Dual Tessellations and Curvature in Regge Calculus

Jonathan R. McDonald SFB/TR7

Institut für Angewandte MathematikFSU Jena

SFB/TR7 Video Seminar2 November 2009

Jonathan R. McDonald

Overview

• What is Regge?

• The Dual and RC

• The Role of the Dual in Defining Curvature

• Diffeomorphism Invariance in RC

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Jonathan R. McDonald

The Why of Regge...

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Binary Black hole initial data. A.P. Gentle, GRG 34, (2002)

2-d Snapshot of a CDT Universe. R. Loll, Nucl.Phys.Proc.Suppl. 94 (2001)

96-107

Quantum Gravity

Numerical Relativity

Regge Calculus is a discretization of GR constructed as a piece-wise flat manifold which

has been used for quantum theory and numerical relativity

Jonathan R. McDonald

Numerics and Regge

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Null-Strut Calculus Thin-Sandwich Formulation

Kheyfets et al., PhysRevD 41(1990) Sorkin Evolution in 3dTuckey , CQG 10 (1990)

Using Regge to study initial-value data dates back to the analysis of Wheeler and Wong on the Icosahedral approximation of the Schwarzschild and Reissner-Nordstrøm geometries. Evolution equations were later introduced by Sorkin (1975) and Miller (1986) and Barrett et al. (1997). Simulations of relativistic, spherical collapse for a perfect fluid were done by Dubal (1990).

Jonathan R. McDonald

The What of Regge...

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1) The geometry is completely determined by coordinate-free Lorentz scalars

2) Local representations of the tangent space are hard-wired into the lattice

3) Evolution schemes are often readily parallelizable

4) Gives a locally simple theory

5) Numerical simulations performed in the 80’s and 90’s have shown approx. 2nd order convergence in test cases (Brill waves, Kasner universe, ...)

Jonathan R. McDonald

The How of Regge...

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PremiseEncode dynamical degrees of freedom such that the geometry is

completely determined by a single set of coordinate-free parameters

Non-rigid

Rigid

The edges in a d-simplex

Jonathan R. McDonald

RC and the Circumcentric Dual

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Every simplicial lattice automatically induces a circumcentric dual lattice

which naturally decomposes the piecewise-flat manifold

For a positive-definite metric, this dual lattice provides a definitive

notion of “closeness” and nearest neighbors in the lattice.

e.g.

Jonathan R. McDonald

An Aside on Voronoi-Delaunay Duality

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There exists a unique triangulation for a set of points (on a positive-definite metric) such that the circumcentric dual defines

a disjoint cover of the geometry.

This triangulation is called the Delaunay triangulation and the circumcentric dual is the Voronoi tessellation.

Jonathan R. McDonald

Decomposition of d-volumes...

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In 2d....

k-simplexThe dual

element to siga

e.g.

This directly results from decomposition of volumes via

restriction to individual simplexes.

Jonathan R. McDonald

Curvature in Regge Calculus

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Curvature is concentrated as a conic singularity on the co-dimension-2 hinges and proportional to the Gaussian Curvature

Ah

A!h

!h!h

2d hinge and loop of parallel transport

Jonathan R. McDonald

Einstein-Hilbert Action in RC

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Notice how all reference to the dual has vanished in the vacuum theory.

*WA Miller. CQG 14 (1997) L199–L204

Construction of the Einstein-Hilbert Action from the continuum theory.

Jonathan R. McDonald

The Regge Equations...

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L

!Lh

We thus have one equation per edge to determine the lattice edge lengths.

Under variation of the action we first notice a remarkable property:

Which gives:

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Cartan Moment of Rotation Trivector and the Einstein Tensor

!G = ! (dP !R)Moment of Rotation Trivector

Moment Arm

Rot’nBivector dx

dydz

Élie Cartan’s moment of rotation trivector provides a clear link for reconstructing the Einstein tensor and studying its

symmetry properties.

Note: There is a freedom to choose the fulcrum as we wish given the ordinary Bianchi identity proven by Regge.

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Constructing the Regge-Einstein Tensor

The Geometric Content of the Einstein-Regge Equations

Gµ! = 8!Tµ!

L

!Lh

L

EffectiveMom. Arm

Rot’n

Jonathan R. McDonald

Contracted Bianchi Identity from the BBP

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The boundary of the boundary of an oriented volume is

automatically zero!

Let topology serve as our guide:

Boundary of a Boundary Principle (BBP)

2-3-4D BBP:

0 !!

Vd!G =

!

!V

!G =!

!!V! (P "R)

1-2-3D BBP

Jonathan R. McDonald

2-3-4d Simplicial BBP

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Jonathan R. McDonald

Kirchhoff-like Conservation

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Gentle, Kheyfets, McDonald, Miller. Class. Quant. Grav. 26 (2009) 015005

The contracted Bianchi identity manifests in RC as a circuit-like approximate conservation equation—for the EOM—which converges to the continuum as the square of the edge lengths or in the linear regime.

Jonathan R. McDonald

The Bianchi Identity Debate?

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What’s the deal with the Regge “conservation” law?

Exact Symmetry Approximate Symmetryvs.

Conservation only results when the rotations are commuted to cancel equal & opposite contributions.

However, there are recent attempts to recover an exact symmetry through modification of the lattice. However, these break the explicit representation of tangent spaces.

Lapse and shift are free to choose in RC﹣4 constraint equations per vertex﹣though they come at a cost with

violations appearing as 2nd order phenomena.

Jonathan R. McDonald

Summary

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This work has been done through discussion and collaborations with Warner Miller, Chris Beetle, Seth Lloyd, Arkady Kheyfets, and Adrian Gentle.

Some of this work was supported by NSF Grant No. 0638662

(I) Regge is a geometric discretization of GR with a locally simple structure(II) The dual lattice construction yields a ‘democratic’ decomposition of

geometric quantities in RC(III) Diffeomorphism invariance is only an approximate symmetry in RC which

approaches exact symmetry in the linear regime or limit of small edge lengths

(IV) Questions that remain:(i) Can we make stronger connections with known numerical methods

on unstructured meshes?(ii) Do the coordinate-free, unstructured meshes provide greater

flexibility and robustness for numerical solutions?