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Home , Albert Einstein, Exotic matter, John Archibald Wheeler, Karl Schwarzschild, Kip Thorne, Nathan Rosen

From Here to and Back: Are Traversable Possible?

Mary Margaret McEachern with advising from Dr. Russell L. Herman Phys. 495 Spring 2009 April 24, 2009 Dedicated in Memory of My Dear Friend and UNCW Alumnus Karen E. Gross (April 11, 1982~Jan. 4, 2009) Do Care, and Why?

 Are wormholes possible?  How can we model to prove or disprove?  Wormholes are being taken seriously  Possible uses?  travel Outline   Models that fail  Desirable traits  primer  Morris-Thorne  Stress  Boundary conditions and embeddings   Examples  Time machines and research What is a Wormhole?

 “Hypothetical shortcut between distant points”

 General relativity Historical Perspective

– general relativity ~ 1916

– first exact solution to equations ~ 1916  Unique spherically symmetric solution

 Ludwig Flamm ~ 1916  solution Historical Perspective

 Einstein and ~ 1935  “Einstein-Rosen Bridge” – first mathematical proof

 Kurt Gödel ~ 1948  Time tunnels possible?

~ 1950’s  Coined , “wormhole”

 Michael Morris and ~ 1988  “Most promising” – wormholes as tools to teach general relativity Traversable Wormholes?

travels from one mouth to other through throat  Never observed  BUT proven valid solution to field equations of general relativity Black Holes Not the Answer

 Tidal too strong  Horizons  One-way membranes  Time slows to stop  “Schwarzschild wormholes”  Fail for same reasons Simple Models – Geometric Traits

 Everywhere obeys Einstein’s field equations  Spherically symmetric, static metric  “Throat” connecting asymptotically flat regions  No horizon  Two-way travel  Finite crossing time Simple Models – Physical Traits

 Small tidal forces  Reasonable crossing time  Reasonable stress- energy tensor  Stable  Assembly possible A Little General Relativity Primer

G  8 GT 1 G  R  Rg   2   Field equations – relate spacetime curvature to matter and energy distribution  Left side – Curvature  Right side – Stress energy tensor  Summation indices More on General Relativity

“ acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.” ~ Misner, Thorne, Wheeler,

“A spacetime is a four-dimensional equipped with a Lorentzian metric…[of signature]…(-,+,+,+). A spacetime is often referred to as having (3+1) .” ~Matt Visser, Lorentzian Wormholes. Representing Spacetimes: Flat

“” 2    Uses differentials ds  g dx dx  Einstein summation  Metric  :

2 2 2 2 2 ds  gtt dt  gxx dx  g yy dy  gzz dz

gtt  1; gxx  1; g yy  1; gzz  1 (cartesian) 2 2 2 gtt  1; grr  1; g  r ; g  r sin  (spherical) Morris-Thorne Wormhole (1988)

1 ds2  e2 dt 2  dr 2  r 2 (d 2  sin2  d 2 )  b 1   r 

 Φ(r) – function  Change in frequency of electromagnetic radiation in

 b(r) – shape function +∞

Properties of the Metric

1 ds2  e2 dt 2  dr 2  r 2 (d 2  sin2  d 2 )  b 1   r  -∞

 Spherically symmetric and static  Radial coordinate r such that circumference of circle centered around throat given by 2πr

 r decreases from +∞ to b=b0 (minimum radius) at throat, then increases from b0 to +∞  At throat exists coordinate singularity where r component diverges  Proper radial distance l(r) runs from - ∞ to +∞ and vice versa Morris-Thorne Metric

1 ds2  e2 dt 2  dr 2  r 2 (d 2  sin2  d 2 )  b 1   r 

e2 0 0 0  gtt 0 0 0  1     b  0 g 0 0  0 1  0 0   rr    g    r   0 0 g 0  2    0 0 r 0  0 0 0 g  2 2      0 0 0 r sin  G  8 GT Determining Curvature (Left Side)

 Do we have the desired shape?  Cartan’s Structure Equations

“Cartan I”

i i j d   j  

“Cartan II” i i i k  j  d j  k   j

Elie Joseph Cartan (1869~1951) Cartan I – Connection One-Forms

 “One-forms” dx, dy, dz pdx  qdy  rdz i j  j  i

,  Take from Morris-Thorne metric:  0  e dt 1  2 2 2 1 2 2 2 2 2  b 2 ds  e dt  dr  r (d  sin  d ) 1  1  dr  b   1  r  r   2  rd  3  r sind Operations with Forms

 “Wedge product” dt  dt  dr  dr  d  d  d  d  0 dt  dr  dr  dt

 “d” Operator: k-form to (k+1)-form 1 form da da a  a(t,r)  da  dt  dr dt dr  Combining:

da da da   a(t,r)dt  d  dt  dt  dr  dt  dr  dt dt dr dr Calculation – Connection One-Forms

1 ds2  e2 dt 2  dr 2  r 2 (d 2  sin2  d 2 )  b 1   r  1  0  e dt  d 0  e dt  dr; dt    0 e 1 1   b 2  b 2 1  1  dr  d1  0; dr  1  1  r   r  1  2  rd  d 2  dr  d; d   2 r 1  3  r sind  d 3  sindr  d  r cosd  d; d   3 r sin Calculation – Connection One-Forms

i i j d   j   i, j  t,r,,

0 0 1 0 2 0 3  d  1    2    3    e dt  dr 1  b 2   0  1  e dt  dr   0  1  e 1  dt  1 1 1  r  1 2 0   b    e 1  dt 1  r  1  b 2   1  e 1  dt 0  r  of One-Forms

 1   b 2  0 e 1  dt 0 0    r    1 1 1  2 2 2    b  b  b  e 1  dt 0 1  d  sin 1  d i  r   r   r   j   1    b 2   0 1  d 0  cosd    r    1   b 2  0 sin1  d cosd 0    r   Calculation – Curvature Two-Forms

i i i k  j  d j  k   j i, j,k,m,n  t,r,, 1   Ri m  n 2 mnj  Computed using matrix of one-forms  Non-zero components of Riemann tensor  Useful for computing geodesics Calculation – Curvature Two-Forms

2 2 2 0 2 1 2  3  d3  0  3  1  3 3   cosd  2   0  0 b d 2  sind  d 0 3  sin d  d 3 r 1  b  1 1  2 3 b 2   sin      2    r  r r sin  1  1  d  r  b   2   3 1 r 3 2 1  b 3   sin1  d   b    R  R  3 r r 1 2 b d    R  R   r   r 3 1 d   3 r sin Riemann Tensor Components

 b 2 br  b Rt  Rr  Rr  Rr  1      rtr ttr ttr trt  r   2r 2 br  b Rr  R  Rr  R  r rr r rr 2r 3 r  r  br  b Rr  Rrr  Rr  Rrr  3  b 2r 1   r  R t  R  R t  R  t tt t tt r  b  1  t  t   r  R t  Rtt  Rt  Rtt  b r R  R  R  R      r 3 Riemann Tensor Properties

Rtr  Rrt  Rtr  Rtr

 Antisymmetric in (t, r)  Antisymmetric in (θ,Φ)  Symmetric in (t, r) and (θ, Φ)  Only 24 independent components  Generally in 4D has 256 components  Governs difference in of two freely falling near each other 1 G  R  Rg   2  Tensor

t r   Rtt  Rttt  Rtrt  Rtt  Rtt  b 21   b 2 br  b  r   1        r    2r 2 r

t r   Rrr  Rrtr  Rrrr  Rrr  Rrr

 b 2 br  b br  b  1        r   2r 2 r 3 1 G  R  Rg   2  Ricci Curvature Tensor

t r   R  Rt  Rr  R  R  b 1   r  br  b b      R r 2r 3 r 3 

t r   R  Rt  Rr  R  R  b 1   r  br  b b      R r 2r 3 r 3  1 G  R  Rg   2  Curvature Scalar

R  Rtt  Rrr  R  R  b 41   b 2 br  b  r  2br  b 2b  21          r    r 2 r r 3 r 3 1 G  R  Rg   2 

 Einstein Tensor G  Gtt ,Grr ,G ,G

 “Ricci” Curvature Tensor R  Rtt , Rrr , R , R

 Curvature Scalar R  Rtt  Rrr  R  R

 Metric g  gtt , grr , g , g 1 G  R  Rg   2  Einstein Tensor – Components

1 b G  R  Rg  tt tt 2 tt r 2 1 b 2  b G  R  Rg    1  rr rr 2 rr r 3 r  r

1  b br  b 2  br  b  G  R  Rg  1        2   G 2  r 2r(r  b) r 2r (r  b)

1  b br  b 2  br  b  G  R  Rg  1        2   G 2  r 2r(r  b) r 2r (r  b) G  8 GT Stress-Energy Tensor

 0 0 0 0  0 0 T     0 0 p 0   0 0 0 p Equations of State

Rearrange, solve….

b    8r 2

1 b   Tension    2(r  b) 8r 2 r 

r  (stress) p  (  )      2 Wormhole Embedding Diagram

 2  r  r  z(r)  b ln     1, b  2 0  b  b   0  0 0   Static (t=constant “slice”)  Assume θ=π/2 (equatorial “slice”)  Only r,Ф variable

1  b ds2  1  dr 2  r 2d 2  r ds2  dz 2  dr 2  r 2d 2 Boundary Conditions - Shape

 b0 – minimum radius at throat  Vertical at throat  Asymptotically flat

r-axis Boundary Conditions – No Horizon

Horizon - “physically nonsingular surface at which g tt vanishes”; defined only for spacetimes containing one or more asymptotically flat regions

 e.g. – coordinate singularity at r=2M 1  2M  2M ds2  1 dt 2  1  dr 2  r 2 d 2  sin2 d2  r   r     Morris-Thorne metric

2 2 2 1 2 2 2 2 2 ds  e dt  dr  r (d  sin  d ) 2  b     e  0 1   r  Other Boundary Conditions

 Crossing time on order of 1  Acceleration and tidal acceleration on order of 1G Geodesics

 “Geodesics” are “extremal worldlines”; equations of that determine geodesics comprise the “ equation” ~ Hartle  Timelike – freefall paths; ds2  0  Null – freefall paths; ds2  0

LOCALLY Variational Principle

1 ds2  e2 dt 2  dr 2  r 2 (d 2  sin2  d 2 )  b 1   r   Lagrangian

1 2 1 2 2 2 dx  dx    dt   b  dr   d   d   2 L  g  e2    1     r 2    sin2                 d d  d r d  d d   Euler-Lagrange Equation 1 d  L  L  AB  Ld     0  geodesics  d  dx    x 0     d  Geodesic Equation

2    1 d x dx dx      g g ,  g ,  g ,    ; 2 d 2 d d ,, ,  t,r,, d  dt    t  e2   0 d  d 

1 1 2 2 2 2 d  b dr   d  b  dr   d   d   dt     r  1     1     r   sin2     e2     0             d  r d  dr r d  d d  d  2 d  d   d     r 2   r 2 sin cos   0 d  d   d  d  d     r 2 sin2    0 d  d 

 Describe curvature in non-  Metric is like first derivative of warp  Christoffel symbol is like derivative of warp  Also called “connection coefficients”

 Non-zero Christoffel symbols are components of a 3-tensor

t t d       1 r 2 rt tr       (r  b)sin  dr r r r    1  r        sin cos   b  r r r r  r br  b cos r  b 2     1 e rr      tt   2r(r  b) sin  r  Morris-Thorne Examples

 Zero tidal forces

limited to throat  “Absurdly benign” wormhole Zero Tidal Solution

  0

b  b(r)  bor  r for b0  1

Equations of State

5 b     16r 2 8r 2 5 1 b      2(r  b)  8r 2 8r 2 r  5 r  p  (  )       2r 2 2 Zero Tidal Force Solution

 Wormhole material extends from throat to proper radial distance +/- ∞

 Density, tension and pressure vanish asymptotically

 Material is everywhere exotic  i.e.,     0 everywhere  Violates energy conditions  Need field theory “Catenoid” Exotic Matter Limited to Throat

2  (r  b0 ) b  b0 1  ;  0 for b0  r  b0  a0  a0  b    0 for r  b0  a0

 Spacetime flat at r > b0 + a0  Tidal forces bearable Absurdly Benign Wormhole  Travel time reasonable  BUT throat radius must be large to have meaningful wormhole Backward

From H.G. Wells,

 Time machine – Any object or system that permits one to travel to the  Not proven possible or impossible  Traveler moves through wormhole at sub-light  Appears to have exceeded light speed to stationary observers  violations and paradoxes (consistency and bootstrap) Summary

 Theoretically reasonable  Morris-Thorne model  No horizons  Exotic matter  violations  Causality violation  Much work has been done and continues to be done in this area; models abound! Possible Questions for Future

 Is necessary topological change even permitted? Exotic matter required?  If so, is it allowed on the quantum level?  If so, can we enlarge to classical size?  Morris and Thorne: “…pulling a wormhole out of the …” THANK YOU!

Karen E. Gross (1982~2009) UNCW Class of 2005 () References (Books and Papers)

 Jaroslaw Pawel Adamiak, Static and Dynamic Traversable Wormholes (Univ. of South Africa, January 2005)  James B. Hartle, -- An Introduction to Einstein's General Relativity (Addison-Wesley 2003)  David C. Kay, Schaum’s Outline of Theory and Problems of Tensor (McGraw-Hill Professional 1988)  Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation (W.H. Freeman and Company 1973)  Michael S. Morris and Kip S. Thorne, Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity (Calif. Inst. of Technology, 17 July 1987)  Thomas A. Roman, Inflating Lorentzian Wormholes (Cent. Conn. St. Univ. 1992)  Matt Visser, Lorentzian Wormholes: From Einstein to Hawking (Amer. Inst. of 1995) References (Websites)

 http://mathworld.wolfram.com/RiemannTensor.html (April 17, 2009)  http://en.wikipedia.org/wiki/%C3%89lie_Cartan (April 10, 2009)  http://en.wikipedia.org/wiki/Exterior_algebra (April 12, 2009)  http://www-gap.dcs.st- and.ac.uk/~history/Biographies/Cartan.html (March 26, 2009)  http://en.wikipedia.org/wiki/Wormhole (January 17, 2009) References (Figures)

 http://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Time_travel_hypothesis_using_ wormholes.jpg/360px-Time_travel_hypothesis_using_wormholes.jpg (April 15, 2009)  http://images.google.com/images?hl=en&um=1&q=tidal+forces+images&sa=N&start=20&nds p=20 (April 12, 2009)  http://www.one-mind-one-energy.com/images/Wormhole.jpg (February 22, 2009)  http://www.scifistation.com/masterpiece2.html (March 5, 2009)  http://www.familycourtchronicles.com/philosophy/wormhole/wormhole-enterprise.jpg (April 2, 2009)  http://www.nysun.com/pics/6255.jpg (March 18, 2009)  http://www.zamandayolculuk.com/cetinbal/VZ/WormholeTimeTravels.jpg (January 31, 2009)  http://www.jedihaven.com/assets/downloads/wallpapers/ds9_wormhole.jpg (February 3, 2009)  http://www.zamandayolculuk.com/cetinbal/astronomyweeklyfacts.htm (April 5, 2009)  http://www.popsci.com/files/imagecache/article_image_large/files/articles/sci1005timeMach_4 85.jpg (April 1, 2009)  http://farm2.static.flickr.com/1332/1171648505_7971058af5.jpg?v=1207757450 (March 9, 2009)  http://zebu.uoregon.edu/1996/ph123/images/antflash.gif (January 19, 2009)  http://cse.ssl.berkeley.edu/bmendez/ay10/2002/notes/pics/bt2lfS314_a.jpg (February 12, 2009) References (Figures, cont’d.)

 http://images.google.com/imgres?imgurl=http://2.bp.blogspot.com/_Vlvi-- zU3NM/RzhJQ4gEThI/AAAAAAAAAE0/Wl8cadtTWho/s320/babyinhand2.jpg&imgre furl=http://goodschats.blogspot.com/2007/11/10-brilliant-christmas-gifts- for.html&usg=__XGt- WxoBX1f1W9MwYX6IwqdLCBs=&h=304&w=320&sz=22&hl=en&start=96&um=1 &tbnid=FVbFR8x2QU9cKM:&tbnh=112&tbnw=118&prev=/images%3Fq%3Dfour %2Bdimensional%2Bmanifold%2Bimage%26ndsp%3D20%26hl%3Den%26sa%3 DN%26start%3D80%26um%3D1 (April 14, 2009)  http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/catenoid.html (April 15, 2009)