Post on 30-Apr-2018
Numerical analysis of Regge Calculus
Snorre H. Christiansen
Department of Mathematics, University of Oslo
Minneapolis, 24. October 2014
Inspiration I
I D. N. Arnold: Differential complexes and numericalstability; Proceedings of the International Congress ofMathematicians, Vol. I, p. 137–157, (Beijing, 2002).
Inspiration II
I D. N. Arnold, R. Winther: Notes on linearized Einsteinequations; Unpublished notes.
Outline
I Crash course on Einstein equations and Regge calculus.
I Linearized Regge calculus in 3D.
I Non-linear Regge calculus in 2D.
Einstein equations
I Unknown is a metric with Lorentzian signature.
I Curvature is a non-linear expression of the metricinvolving second order derivatives.
I Vacuum: Einstein curvature is zero.
I Diffeomorphism invariance. Gauge freedom and constraints.
I Einstein-Hilbert action:
A(g) =
∫κ(g)µ(g). (1)
I 2D: all metrics are solutions,3D: only flat space (up to diffeomorphism),4D: interesting.
Regge calculus
I Regge, T.: General relativity without coordinates; NuovoCimento (10), Vol. 19, p. 558–571,1961.
I Manifold represented by simplicial complex.Metric determined by edge lengths.
I Curvature defined by deficit angle on codimension 2 simplices.
I Combinatorial action:sum over hinges of deficit angles times area.
I Discrete variational principle:Find critical points of action on space.
2D Regge calculus
I Euler-Poincare characteristic for simplicial complex:
# V − # E + # F = χ. (2)
I Gauss-Bonnet: ∫κ(g)µ(g) = 2πχ. (3)
I Regge: ∑V
(2π −∑
θ) = 2π # V − π # F (4)
= 2π(# V − # E + # F ) (5)
(using 2 # E = 3 # F ).
I Regge calculus defines a local curvature where possible,with right global property.
Elasticity complex
C∞(S ,V)def //
I 0h��
C∞(S ,S)curlt curl //
I 1h��
C∞(S ,S)div //
I 2h��
C∞(S ,V)
I 3h��
X 0h
def // X 1h
curlt curl // X 2h
div // X 3h
(6)
I Arnold, D. N. and Falk, R. S. and Winther, R.:Differential complexes and stability of finite element methods.II. The elasticity complex; Compatible spatial discretizations,IMA Vol. Math. Appl., Vol. 142, p. 47–67, Springer, 2006.
I Good FE spaces X 2h ,X
3h presuppose good FE spaces X 1
h .
I Related to de Rham complex by BGG construction.
Elasticity complex: relativity
C∞(S ,V)def //
I 0h��
C∞(S ,S)curlt curl //
I 1h��
C∞(S ,S)div //
I 2h��
C∞(S ,V)
I 3h��
X 0h
def // X 1h
curlt curl // X 2h
div // X 3h
(7)
I SHC: On the linearization of Regge calculus; Numer. Math.,Vol. 119, No. 4, p. 613–640, Springer, 2011.
I Complex encodes:– 1: linearized diffeomorphism invariance,– 2: linearized Bianchi identity (energy momentum).
Elasticity complex: Regge
C∞(S ,V)def //
I 0h��
C∞(S ,S)curlt curl //
I 1h��
C∞(S ,S)div //
I 2h��
C∞(S ,V)
I 3h��
X 0h
def // X 1h
curlt curl // X 2h
div // X 3h
(8)
I X 0h : Continuous piecewise affine vectorfields.
I X 1h : TT -continuous piecewise constant metrics (Regge).
I X 2h : Dirac deltas on edges: τe ⊗ τeδe .
I X 3h : Dirac deltas on vertices: Vδv .
Is Regge calculus conforming?
I Definition of curvature in Regge calculus seems plausible,but is it the true curvature or merely a consistentapproximation?
I The Einstein-Hilbert action and the Regge actionhave the same second variation (linearization),namely u 7→ 〈curlt curl u, u〉.
I Proof:combinatorial formula for second variation in Regge calculusmatches combinatorial formula for curlt curl(computed in the sense of distributions).
Can Regge calculus be justified for linearized GR?
I Linearized GR is a wave equation with curlt curl in space.But constraints involving trace and divergence.
I Convergent eigenvalue problem for curlt curl in RC.
I Proof inspired by Maxwell eigenvalue problem:– Boffi, D.: Finite element approximation of eigenvalueproblems; Acta Numer., Vol. 19, p. 1–120, 2010.– SHC and Winther, R.: On variational eigenvalueapproximation of semidefinite operators; IMA J. Numer.Anal., Vol. 33, No. 1, p. 164–189, 2013.
I Commuting projections (nice kernel) but:– Non-conforming: L2 metrics with curlt curl in H−1,– Non semi-definite, need for inf-sup.
Is Regge calculus conforming? (bis)
I SHC: Exact formulas for the approximation of connectionsand curvature; arXiv:1307.3376.
I Regularize Regge metric by convolution:
gε = g ∗ φε. (9)
I Then curvatures converge to Regge curvature for ε→ 0:
κ(gε)µ(gε)→ (2π −∑
θ)δ, (10)
in the sense of measures.
Proof
I Claim: Fix ε, then κ(gε)µ(gε) has bounded support and:∫κ(gε)µ(gε) = 2π −
∑θ. (11)
I In orthonormal frame, connection 1-form A:
dA = Rie = Jκµ, J =
[0 1−1 0
]. (12)
Integrate and take exponentials, use Stokes on LHS:
Hol(A, ∂T ) = exp(
∫∂T
A) = exp(J
∫Tκµ). (13)
Determine LHS as rotation matrix by angle −∑θ.
Gives result mod 2π, conclude by a continuity argument.
I gε related to gε′ by pullback and scaling.
Prescribed densitized scalar curvature
I Funny identity:
(κµ)(exp(2u)g0) = (∆0u)µ0. (14)
I Discrete conformal tranformations:
C(u) : gij 7→ exp(ui + uj)gij . (15)
I Given f find u sutch that:
(κµ)(C(u, g0)) =∑
fiδi . (16)
I Linearize around u = 0 gives: Find u such that for all v∫grad u · grad v =
∑fivi . (17)
Laplace equation with P1 elements (for g0).