dual tessellations and curvature in regge calculus · dual tessellations and curvature in regge...
TRANSCRIPT
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:
Jonathan R. McDonald13
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
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?