MWRM 16, Wash. U. in St. Louis 18 November 2006 Nontrivial Spacetimes and the (Cosmological) Casimir Effect Paul Matthew Sutter University of Illinois at Urbana-Champaign With: Tsunefumi Tanaka Humboldt State University, Arcata, CA (M.C. Escher) preprint available at: gr-qc/

2 MWRM 16, Wash. U. in St. Louis 18 November 2006 The (Non-cosmological) Casimir Effect Boundary conditions affect vacuum energy density between plates (Courtesy of CIPA)

3 MWRM 16, Wash. U. in St. Louis 18 November 2006 Constructing 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)

4 MWRM 16, Wash. U. in St. Louis 18 November 2006 A Smorgasbord of (Flat) Spaces (Roboucas and Gomero, 2004)

5 MWRM 16, Wash. U. in St. Louis 18 November 2006 A Smorgasbord of (Flat) Spaces (Roboucas and Gomero, 2004) Which one is our universe ??

6 MWRM 16, Wash. U. in St. Louis 18 November 2006 Sutter’s Seven Simple Steps to Success 1)Choose field… )Choose geometry… )Choose topology… )Determine spacetime interval……………………... 5)Try to evaluate ….. 6)Renormalize with Method of Images… )Publish! massless scalar flat! Klein space, 3-Torus, etc… Sutter, P.M. and Tanaka, T. Phys. Rev. D 74, (2006)

7 MWRM 16, Wash. U. in St. Louis 18 November 2006 Flips and Position-Dependence x y

8 MWRM 16, Wash. U. in St. Louis 18 November 2006 Third-Turn Space Hexagonal Cross-Section Quarter-Turn Space Rectangular Cross-Section

9 MWRM 16, Wash. U. in St. Louis 18 November 2006 Hantzsche-Wendt Space z = 0.0 z = 0.5z = 1.0

10 MWRM 16, Wash. U. in St. Louis 18 November 2006 The Dominance of the FP Quarter-Turn Half-Turn Third-Turn Sixth-Turn

11 MWRM 16, Wash. U. in St. Louis 18 November 2006 The FP and Energy Density Fundamental PolyhedronEnergy 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

12 MWRM 16, Wash. U. in St. Louis 18 November 2006 You are here?

13 MWRM 16, Wash. U. in St. Louis 18 November 2006 Lessons Learned “Closed” dimensions restrict modes, just like Casimir plates Can calculate relative to plain Euclidean space Interesting effects: Energy density ~ size -4 Closing dimensions => reduction in energy density Adding flips/rotations => position-dependence Chosen FP geometry => pattern in position-dependence Adding rotations => off-diagonal elements in