From Here to Eternity and Back: Are Traversable Wormholes Possible?
Mary Margaret McEachernwith advising fromDr. Russell L. HermanPhys. 495 Spring 2009April 24, 2009Dedicated 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
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 stress-energy tensor
Stable
Assembly possible
A Little General Relativity Primer
G GT 8
Field equations – relate spacetime curvature to matter and energy distribution
Left side – Curvature
Right side – Stress energy tensor
Summation indices
G R Rg 1
2
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 dx2
ds g dt g dx g dy g dz
g g g g cartesian
g g g r g r spherical
tt xx yy zz
tt xx yy zz
tt rr
2 2 2 2 2
2 2 2
1 1 1 1
1 1
; ; ; ( )
; ; ; sin ( )
Morris-Thorne Wormhole (1988)
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
Φ(r) – redshift function
Change in frequency of electromagnetic radiation in gravitational field
b(r) – shape function
Properties of the Metric
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
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
-∞
+∞
Morris-Thorne Metric
g
g
g
g
g
e
b
r
r
r
tt
rr
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 1 0 0
0 0 0
0 0 0
2
1
2
2 2
sin
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
Determining Curvature (Left Side)
Do we have the desired shape?
Cartan’s Structure Equations
Elie Joseph Cartan(1869~1951)
d
d
i
j
i j
j
i
j
i
k
i
j
k
G GT 8
“Cartan I”
“Cartan II”
Cartan I – Connection One-Forms
“One-forms”
Take from Morris-Thorne metric:,
dx dy dz
pdx qdy rdz
j
i
i
j
, ,
0
1
1
2
2
3
1
e dt
b
rdr
rd
r d
sin
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
Operations with Forms
“Wedge product”
“d” Operator: k-form to (k+1)-form
Combining:
dt dt dr dr d d d d
dt dr dr dt
0
1
form
a a t r dada
dtdt
da
drdr( , )
a t r dt dda
dtdt dt
da
drdr dt
da
drdr dt( , )
Calculation – Connection One-Forms
0 0 0
1
1
21
1
21
2 2 2
3 3 3
1
1 0 1
1
1
e dt d e dt dr dte
b
rdr d dr
b
r
rd d dr d dr
r d d dr d r d d dr
;
;
;
sin sin cos ;sin
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
Calculation – Connection One-Forms
d e dt dr
e dt dr eb
rdt
eb
rdt
eb
rdt
0
1
0 1
2
0 2
3
0 3
1
0 1
1
0 1
1
21
1
0
1
2
0
1
1
2
1
1
1
d
i j t r
i
j
i j
, , , ,
Matrix of One-Forms
j
i
eb
rdt
eb
rdt
b
rd
b
rd
b
rd d
b
rd d
0 1 0 0
1 0 1 1
0 1 0
0 1 0
1
2
1
2
1
2
1
2
1
2
1
2
sin
cos
sin cos
Calculation – Curvature Two-Forms
Computed using matrix of one-forms
Non-zero components of Riemann tensor
Useful for computing geodesics
j
i
j
i
k
i
j
k
mnj
i m n
d
R
1
2
i j k m n t r, , , , , , ,
Calculation – Curvature Two-Forms
3
2
3
2
0
2
3
0
1
2
3
1
2 3
3
2 3
3
3
1 1
d
b
rd d
b
r r r
b
r
R Rb
r
R Rb
r
sin
sinsin
3
2
3
2
cos
sin
d
d d d 0
2
3
0 0
1
2
1
2
3
1
1
2
1
1
b
rd
b
rdsin
dr
dr
1
1
2
3
sin
Riemann Tensor Components
R R R Rb
r
b r b
rrtr
t
ttr
r
ttr
r
trt
r
1
2
2
2
R R R R
b
r
rt
t
t t t
t
tt
1
R R R R
b
r
rt
t
t t t
t
tt
1
R R R Rb r b
rr
r
r r r
r
rr
2 3
R R R Rb r b
rr
r
r r r
r
rr
2 3
R R R Rb
r
3
Riemann Tensor Properties
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
R R R Rtr rt tr tr
Ricci Curvature Tensor
R R R R R
b
r
b r b
r
b
r
r
tt ttt
t
trt
r
t t t t
12
2 12
2
R R R R R
b
r
b r b
r
b r b
r
rr rtr
t
rrr
r
r r r r
12
2
2 3
G R Rg 1
2
Ricci Curvature Tensor
R R R R R
b
r
r
b r b
r
b
rR
t
t
r
r
1
2 3 3
R R R R R
b
r
r
b r b
r
b
rR
t
t
r
r
1
2 3 3
G R Rg 1
2
Einstein Tensor
Einstein Tensor
“Ricci” Curvature Tensor
Curvature Scalar
Metric
G R Rg 1
2
R R R R Rtt rr , , ,
G G G G Gtt rr , , ,
R R R R Rtt rr
g g g g gtt rr , , ,
Einstein Tensor – Components
G R Rgb
rtt tt tt
1
2 2
G R Rgb
r r
b
rrr rr rr
1
2
21
3
G R Rgb
r
b r b
r r b r
b r b
r r bG
1
21
2 2
2
2
( ) ( )
G R Rgb
r
b r b
r r b r
b r b
r r bG
1
21
2 2
2
2
( ) ( )
G R Rg 1
2
Equations of State
Energy density
Tension
Pressure (stress)
Rearrange, solve….
b
r
r
b
rr b
pr
8
1
82
2
2
2( )
( )
Wormhole Embedding Diagram
Static (t=constant “slice”)
Assume θ=π/2 (equatorial “slice”)
Only r,Ф variable
dsb
rdr r d2
1
2 2 21
ds dz dr r d2 2 2 2 2
z r br
b
r
bb( ) ln ,
0
0 0
2
01 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 vanishes”; defined only for spacetimes containing one or more asymptotically flat regions
e.g. Schwarzschild metric – coordinate singularity at r=2M
Morris-Thorne metric
gtt
e2 0
dsM
rdt
M
rdr r d d2 2
1
2 2 2 2 212
12
sin
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
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;
Null – Light freefall paths;
LOCALLY
ds2 0
ds2 0
Variational Principle
Lagrangian
Euler-Lagrange Equation
ds e dtb
r
dr r d d2 2 2 2 2 2 2 21
1
( sin )
L gdx
d
dx
de
dt
d
b
r
dr
dr
d
d
d
d
2
2 1 2
2
2
2
2
1
2
1 sin
AB Ldd
d
L
dx
d
L
xgeodesics
0
1
0
Geodesic Equation
d x
d
dx
d
dx
d
2
2
;
1
2g g g g
t r
, , ,
, , , , , ,
td
de
dt
d
rd
d
b
r
dr
d
d
dr
b
r
dr
dr
d
d
d
de
dt
d
d
dr
d
d
2
1 1 2 2
2
2
2
2
2
0
1 1 0
sin
rd
d
d
dr
d
d
2
2
2 2
0
0
sin cos
sin
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
rt
t
tr
t
r r
d
dr
r
1
cos
sin
r r
tt
r
r
b
re
1
1 2
sin cos
r
r
rr
r
r b
b r
b r b
r r b
( ) sin
( )
2
2
Zero Tidal Force Solution
Equations of State
0
10b b r b r r for bo( )
b
rr
r
b
rr b r
pr
r
816
1
82 8
22
2
5
2
2
5
2
5
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., everywhere
Violates energy conditions
Need quantum field theory“Catenoid”
0
Exotic Matter Limited to Throat
Spacetime flat at r > b0 + a0
Tidal forces bearable
Travel time reasonable
BUT throat radius must be large to have meaningful wormhole
b br b
aforb r b a
b for r b a
00
0
2
0 0 0
0 0
1 0
0
( );
Absurdly Benign Wormhole
Backward Time Travel
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)
From H.G. Wells,The Time Machine
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…”
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.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&ndsp=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&imgrefurl=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%3DN%26start%3D80%26um%3D1 (April 14, 2009)
http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/catenoid.html (April 15, 2009)