Pennsylvania State University

Joint work at Southampton University

Ulrich Sperhake

Ray d’Inverno

Robert Sjödin

James Vickers

Cauchy characteristic matchingIn 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: ijij K,

Gauge variables: i ,

jiij

ii

ii dxdxdtdxdtds 2)( 222

Field equations: 6 evolution Eqs. K ,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, http://www.livingreviews.org

How does it work in practice?

Cylindrically symmetric line element

2222222)(22 )()( dzdederdrdteds

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:

)(1 2

,2,

2,

2, trrt

□

)(2

,,,, rrtt

□

)(4

2,

2,

2,

2,2, trtrr

r

rrrt

1

2

2

2

2

□

)(1

22

22

urrrru

□Characteristic region:

Compactification:r

y1

=> Null infinity at 0y

rtu ,

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.)