From Here to Eternity and Back:
Are Traversable Wormholes
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 Physicists Care, and Why?
 Are wormholes possible?
 How can we model to prove or disprove?
 Wormholes are being taken seriously
 Possible uses?
 Interstellar travel
 Time travel
Outline





History
Models that fail
Desirable traits
General relativity primer
Morris-Thorne wormhole





Curvature
Stress energy tensor
Boundary conditions and embeddings
Geodesics
Examples
 Time machines and future research
What is a Wormhole?
 “Hypothetical
shortcut between
distant points”
 General relativity
Historical Perspective
 Albert Einstein – general
relativity ~ 1916
 Karl Schwarzschild – first
exact solution to field
equations ~ 1916
 Unique spherically
symmetric vacuum
solution
 Ludwig Flamm ~ 1916
 White hole solution
Historical Perspective
 Einstein and Nathan Rosen ~ 1935
 “Einstein-Rosen Bridge” – first mathematical proof
 Kurt Gödel ~ 1948
 Time tunnels possible?
 John Archibald Wheeler ~ 1950’s
 Coined term, “wormhole”
 Michael Morris and Kip Thorne ~ 1988
 “Most promising” – wormholes as tools to teach
general relativity
Traversable Wormholes?
 Matter 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 forces 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
spacetime regions
 No event horizon
 Two-way travel
 Finite crossing time
Simple Models – Physical Traits
 Small tidal forces
 Reasonable
crossing time
 Reasonable stressenergy 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
“Space acts on matter, telling it how to move. In turn,
matter reacts back on space, telling it how to curve.” ~
Misner, Thorne, Wheeler, Gravitation
Spacetimes
“A spacetime is a four-dimensional
manifold equipped with a Lorentzian
metric…[of signature]…(-,+,+,+). A
spacetime is often referred to as
having (3+1) dimensions.” ~Matt
Visser, Lorentzian Wormholes.
Representing Spacetimes: Flat
“Line Element”




Uses differentials
Einstein summation
Metric
Minkowski space:

ds  g dx dx
2
ds 2  gtt dt 2  g xx dx 2  g yy dy 2  g zz dz 2
gtt  1; g xx  1; g yy  1; g zz  1 (cartesian)
gtt  1; grr  1; g  r 2 ; g  r 2 sin 2  ( spherical )

Morris-Thorne Wormhole (1988)
2
ds  e dt 
2
2
1
 b
1  
 r
dr  r (d  sin  d )
2
 Φ(r) – redshift function
 Change in frequency of
electromagnetic radiation in
gravitational field
 b(r) – shape function
2
2
2
2
+∞
Properties of the Metric
ds 2   e 2  dt 2 
1
b

1  

r
dr 2  r 2 (d 2  sin 2  d 2 )
-∞
 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
ds 2   e 2  dt 2 
 gtt 0 0
0 g
0
rr
g  
 0 0 g

0 0 0
1
b

1  

r
dr 2  r 2 (d 2  sin 2  d 2 )
2


e
0
0
0 
0
1


 b

0
0
1

0
0





 r


0
2

0
0
r
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”
d i   ij   j
“Cartan II”
 ij  d ij   ki   kj
Elie Joseph Cartan
(1869~1951)
Cartan I – Connection One-Forms
 “One-forms”
dx, dy, dz
pdx  qdy  rdz
 ij  i j
,
 Take from Morris-Thorne metric:
 0   e  dt
ds 2   e 2  dt 2 
1
 b
1  
 r
dr 2  r 2 (d 2  sin 2  d 2 )
 b
1  1  
 r

1
2
 2  rd
 3  r sin  d
dr
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
a  a (t , r )  da 

da
da
dt 
dr
dt
dr
Combining:
  a(t , r )dt  d 
da
da
da
dt  dt  dr  dt  dr  dt
dt
dr
dr
Calculation – Connection One-Forms
ds   e
2
2
dt 
2
1

1 

b

r
dr 2  r 2 (d 2  sin 2  d 2 )
 0   e  dt  d 0   e  dt  dr; dt  
 b
1  1  

r

1
2
1 0


e
1
2
 b
dr  d 1  0; dr   1    1

r
 2  rd  d 2  dr  d ; d 
1 2

r
1
  r sin  d  d  sin  dr  d  r cos d  d; d 
3
r sin 
3
3
Calculation – Connection One-Forms
d i   ij   j
i , j  t , r , , 
d 0  10   1   20   2   30   3   e  dt  dr
1
2
b

 10   1   e  dt  dr  10   1    e   1   dt   1

r
1
2
b

 10    e   1   dt

r
1
2
b

  01   e   1   dt

r
Matrix of One-Forms


0

1

b 2



e
1



 dt


r
 ij  

0




0

1
2
 b
  e  1   dt
 r

0
1
2
 b
 1   d
 r
0
1
2
 b
 1   d
 r
1
2
 b
sin   1   d
 r
0
cos d


0

1

2
 b

 sin   1   d 
 r


 cos d




0

Calculation – Curvature Two-Forms
 ij  d ij   ki   kj
1 i m
n
  Rmnj   
2
i , j , k , m, n  t , r , , 
 Computed using matrix of one-forms
 Non-zero components of Riemann
tensor
 Useful for computing geodesics
Calculation – Curvature Two-Forms
   cos d
2
3
 02   30  0
d  sin  d  d
2
3


b
r
1
2
12   1   d


b
r
1
2
 31   sin   1   d
1
d   2
r
1
d 
3
r sin 
 23  d 32   02   30  12   31
b
 sin  d  d
r
b
  1  1  2
3
  sin     
 
r
  r   r sin  
b
 3 2  3
r
b


 R
 R
 3
r
b


 R
 R
 3
r
Riemann Tensor Components
br  b
2
 b
R  R   R   R  1      
 r
2r 2
t
rtr
r
r
R
r
ttr
r
ttr

rr
R

r
trt
 R
r
r

rr
 R
b r  b

2r 3

Rrr  Rrr   Rr r   Rrr  
Rt t  Rtt   Rtt
Rt t  Rtt   Rtt
 b
  1  
 r
  Rtt 
r
b r  b
2r 3
 b
  1  
 r

  Rtt 
r




R
 R
  R
  R

b
r3
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 acceleration of two
freely falling particles near each other
G
1
 R  Rg
2
Ricci Curvature Tensor
Rtt  Rtttt  Rtrtr  Rtt  Rtt

2
 b
  1       
 r

 b
2   1  
 b r  b
 r


r
2r 2
r
Rrr  Rrtrt  Rrrr
 Rrr  Rrr
 b r  b b r  b
2
 b
 1        
 3
2
 r
2r
r


1
G  R  Rg
2
Ricci Curvature Tensor


R  Rtt  Rrr  R
 R
 b
  1  

r  b r  b b


 3  R
3
r
2r
r


R  Rtt  Rrr  R
 R
b

  1  

r  b r  b b


 3  R
3
r
2r
r
1
G  R  Rg
2
Curvature Scalar
R   Rtt  Rrr  R  R

2
 b
 2 1        
 r

 b
4   1  
 b r  b
 r  2b r  b 2b



 3
2
3
r
r
r
r
1
G  R  Rg
2
Einstein Tensor
 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
G
1
 R  Rg
2
Einstein Tensor – Components
Gtt  Rtt 
1
b
Rgtt  2
2
r
1
b 2   b 
Grr  Rrr  Rgrr   3 
1  
2
r  r
r
1
br  b
  br  b 
2
 b 
G  R  Rg  1     
       2
 G



2
r 
2r(r  b)
r 2r (r  b) 
1
br  b

br  b 
2
 b 
G  R  Rg  1     
       2
 G



2
r 
2r(r  b)
r 2r (r  b) 
G  8 GT
Stress-Energy Tensor
T

0

0

0
0 0
 0
0 p
0 0
0
0

0

p
Equations of State
Rearrange, solve….
 Energy density

b
8 r 2
 Tension

1
8 r 2
 Pressure (stress)
p
r
(    )      
2
b


2
(
r

b
)


 r

Wormhole Embedding Diagram

r
z (r )  b0 ln

 b0

2

 r
   1 , b0  2
 b0 

 Static (t=constant
“slice”)
 Assume θ=π/2
(equatorial “slice”)
 Only r,Ф variable
1
 b
ds 2  1   dr 2  r 2 d 2
 r
ds 2  dz 2  dr 2  r 2 d 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. Schwarzschild metric – coordinate singularity at
r=2M
1
2
M
2
M




2
2
2
2
2
2
ds2  1 
 dt  1 
 dr  r d  sin  d


r 
r 


 Morris-Thorne metric
ds 2   e 2  dt 2 
1
b

1  

r
dr 2  r 2 (d 2  sin 2  d 2 )
    e 2  0
Other Boundary Conditions
 Crossing time on order of 1 year
 Acceleration and tidal acceleration on
order of 1G
Geodesics
 “Geodesics” are “extremal proper time worldlines”;
equations of motion that determine geodesics comprise
the “geodesic equation” ~ Hartle
 Timelike – Particle freefall paths; ds2  0
 Null – Light freefall paths; ds2  0
LOCALLY
Variational Principle
ds 2   e 2  dt 2 
1
b

1  

r
dr 2  r 2 (d 2  sin 2  d 2 )
 Lagrangian

dx  dx   2   dt 
 b   dr 
 d 
2  d 
 e     1      r    sin 2   

 d 
d d
r   d 
  d 
 d 
1
2
L   g
2
2
 Euler-Lagrange Equation
1
 AB   Ld
0

d
d
L
L

 0  geodesics


 dx   x


 d 
2
 

 
1
2
Geodesic Equation
t

1 
g
g ,  g ,  g  ,
2
 ,  ,  ,   t , r , , 


d 2 x
 dx dx
 
;
2
d d
d




d  2  dt 
e
 0
d 
d 
1
1
2
2
2

 d  2

d  b  dr   d  b   dr 
2  d 
2   dt 
  r   1  
    1      r    sin       e     0
 d  
 d  
d  r  d   dr  r   d 
 d 

d  d 
 d 
     r 2   r 2 sin  cos    0
 d 
d  d 
2
 
d  2 2 d 
 r sin    0
d 
d 
Christoffel Symbols

Describe curvature in non-Euclidean space
 Metric is like first derivative of warp
 Christoffel symbol is like second derivative of warp
 Also called “connection coefficients”

Non-zero Christoffel symbols are components of a 3-tensor
d
dr
1

r
cos

sin 
rtt  trt 
r  r



 
1
r
   sin  cos
r  r 
 b
   1    e 2 
 r
r
tt
r  (r  b) sin 2 
r  b  r
b r  b
 
2r ( r  b)
r
rr
Morris-Thorne Examples
 Zero tidal forces
 Exotic matter limited to throat
 “Absurdly benign” wormhole
Zero Tidal Force Solution
0
b  b( r ) 
bo r 
r for b0  1
Equations of State

b

 16 r 2
2
8 r
5
1

8 r 2
5

b

2

2
(
r

b
)


8

r

 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 quantum field theory
“Catenoid”
Exotic Matter Limited to Throat
2
 (r  b0 ) 
b  b0 1 
 ;  0 for b0  r  b0  a 0
a0 

b    0 for r  b0  a 0




Spacetime flat at r > b0 + a0
Tidal forces bearable
Travel time reasonable
BUT throat radius must be large
to have meaningful wormhole
Absurdly Benign Wormhole
Backward Time Travel
From H.G. Wells,
The Time Machine





Time machine – Any object or system that permits one to travel to
the past
Not proven possible or impossible
Traveler moves through wormhole at sub-light speed
Appears to have exceeded light speed to stationary observers
Causality violations and paradoxes (consistency and bootstrap)
Summary
 Theoretically reasonable
 Morris-Thorne model




No horizons
Exotic matter
Energy condition 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 quantum foam…”
THANK YOU!
Karen E. Gross (1982~2009)
UNCW Class of 2005 (Mathematics)
References (Books and Papers)







Jaroslaw Pawel Adamiak, Static and Dynamic Traversable
Wormholes (Univ. of South Africa, January 2005)
James B. Hartle, Gravity -- An Introduction to Einstein's
General Relativity (Addison-Wesley 2003)
David C. Kay, Schaum’s Outline of Theory and Problems of
Tensor Calculus (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 Physics 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.stand.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-giftsfor.html&usg=__XGtWxoBX1f1W9MwYX6IwqdLCBs=&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)