Regge Calculus as a Dynamical Theory: The Fundamental Role of Dual Tessellations

Transcription

1 Regge Calculus as a Dynamical Theory: The Fundamental Role of Dual Tessellations Warner A. Miller FAU SpaceTime Physics (FAUST) Group Florida Atlantic University (John Wheeler, Arkady Kheyfets, Ruth Williams, Adrian Gentle, and Jonathan McDonald) Tullio Regge presented with Dirac Medal Presented at the Workshop on Unstructured Meshes in Dynamical Spacetimes August 2010, Friedrich Schiller University, Jena 1

2 Regge Calculus (RC) In Regge Calculus the spacetime geometry is represented by a simplicial lattice. The discrete geometry is built of internally flat 4- dimensional triangles (simplices). Edgelengths define the skeletal geometry! Hilbert Action I = (4) R d (4) V proper Regge-Hilbert Action I R = 1 8π triangle hinges, h h A h 2

3 Why Regge Calculus? If nature is indeed fundamentally discrete; built out of a finite number of elementary quantum phenomena. 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 glean some of the fundamental features of the discretization that may yield way points to a true understanding of the basic building blocks of nature. We are not aware of a more pure and geometrically-based discrete model of gravity than Regge Calculus. The laws of gravitation appear to be encoded locally on the lattice in with less complexity than in the continuum. 3

4 Regge Calculus in Numerical Relativity Better computing algorithms can be expected from better understanding of the structure that underlies geometrodynamics. This structure we suggest will: 1.show more clearly when examined in the language of geometry rather than in the language of differential equations; 2.show more readily in Regge calculus than in Ricci calculus; and 3.show more directly from a geometry idealized to be blockwise flat than from functions idealized as piecewise linear. 4

5 Outline 1. Curvature, Action, Equations and the Degrees of Freedom. The Role of Dual Tessellations in Regge Calculus 2. The Initial-Value Prescription in Regge Calculus. Regge Calculus as a Dynamical Theory. 3. Numerical Applications. Null-Strut and Regge Geomerodynamics. 5

6 Part 1 1. Curvature, Action, Equations and the Degrees of Freedom. The Role of Dual Tessellations in Regge Calculus 2. The Initial-Value Prescription in Regge Calculus 3. Numerical Applications. Null-Strut and Regge Geomerodynamics 6

7 Curvature in RC (2 D) Edgelengths define the skeletal geometry! θ θ 1 A 2 h θ 3 θ 4 θ 5 5 h = 2π i=1 θ i Def icit Angle Angle V ector Rotates K h = h A h 7

8 Scalar Curvature Example K h = h A h

9 Curvature in RC (3-D) The Building block is a tetrahedron. The hinge is the edge common to the five tetrahedrons. h (3)K h = h A h 9

10 Curvature in RC (4-D) The Building block is a simplex and the co-dimension 2 hinge (h) is the triangle common to all the 5 simplicies A h A h One of the five simplexes hinging on hinge, h (4)K h = h A h 10

11 Hinge-Based Hybrid Block: Coupling the Voronoi and Delaunay Lattices d (D) V proper = 2 D(D 1) A h A h A co-dimension 2 version of 1 2 base altitude 11

12 Scalar Curvature of the Hybrid Block Curvature is concentrated on co-d-2 hinge, h Rotation in plane perpendicular to the hinge, h. Voronoi polygon A h is perpendicular to hinge, h Locally the Regge spacetime is an Einstein space (D)R h = D(D 1) (D) K h = D(D 1) h A h 12

13 Hilbert Action I = 1 16π (4) R d (4) V proper 1 D(D 1)h 2 16π hinges, h A h D(D 1) A h A h I R = 1 8π triangle hinges, h h A h Regge Action 13

14 Curvature in Regge Calculus: The Reduced Hybrid Blocks R h = d(d 1) h h A h h = d(d 1) h A h Ric l = d(d 1) h l A h l R v = d(d 1) h v A h v Q h v h v Q h A hv h v A hv 14

15 Regge Equations δ(i R ) = δ hinges, h A h h = δ(a h ) h + A h δ( h ) = 0 hinges, h δ(a h ) = A h L δ(l) = 1 2 L cot (θ h) hinges, h 1 2 L cot (θ h) h = 0 sharing edge L Regge Equation L θ h 15

16 Part 2 1. Curvature, Action, Equations and the Degrees of Freedom. The Role of Dual Tessellations in Regge Calculus 2. The Initial-Value Prescription in Regge Calculus: Regge Calculus as a Dynamical Theory 3. Numerical Applications. Null-Strut and Regge Geomerodynamics 16

17 The Dynamical Degrees of Freedom of a Discrete Spacetime There is one Regge equation for each edge in the skeleton geometry. It might seem that there is no freedom whatsoever in the edgelengths. However, exactly such freedom is an essential part of the idea of approximating a smooth geometry via skeletonization. We have found no better way to illustrate this point by applying the Cartan 0 principle to the discrete spacetime, in it s Dimensional formulation. Number of determinable edgelengths N 1 4N 0 = Number of Regge equations N 1 + Number of contractedbianchi identities 4N 0 17

18 Constraints, Contracted Bianchi Identity & Diffeomorphism Invariance 1 δ (I) = δ (4) R d (4) V proper = 0 16π g µν g µν + (ξ µ;ν +ξ ν;µ ) x µ new = x µ ξ µ G µν ;ν = 0 1 π δ ijġ ij NH(π ij, g ij) N i H i (π ij, g ij) d x 4 = 0 16π N N + δn N i N i + δn i H = g 1 2 T rπ 2 12 (T rπ)2 g 1 2 R = 0 H i = 2 π ik k = 0 Hamiltonian & Momentum Constraints Lapse & Shift Freedom Diffeomorphism No creation of source D form Ω d G = = = Ω G = Ω the eight 3-cubes that bound Ω T he eight bounding 3-cubes (dp R) moment of rotation 3-cube (dp R) associated with specif ied 3-cube T he six faces bounding the 3-cube Contracted Bianchi Identity moment of rotation (P R) face associated with specified face of 0 specif ied 3-cube What s the story in discrete or discretized representations? 18

19 The Entourage of Edge L = AB in a 4-D Simplicial Geometry 19

20 The Entourage of Edge L = AB in a 4-D Simplicial Geometry 20

21 The Moment Arm in Regge Calculus OC AB =0 Moment Arm = OC = 1 2 AB cot θ OC = 1 2 L cot θ OC 21

22 Cartan Moment of Rotation Trivector M oment of Rotation Bivectorf or T riangle ABN Fig. 6. Moment of rotation associated Rotation Bivectorf or T riangle ABN = dp R = = R = 1 AB cot θ 2 NB NA 2 ABN NB NA 2 ABN epsilon = 1 2 L cot (θ) 22

23 Cartan Moment-of-Rotation in RC N L * Moment Arm C dplh O dp L V G LL V L = hinges, h sharing edge L 1 2 L cot (θ h) Moment Arm h Rot n Einstein Field Equation is Diagonal, and Directed along L! Identical to the variational equation! 23

24 Ordinary Bianchi Identity In Regge Calculus 2 LABN = e i NB NA 2 ABN ABN =1l Finite Rotations Do Not Ordinarily Commute; therefore, we have leaky boxes in Regge Calculus 24

25 Contracted Bianchi Identity In Regge Calculus: Kirchhoff-like Conservation Law T he eight bounding 3-cubes T he six faces bounding the 3-cube moment of rotation (P R) face associated with specified face of specif ied 3-cube ulus 0 1 L 2 cot(θ h ) h A h edges L v Hinges h L 0 sharing vertex v sharing edge L δ δ 0 4 Per Vertex! Diffeomorphic Degrees of Freedom Bia 25

26 Thin Sandwich Initial Value Data D g Ω α β i Σ 1 g Σ 0 g base Ω H =0 H i =0 26

27 Thin Sandwich Regge Spacetime TET 1 TET 0 27

28 The 4 Simplexes of a Thin Sandwich Prism 28

29 Quantity Production Simplicial 3-Geometry and its Dual 29

30 Regge Calculus Initial Value Data: Quantity Production Lattice 3N 0 3N 0 N 0 N 0 7N 0 12N 0 6N 0 TET 1 N 0 7N 0 3N 0 4N 0 N 2 = 12N TET 0 0 N 1 =7N N 0 3 =6N 0 N 0 6N 0 N 0 Solve 8 First-Order Regge Equations for 8 Remaining Unknowns 30

31 Lapse and Shift Conditions: Geometric Insight Avoidance of elongated cells: waste minimization W AST E = triangles, h log area of best triangle area of triangle h Minimize simplex volume in high curvature regions, minimize in low curvature regions (Curvature) (Area) (Def icit) Constant Mean Curvature, Minimal Shear, etc. I CMC =(4 V olume) λ (3 V olume) =(extremum) I MS = 2 dv 31

32 Evolution Algorithm: Canonical and Sorkin We have freedom to specify 4 lapse/shift conditions per vertex across the 3- geometry Four of the first-order Regge equations can be treated as constraints, provided consistent and uniform data structure. Can substitute evolution equations with constraint and or momenergy conservation equations. Sorkin evolution decouples evolution equations, and they can be easily parallelized (e.g. 11 equations for 11 unknowns in QPL). 32

33 An Application: Null-Strut RC V D φ = 0 N ull Strut Regge Equation Moment arms are all constant and equal for all the six triangles hinging on each null-strut edge! 33

34 Part 3 1. Curvature, Action, Equations and the Degrees of Freedom. The Role of Dual Tessellations in Regge Calculus 2. The Initial-Value Prescription in Regge Calculus 3. Numerical Applications. Null-Strut and Regge Geomerodynamics. 34

35 Numerical Applications of Regge Calculus Early applications of RC were largely analytic calculations in high symmetry Null Strut Calculus (Miller et al ) Simple Cosmological models: 1. Taub (Tuckey & Williams 1988) Early applications of RC were largely analytic calculations in high symmetry 2. FRW (Brewin 1987, Gentle 1999) Spherical symmetric evolution (Porter 1987) Construction of initial data : 1. Schwarzschild, Reisser-Nordstromblack holes (Wong 1971) 2. Axisymmetric Brill waves (Gentle & Miller 1999) 3. Binary black holes (Gentle 2002) Binary Black HoleConformal Factor Brill Wave Conformal Factor 35

36 Null-Strut Calculus: The First Test ds 2 = dt 2 + t 2p l dl 2 + t 2p w dw 2 + t 2p h dh 2 p l + p w + p h = p 2 l + p 2 w + p 2 h =1 G 1 1 G 2 2 G 3 3 G 0 0 Kasner: Theory = Ẅ W 2 = L L 2 Ḧ H 2 Ḧ H 2 = L L 2 Ẅ W 2 Ẇ Ḣ WH LḢ LH LẆ LW = LẆ LW LḢ LH Ẇ Ḣ WH 175,000 Time Steps Kasner: Null Strut Calculus w h + h w +2l ( l + l ) = 0 l h + h l +2w ( w + w) = 0 l w + w l +2h ( h + h) = 0 l + l + w + w + h + h =0 36

37 Null-Strut Calculus (3+1)-D IVP, 1989 Data Structure V D = 0 N ull Strut Regge Equation 216 Dimensional Solution Space 37

38 Kasner Evolution in 3-D 38

39 Towards Realistic RC Simulations Parallelizable implicit evolution scheme by Barrett et al. (1997) made realistic numerical simulations feasible. Applications include: Low-resolution 3-D Kasner evolution (Gentle & Miller 1998) FRW cosmology with 600-cell lattice (Gentle 1999) Evolution of Brill wave initial data (Gentle 2002) Metric function evolution: t=0 to t= Metric function evolution: t=5.0 to t= psi psi psi rho rho Deviation in simplicial conformal factor with time in Brill wave evolution.(a) Slices for 0<t<2, while (b) covers 5<t<6. Evolution of TrK at a point in Brill wave spacetime 39

40 IMPORTANT POINTS No one knows the relative strengths and weaknesses of RC to that of the finite difference approaches... but we can say that: Principles of general relativity applied directly to the lattice geometry Provides a true finite representation of the theory based on the underlying physical principles Voronoi-Delaunay duality appears to be a salient feature of Regge Calculus, providing a new fundamental building block. The underlying discrete theory appears more austere via the underlying orthogonality; however, the full theory is recovered by convergence in mean. Can the Voronoi-Delaunay structure provide a platform for complementarity in quantum gravity? 40

41 IMPORTANT OBSERVATION 41