Pennsylvania State University Joint work at Southampton University Ulrich Sperhake Ray d’Inverno Robert Sjödin James Vickers Cauchy characteristic matching In cylindrical symmetry

 ADM “3+1” formulation  Characteristic formulation  CCM in cylindrical symmetry Overview o 3+1 formulation o Characteristic formulation o Interface

Ideal numerical code  fully non-linear 3D field + matter Eqs.  long term stability  exact boundary conditions (infinity)  proper treatment of singularities (excision, avoidance)  detailed description of matter (microphysics)  exact treatment of hydrodynamics (shock capturing)  high accuracy for signals with arbitrary amplitude   extraction of grav. waves at infinity 

ADM “3+1” formulation Arnowitt, Deser and Misner (1961) Foliate spacetime into 1-par. family of 3-dim. spacelike slices

”3+1” ADM formulation  Initial value problem  Dynamic variables:  Gauge variables:  Field equations:6 evolution Eqs. 3+1 constraints (conserved)

Advantages and drawbacks  “3+1” formulations preferred in regions of strong curvature  non-hyperbolicity of ADM  unclear stability properties => Modifications: introduce auxiliary variables => “BSSN”, hyperbolic formulations: appear to be more stable  Not clear how to compactify spacetime => 1) Interpretation of grav. waves at finite radii, 2) artificial boundary conditions at finite radii => spurious reflections, numerical noise

Spurious reflections

Characteristic formulation Bondi, Sachs (1962)  Foliate spacetime into 2-par. family of 2-dim. spacelike slices  One of the 2 families of curves threading the slices is null

Characteristic formulation  Field equations:2 evolution Eqs. 4 hypersurface Eqs. (in surfaces u=const) 3 supplementary Eqs., 1 trivial Eq.  compactification => 1) description of radiation at null infinity 2) Exact boundary conditions  Problem: Caustics in regions of strong curvature => Foliation breaks down “3+1” and char. formulation complement each other !

Cauchy characteristic matching  “3+1” in interior region  char. In the outer region  interface at finite radius J. Winicour, Living Reviews,

How does it work in practice?  Cylindrically symmetric line element  Factor out z-Killing direction (Geroch decomposition)  Describe spacetime in terms of 2 scalar fields on 3-dim. quotient spacetime: Norm of the Killing vector Geroch potential

Field equations Cauchy region: □ □ □ □ Characteristic region: Compactification: => Null infinity at,

The interface

Testing the code Xanthopoulos (1986)

Cylindrical Gravitational Waves CCM ORC (r=1) ORC (r=5) ORC (r=25) CCM versus ORC (Outgoing Radiation Condition)

Where to go from here? CCM in higher dimensions – axisymmetry (d’Inverno, Pollney) – 3 dim. (Bishop, Winicour et al.)