GÖDEL SPACETIME A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY MEHMET KAVUK

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1 GÖDEL SPACETIME A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY MEHMET KAVUK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN PHYSICS AUGUST 5

2 Approval of th Grauat School of Natural an Appli Scincs. Prof. Dr. Canan Özgn Dirctor I crtify that this thsis satisfis all th rquirmnts as a thsis for th gr of Mastr of Scinc. Prof. Dr. Sinan Bilikmn Ha of Dpartmnt This is to crtify that w hav ra this thsis an that in our opinion it is fully aquat in scop an quality as a thsis for th gr of Mastr of Scinc. Assoc. Prof. Dr. Özgür Sarıoğlu Suprvisor Eamining Committ Mmbrs Prof. Dr. Ayş Karasu METU PHYS Assoc. Prof. Dr. Özgür Sarıoğlu METU PHYS Assoc. Prof. Dr. Yusuf İpkoğlu METU PHYS Assist. Prof. Dr. Hakan Öktm METU IAM Assoc. Prof. Dr. Bayram Tkin METU PHYS

3 I hrby clar that all information in this ocumnt has bn obtain an prsnt in accoranc with acamic ruls an thical conuct. I also clar that as rquir by ths ruls an conuct I hav fully cit an rfrnc all matrial an rsults that ar not original to this work. Mhmt Kavuk iii

4 ABSTRACT GÖDEL SPACETIME Kavuk Mhmt MS. Dpartmnt of Physics Suprvisor: Özgür Sarıoğlu August 5 56 pags. In this thsis proprtis of th Göl spactim ar analyz an it is plicitly shown that thr ist clos timlik curvs in this spactim. Gosic motions for massiv particls an light rays ar invstigat. On obsrvs th focusing ffct as a rsult of th solution of th gosic quations. Th tim it taks for a fr particl rlas from a point to com back to its starting point is calculat. A gomtrical intrprtation of th Göl spactim is givn an th qustion of what th Göl spactim looks lik is answr. Kywors: Göl spactim Null gosics Timlik gosics Tim travl Rotation of spactim. iv

5 ÖZ GÖDEL UZAY-ZAMANI Kavuk Mhmt Yüksk Lisans Fizik Bölümü Tz Yönticisi: Özgür Sarıoğlu Ağustos 5 56 sayfa Bu tz Göl uzay-zamanının özlliklri çözümlni v kapalı zamansı ğrilrin varlığı göstrili. Jozik nklmi çözülrk kütlli parçacıkların v ışık mtlrinin jozik harktlri inclni v oaklanma tkisi görülü. Bir noktaan yollanan bir ışık ışınının v srbst bir parçacığın n kaar sür aynı yr öncği hsaplanı. Göl vrninin gomtrik bir yorumu vrili v bu vrnin ny bnziği sorusu cvaplanı. Anahtar klimlr: Göl uzay-zamanı Null joziklr Zamansı joziklr Zamana yolculuk Uzay zamanın önmsi. v

6 To my family an to my girlfrin Görkm Öcalan vi

7 ACKNOWLEDGMENTS Th author wishs to prss his pst gratitu to his suprvisor Özgür Sarıoğlu for his guianc avic criticism ncouragmnts an insight throughout th rsarch. Thank you sincrly. Thr ar no wors to scrib th apprciation an gratitu I fl for my girlfrin Görkm Öcalan. I thank hr for hr ncouragmnt an support. vii

8 TABLE OF CONTENTS PLAGIARISM...iii ABSTRACT...iv ÖZ...v DEDICATION...vi ACKNOWLEDGMENTS...vii TABLE OF CONTENTS...viii CHAPTER. INTRODUCTION.... DERIVATION OF GÖDEL SPACETIME...4. Th Mtric...4. Th Choic of Intgration Constants PROPERTIES OF THE GÖDEL SPACETIME Transformation to Cylinrical Coorinats Killing Vctors Rotation Shar an Epansion GEODESICS Null Gosics Timlik Gosics Th Structur of Light Cons Som Rmarks on Göl Spactim CONCLUSION...39 viii

9 REFERENCES...4 APPENDICES...43 A. Th Componnts of th Einstin Tnsor...43 B. Transformation to Cylinrical Coorinats...44 C. Proof of D. Calculations of th Intgrals..5 i

10 LIST OF FIGURES Figur 3. Th circl C on th t plan which has a tangnt vctor....4 Figur 3. t plan. Lightcons tip ovr an opn out as w proc away from A along concntric circls. Initially such circls ar clos spaclik curvs lik S. Evntually lightcons tip to crat clos null curvs lik N an thn tip furthr to crat clos timlik curvs lik T. Sinc spactim is stationary an rotationally symmtric about ach mattr worllin all obsrvrs s what A ss; all s mattr spinning about thm... Figur 4. Göl s spactim with th irrlvant coorinat z supprss. Th spac is rotationally symmtric about any point; th iagram rprsnts corrctly th rotational symmtry about th r= ais an th tim invarianc. Th light con opns out an tips ovr as r incrass s lin L rsulting in clos timlik curvs. Th iagram os not corrctly rprsnt th fact that all points ar in fact quivalnt...6 Figur 4.: Path of th photon in th r plan for. For vry valu thr is a iffrnt llips passing through th origin. Ths llipss form an nvlop which is a circl with raius r ln aroun th origin...35

11 CHAPTER INTRODUCTION Gnral Rlativity thory hlps us to trmin possibl mols for th larg scal structur of spactim. Each of th mols is a pair M g. Hr M is a four imnsional manifol which rprsnts th totality of all point-vnt locations; an g is a gomtric objct which rprsnts th mtric structur of spactim. W can think of it simply as a function which assigns lngths to vctors at points of M but which assigns ngativ an zro lngths to vctors as wll as positiv ons. It thus partitions th vctors at any point into thr classs an trmins a light con structur. A vctor is sai to b timlik null or spaclik accoring to whthr its lngth is ngativ zro or positiv. Timlik vctors fall insi th con whras null ons fall on th bounary. Of cours ths null cons hav an immiat physical significanc. It is absolutly funamntal to Gnral Rlativity thory that thr is an uppr boun to th sps with which particls can travl as masur by any obsrvr. If w think of vctors at a point as vlocity vctors thn th light con can b intrprt as a marking of that uppr boun []. Massiv particls must hav timlik vlocity vctors; thos with zro mass must hav null ons. Th most important gomtrical notion that turns out to b th tool to travl back in tim is clos timlik curvs. Thy ar simply clos curvs on th spactim manifol whos tangnt vctors at vry point ar timlik an point to th futur. Ths futur irct timlik curvs of cours hav various magnitus u to th spactim mtric in particular lngth an acclration. Th lngth is usually call laps propr tim. Timlik curvs with no accrlation ar call timlik gosics []. In Gnral Rlativity Göl spactim is important bcaus it is th first ampl of a spactim that is a solution of Einstin s fil quations an that has clos timlik curvs which ar classically forbin. In this thsis w will rviw th Göl spactim. In th lat 94s Kurt Göl took an intrst in Einstin's thory of Gnral Rlativity. H was looking for an answr to th qustion of what th natur of tim

12 is. Evntually h succ in fining a nw solution to th fil quations of Einstin s thory of Gnral Rlativity in 949 [] which influnc all th notions about tim known thus far. This nw solution suggsts a spactim with strang proprtis. First of all th spactim is fill with incohrnt prssurlss flui mattr in a stat of uniform rigi rotation. Sconly th particls with no forc on thm follow a path similar to th paths that boomrangs follow. This mans that th particls rlas from a point of th spactim rc from thir starting point until thy rach a critical istanc an thn mov back to thir starting point at a latr tim. Finally an th most important of all thr is a possibility of travling back in tim in th Göl spactim. As a mattr of fact all of us ar travlling in tim uring our aily lif on arth but hr by travlling w man that a particl starting its motion from a spactim point is abl to com back to its starting point again. So thr is no such finit notion as past futur an prsnt in th Göl spactim. In fact Göl s work i not attract much attntion in th bginning. In 956 Kunt [3] in Hamburg Grmany calculat th gosic worl lins in Göl s spactim. H us Killing vctors to fin th first intgrals of th gosic quations as suggst by Fli Pirani. Thn in 96 S. Chanraskhar an J.P. Wright [4] calculat th timlik gosics of th Göl spactim an conclu that thr wasn t any clos timlik gosic so that Göl s rmark on this issu was incorrct. Howvr Göl ha nvr claim that thr wr clos timlik curvs which wr gosics. Noboy notic this misunrstaning until 97 whn Howar Stin [5] show that Chanraskhar an Wright ha bn mistakn. In fact all th calculations on to fin th gosic worl lin of th Göl spactim bgan with ithr th mtric givn in.6 whos solution of th gosic quations has a complicat form an is not vry nlightning to unrstan th gomtrical pictur of th spactim or th mtric conformal to th Göl on. Actually Göl himslf appli a transformation to his mtric.6 to writ it in cylinrical coorinats so as to show th istnc of clos timlik curvs asily. In this thsis w calculat th timlik an null gosics of th Göl spactim by using th Göl mtric writtn in cylinrical coorinats. Thr was no gomtrical figur showing th bhavior of th gosics of th Göl spactim until 973 whn Hawking an Ellis pictur Göl spactim in

13 thir monograph ntitl Th Larg Scal Structur of Spactim [6]. This was th first tim that gomtrical insight was mploy in unrstaning th proprtis of th Göl spactim. In 3 I. Ozsvath an E. Schucking [7] publish a papr whr thy ask th qustion as to how Hawking an Ellis rw this pictur i.. what ar th quations of th null gosics that la to this pictur? Thy bliv that som intrigu ha bn ma whil rawing this pictur. By using a mtric conformal to th Göl mtric thy calculat th null gosics an rconstruct th pictur of Hawking an Ellis. Evn though thy stat a problm rlat with th mattr worl lin which is rawn as spaclik in th pictur of Hawking an Ellis s Figur 4. thy i not giv th quations of th timlik gosics showing that thy rally must b insi th critical raius. In this thsis th problms of th pictur in [6] ar stat again whn ncssary an th ways to corrct it ar shown. In th scon chaptr w bgin with an ansatz an fin th conitions unr which this ansatz satisfis th fil quations of cosmological Einstin s thory of Gnral Rlativity coupl with a prssurlss prfct flui. W first gt th most gnral form of th Göl mtric with thr intgration constants. Howvr A. K. Raychauhuri an S. N. Thakurta [8] show in thir papr that all th solutions obtain for iffrnt valus of th intgration constants can b transformabl to th Göl mtric. Hnc w ci on how to choos th intgration constants so as to gt th original mtric that Göl us in his calculations. In th thir chaptr w invstigat som physical proprtis of th Göl spactim an choos a coorinat systm to transform th mtric into cylinrical coorinats which is usful in monstrating th istnc of clos timlik curvs. W also fin th Killing vctors of th Göl spactim. In th fourth chaptr null an timlik gosics ar calculat an it is shown that ths quations ar th ons from which Figur 4. can b obtain. 3

14 CHAPTER DERIVATION OF THE GÖDEL SPACETIME In this chaptr w bgin with an ansatz an a spcific nrgy-momntum tnsor namly that of a prssurlss prfct flui sourc for th spactim an fin unr what conitions this ansatz satisfis th fil quations of cosmological Einstin s thory of Gnral Rlativity. W n up with th gnral solution of th fil quations with thr intgration constants. As it has bn stat bfor in [8] it is shown that all th solutions obtain for iffrnt valus of th intgration constants can b transformabl to th Göl mtric. By appropriat choic of constants w gt th mtric that Göl foun in his papr.. Th Mtric In 949 Göl foun a nw cosmological solution of Einstin s fil quations of th form [] s g t z u y v t y.. In this mol spactim is full of a prssurlss prfct flui which has an nrgymomntum tnsor of th form T. Hr is th nrgy nsity of th VV flui an V is th flui four-vlocity. To fin unr what conitions. is a solution to th fil quations of cosmological Einstin s thory of Gnral Rlativity with th corrsponing nrgy-momntum tnsor stat i.. to fin th unknown functions u an v w hav to start by calculating th componnts of th Christoffl symbol an th Ricci tnsor corrsponing to this mtric. Th mtric can b writtn in matri form as 4

15 g v v u with an invrs u v v u v u g v. v u v u By using th finition of th Christoffl symbol g g g g. w can calculat th nontrivial componnts of it asily as v v v u vu uv v v u v v u vv u v u u. Hr ot nots rivativ with rspct to. W will us th plicit form of it / whn ncssary. In th sam mannr by using th finition of th Ricci tnsor [9] R.3 w can asily obtain th nontrivial componnts of th Ricci tnsor as R v.4 v u 5

16 R v u v 4 v u vv 4vvu v 4 v u u u u.5 R vu R v 4 v u R u uv vvu u..6 4 v u Ths componnts of th Christoffl symbol an of th Ricci tnsor ar th only ons that o not vanish. Th componnts of th Einstin tnsor ar givn in Appni A. Th Ricci scalar is obtain asily v 3uv 4 v u u u 8vvu 8v v u v R R g..7 4 v u If w put all of ths things togthr in th fil quations of Einstin s Gnral Rlativity with a cosmological constant 8G VV.8 c G g R Rg g an us a co-moving coorinat systm for th nrgy-momntum tnsor i.. tak V.9 so that Hr G is Nwton s gravitational constant c is th sp of light an is th cosmological constant. For simplicity w us units such that G c. 6

17 V g V v g. w gt first of all that G33 R33 Rg33 8 V3V 3 g33 R. whr w us th fact that R 33 an V 3 ar zro. If w us. an put it into.8 w obtain R 8.. VV Thn w can asily obtain th rst of th quations R V V R R V V R.4 8 R v 8VV R 8.5 R 8 V V R R 8v..7 From.5 an.4 w obtain k k k v.7 whr k v u. W also fin from.4 an.3 that 7

18 v k 8..8 Combining.7 togthr with.8 w gt k k 6k..9 k Now lt us solv.9. First not that k / 3/ k k kk / k k k /. Thn.9 bcoms k / kk / k 6k / an multiplying this by k on fins k / / / k k k 6k which can b writtn as k k c k / /. whr c 6. Lt k / p. 8

19 thn. bcoms c p p which is asy to hanl sinc it is a scon orr homognous quation with constant cofficints. Th most gnral solution is givn by p c / c / c c whr c an c ar intgration constants. So by using. w gt k c / c / c c.. From.8 w hav c / c / v c c c whos solution is simply v c / c / c c c3.3 whr c 3 is anothr intgration constant. Lt us fin th paramtr c / 4 4 thn.3 bcoms v c c..4 c 3 9

20 Th maning of will bcom clar in th nt sction. In fact with. an.4 w now hav all w n sinc u can asily b foun by using k v u. Th mtric functions u an v constraints on th intgration constants. satisfy th fil quations with no furthr. Th Choic of th Intgration Constants Now w hav a solution with thr intgration constants. W now limit our consirations to th spcific choic of intgration constants to gt th mtric that Göl prsnt in his papr. In fact Raychauhuri an Thakurta [8] show that all th solutions obtain for iffrnt valus of c c an c 3 can b transformabl to th Göl mtric. Sinc w ar aftr th Göl spactim w will choos appropriat constants so as to gt th Göl mtric. If w tak th ngativ valu of v an choos intgration constants as c c c3 thn. an.4 bcom k v. Rmmbr that k v u so u is asily obtain as u. Now w hav th mtric that Göl us an corrsponing to this mtric i.. all th rlvant quantitis

21 s t z y ty.5 R 4 4 whr th scon quation in th last lin follows immiatly from. an. i.. R g R 8 g V V 8V V 8. This four imnsional spactim can b viw as th sum of a thr imnsional spactim M whos mtric g is givn by s t y ty.6 on th gomtry M fin by th coorinats t y an a on imnsional 3 R spaclik lin with a mtric g givn by s z on th manifol M R fin by th coorinat z. Th mtric on M os not pn on z an th z constant lins ar all orthogonal to M. Sinc clos

22 timlik lins ist for constant z in th M w o not n to consir th full four imnsional spactim insta w can rstrict ourslvs to just M [67]

23 CHAPTER 3 PROPERTIES OF THE GÖDEL SPACETIME In this chaptr w invstigat som physical proprtis of th Göl spactim an choos a coorinat systm to transform th mtric into cylinrical coorinats which is usful in monstrating th istnc of clos timlik curvs. 3. Transformation to Cylinrical Coorinats In his papr Göl ma th transformation to his mtric to monstrat th cylinrical symmtry of th spactim mor plicitly. Th istnc of clos timlik curvs can asily b shown for th mtric writtn in cylinrical coorinats. With th transformation Coshr Cos Sinhr y Sin Sinhr r Tan t t Tan t t th mtric.6 can b transform to th form s Appni B s t r Sinh 4 r Sinh r Sinh r t 3. whr t r an an is intifi with. W in fact hav writtn th mtric in a form which is plicitly cylinrically symmtric sinc it os not pn on. Lt us us th avantag of this symmtry conition an tak a circl r C r const. in th t plan to invstigat th charactristic of such a circl; i.. lt us amin unr what circumstancs it is timlik spaclik or null. 3

24 Figur 3.: Th circl C on th t plan which has a tangnt vctor. W s that th tangnt vctor to th circl C has th lngth squar g g Sinh r Sinh r.. 3. Thn if Sinh r thn C is timlik thn C is null thn C is spaclik s Figur 3.. Th acclration vctor is givn by ; for th timlik unit vctor [7] / g. An plicit calculation yils g / / / g g g 4

25 g g r. r W thn gt th magnitu of th acclration vctor fin as to b Coshr Sinh r. 3.3 Sinhr Sinh r It is asily sn that os not vanish for Sinh r. Thus with th hlp of 3. w s that w hav clos timlik curvs but not gosics sinc its acclration is not zro for r ln. 3. Killing vctors It is asily sn from th mtrics.5 an 3. that th formr amits thr translations along th y t an z as whil th lattr amits rotation aroun th ais. As a mattr of fact by using th Killing quation in Cartsian coorinats 3.4 ; ; w can asily obtain four of th Killing vctors as A t B z C y D y. y 5

26 W also hav a Killing vctor th Cartsian coorinats; in cylinrical coorinats. Transforming back to t y t y w gt th fifth Killing vctor as E t y y y in Cartsian coorinats. Now w hav fiv inpnnt Killing vctors with th following Li algbra. A B A C A D E A B C B D B E C D C C E D D E E. A B an C ar th trivial gnrators corrsponing to th translation invarianc of th mtric.5 in th t z an y irctions rspctivly. W s that A an B commut with vry Killing vctor. Anothr fatur is that th commutator of D with th Killing vctors C an D is proportional to th Killing vctor itslf. Sinc w hav fiv Killing vctors thr is a fiv imnsional group of isomtris in this spactim which is also transitiv. Transitiv mans that no point rmains unchang by a finit transformation. Hnc Göl spactim is homognous an amits th following four paramtr group of transformations: 6

27 i ii iii iv t t y y z z whr an ar th paramtrs of th group. 3.3 Rotation Shar an Epansion If V is th four-vlocity of a flui thn V can b compos as V P 3 a V ; 3.5 whr a is th four-acclration of th flui a V ; V 3.6 is th pansion of th flui worl lins V V ; 3.7 is th rotation -form of th flui an is th shar tnsor V ; P V ; P 3.8 V ; P V ; P P

28 Hr P is th projction tnsor P g VV P V V 3. which projcts a vctor onto th 3-surfac prpnicular to V []. Rotation -form an th shar tnsor ar antisymmtric an symmtric rspctivly. Now sinc w ar co-moving obsrvrs sitting on th flow of th flui our four-vlocity tangnt to th worl lins of th flui has th simplst form givn in.9 an.: V g. V First lt us amin th acclration by using 3.6. Straightforwar calculations show that it is zro: a V V g g. This mans that th flow of th flui movs on th gosics. Th pansion is also asy to calculat with th hlp of 3.7 V. So Göl spactim is not paning which is iffrnt than th obsrvations of our spactim. Our spactim is paning as it is obvious from th r-shift light rays coming towars us from th istant galais. Now a littl bit of work is rquir to fin th rotation -form of th flui: By using th finition of th covariant rivativ V V V ; 3. 8

29 an. w gt V ; g g. From th finition of th Christoffl symbol. w fin g g g g. 3. Multiplication of 3. an 3. givs V P g g g g ; whr th scon trm in th right han si is zro bcaus thr is no tim pnncy in th mtric. Thn w hav V P g g 3.3 ; V P g g 3.4 ; whr in th last lin w hav intrchang th inics an. Now subtract ths two quations an put thm into 3.9 to gt g g g g. 3.5 From 3. w fin g 3.7 9

30 whr w us th fact that g constant. Thn 3.5 bcoms g g. Now w s that th only componnt which survivs is g g an th magnitu of th rotation -form is.

31 Figur 3.: t plan. Light cons tip ovr an opn out as w proc away from A along concntric circls. Initially such circls ar clos spaclik curvs lik S. Evntually light cons tip to crat clos null curvs lik N an thn tip furthr to crat clos timlik curvs lik T. Sinc spactim is stationary an rotationally symmtric about ach mattr worl lin all obsrvrs s what A ss; all s mattr spinning about thm. Now can b intrprt as th rotation of th prssurlss mattr flui in th Göl spactim. In sction 4. w will show that th cntral mattr worl lin is a timlik gosic so if w mov frly on this worl lin w s that th mattr flui aroun us is rotating with th angular vlocity. In fact all th flui mattr contnt is rotating with rspct to vry point in spac bcaus of th homognity of th Göl spactim. Now lt us igrss hr a littl bit an think about th maning of having nonvanishing rotation. Actually this spactim is stationary sinc it amits a timlik Killing vctor fil but it is not static bcaus in orr for this spactim to b static in aition to bing stationary thr shoul ist a hyprsurfac which is

32 orthogonal to th congrunc of timlik Killing vctor fils. Howvr a straightforwar calculation shows that if an only if s Appni C for th proof V [ V ;] = 3.8 which is th conition for hyprsurfac orthogonality. Hnc w can conclu that our worl lins ar not hyprsurfac orthogonal sinc w hav non-vanishing rotation. Lt us now show that th conition 3.8 is ncssary an sufficint for a vctor to b orthogonal to a on-paramtr hyprsurfac. Suppos that thr is a on-paramtr family f of hyprsurfacs givn by f constant. A normal vctor to a hyprsurfac can b writtn as follows; f hf 3.9 whr h is an arbitrary scalar function. Now lt us construct from an arbitrary vctor th compltly antisymmtric tnsor a [ ;] ; ; ; ; ; ;. 3! This tnsor is actually zro for th spcific vctor givn by 3.9 as can b shown as follows: ; h f hf ; ; hf h f h f f ;

33 } { 3! ; ; ; ; ; ; f f h f f h f f h f f h f f h f f h f h hf f h hf f h hf f h hf f h hf f h hf a whr th first trm in th parnthss cancls out with th scon th thir trm with th fourth tc... Thus a ncssary conition for a vctor fil to b vrywhr orthogonal to a on-paramtr family f of thr imnsional hyprsurfacs is a [ ;] =. Hnc our worl lins ar not hyprsurfac orthogonal bcaus 3.8 os not hol for our situation. Thrfor w hav a stationary but non-static spactim. To fin th shar tnsor w just a 3.3 an 3.4 up an fin g g g g. 3. Th last two trms ar obviously zro u to 3.7. By using 3.6 w gt g g g. If w put this quation into 3. w fin. 3. Non of th componnts of th shar tnsor survivs so its magnitu is zro. 3. rprsnts th fact that thr is no istortion in th shap of our collction of tst particls. That is suppos som portion of our tst particls occupy a volum in 3

34 spac in th shap of a sphr thn as tim gos on th sphr will rmain as a sphr as oppos to th cas whr thr is a non-vanishing shar. In th cas of a non-vanishing shar th sphr woul hav bcom an llipsoi as tim pass by. Hnc thr is only a rotation in Göl spactim which actually prvnts us from choosing a cosmic tim coorinat to agr on vnts that ar simultanous. 4

35 CHAPTER 4 GEODESICS In this chaptr gosic motion in Göl spactim is analyz an th gomtrical pictur of th spactim is givn. 4. Null Gosics To stuy th null gosics w us th variational mtho for th mtric givn in 3. an rcapitulat th argumnts that la to Figur 4. blow. This figur is rprouc from [6]. In thir papr [7] Ozsvath an Schucking rw attntion to th fact that th quations us in rawing Figur 4. ar not givn an stat thir blif that an intrigu has bn ma uring th rawing of it. So thy calculat th null gosics of th Göl spactim by using th mtric conformal to th Göl mtric to rconstruct it again. Sinc null gosics ar not affct by such a conformal transformation an sinc th timlik or null natur of a vctor is lft invariant unr such a transformation thr is no loss of gnrality in oing so. Hr w fin th paramtric quations of th null gosics by using th original Göl mtric an try to show that thy ar th ons us in rawing Figur 4.. 5

36 Figur 4.: Göl s spactim with th irrlvant coorinat z supprss. Th spac is rotationally symmtric about any point; th iagram rprsnts corrctly th rotational symmtry about th r= ais an th tim invarianc. Th light con opns out an tips ovr as r incrass s lin L rsulting in clos timlik curvs. Th iagram os not corrctly rprsnt th fact that all points ar in fact quivalnt. 6

37 Consir th following trmization rlat to th thr imnsional mtric 3. which yils th gosics: L L g t 4 r Sinh r Sinh r Sinh r t whr t t r r with th affin paramtr that can b us for null gosics. Sinc th Lagrangian os not pn on t an w hav from th Eulr-Lagrang quations th two intgrals L t t Sinh r a constant 4. L Sinh 4 r Sinh r Sinh r t b constant. 4. Hr a can b intrprt as th total nrgy an b is intrprt as th total angular momntum of th photons. Sinc w ar trying to fin th null gosics w also hav L t r Sinh If w substitut 4. into 4.3 w fin 4 r Sinh r Sinh r t. 4.3 r Cosh r Sinh r a

38 Morovr from 4. an 4. w also gt a b a Sinh r t 4.5 Cosh r b a Sinh r. 4.6 Sinh r Cosh r If w put 4.6 into 4.4 w fin th intgral Coshr Sinhr r /. 4.7 a Cosh r Sinh r b a Sinh r 4.7 givs s Appni D Sinh r Sin a u whr b a 4 4 b a b u an is an intgration constant. a If w put this solution into 4.5 an 4.6 w gt th intgrals t { a Sin a Sin a u a u b} 4.8 Sin a a u { Sin a u b }{ Sin a. u } 8

39 Th first on of ths two intgrals is asir to hanl compar to th scon on so first w fin th solution of th first intgral an thn al with th scon on latr. Aftr som calculations w gt th solution of th first intgral as follows s Appni D: a b u t t a ArcTan Tan a a u u u. 4.9 Now w can invstigat th gosic motion for thr istinct cass: b b an b. Howvr sinc w ar aftr Figur 4. an sinc all th null gosics rawn in it com from r w must choos b. Now it is much simplr to intgrat th scon quation bcaus aftr insrting b th rmaining intgration bcoms trivial. Notic that if w st b thn u an 4. bcoms t t a ArcTan Tan a whras th scon intgral bcoms a. 4. Sin a 4. givs ArcTan Tan a. If w writ thm togthr w now hav th paramtric rprsntation of th null gosics paramtriz by : 9

40 Sinhr Sin a 4. t t a ArcTan Tan a 4. ArcTan Tan a 4.3 whr t an ar intgration constants. Ths quations giv th corrct rprsntation of th null gosics rawn in Figur 4.. For vry valu w hav a iffrnt null gosic. Suppos that on sns out a light signal from point p at som tim of t p in th irction quals zro an lt th constants an t as wll. Thn as th angl incrass th istanc btwn th light signal an th worl lin r incrass until th light signal rachs its maimum istanc r ln from th worl lin r in which cas th angl is qual to /. On can tak th tim th signal rachs th point r ln as t. Continuing on its path th light signal rturns to its starting point at th tim t p. At point p r an 4. givs whras 4. givs t p a. Whn th light signal rachs th point r ln at tim t 4. givs a /. Thn from 4. w fin a. Th tim t p bcoms t p. 3

41 In fact this is th tim for half of th trip. Whn th light signal rturns to its starting point p at a latr tim t p 4. givs a an 4. yils t p a i.. t p. All th rays snt out simultanously in iffrnt irctions of th plan rturn latr at th sam momnt. At th critical istanc r ln from th worl lin r w hav circls for t constant bcaus vry null gosic touchs th t constant plans in on point. In thir paprs Ozsvath an Schucking [7] an Malamnt [] stat that th iagram of th Göl mol in [6] is not corrct about th mattr lins sinc thy ar rawn as spaclik. Howvr thy o not giv th timlik gosics quations which show that all th timlik gosics must b in th circl with raius r ln. In th nt sction w show this proprty of th timlik gosics. Morovr th structur of th light cons i.. what happns to th light cons as w mov away from th worl lin r is givn in sction Timlik Gosics Timlik gosics can b calculat with th sam mtho us in calculating th null gosics but this tim our Lagrangian must b qual to. Obviously 4. an 4. hol also for this situation. Howvr this tim th affin paramtr can b chosn as th propr tim masur by a co-moving obsrvr. Again w st th valu of b to zro bcaus of th sam rason us in th prvious sction. Hnc w hav 3

42 a 4.4 Cosh r a Sinh r t 4.5 Cosh r L t r Sinh 4 r Sinh r Sinh r t. 4.6 If w put 4.4 an 4.5 into 4.6 w obtain Coshr r 4.7 a a Sinh r Cosh r / whos solution is s Appni D Sinhr Sin 4.8 whr a an a with th conition a. If w substitut 4.8 into 4.4 an 4.5 w gt th intgrals { a a Sin Sin } t 4.9 a. 4. Sin Ths ar asily valuat as 3

43 a ArcTan Tan a 4. a a t t a ArcTan Tan a. 4. a For vry valu of a w hav a iffrnt timlik gosic. Whn a an 4. bcom Sinhr r t t rspctivly which corrspons to th cntral mattr worl lin. Suppos that at tim t t p a particl is rlas from point p in Figur 4. in th irction. Thn this particl will mov on th cntral mattr worl lin notic that this particl movs only in tim an will rach th point O at tim t whn. At th n th particl rachs th point pat tim asily obtain t t whn p. Thn on can t p t. p 4 4 For all valus of a r is always smallr than ln. Sinc a is intrprt as th nrgy pr unit mass thn no massiv particl no mattr what its nrgy is can rach this critical raius. All th particls rlas from th point p with iffrnt nrgis will rach som maimum valu of r smallr than ln an 33

44 thn rconvrg to p which shows that all th mattr lins must b insi th cylinr with raius ln. 4.3 Th Structur of th Light Cons To stuy th structur of th light cons w n to fin th clos-form solution of th null gosics quations. Aftr stting b w now com back to an 4.6. Tak 4.6 a Cosh r an substitut this prssion into 4.4 to obtain r Coshr Sinh r givs Sin Sinhr 4.4 Cos with th intgration constant. 34

45 Figur 4.: Path of th photon in th r plan. For vry valu of thr is a iffrnt llips passing through th origin. Ths llipss form an nvlop which is a circl with raius r ln aroun th origin. W hav from 4. that t Sinh r 4.5 or using 4.4 t Tan. 4.6 If w intgrat 4.6 w gt 35

46 t t ArcTan Tan. 4.7 W hav two quations to think about which ar 4.4 an 4.5. From ths two quations w can of cours obtain Figur 4. again. Although all of th commnts ma in sction 4. hol also for ths two quations sinc thr is no istinction btwn thm it will b usful to consir th situation again. For vry valu w hav a iffrnt null gosic. Lt us s what happns to th light rays originating from point p in Figur 4.. Lt us think of th cas whr is zro. W s that th lft han si of 4.4 is maimum whn / th null gosic rachs its largst istanc r ln ; an its istanc is zro whn an. Th light ray starts to mov from point p whn an t t an rachs to its largst istanc from th worl lin r an thn gos to th point p. Th tim takn by th light ray uring th first half of its trip is t / t. All th light rays originat from point p in iffrnt irctions rturn latr at th sam momnt to th point p. Of cours th tim intrval btwn th point p an p is t t. At th istanc r ln from th worl lin r w hav circls for t constant. Ths circls ar null curvs as w s from 3.. Howvr thy ar not gosics. Thy form a cylinr with an ais at r. What happns to th light cons whn w mov away from th ais r? To answr this qustion w n r / t an it is sufficint to consir on raial irction s lin L in Figur 4.. From 4.3 an 4.5 w obtain 36

47 r t r t Coshr Sinh r / Sinh r Coshr Sinh r /. At th origin r th slop of r t is qual to on. This mans that w hav a light con with an angl / 4. That is on th cntral mattr lin th cons ar upright an orint at / 4 to th inicat horizontal plan. Th last prssion bcoms infinit for r ln. It mans as on movs outwar that th cons opn up an tilt ovr in a countr-clockwis irction. At a critical raius r ln thy bcom tangnt to th plan. 4.4 Som Rmarks About th Göl Spactim For a givn point p in th Göl spactim thr is a cylinr associat with p which sparats th spactim into two rgions in a givn coorinat systm in which p rsts on th r lin. All of th points lying insi this cylinr hav raial istancs from p which is lss than ln. In fact this cylinr is construct by th circls with raius r ln which ar also null curvs. Circls with raius r ln ar clos timlik curvs. For any valu of nrgy a th photons originating from p ar confin insi this cylinr. This rsult hols also for massiv particls. Howvr massiv particls ar nvr abl to rach th cylinrical surfac whras th photons ar. Sinc thr ar no clos timlik curvs within this cylinr tim t always incrass along vry futur irct timlik curv. Byon r ln th tim t runs backwar which can b sn from 4.5 a Sinh r t. 4.5 Cosh r 37

48 This prssion is zro for r ln for which th tim t changs its sign. For r ln 4.5 bcoms ngativ which mans t crass. This shows that thr can not b a cosmic tim coorinat t in this spactim which incrass along vry futur irct timlik or null curv. Morovr our obsrvr sitting on th r lin will obsrv only som part of th spactim sinc all of th gosics ar confin insi th cylinr. That is you can not sn a light signal byon th raius r ln. Sinc thr is no way to go byon that cylinr thn thr is no prmission for th light rays coming from istant galais to ntr insi th cylinr ithr. 38

49 CHAPTER 5 CONCLUSION W hav shown that by using th original Göl mtric in calculating th null gosics of Göl spactim Figur 4. can b obtain vry asily. This answrs th qustion that Ozsvath an Schucking hav pos in thir papr [7]. Morovr th problm of Figur 4. that is point out by Ozsvath an Schucking [7] an Malamnt [] is corrct with th hlp of timlik gosics quations. By invstigating th structur of light cons w saw how th clos timlik curvs ar crat in Göl spactim. Göl spactim is intrsting in two faturs: Firstly all th light rays an th fr particls follow th trajctoris which ar clos in spac but not in spactim of cours. This is call th boomrang ffct. As it is asily sn from Figur 4. for th cas whr if w stan at th origin an hol a flashlight pointing to th ast thn all th light rays follow a clos path an com back to th origin from th wst. Evrything woul b quit mssy if w wr to liv in a Göl spactim. Sconly th istnc of clos timlik curvs was first shown in Göl spactim an such curvs allow on to travl back in tim. So far w hav not rfrr to th notion of tim travl vn though th istnc of clos timlik curvs has bn plicitly shown in Göl spactim. Sinc ths clos timlik curvs ar not gosics th acclration is not zro on has to o an acclrat motion to follow ths curvs. Th nrgy rquir for a tim travlr to mov on a clos timlik curv was calculat by Malamnt []. Sinc this nrgy issu is byon th scop of this thsis w hav not stui numrical rsults of his calculations. It is just nough to say that a hug amount of nrgy that is impossibl to prouc with our prsnt tchnology is rquir for such an acclrat motion to travl back in tim. Howvr th consquncs of tim travl woul b important. Thr is no satisfactory answr to th qustion of what happns if on can travl backwars in tim an mt his/hr youngr slf. Th issu rmains obscur vn though thr ar lots of scnarios to liminat th problm that such a mting can caus. Thr woul crtainly b a parao if th prson kill 39

50 his/hr youngr slf. Som popl tn to lav th Göl solution asi u to such a possibility. W can not clu th Göl spactim immiatly just bcaus it allows for tim travl an th fact that th rsults of tim travl ar paraoical to our mins. Th scon rason for th tnncy of laving Göl spactim asi is that it is not paning whil our univrs is. This rsult alon maks it possibl to rul out th Göl spactim as a possibl mol for scribing our univrs. Howvr th issus that aris by th istnc of clos timlik curvs is still staning. 4

51 REFERENCES [] Malamnt D. 984 Tim travl in th Göl Spactim in Procings of th Binnial Mting of th Philosophy of Scinc Association Issu Volum Two: Symposia an Invit Paprs 9-. [] Göl K. 949 An Eampl of a Nw Typ of Cosmological Solutions of Einstin s Fil Equations of Gravitation Rviws of Morn Physics [3] Kunt W. 956 Träghitsbahnn in inm von Göl anggbnn kosmologischn Moll Zitschrift für Physik [4] Chanraskhar S. an Wright J. P. 96 Th Gosics in Göl s Univrs Procings of th National Acamy of Scincs [5] Stin H. 97 On th Paraoical Tim-Structurs of Göl Philosophy of Scinc [6] Hawking S. W. an Ellis G. F. R. 973 Th Larg Scal Structur of Spac-tim Cambrig: Cambrig Univrsity Prss. [7] Ozsvath I. an Schucking E. 3 Göl s Trip Amrican Journal of Physics [8] Raychauhuri A. K. an Thakurta S. N. G. 98 Homognous spactims of th Göl typ Physical Rviw D [9] Wal R. 984 Gnral Rlativity Chicago: Th Univrsity of Chicago Prss. 4

52 [] Lightman A. Prss W. H. Pric R. H. an Tukolsky S. A. 975 Problm Book in Rlativity an Gravitation Princton Nw Jrsy: Princton Univrsity Prss. [] Malamnt D. 985 Minimal Acclration Rquirmnts for Tim Travl in Göl Spactim Journal of Mathmatical Physics

53 APPENDIX A THE COMPONENTS OF THE EINSTEIN TENSOR For th sak of compltnss hr w list th nontrivial componnts of th Einstin tnsor of th Göl spactim corrsponing to th mtric.. G 3v u v u v u u 4vvu 4 v 4 v u 3 uv v G v 4 u v 3v u uv 3uv v 4 v u 3 v v 3 uv u vu G v 4 v u G v 4 uv u v u v 3 uv uv 3uv 4 v u u v 4 uv 3 u v v G 33 3uv u v u u 4vvu 4v v 4 v u 3 uv v v. 43

54 APPENDIX B TRANSFORMATION TO CYLINDRICAL COORDINATES In this appni w show th tails of th calculation that transforms th thr imnsional part of th Göl mtric.6 to its form 3. in cylinrical coorinats. For this purpos lt us start with.6 s t y ty..6 Lt us writ.6 as s t y y B. an thn apply th following transformations to B. Coshr Cos Sinhr B. y Sin Sinhr B.3 r Tan t t Tan t t. B.4 Lt us first fin nw coorinats such that r. Thn B. B.3 an B.4 can simply b writtn as Cos Sin B.5 y Sin Cos B.6 44

55 Tan t t Tan. B.7 Now lt us iffrntiat B.5: Cos Sin Sin Cos i.. Cos Sin Sin Cos. B.8 Manwhil th iffrntiation of B.6 givs:. Cos Sin Sin Cos y y B.9 If w us B.8 in B.6 w gt B. Now subtract B. from B.9 to obtain Sin Cos Sin Cos y. B. Taking th squar of B. givs Cos Sin Cos Sin Sin Cos y 45

56 . 4 4 Sin Cos Sin Cos Sin Cos Sin Cos y B. In th sam mannr taking th squar of B.8 givs Cos Sin Sin Cos Cos Sin Sin Cos r r 4 B.3 Th combination y in th mtric B. can asily b calculat as notic that th trms in B. an B.3 ar qual an opposit y. B.4 Now th iffrntiation of B.7 givs: Sc r Tan t t t t Tan B.5 an by using B.8 B.6 can b writtn as 46

57 Tan Sc Tan Tan t t. Nt lt us multiply an ivi th right han si of th abov quation with Cos : Sin Cos Sin Cos Cos Sin t t i.. t Cos Sin t. B.6 Now a B. an B.6 up Sin Cos t y t i.. Sin Cos t y t an w hav t y t / /. B.7 47

58 Taking th squar of B.7 givs B.8 Now lt us put B.4 an B.8 into B.. / / 4 / / t t s If w writ th abov quation in trms of r an t again w fin It is now straightforwar to obtain 3. 4 t r Sinh r Sinh r Sinh r t s. 3.. / / 4 / / t t y t. 8 4 t t r s r r r r r r 48

59 APPENDIX C PROOF OF 3.8 In this appni w show th proof of th statmnt that if an only if V [ V ;] =. For this purpos lt us first us 3. an thn 3. to obtain V ; P V ; V V i.. V ; P V ; V VV ;. C. Now substituting C. into 3.8 V ; V VV ; V ; V VV ; w can obtain V VV ; VV; VV; VV ; VV ; V V; +V [α;β] = C. whr w hav a an subtract som trms. C. can b writtn as V [α;β] V V V [α;μ] +V V [μ;β] + V V μ;α V μ;β V i.. 49

60 V [α;β] V V V [α;μ] V [μ;β] V V V V μ;α V V V μ;β. Notic that th trms in parnthss can b writtn as follows: V [α;β] V V V [ β;α] 3 V [V μ;β] V V V μ;α V V V μ;β or 3V V [V μ;β] V V V μ;α V V V μ;β. C.3 Rcall that V V i.. w gt thn C.3 bcoms V is timlik thn V V V V. If w tak th covariant rivativ of V V V V V V 3V V [V μ;β]. Thus if an only if V [ V ;] =. 5

61 D. How to Intgrat 4.7 APPENDIX D CALCULATIONS OF THE INTEGRALS Rcall Coshr Sinhr r a Cosh r Sinh r b a Sinh r /. 4.7 If w mak th substitution u Sinh r D. w gt / a u u b b u a a. Now lt us fin th iscriminant of th quaratic quation in th nominator b b b 4 4 a a a 4 b. a Hnc th roots of th nominator ar b u a b u. a 5

62 W now hav u / /. D. a u u u u Now w n anothr substitution u u u u Sin. D.3 Diffrntiation of D.3 givs u u u u Sin u / u u /. D.4 Thn by using D.3 an D.4 D. can b writtn as a. a Aftr using th invrss of th substitutions D.3 an D. rspctivly w gt th sir rsult Sinh r Sin a u. D. How to Intgrat 4.8 Rcall t { a Sin a Sin a u a u b}

63 4.8 can b writtn as a b t a. D.5 Sin a u Aftr th substitution q Tan a D.6 D.4 bcoms t a a b a q q q q u a b q t a. D.7 a u q u Now w n anothr substitution q u Tan D.8 u u q Tan u which taks D.7 to 53

64 a b t a a u u a b t t a. a u u Aftr using th invrss of th substitutions D.8 an D.6 rspctivly w gt a b u t t a ArcTan Tan a a u u u. D.3 How to Intgrat yils Coshr r /. a a Sinh r Cosh r Th substitution n Sinhr D.9 yils 54

65 n / a n a which can b writtn as n / / a a n a. Thn aftr th scon substitution a n Sin D. a w hav / a. a Now using th invrss of th substitutions D. an D.9 rspctivly w gt th sir rsult a Sinhr Sin a. a 55