mwrm 16, wash. u. in st. louis 18 november 2006 [email protected] nontrivial spacetimes and the...
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
Nontrivial Nontrivial SpacetimesSpacetimes
and the and the (Cosmological) (Cosmological) Casimir EffectCasimir EffectPaul Matthew Sutter
University of Illinois at Urbana-Champaign
With:Tsunefumi Tanaka
Humboldt State University, Arcata, CA
(M.C. Escher)preprint available at:
gr-qc/0610051
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
The (Non-cosmological) Casimir The (Non-cosmological) Casimir EffectEffect
Boundary conditions affect vacuum energy density between plates
(Courtesy of CIPA)
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
Constructing a UniverseConstructing a Universe
We are going to construct “multiply-connected” topologies:
Take a basic geometric object (a “Fundamental Polyhedron”, or FP)
and identify opposite sides.
“Multiply-connected”: more than one path between x and x’
The “curvature” of a cylinder is extrinsic(i.e. not a property of the space itself)
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
A Smorgasbord of (Flat) SpacesA Smorgasbord of (Flat) Spaces
(Roboucas and Gomero, 2004)
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
A Smorgasbord of (Flat) SpacesA Smorgasbord of (Flat) Spaces
(Roboucas and Gomero, 2004)
Which one is our universe??
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
Sutter’s Seven Simple Steps to Sutter’s Seven Simple Steps to SuccessSuccess
1) Choose field….......................
2) Choose geometry…..............
3) Choose topology…...............
4) Determine spacetime
interval……………………...
5) Try to evaluate …..
6) Renormalize with Method of
Images…............................
7) Publish!...................................
massless scalar
flat!
Klein space, 3-Torus, etc…
Sutter, P.M. and Tanaka, T. Phys. Rev. D 74, 024023 (2006)
...)()( 2
02
02 nLxxtt
0||0 T
MTTT 0||00||0
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
Flips and Position-DependenceFlips and Position-Dependence
x
y
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
PatternsPatterns
Third-Turn SpaceHexagonal Cross-Section
Quarter-Turn SpaceRectangular Cross-Section
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
Hantzsche-Wendt SpaceHantzsche-Wendt Space
z = 0.0 z = 0.5 z = 1.0
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
The Dominance of the FPThe Dominance of the FP
Quarter-TurnHalf-Turn
Third-TurnSixth-Turn
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MWRM 16, Wash. U. in St. Louis18 November [email protected]
The FP and Energy DensityThe FP and Energy Density
Fundamental Polyhedron Energy Density
One-Torus -0.11
Two-Torus -0.31
Rectangular Three-Torus
-0.83
Hexagonal Prism -0.99
Klein Space -2.39
Hantzsche-Wendt Space
-0.32
