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I D. N. Arnold, R. Winther: Notes on linearized Einsteinequations; Unpublished notes.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).