Initial Value and Evolution Structure of

Transcription

1 Present State and Future Directions of Initial Value and Evolution Structure of Regge Calculus Warner A. Miller Florida Atlantic University & LANL In Regge calculus the principles of GR are applied directly to the lattice spacetime, and we are left with a truly finite representation of the theory based on the underlying physical principles. This geometric foundation may provide insight into how we can best discretize Einstein s theory of gravitation. Collaborators: John Wheeler, Adrian Gentle, Arkady Kheyfets Thank: Ruth Williams and the committee, James York Presented to PSU CGPG, A decennial Perspective 8-11 June 2003

2 Present Status of Regge Calculus Independent and geometric 3+1 numerical technique Fundamental structure of Regge calculus understood (York thin sandwich IVP, evolution, dynamical degrees of freedom, simplicial diffeomorphism etc.) Dual (Voronoi/Delaunay) lattice structure appears to be a fundamental feature of Regge calculus. Suite of tested 1, 2 and 3-D discrete time evolution and initial-value codes. Simple vacuum cosmology models and simple linearized gravity waves propagated on lattice. 2nd order accuracy and simplicial diffeomorphism freedom understood and numerically illustrated.

3 Regge Calculus In Regge calculus the spacetime geometry is represented by a simplicial lattice -- a discrete geometry build of internally-flat 4-dimensional triangles (simplexes) each with signature (-,+,+,+). Why Simplexes? The squared edge lengths and only edge lengths determine the geometry. C q A B sin(θ) = 2 ABC AB AC θ 2 θ 1 θ 1 θ 2 Not Rigid

4 Regge Calculus in 2-D Building Block (Triangle) εh θ5 θ1 θ4 θ2 θ3 2εh Rh = Ah εh = 2π 5! θi i=1 " #$ % F unctionof Edges

5 Regge Calculus in 3-D Building Block (Tetrahedron) q1 q4 sin (θ i ) = 3 (3) V L q2 q3 ε h = 2π 4 i=1 θ i }{{} F unctionof Edges R h = 6ε h A h

6 Regge Calculus in 4-D Basic Building Block (Simplex) R h = 12ε h A h sin (θ i ) = 4 (4) V h 3 (3) V 1 (3) V 2 ε h = 2π 6 i=1 θ i }{{} F unctionof Edges

7 Regge Equation G ll V l = h l l 2 cot (θ h) }{{} moment ε h }{{} rot n

8 More than One way to the Regge Equations δ ( 1 8π Hilbert action: A h ε h + ) L m = 0 h E. Cartan moment of rotation: dp R l 2 l 2 + δl 2 P = l 2 cot (θ h) R = Â h ε h l 2 h l cot (θ h ) ε h = 8π T ll V l Variational integrator, yes, but local derivation too!

9 Contracted Bianchi Identity co-boundary of a co-boundary in dimensions creation = Ω G 0 (dp R) = 0 Kirchhoff-like Law l v h l }{{} δ δ 0 4 simplicial diffeomorphic degrees Of freedom per vertex (lapse + shift)

10 Generic Thin-Sandwich Initial Value Data for Regge Calculus 2N 0 N 0 3N 0 N 0 N 0 N 1 =7N 0 N 2 =12N 0 N 3 =6N 0 N 0 7N 0 4N 0 lapse+shift 4N 0 tetrad relations N 0 N 1 =7N 0 N 2 =12N 0 N 3 =6N 0 2N 0 N 0 3N 0 N 0 Left with 8 equations for 8 unknowns.

11 Sorkin Evolution in 1-D Decouples evolution equations Provides lapse and shift conditions Recasts constraint equations as evolution equations Clear illustration of diffeomorphism in Regge calculus Generalizable to any dimension and admits parallel implementation.

12 Sorkin Evolution in 2+1 Gravity The calculations to raise the tent in each color group can be done independently All the vertices can be raised to yield a new simplicial spacelike surface in 4 steps. This algorithm is highly parallelizable, yields many sets of 4 equations for 4 unknowns.

13 A Numerical Example (Kasner) ds 2 = dt 2 + t 2p 1 dx t 2p 2 dx t 2p 3 dx 2 3 p 1 + p 2 + p 3 = p p p 2 3 = 1 Axisymmetric Kasner p 1 =p 2 =2/3, p 3 =-1/3 10,000 time steps, dv/v=10-5 Preserve homogneity to less than 1:10-10 Explicit display of diffeomorphism freedom in Regge calculus Explicit display of 2nd order convergence

14 Future Directions for Regge Calculus Generic coupling of stress-energy. Robust benchmarking of Regge calculus against analytic and finite difference calculations. Examine stability, convergence, dispersion and accuracy of the Regge approach. Map structure of Regge calculus in the continuum and numerically. Further development of classical and quantum gravity algorithms.

15 Why Regge Calculus What captures my interest in Regge calculus? If nature is indeed fundamentally discrete; built out of a finite number of elementary quantum phenomenon. Then one may hope that by studying the discrete representations of the most beautiful geometric theory of nature we know, gravitation, one may be able to distill out some of the fundamental features of the discretization that may yield waypoints to a true understanding of the basic building blocks of nature. Regge calculus provides the purest, geometrically-based discretization of gravity I know of.

16 Clues Voronoi-Delaunay duality arises naturally This provides a natural platform for complementary. This inherent orthogonality provides a beautiful factorization of the phase space. Einstein s theory is encoded in a less complex fashion; it is locally an Einstein spacetime geometry. Convergence in mean to continuum Prevention of coherent fluctuations may be a clue to Wheeler s question Why the Quantum?

17 Answers? Golden Rule: 1 st: 2 nd: 3 rd: 4 th: Begin with discrete structure Apply basic principles directly to discrete structure Analyze map/links between discrete and continuum Distill out novel features