IDLinearWaves

Gabrielle Allen, Tom Goodale, Gerd Lanfermann, Joan Masso,
Mark Miller, Malcolm Tobias, Paul Walker

\( \)Date\( \)

Abstract

Provides gravitational wave solutions to the linearized Einstein equations

1 Purpose

There are two different linearized initial data sets provided, plane waves and Teukolsky waves.

2 Plane Waves

A full description of plane waves can be found in the PhD Thesis of Malcolm Tobias, The Numerical Evolution of Gravitational Waves, which can be found at http://wugrav.wustl.edu/Papers/Thesis97/Thesis97.html.

Plane waves travelling in arbitrary directions can be specified. For these plane waves the TT gauge is assumed (the metric perturbations are transverse to the direction of propagation, and the metric is traceless). In the case of waves travelling along the \(z-\)direction this would give the plus solution

hxx = − hyy = f (t± z),hxy = hxz = hyz = hzz = 0

and the cross solution

hxy = hyx = f(t ± z),hyz = hxx = hyy = hzz = 0

This thorn implements the plus solution, with the waveform \(f(t\pm z)\) having the form of a Gaussian modulated sine function. Now working with a general direction of propagation \(k\) we have the plane wave solution:

p i 2 p i 2 f(t,x,y,z) = Aine−(kix+ ωp(t−ra)) cos(kixi + ωt)+ Aoute−(kix−ωp(t−ra))cos(kixi − ωt)

and

\begin {eqnarray*} g_{xx}&=& 1 + f[\cos ^2\phi - \cos ^\theta \sin ^2\phi ] \\ g_{xy}&=& - f \sin ^2 \theta \sin \phi \cos \phi \\ g_{xz} &=& f \sin \theta \cos \theta \sin \phi \\ g_{yy} &=& 1+f [\sin ^2\phi - cos^2\theta \cos ^2\phi ] \\ g_{yz} &=& f \sin \theta \cos \theta \cos \phi \\ g_{zz} &=& 1-f\sin ^2\theta \end {eqnarray*}

The extrinsic curvature is then calculated from \begin {equation} K_{ij} = - \frac {1}{2\alpha } \dot {g}_{ij} \end {equation}

3 Teukolsky waves

Teukolsky waves are quadrupole wave solutions to the linearized Einstein equations. For a full description, see: PRD 26:745 (1982).

4 Comments

The extrinsic curvature is initialized assuming the initial lapse is one.

5 Parameters

amplitude
 Scope: private REAL
Description: Amplitude of the wave: both for teuk and plane
Range Default: 0.001
0:
positive amplitude

mvalue
 Scope: private INT
Description: m value for teukwaves waves: integer from -2 to 2
Range Default: (none)
-2:2
implemented : m = -2..2

packet
 Scope: private KEYWORD
Description: Packet for teukwaves: eppley,evans,square
Range Default: eppley
eppley
Eppley type
evans
Evans type
square
Square type

parity
 Scope: private KEYWORD
Description: Parity for teukwaves: even or odd
Range Default: even
even
even parity
odd
odd parity

teuk_no_vee
 Scope: private KEYWORD
Description: Initialize Teuk. waves with V=0?
Range Default: no
no
Bona Masso setting
yes
Bona Masso setting

wavecenter
 Scope: private REAL
Description: linears waves thingie
Range Default: 0.0
:

wavelength
 Scope: private REAL
Description: linearwaves wave length
Range Default: 2.0
0:
positive wavelength

wavephi
 Scope: private REAL
Description: Phi angle for planewaves
Range Default: 0.0
:

wavepulse
 Scope: private REAL
Description: planewaves thingy for the gaussian pulse
Range Default: 1.0
0:
positive pulse

wavesgoing
 Scope: private KEYWORD
Description: in and outgoing waves...
Range Default: both
in
Ingoing wave
out
Outgoing wave
both
In and outgoing wave

wavetheta
 Scope: private REAL
Description: Theta angle for planewaves
Range Default: 0.0
:

conformal_storage
 Scope: shared from STATICCONFORMALKEYWORD

6 Interfaces

General

Implements:

idlinearwaves

Inherits:

admbase

staticconformal

grid

7 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/IDLinearWaves. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

NONE

Scheduled Functions

CCTK_PARAMCHECK (conditional)

  idlinearwaves_paramchecker

  check that the metric_type is recognised

 

  Language: c
  Options: global
  Type: function

ADMBase_InitialData (conditional)

  idlinearwaves_planewaves

  construct linear planewave initial data

 

  Language: fortran
  Type: function

CCTK_PARAMCHECK (conditional)

  idlinearwaves_paramchecker

  check that the metric_type is recognised

 

  Language: c
  Options: global
  Type: function

ADMBase_InitialData (conditional)

  idlinearwaves_standwaves

  construct linear planewave initial data

 

  Language: fortran
  Type: function

CCTK_PARAMCHECK (conditional)

  idlinearwaves_paramchecker

  check that the metric_type is recognised

 

  Language: c
  Options: global
  Type: function

ADMBase_InitialData (conditional)

  idlinearwaves_teukwaves

  construct linear teukolsky wave initial data

 

  Language: fortran
  Type: function
  Writes: admbase::metric(everywhere)
    admbase::curv(everywhere)

CCTK_PARAMCHECK (conditional)

  idlinearwaves_paramchecker

  check that the metric_type is recognised

 

  Language: c
  Options: global
  Type: function

ADMBase_InitialData (conditional)

  idlinearwaves_sineplanewaves

  construct linear plane wave initial data

 

  Language: fortran
  Type: function