From Here to Eternity and Back: Are Traversable Wormholes Possible?

Transcription

1 Fom Hee to Etenity and Back: Ae Tavesable Womholes Possible? May Magaet McEachen with advising fom D. Russell L. Heman Phys. 495 Sping 009 Apil 4, 009 Dedicated in Memoy of My Dea Fiend and UNCW Alumnus Kaen E. Goss (Apil, 98~Jan. 4, 009)

2 Do Physicists Cae, and Why? Ae womholes possible? How can we model to pove o dispove? Womholes ae being taken seiously Possible uses? Intestella tavel Time tavel

3 Outline Histoy Models that fail Desiable taits Geneal elativity pime Mois-Thone womhole Cuvatue Stess enegy tenso Bounday conditions and embeddings Geodesics Examples Time machines and futue eseach

4 What is a Womhole? Hypothetical shotcut between distant points Geneal elativity

5 Histoical Pespective Albet Einstein geneal elativity ~ 96 Kal Schwazschild fist exact solution to field equations ~ 96 Unique spheically symmetic vacuum solution Ludwig Flamm ~ 96 White hole solution

6 Histoical Pespective Einstein and Nathan Rosen ~ 935 Einstein-Rosen Bidge fist mathematical poof Kut Gödel ~ 948 Time tunnels possible? John Achibald Wheele ~ 950 s Coined tem, womhole Michael Mois and Kip Thone ~ 988 Most pomising womholes as tools to teach geneal elativity

7 Tavesable Womholes? Matte tavels fom one mouth to othe though thoat Neve obseved BUT poven valid solution to field equations of geneal elativity

8 Black Holes Not the Answe Tidal foces too stong Hoizons One-way membanes Time slows to stop Schwazschild womholes Fail fo same easons

9 Simple Models Geometic Taits Eveywhee obeys Einstein s field equations Spheically symmetic, static metic Thoat connecting asymptotically flat spacetime egions No event hoizon Two-way tavel Finite cossing time

10 Simple Models Physical Taits Small tidal foces Reasonable cossing time Reasonable stessenegy tenso Stable Assembly possible

11 A Little Geneal Relativity Pime G 8 GT G R Rg Field equations elate spacetime cuvatue to matte and enegy distibution Left side Cuvatue Right side Stess enegy tenso Summation indices

12 Moe on Geneal Relativity Space acts on matte, telling it how to move. In tun, matte eacts back on space, telling it how to cuve. ~ Misne, Thone, Wheele, Gavitation

13 Spacetimes A spacetime is a fou-dimensional manifold equipped with a Loentzian metic [of signatue] (-,+,+,+). A spacetime is often efeed to as having (3+) dimensions. ~Matt Visse, Loentzian Womholes.

14 Repesenting Spacetimes: Flat Line Element Uses diffeentials Einstein summation Metic Minkowski space: ds g dx dx ds g dt g dx g dy g dz tt xx yy zz g ; g ; g ; g ( catesian) tt xx yy zz g ; g ; g ; g sin ( spheical) tt

15 Mois-Thone Womhole (988) ds e dt b d ( d sin d ) Φ() edshift function Change in fequency of electomagnetic adiation in gavitational field b() shape function

16 + Popeties of the Metic ds e dt b d ( d sin d ) - Spheically symmetic and static Radial coodinate such that cicumfeence of cicle centeed aound thoat given by π deceases fom + to b=b 0 (minimum adius) at thoat, then inceases fom b 0 to + At thoat exists coodinate singulaity whee component diveges Pope adial distance l() uns fom - to + and vice vesa

17 Mois-Thone Metic ds e dt b d ( d sin d ) g gtt g g g e b sin

18 G 8 GT Detemining Cuvatue (Left Side) Do we have the desied shape? Catan s Stuctue Equations Catan I d i i j j i j Catan II i i k d j k j Elie Joseph Catan (869~95)

19 Catan I Connection One-Foms, One-foms dx, dy, dz pdx qdy dz Take fom Mois-Thone metic: ds e dt b i j j i d ( d sin d ) 0 3 e dt d b sind d

20 Opeations with Foms Wedge poduct dt dt d d d d d d 0 dt d d dt d Opeato: k-fom to (k+)-fom fom a a ( t, ) da da dt dt Combining: da d d a ( t, ) da dt d dt dt dt da d d dt da d d dt

21 Calculation Connection One-Foms ds e dt e dt d e dt d; dt e b 0 b d d ; d b d ( d sin d ) d d d d; d sind d sind d cos d d; d sin 3 3 3

22 Calculation Connection One-Foms d i i j i, j t,,, j d e dt d 0 0 b e dt d e dt 0 e 0 e b b dt dt 3

23 Matix of One-Foms j i e b dt e b dt b d b d b d d b d d sin cos sin cos

24 Calculation Cuvatue Two-Foms i j d i j R i k k j i m n mnj i, j, k, m, n t,,, Computed using matix of one-foms Non-zeo components of Riemann tenso Useful fo computing geodesics

25 Calculation Cuvatue Two-Foms 3 cosd d sind d 3 d b 3 sin d d d b sin d b sin d d b sin sin b 3 3 b R R 3 b R R 3 3

26 Riemann Tenso Components b R R R R t b b t tt tt tt b b R R R R 3 t t R R R R t tt t tt b b b R R R R 3 b t t Rt Rt t Rt Rtt b R R R R 3

27 Riemann Tenso Popeties R R R R t t t t Antisymmetic in (t, ) Antisymmetic in (θ,φ) Symmetic in (t, ) and (θ, Φ) Only 4 independent components Geneally in 4D has 56 components Govens diffeence in acceleation of two feely falling paticles nea each othe

28 G R Rg Ricci Cuvatue Tenso t R R R R R tt ttt tt tt tt b b b b t R R R R R t b b b b b 3

29 G R Rg Ricci Cuvatue Tenso t R R R R R t b b b b 3 3 R t R R R R R t b b b b 3 3 R

30 G R Rg Cuvatue Scala R R R R R tt b b b b b 4 b b 3 3

31 G R Rg Einstein Tenso Einstein Tenso G G, G, G, G tt Ricci Cuvatue Tenso R R, R, R, R tt Cuvatue Scala R R R R R tt Metic g g, g, g, g tt

32 G R Rg Einstein Tenso Components b Gtt Rtt Rgtt b G R Rg 3 b b b b b b G R Rg G b b ( ) ( ) b b b b b G R Rg G b b ( ) ( )

33 G 8 GT Stess-Enegy Tenso T p p

34 Equations of State Reaange, solve. Enegy density b 8 Tension b ( b) 8 Pessue (stess) p ( )

35 Womhole Embedding Diagam z( ) b ln, b b 0 0 b 0 0 Static (t=constant slice ) Assume θ=π/ (equatoial slice ) Only,Ф vaiable ds b d d ds dz d d

36 Bounday Conditions - Shape b 0 minimum adius at thoat Vetical at thoat Asymptotically flat -axis

37 Bounday Conditions No Hoizon Hoizon - physically nonsingula suface at which g tt vanishes ; defined only fo spacetimes containing one o moe asymptotically flat egions e.g. Schwazschild metic coodinate singulaity at =M M M ds dt d d d sin Mois-Thone metic ds e dt b d ( d sin d ) e 0

38 Othe Bounday Conditions Cossing time on ode of yea Acceleation and tidal acceleation on ode of G

39 Geodesics Geodesics ae extemal pope time woldlines ; equations of motion that detemine geodesics compise the geodesic equation ~ Hatle Timelike Paticle feefall paths; Null Light feefall paths; ds 0 ds 0 LOCALLY

40 Vaiational Pinciple Lagangian Eule-Lagange Equation ds e dt b d d d ( sin ) L g dx d dx d e dt d b d d d d d d sin AB Ld d d L dx d L x geodesics 0 0

41 Geodesic Equation ;,,, d x dx dx d d d g g g g,,, t,,, d dt t e 0 d d d b d d b d d d dt sin e 0 d d d d d d d d d d d d d sin 0 d d d sin cos d 0

42 Chistoffel Symbols Descibe cuvatue in non-euclidean space Metic is like fist deivative of wap Chistoffel symbol is like second deivative of wap Also called connection coefficients Non-zeo Chistoffel symbols ae components of a 3-tenso t t t t d d cos sin sin cos b tt e ( b)sin b b b ( b)

43 Mois-Thone Examples Zeo tidal foces Exotic matte limited to thoat Absudly benign womhole

44 Zeo Tidal Foce Solution 0 b b( ) bo fo b 0 Equations of State b b ( b) 8 8 p ( ) 5 5

45 Zeo Tidal Foce Solution Womhole mateial extends fom thoat to pope adial distance +/- Density, tension and pessue vanish asymptotically Mateial is eveywhee exotic i.e., 0 eveywhee Violates enegy conditions Need quantum field theoy Catenoid

46 Exotic Matte Limited to Thoat ( b0 ) b b0 ; 0 fo b b a a0 b 0 fo b a Spacetime flat at > b 0 + a 0 Tidal foces beaable Tavel time easonable BUT thoat adius must be lage to have meaningful womhole Absudly Benign Womhole

47 Backwad Time Tavel Fom H.G. Wells, The Time Machine Time machine Any object o system that pemits one to tavel to the past Not poven possible o impossible Tavele moves though womhole at sub-light speed Appeas to have exceeded light speed to stationay obseves Causality violations and paadoxes (consistency and bootstap)

48 Summay Theoetically easonable Mois-Thone model No hoizons Exotic matte Enegy condition violations Causality violation Much wok has been done and continues to be done in this aea; models abound!

49 Possible Questions fo Futue Is necessay topological change even pemitted? Exotic matte equied? If so, is it allowed on the quantum level? If so, can we enlage to classical size? Mois and Thone: pulling a womhole out of the quantum foam

50 THANK YOU! Kaen E. Goss (98~009) UNCW Class of 005 (Mathematics)

51 Refeences (Books and Papes) Jaoslaw Pawel Adamiak, Static and Dynamic Tavesable Womholes (Univ. of South Afica, Januay 005) James B. Hatle, Gavity -- An Intoduction to Einstein's Geneal Relativity (Addison-Wesley 003) David C. Kay, Schaum s Outline of Theoy and Poblems of Tenso Calculus (McGaw-Hill Pofessional 988) Chales W. Misne, Kip S. Thone, John Achibald Wheele, Gavitation (W.H. Feeman and Company 973) Michael S. Mois and Kip S. Thone, Womholes in Spacetime and Thei Use fo Intestella Tavel: A Tool fo Teaching Geneal Relativity (Calif. Inst. of Technology, 7 July 987) Thomas A. Roman, Inflating Loentzian Womholes (Cent. Conn. St. Univ. 99) Matt Visse, Loentzian Womholes: Fom Einstein to Hawking (Ame. Inst. of Physics 995)

52 Refeences (Websites) (Apil 7, 009) (Apil 0, 009) (Apil, 009) (Mach 6, 009) (Januay 7, 009)

53 Refeences (Figues) womholes.jpg/360px-time_tavel_hypothesis_using_womholes.jpg (Apil 5, 009) p=0 (Apil, 009) (Febuay, 009) (Mach 5, 009) (Apil, 009) (Mach 8, 009) (Januay 3, 009) (Febuay 3, 009) (Apil 5, 009) 85.jpg (Apil, 009) (Mach 9, 009) (Januay 9, 009) (Febuay, 009)

54 Refeences (Figues, cont d.) zu3nm/rzhjq4gethi/aaaaaaaaae0/wl8cadttwho/s30/babyinhand.jpg&imge ful= fo.html&usg= XGt- WxoBXfW9MwYX6IwqdLCBs=&h=304&w=30&sz=&hl=en&stat=96&um= &tbnid=fvbfr8xqu9ckm:&tbnh=&tbnw=8&pev=/images%3fq%3dfou %Bdimensional%Bmanifold%Bimage%6ndsp%3D0%6hl%3Den%6sa%3 DN%6stat%3D80%6um%3D (Apil 4, 009) (Apil 5, 009)