Obstacles in Numerical Calculations Erik Schnetter Paris, November - PowerPoint PPT Presentation

Obstacles in Numerical Calculations Erik Schnetter Paris, November 2006

Obstacles in Numerical Calculations General Numerical Relativity Analysis Numerical Relativity

Layout • Hawking Energy • Ricci tensor and higher derivatives • Dynamical and Isolated Horizons • Coordinates

Hawking Energy • Interesting problem: - assuming conformal determine amount of flatness outside energy contained in the domain simulated domain - approximate Bartnik mass? • People usually calculate the ADM mass or related • Question: Why don’t quantities: people calculate the - ADM mass at finite Hawking energy? distance - as volume integral

Hawking Energy � � � E H = R Simple definition: Θ ( ℓ ) Θ ( n ) 1 − 2 Unfortunately, this equation is numerically not well-posed. Good definition: � � E H = R σ ¯ λ − Ψ 2 − ¯ � σλ + ¯ Ψ 2 + 2 Φ 11 + 2 Λ 2 [LRR 2004 4]

Hawking Energy � � � E H = R Θ ( ℓ ) Θ ( n ) 1 − 2 Asymptotic behaviour With numerical error: (large r): 1 − 1 � � 1 − 1 r + O ( ǫ ) Θ ( ℓ ) Θ ( n ) ∼ Θ ( ℓ ) Θ ( n ) ∼ r � � 1 − 1 �� ¯ � � �� 1 − 1 1 − r + O ( ǫ ) E H r ∼ E H r 1 − ∼ r ¯ E H + O ( r ǫ ) E H ∼

Noise through Derivatives • Numerical simulations Mass quadrupole contain noise. 0 M 2 -0.5 Derivatives amplify noise. M 2 for Kerr -1 M 2 -1.5 • Formally, loses n d n /dx n -2 orders of accuracy -2.5 125 135 145 155 • Empirically, higher than Angular momentum octupole t 0 second derivatives are J 3 -2 J 3 for Kerr difficult (... with current -4 -6 J 3 methods) -8 -10 -12 • In 3+1 D, resolution is -14 125 135 145 155 always a problem t

Noise through Derivatives • Goal: Define angular • Problem: This would momentum on non- require at least n=4 axisymmetric horizons derivatives • Requires: Find a • Which would therefore generalisation of a Killing not work in spacetimes vector field on a horizon with matter • Idea (Ashtekar?): Use isocontour lines of a 2- scalar on the horizon

From DH To IH • Intuitively, a dynamical • Numerically, the horizon horizon will become will be indistinguishable “more and more null” at from a null surface at late times, becoming some time, and the isolated “at late times”. transition must be handled. • Mathematically, this transition from spacelike S 2 to null is not smooth, and � a does not happen. H τ a ˆ T a n a r a ˆ R a Σ S [PRD 74 024028] S 1

From DH To IH Relation between normals: ℓ = T + R n = T − R ˆ r n = ˆ r ℓ = ˆ τ + ˆ τ − ˆ ˆ ℓ = α ˆ n = ˆ n/ α ℓ S 2 � a H τ a ˆ T a n a r a ˆ R a Σ S S 1

Coordinates • In numerical work, • Transformations between everything is expressed in domains (e.g. from a 3D terms of coordinates hypersurface to a 2D (basis, gauge): surface) require interpolation, which is - domain (grid points) inaccurate - tensors (components) • Coordinate systems can 4 3 have singularities; handling multiple maps requires 2 much additional work 1 [CQG 20 4719] 0 0 1 2 3 4 ( a ) ( b )

Coordinates • In a 3+1 time evolution, • There would be the foliation is interesting questions: In a determined by the gauge different slicing, conditions, which is - how do the trapped chosen according to surfaces look? what is stability properties the total trapped region? • No one (afaik) has - do extracted waves analysed a 3+1 spacetime in a foliation different change much? than the given one - do different codes converge pointwise?

Final Thoughts • There are also some tasks which are easier numerically: - Represent arbitrary functions - Solve ODEs - Integrate (over a given domain) • I don’t want to be blinded by my numerical glasses