mwrm 16, wash. u. in st. louis 18 november 2006 psutter2@uiuc.edu nontrivial spacetimes and the...

MWRM 16, Wash. U. in St. Louis18 November 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

Flips and Position-DependenceFlips and Position-Dependence

x

y

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MWRM 16, Wash. U. in St. Louis18 November 2006psutter2@uiuc.edu

PatternsPatterns

Third-Turn SpaceHexagonal Cross-Section

Quarter-Turn SpaceRectangular Cross-Section

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MWRM 16, Wash. U. in St. Louis18 November 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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 2006psutter2@uiuc.edu

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

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MWRM 16, Wash. U. in St. Louis18 November 2006psutter2@uiuc.edu

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