive paricles wih small and large negaive and posiive iniial velociies. When hey leave he bub-

10 1 1 bubble. The blueshif for paricle ineracions in he P and B regions are given by b = 1 + v s v p v s (γ c 1) and 1 (1 γ c ) 2 + γpv 2 s 2 b = γ c γpv 2 s 2 + γ p (1 γ c ) 2 ( ) + vs 2 γ 2 p γc FIG. 14. As in Figure 13 for massive paricles wih nonzero iniial velociy. The paricles in he upper lef panel have velociy -.1, he upper righ -.1, he lower lef.1 and he lower righ.1. ble again hey are spaially displaced in he direcion of heir iniial velociy. Thus afer he ship has made he round rip, negaive velociy paricles are shifed owards he origin of he rip in line wih how early in he rip he bubble picks hem up. This effec could never shif hem all o he origin, as i can be hough of as a srech funcion wih he limiing value of he origin of he rip, no amoun of sreching will remove all he negaive velociy paricles. Posiive velociy paricles are capured and asympoe oward he fron horizon. The closer he massive paricles approach he fron horizon, he larger heir velociy upon leaving he bubble. This resuls in a large void beween he origin of he journey and par way owards he furhes poin away from he origin reached by he bubble, which conains almos none of he massive paricles in quesion. As wih he one way rips discussed earlier, a very small amoun of massive paricles wih grealy reduced velociy would be sparsely disribued hrough his region. The magniude of hese effecs scale wih he iniial velociy of he paricles and he ime spen in he bubble, hus paricles picked up by he bubble earlier also receive a larger effec. IV. CONCLUSIONS We have eamined he pahs of null and massive paricles wih a range of iniial velociies from -c o c ineracing wih an Alcubierre warp bubble ravelling a a range of globally subluminal and superluminal velociies on boh consan and variable velociy pahs. The key resuls for null paricles mach wha would be epeced of massive es paricles ravelling a ±1. When ineracing wih a consan velociy warp bubble, paricles in he regions N +, N, and B + are blueshifed by b = 1 v s v p a he ship, bu ei he opposie side of he bubble wih heir original velociy and energy. Paricles in he regions P and B are acceleraed and deceleraed respecively and ejeced ou he same side of he bubble ha hey enered such ha here is a bijecive map beween he regions for he iniial and final paricle velociy and energy of he paricle when ineracing wih a warp respecively. Paricles in he P + region obain eremely high blueshifs and become ime locked for he duraion of heir ime in he bubble, eperiencing very lile proper ime beween enering and evenually leaving he bubble. When ineracing wih an acceleraing bubble, any paricles wihin he bubble a he ime receive a velociy boos ha increases or decreases he magniude of heir velociy if he paricle is moving owards he fron or rear of he bubble respecively. If he bubble is deceleraing, he opposie effec is observed. Thus Eulerian maer is unaffeced by bubble acceleraions/deceleraions. Furhermore, if he paricle has nonzero iniial velociy, hen upon leaving he bubble i will sill have nonzero velociy. The velociy of null paricles canno be alered by hese effecs, however he energy of he null paricles ransforms in he same way as for massive paricles. The magniude of he velociy booss scales wih he magniude of he bubble acceleraion/deceleraion. Since hese effecs occur during he ime he paricle is inside he bubble, hey will have a greaer effec on paricles ha spend more ime wihin he bubble. Hence paricles wih small negaive velociies obain a larger effec han paricles wih large negaive velociies. However, he magniude of he velociy booss due o bubble acceleraion/deceleraion are comparaively small o he bubble ineracion effecs discussed for posiive velociy paricles, especially when compared o blueshifs found for he P + region. The region of space behind a superluminally ravelling warp bubble is almos enirely devoid of forward ravelling paricles, however i conains a sparse disribuion of paricles wih grealy reduced energy. Meanwhile he region of space infron of a ship deceleraing from superluminal velociy o subluminal velociy is blased wih a concenraed beam of eremely high energy paricles. These resuls sugges ha any ship using an Alcubierre warp drive carrying people would need shielding o proec hem from poenial dangerously blueshifed paricles during he journey, and any people a he desinaion would be gamma ray and high energy paricle blased ino oblivion due o he ereme blueshifs for P + region paricles. While in one way journeys paricles ravelling owards he origin are poenially dangerously blueshifed, heiupposed disance from he origin would render hem oo sparse o be of major concern by he ime hey reached he origin.

11 11 ACKNOWLEDGMENTS The auhors would like o hank he Universiy of Sydney for heiuppor via he Physics Honours Scholarship. [1] M. Alcubierre, Classical and Quanum Graviy 11, L73 (1994). [2] C. Barceló, S. Finazzi, and S. Liberai, ArXiv e-prins (21), arxiv: [gr-qc]. [3] T. H. Anderson, T. G. Mackay, and A. Lakhakia, Journal of Opics 13, 17 (211). [4] H. Ruder, D. Weiskopf, H.-P. Noller, and T. Mller, New Journal of Physics 1, 1214 (28). [] D. Kobras, D. Weiskopf, H. Ruder, and T. Asrophysik, Foundaions and Trends in Compuer Graphics and Vision 2, 173 (26). [6] D. Weiskopf, in Visualizaion 2. Proceedings (2) pp [7] C. Clark, W. A. Hiscock, and S. L. Larson, Classical and Quanum Graviy 16, 396 (1999). [8] M. J. Pfenning and L. H. Ford, Classical and Quanum Graviy 14, 1743 (1997). [9] F. S. N. Lobo and M. Visser, Classical and Quanum Graviy 21, 871 (24). [1] F. S. N. Lobo, ArXiv e-prins (27), arxiv: [gr-qc]. [11] A deailed discussion of eoic maer and he validiy of warp drive spaceimes in general is given by E. W. Davis, in Collecion of Technical Papers - AIAA/ASME/SAE/ASEE 42nd Join Propulsion Conference, Vol. 7 (26) pp [12] M. S. Morris and K. S. Thorne, American Journal of Physics 6, 39 (1988). [13] D. H. Coule, Classical and Quanum Graviy 1, 223 (1998). [14] K. D. Olum, Phys. Rev. Le. 81, 367 (1998). [1] M. Visser, B. A. Basse, and S. Liberai, Nuclear Physics B Proceedings Supplemens 88, 267 (2), arxiv:gr-qc/ [16] R. J. Low, Classical and Quanum Graviy 16, 43 (1999), arxiv:gr-qc/ [17] These equaions were numerically inegraed o find,, u and u of he paricles using he inbuil Ordinary Differenial Equaion (ODE) solver ode23s in Malab. Only one pair of hese variables ( and u, or and u ) needed o be solved for, however boh were found and he normalisaion of he 4 velociy was used as a check on he accuracy of he inegraion. In cases where u and u diverge, he equaions were numerically inegraed o find and d using ode4. d [18] J. V. Narlikar, American Journal of Physics 62, 93 (1994).